Faculty of Mathematics Part III Essays: 2011 - 2012

Combinatorial Algorithms in the Representation Theory of Symmetric. Groups . ..... describes the essay topic and answers general questions. ... Then take your essay and the statement to the Faculty Office at the Centre for Mathematical ... You should include on the informative cover sheet the following declaration: I declare ...

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Report

Faculty of Mathematics Part III Essays: 2011 - 2012

Titles 1 – 52 Department of Pure Mathematics & Mathematical Statistics

Titles 53 – 88 Department of Applied Mathematics & Theoretical Physics

Titles 89 – 116 Additional Essays

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Contents Introductory Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1. Zariski Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2. Geometric Group Theory, Measured Foliations, and Train-Tracks . . . . . . . . . 15 3. Hilbert’s Third Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4. Quasimorphisms and Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 5. Values of the Riemann Zeta Function at the Odd Positive Integers . . . . . . . . 17 6. Constructing Elliptic Curves of Large Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 7. Counting Cubic and Quartic Number Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 8. Wellquasiorders and Betterquasiorders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 9. Set Theory without the Axiom of Foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 10. Fraenkel-Mostowski Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 11. Expansion in Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 12. Hilbert’s Fifth Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 13. Yau’s Solution of the Calabi Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 14. Stable Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 15. K3 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 16. Canonical Ramsey Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 17. Pursuit on Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 18.

Combinatorial Algorithms in the Representation Theory of Symmetric Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

19. Non-Commutative Character Theory of the Symmetric Group . . . . . . . . . . . 27 20. Symmetric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 21. Penrose Stability Criterion for Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

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22. The Calder´ on Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 23. Branched Double Covers of Alternating Knots . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 24. Khovanov Homology for Tangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 25. K¨ ahler-Einstein Metrics on Fano Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 26. Lagrangian Klein Bottles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 27. Non-K¨ ahler Calabi-Yau Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 28. Ramsey Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 29. Tutte’s Flow Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 30. Auslander–Gorenstein Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 31. Complex Moduli Spaces for Calabi–Yau Manifolds . . . . . . . . . . . . . . . . . . . . . . . 34 32. Local Langlands Correspondences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 33. Etale Cohomology and Galois Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 34. The Cosmic Censorship Conjectures in General Relativity . . . . . . . . . . . . . . . 37 35. The Gaussian Free Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 36. Inference about Mechanistic Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 37. Causal Statistical Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 38. Forensic Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 39. Information and Decision Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 40. The Rate of Convergence of Fictitious Play . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 41. Positive and Negative Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 42. Lorentz/Ehrenfest Wind–Tree Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 43. Nonparametric Statistics on Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 44. Statistical Inference for L´ evy Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 45. Multiplicative Coalescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3

46. Independent Component Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 47. Model Misspecification in Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 48. Statistical Challenges with EEG Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 49. Dimension Reduction in Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 50. Nonparametric Shape Restricted Regression with Multiple Covariates . . . 49 51. Bond Market Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 52. Ranking, Reputation and Recommender Systems . . . . . . . . . . . . . . . . . . . . . . . . 51 53. h → γγ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 54. Effective Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 55. Ab initio Definite Quantities for Quantum Theory . . . . . . . . . . . . . . . . . . . . . . . 53 56. Information-Theoretic Aspects of Quantum Foundations . . . . . . . . . . . . . . . . . 54 57. Varieties of Locality for Quantum Fields and Strings . . . . . . . . . . . . . . . . . . . . . 55 58. Philosophical Aspects of Spontaneous Symmetry Breaking . . . . . . . . . . . . . . . 56 59. The Effect of Boundary Layers in Aero-Acoustics . . . . . . . . . . . . . . . . . . . . . . . . 56 60. Numerical Techniques for Computational Aero-Acoustics . . . . . . . . . . . . . . . . 57 61. Gravity Currents Flowing Over Corrugated Boundaries . . . . . . . . . . . . . . . . . . 58 62. How Many Puffs Does it Take to Make a Jet? . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 63. Supersymmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 64. The Two-State Vector Formalism in Quantum Mechanics . . . . . . . . . . . . . . . . 60 65. FreeFem++ Applied to the Driven Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 66. Numerical Simulations of Incompressible Two-Dimensional Turbulence . . 61 67. The Boundary Integral Technique for Potential Flows . . . . . . . . . . . . . . . . . . . . 62 68. Spontaneous Symmetry Breaking and Phase Transitions in Two Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

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69. Highly Oscillatory Quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 70. Homological Techniques for Finite Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 71. Quantum Complexity of Local Hamiltonian Problems . . . . . . . . . . . . . . . . . . . . 65 72. Theory and Constraints on Lorentz Symmetry Violation . . . . . . . . . . . . . . . . . 66 73. Fluid Flow and Elastic Flexure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 74. Eccentric Astrophysical Discs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 75. Accretion Discs and Planet Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 76. Global Modes in Shear Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 77. Generalised Geometry from M-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 78. Localised Convective Cells in Subcritical Convection . . . . . . . . . . . . . . . . . . . . . 70 79. The Backus-Gilbert Method for the Solution of Inverse Problems . . . . . . . . 71 80. Sampling Methods in Inverse Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 81. Black Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 82. Nonlinear Diffusion Equations for Image Enhancement . . . . . . . . . . . . . . . . . . 74 83. Feynman Diagrams in String Theory: the Topological Vertex . . . . . . . . . . . . 75 84. Small Thermal Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 85. BPS States in Supersymmetric Gauge Theories . . . . . . . . . . . . . . . . . . . . . . . . . . 76 86. Global Isometric Embeddings and their Application to Black Hole Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 87. Segregated Granular Collapse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 88. Walk, Don’t Run! – A Story of Broken Symmetry (Presented in Technicolour) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 89. Primordial Non-Gaussianity and Large-Scale Structure . . . . . . . . . . . . . . . . . . 79 90. Categorical Galois Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 91. Homology in Semi-abelian Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5

92. Complex Multiplication of Abelian Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 93. p-adic Uniformisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 94. Smoothness Results in Deformation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 95. Jacobians and Prym Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 96. Normal Subgroups of the Cremona Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 97. Analysis of a Large and Complex Data Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 98. Atiyah–Singer Index Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 99. Algebraic Groups, Geometry of the Flag Variety, Combinatorics of the Weyl Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 100. Hodge Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 101. Rational Homotopy Theory, and Mapping Spaces . . . . . . . . . . . . . . . . . . . . . . . 87 102. Topos-Theoretic Approaches to Quantum Theory . . . . . . . . . . . . . . . . . . . . . . . 88 103. Exotic 4-Spheres and Khovanov Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 104. Regularity of Energy Minimizing Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 105. Implied Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 106. The Potential Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 107. General Covariance and Background Independence in Quantum Gravity

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108. Theories of Spiral Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 109. Instabilities in Developing Ocean Fronts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 110. Turbulence and Plankton Ecosystem Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 93 111. Statistical Methods for Cancer Genomics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 112. B-Spline Basis Condition Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 113. Non-Archimedean Amoebae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 114. The Multiplication Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

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115. Entanglement and Entropies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 116. Instabilities in Dilute Suspension of Self-Propelled Microscopic Swimmers 98

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Introductory Notes General advice. Before attempting any particular essay, candidates are advised to meet the setter in person. Normally candidates may consult the setter up to three times before the essay is submitted. The first meeting may take the form of a group meeting at which the setter describes the essay topic and answers general questions. Choice of topic. The titles of essays appearing in this list have already been announced in the Reporter. If you wish to write an essay on a topic not covered in the list you should approach your Part III Adviser or any other member of staff to discuss a new title. You should then ask your Director of Studies to write to the Secretary of the Faculty Board, c/o the Faculty Office at the CMS (Room B1.28) no later than 1 February. The new essay title will require the approval of the Examiners. It is important that the essay should not substantially overlap with any course being given in Part III. Additional Essays will be announced in the Reporter no later than 1 March and are open to all candidates. Even if you request an essay you do not have to do it. Essay titles cannot be approved informally: the only allowed essay titles are those which appear in the final version of this document (on the Faculty web site). Originality. The object of a typical essay is to give an exposition of a piece of mathematics which is scattered over several books or papers. Originality is not usually required, but often candidates will find novel approaches. All sources and references used should be carefully listed in a bibliography. Length of essay. There is no prescribed length for the essay in the University Ordinances, but the general opinion seems to be that 6,000-10,000 words is a normal length. If you are in any doubt as to the length of your essay please consult your adviser or essay setter. Presentation. Your essay should be legible and may be either hand written or produced on a word processor. Candidates are reminded that mathematical content is more important than style. Usually it is advisable for candidates to write an introduction outlining the contents of the essay. In some cases a conclusion might also be required. It is very important that you ensure that the pages of your essay are fastened together in an appropriate way, by stapling or binding them, for example. Credit. The essay is the equivalent of one three-hour exam paper and marks are credited accordingly. Final decision on whether to submit an essay. You are not asked to state which papers you have chosen for examination and which essay topic, if any, you have chosen until the beginning of the third term (Easter) when you will be sent the appropriate form to fill in and hand to your Director of Studies. Your Director of Studies should counter-sign the form and send it to the Chairman of Examiners (c/o the Faculty Office, Centre for Mathematical Sciences) so as to arrive before the second Friday in Easter Full Term, which this year is Friday 4 May 2012. Note that this deadline will be strictly adhered to.

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Date of submission. You should submit your essay, through your Director of Studies, to the Chairman of Examiners (c/o Faculty Office, CMS). Your essay should be sent with a cover note along the following lines: To the Chairman of Examiners for Part III Mathematics. Dear Sir, I enclose the Part III essay of ....................................... Signed ........................... (Director of Studies) Then take your essay and the statement to the Faculty Office at the Centre for Mathematical Sciences so as to arrive not later than the second Friday in Easter Full Term, which this year is Friday 4 May 2012. Note that this deadline will be strictly adhered to. Cover sheets. You should submit TWO title pages with your essay. The first should bear (1) the title of your essay, (2) your name, (3) your college, (4) a signed declaration (see below) and (5) your home address. The second should bear ONLY the essay title. Signed declaration. You should include on the informative cover sheet the following declaration: I declare that this essay is work done as part of the Part III Examination. I have read and understood the Statement on Plagiarism for Part III and Graduate Courses issued by the Faculty of Mathematics, and have abided by it. This essay is the result of my own work, and except where explicitly stated otherwise, only includes material undertaken since the publication of the list of essay titles, and includes nothing which was performed in collaboration. No part of this essay has been submitted, or is concurrently being submitted, for any degree, diploma or similar qualification at any university or similar institution. Signed ..................................... Important note. The Statement on Plagiarism for Part III and Graduate Courses issued by the Faculty of Mathematics is reproduced starting on page 11 of this document. If you are in any doubt as to whether you will be able to sign the above declaration you should consult the member of staff involved in the essay. If they are unsure about your situation they should consult the Chairman of the Examiners as soon as possible. The examiners have the power to examine candidates viva voce (i.e. to give an oral examination) on their essays, although this procedure is not often used. However, you should be aware that the University takes a very serious view of any use of unfair means (plagiarism, cheating) in University examinations. The powers of the University Court of Discipline in such cases extend to depriving a student of membership of the University. Fortunately, incidents of this kind are very rare. Return of essays. It is not possible to return essays. You are therefore advised to make your own copy before handing in your essay.

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Further advice. It is important to control carefully the amount of time spent writing your essay since it should not interfere with your work on other courses. You might find it helpful to construct an essay-writing timetable with plenty of allowance for slippage and then try your hardest to keep to it. Research. If you are interested in going on to do research you should, if possible, be available for consultation in the next few days after the results are published. If this is not convenient you should leave a note of how you can be contacted (address, telephone number, email address, etc.) with the Research Student Secretary, Room B1.29, CMS or email [email protected].

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Faculty of Mathematics Statement on Plagiarism for Part III and Graduate Courses The Board of Examinations publishes information on plagiarism at http://www.admin.cam.ac.uk/univ/plagiarism/

In particular, the definition and scope of plagiarism, together with ways to avoid it, are given in the University statement on plagiarism for undergraduate and graduate students at http://www.admin.cam.ac.uk/univ/plagiarism/students/

There is a reference to this University statement in the Part III Essay booklet and in each M.Phil. course booklet. Please read this statement carefully. It is your responsibility to read and abide by the University statement. The guidelines below are provided by the Faculty to help students interpret what the University statement means for Mathematics. However neither the University nor the Faculty set of guidelines supersede the University’s Regulations as set out in the Statutes and Ordinances. If you are unsure as to the interpretation of either set of guidelines, or the Statutes and Ordinances, you should ask your course director.

What is plagiarism? Plagiarism can be defined as the unacknowledged use of the work of others as if this were your own original work. In the context of any University examination, this amounts to passing off the work of others as your own to gain unfair advantage. Such use of unfair means will not be tolerated by the University or the Faculty. If detected, the penalty may be severe and may lead to failure to obtain your degree or certificate. This is in the interests of the vast majority of students who work hard for their degree through their own efforts, and it is essential in safeguarding the integrity of the degrees and certificates awarded by the University.

Checking for plagiarism Faculty Examiners will routinely look out for any indication of plagiarised work. They reserve the right to make use of specialised detection software if appropriate (the University subscribes to Turnitin Plagiarism Detection Software). 1 For information on the procedures that will be followed for handling suspected cases of plagiarism • in Part III, please see the Board of Examinations statement on Plagiarism and Collusion available at http://www.admin.cam.ac.uk/univ/plagiarism/examiners/

• in graduate courses, please see the Board of Graduate Studies guide Information for Degree Committees and Departments available at 1

For more information see the University Policy Statement available at http://www.admin.cam.ac.uk/univ/plagiarism/examiners/detection.html

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http://www.admin.cam.ac.uk/offices/gradstud/staff/dc/

The scope of plagiarism Plagiarism may be due to • copying (this is using another person’s language and/or ideas as if they are your own); • collusion (this is collaboration either where it is forbidden, or where the extent of the collaboration exceeds that which has been expressly allowed).

How to avoid plagiarism Your Part III essay, or your M.Phil. projects and course-work, are marked on the assumption that it is your own work: i.e. on the assumption that the words, diagrams, computer programs, ideas and arguments are your own. Plagiarism can occur if, without suitable acknowledgement and referencing, you take any of the above (i.e. words, diagrams, computer programs, ideas and arguments) from books or journals, obtain them from unpublished sources such as lecture notes and handouts, or download them from the web. Plagiarism also occurs if you submit work that has been undertaken in whole or part by someone else on your behalf (such as employing a ‘ghost writing service’). Furthermore, you should not deliberately reproduce someone else’s work in a written examination. These would all be regarded as plagiarism by the Faculty and by the University. In addition you should not submit any work that is substantially the same as work you have submitted, or are concurrently submitting, for any degree, diploma or similar qualification at any university or similar institution. However, it is often the case that parts of Part III essays and M.Phil. projects and course-work will be based on what you have read and learned from other sources, and it is important that in your essay or project or course-work you show exactly where, and how, your work is indebted to these other sources. The golden rule is that the Examiners must be in no doubt as to which parts of your work are your own original work and which are the rightful property of someone else. A good guideline to avoid plagiarism is not to repeat or reproduce other people’s words, diagrams or computer programs. If you need to describe other people’s ideas or arguments try to paraphrase them in your own words (and remember to include a reference). Only when it is absolutely necessary should you include direct quotes, and then these should be kept to a minimum. You should also remember that in an essay or project or course-work, it is not sufficient merely to repeat or paraphrase someone else’s view; you are expected at least to evaluate, critique and/or synthesise their position. In slightly more detail, the following guidelines may be helpful in avoiding plagiarism. Quoting. A quotation directly from a book or journal article is acceptable in certain circumstances, provided that it is referenced properly: • short quotations should be in inverted commas, and a reference given to the source; • longer pieces of quoted text should be in inverted commas and indented, and a reference given to the source.

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Whatever system is followed, you should additionally list all the sources in the bibliography or reference section at the end of the piece of work, giving the full details of the sources, in a format that would enable another person to look them up easily. There are many different styles for bibliographies. Use one that is widely used in the relevant area (look at papers and books to see what referencing style is used). Paraphrasing. Paraphrasing means putting someone else’s work into your own words. Paraphrasing is acceptable, provided that it is acknowledged. A rule of thumb for acceptable paraphrasing is that an acknowledgement should be made at least once in every paragraph. There are many ways in which such acknowledgements can be made (e.g. “Smith (2001) goes on to argue that . . . ” or “Smith (2001) provides further proof that . . . ”). As with quotation, the full details of the source should be given in the bibliography or reference list. General indebtedness. When presenting the ideas, arguments and work of others, you must give an indication of the source of the material. You should err on the side of caution, especially if drawing ideas from one source. If the ordering of evidence and argument, or the organisation of material reflects a particular source, then this should be clearly stated (and the source referenced). Use of web sources. You should use web sources as if you were using a book or journal article. The above rules for quoting (including ‘cutting and pasting’), paraphrasing and general indebtedness apply. Web sources must be referenced and included in the bibliography. Collaboration. Unless it is expressly allowed, collaboration is collusion and counts as plagiarism. Moreover, as well as not copying the work of others you should not allow another person to copy your work.

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Table 1: A Timetable of Relevant Events and Deadlines Wednesday 1 February

Deadline for Candidates to request additional essays.

Friday 4 May

Deadline for Candidates to return form stating choice of papers and essays.

Friday 4 May

Deadline for Candidates to submit essays.

Thursday 31 May

Part III Examinations begin.

Thursday 21 June

Classlist read out in Senate House, 9.00 am.

Comments. If you feel that these notes could be made more helpful please write to The Chairman of Examiners, c/o the Faculty Office, CMS. Further information. Professor T.W. Körner (DPMMS) wrote an essay on Part III essays which may be useful (though it is slanted towards the pure side). It is available via his home page http://www.dpmms.cam.ac.uk/˜twk/

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1. Zariski Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr C. Birkar The Zariski decomposition problem is about decomposing a pseudo-effective divisor D on a smooth projective variety into D = P + N where P is supposed to be positive in some sense and N ≥ 0. This problem is closely related to some important problems in birational geometry such as the minimal model conjecture and the finite generation conjecture. The candidate is asked to elaborate on the various definitions of Zariski decomposition and their relations with other problems, and give an account of some of the known results and counter-examples. Zariski decompositions have also been introduced in the context of arithmetic geometry, and the candidate may devote a part of the essay to this direction. Relevant Courses Essential: Algebraic Geometry References [1] Yu. Prokhorov; On Zariski decomposition problem. Proc. Steklov Inst. Math., 240:37–65, 2003. [2] N. Nakayama; Zariski decomposition and abundance. MSJ Memoirs 14, Tokyo (2004). [3] C. Birkar; On existence of log minimal models and weak Zariski decompositions. arXiv:0907.5182v1. [4] Y. Kawamata; The Zariski decomposition of log-canonical divisors. Proc. Symp. Pure Math. 46 (1987), Amer. Math. Soc., 425–433. [5] A. Moriwaki; Birational Arakelov geometry. Slides of a talk.

2. Geometric Group Theory, Measured Foliations, and Train-Tracks . . . . . . . Professor D.C. Calegari The theory of train tracks, introduced by Thurston, reduces the study of surface automorphisms to endomorphisms of (quite simple) 1-dimensional objects. This theory is locally linear, and gives rise to a piecewise integral projective structure on the Thurston boundary of Teichmuller space. A similar theory exists for automorphisms (endomorphisms) of free groups, and via the Rips machine is connected to Bass-Serre theory, foliated complexes, Makanin-Razborov diagrams, Sela’s theory, etc. The essay should aim to explain how train tracks can be used to compute and characterize dilatations of surface or free group automorphisms, how such dilatations vary in families, and to survey what is known about the complementary question of realizing algebraic numbers as dilatations. Background: no special background should be necessary to do any of these essays. Relevant Courses Useful: Introductory Algebraic Topology

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References [1] Thurston, W: On the geometry and dynamics of automorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417-431. [2] Bestvina, M: R-trees in topology, geometry and group theory, available from the author’s website [3] Matsumoto, S: Topological entropy and Thurston’s norm of atoroidal surface bundles over the circle, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987), no. 3, 763-778. [4] Thurston, W: Entropy in dimension one, preprint

3. Hilbert’s Third Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Professor D.C. Calegari Hilbert’s third problem asked to prove that it was impossible to decompose a cube into a finite collection of polyhedra and reassemble them into a regular tetrahedron of the same volume. This was solved by Dehn in the same year, who introduced the Dehn invariant of the scissors congruence class of a polyhedron. Sydler showed that equality of volume and Dehn invariant is sufficient for equidecomposibility; his proof was clarified by Jessen, and then translated into homological algebra by Sah, and the computation of the homology of classical groups made discrete. More recent work of Dupont-Sah, Neumann and Goncharov has connected the theory of Dehn invariants to central conjectures in number theory. This essay asks to give an explicit and elementary exposition of the Dehn-Sydler theorem in terms of the homology of low dimensional orthogonal groups, following Sah and Dupont-Sah, and to explain the issues that stand in the way of progress on the analogous problems for 3-dimensional hyperbolic or spherical geometry. Background: no special background should be necessary to do any of these essays. Relevant Courses Useful: Introductory Algebraic Topology References [1] Jessen, B: The algebra of polyhedra and the Dehn-Sydler theorem. Math. Scand. 22 1968 241-256 (1969). [2] Sah, C: Hilbert’s third problem: scissors congruence. Research Notes in Mathematics, 33. Pitman (Advanced Publishing Program), Boston, Mass.-London, 1979. [3] Dupont, J and Sah, C: Homology of Euclidean groups of motions made discrete and Euclidean scissors congruences. Acta Math. 164 (1990), no. 1-2, 1-27. [4] Neumann, W and Yang, J: Bloch invariants of hyperbolic 3-manifolds. Duke Math. J. 96 (1999), no. 1, 29-59. [5] Goncharov, A: Volumes of hyperbolic manifolds and mixed Tate motives. J. Amer. Math. Soc. 12 (1999), no. 2, 569-618. [6] Dupont, J: Algebra of polytopes and homology of flag complexes. Osaka J. Math. 19 (1982), no. 3, 599-641.

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4. Quasimorphisms and Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Professor D.C. Calegari The homological theory of transformation groups was brought to a high point by Mather and Thurston in the 70’s, and developed further by Banyaga and others. One negative outcome of this theory is the fact that transformation groups tend to have very few normal subgroups, and therefore essentially no interesting homomorphisms to groups small enough to understand or compute with. This deficit is rectified by the observation that many of these groups turn out to have (uncountably) many quasimorphisms — maps to the reals which are “almost” homomorphisms, and which capture a huge amount of dynamics. The essay should begin with an abstract discussion of quasimorphisms and their dual relationship to stable commutator length, move on to a discussion of various constructions of quasimorphisms of a dynamical origin and their properties, and end by summarizing what kinds of dynamical information can and can not be extracted from this point of view. Background: no special background should be necessary to do any of these essays. Relevant Courses Useful: Introductory Algebraic Topology References [1] Ruelle, D: Rotation numbers for diffeomorphisms and flows. Ann. Inst. H. Poincaré Phys. Théor. 42 (1985), no. 1, 109-115. [2] Ghys, E: Groupes d’homéomorphismes du cercle et cohomologie bornée. The Lefschetz centennial conference, Part III (Mexico City, 1984), 81-106, Contemp. Math., 58, III, Amer. Math. Soc., Providence, RI, 1987. [3] Entov, M and Polterovich, L. Calabi quasimorphism and quantum homology. Int. Math. Res. Not. 2003, no. 30, 1635-1676. [4] Gambaudo, J. and Ghys, E: Commutators and diffeomorphisms of surfaces. Ergodic Theory Dynam. Systems 24 (2004), no. 5, 1591-1617. [5] Ghys, E: Knots and dynamics. International Congress of Mathematicians. Vol. I, 247277, Eur. Math. Soc., Z¨ urich, 2007. [6] Calegari, D: scl. MSJ Memoirs, 20. Mathematical Society of Japan, Tokyo, 2009.

5. Values of the Riemann Zeta Function at the Odd Positive Integers . . . . . . Professor J.H. Coates The values of the Riemann zeta function at the odd positive integers have long fascinated number theorists, going back at least as far as Euler, and sum of their key properties are still unknown (for example, even whether or not they are all irrational numbers). Over the last 40 years, there has been a considerable body of work done on them via the methods of arithmetic geometry, after Borel’s proof that ζ(n), for n an odd integer > 1, is a rational number times a regulator coming from K2n−1 (Z). In particular, Bloch and Kato proposed a Tamagawa number type conjecture for these values, and made significant progress towards proving it. Their ideas were then taken up by Soule and Huber and Kings, using the methods of Iwasawa theory and 17

some difficult geometric techniques. The aim of the essay will be to write up at least part of this modern theory, explaining rather fully the background of whatever machinery from arithmetic geometry is used. See John Coates for further details. Relevant Courses Essential: An Introduction to Iwasawa Theory References [1] A. Huber, G. Kings, “Bloch-Kato conjecture and main conjecture of Iwasawa theory for Dirichlet characters, Duke Math. J. 119 (2003), 393-464. [2] C. Soule, “K-theorie des anneaux d’entiers de corps de nombres et cohomologie etale”, Invent. Math. 55 (1979), 251-295. [3] C. Soule, “On higher p-adic regulators”, Springer LNM 854 (1981), 372-401. [4] S. Bloch, K. Kato, L-functions and Tamagawa numbers of motives, Grothendieck Festschrift I, 333-400, Progress in Mathematics 86 (1990), Birkhauser.

6. Constructing Elliptic Curves of Large Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr T.A. Fisher Let E be an elliptic curve over Q. By Mordell’s theorem, the group of rational points E(Q) is a finitely generated abelian group. This means that E(Q) ∼ = T × Zr where T is a finite group. By a theorem of Mazur, there are only 15 possibilities for T . On the other hand, it is not known what values of the rank r are possible. The “folklore” conjecture is that the rank can be arbitrarily large. The current record is an example of an elliptic curve with rank ≥ 28, found by Elkies [2] in 2006 (the previous record was rank ≥ 24, found by Martin and McMillen in 2000, based on earlier work of Mestre [3]). This essay should describe some of the methods used to construct elliptic curves of large rank over Q(t). (The records for elliptic curves over Q are obtained by then specialising t to a rational number.) Suitable references may be found by consulting [1] or [4]. Relevant Courses Essential: Elliptic Curves Useful: Algebraic Geometry References [1] A. Dujella, History of elliptic curves rank records (website), http://web.math.hr/∼duje/tors/rankhist.html [2] N. D. Elkies, Three lectures on elliptic surfaces and curves of high rank, (July 2007), http://arxiv.org/abs/0709.2908 [3] J.-F. Mestre, Courbes elliptiques de rang ≥ 11 sur Q(t), C. R. Acad. Sci. Paris Sér. I Math. 313 (1991), no. 3, 139–142. [4] K. Rubin, A. Silverberg, Ranks of elliptic curves, Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 4, 455–474. 18

7. Counting Cubic and Quartic Number Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr T.A. Fisher Let Nd (X) be the number of degree d number fields whose discriminant has absolute value at most X. It is conjectured that Nd (X) ∼ cd X for some constant cd as X → ∞. The essay should start by proving this in the case d = 3 following the paper of Davenport and Heilbronn [6]. Their method relies on a connection between binary cubic forms and cubic number fields. The remainder of the essay should describe applications of these ideas to computational number theory (see [1],[2]) and explain how they generalise to the case d = 4 (see [3],[4]). Relevant Courses Essential: Local Fields Useful: Class Field Theory References [1] K. Belabas, A fast algorithm to compute cubic fields, Math. Comp. 66 (1997), no. 219, 1213–1237. [2] K. Belabas, On quadratic fields with large 3-rank, Math. Comp. 73 (2004), no. 248, 2061–2074. [3] M. Bhargava, Higher composition laws. III, The parametrization of quartic rings, Ann. of Math. (2) 159 (2004), no. 3, 1329–1360. [4] M. Bhargava, The density of discriminants of quartic rings and fields, Ann. of Math. (2) 162 (2005), no. 2, 1031–1063. [5] H. Cohen, F. Diaz y Diaz and M. Olivier, Counting discriminants of number fields, J. Théor. Nombres Bordeaux 18 (2006), no. 3, 573–593. [6] H. Davenport, and H. Heilbronn, On the density of discriminants of cubic fields. II, Proc. Roy. Soc. London Ser. A 322 (1971), no. 1551, 405–420.

8. Wellquasiorders and Betterquasiorders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr T.E. Forster A well-quasi-order is a reflexive transitive relation with no infinite descending chains and no infinite antichains. Although this may not sound natural there are many natural examples, at least one of which is famous: the theorem of Seymour and Robertson that finite graphs under the graph minor relation form a WQO. There is Laver’s theorem that the isomorphism types of scattered total orders (orders in which the rationals cannot be embedded) form a WQO. Finite trees with nodes labelled with elements of a WQO are also WQO-ed. The class of WQO’s lacks certain nice closure properties and this leads to a concept of Better-quasi-ordering. The class of BQOs is algebraically nicer. These combinatorial ideas have wide ramifications in graph theory, logic and computer science (lack of infinite descending chains is always liable to be connected with termination of processes) and the area has a good compact literature and some meaty theorems. Recommended for those of you who liked the Logic course and the Combinatorics course.

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A Big plus for this topic is that there is no textbook—tho’ I am trying to write one. There is a wealth of literature, of some of which I have photocopies. Interested students should discuss this with me. In 2009/10 we organised a reading group on this topic, and it was a huge success. Several participants of that group will be in Cambridge in 2011/12 and have expressed a willingness to repeat the exercise. I for one would be more than happy to repeat it. Relevant Courses Useful: Combinatorics, If there is a reading course in Set Theory that might be useful too

9. Set Theory without the Axiom of Foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr T.E. Forster The independence of the axiom of foundation can be easily established. Various axioms have been proposed which contradict foundation and these should be discussed and the consistency of a few proved. Asking oneself whether or not one believes the axiom of foundation prompts one to reflect on what one thinks sets are and what a formalised set theory is for. Various people have advocated set theory without foundation as a way of modelling other phenomena which exhibit illfoundedness. Examples are situation semantics (Barwise and Perry et al.) and Robin Milner’s work on concurrency. The appropriateness of this use of set theory is controversial and could be discussed. Typically the antifoundation axioms provoked by these motivations do not involve the existence of a universal set, and their consistency is generally unproblematic. In contrast set theories with a universal set have deep and poorly-understood connections with Type theory and Polymorphism. I would be prepared to discuss this matter with interested students and supply copies of relevant literature. I am also giving a graduate course in Michaelmas after the division of term which would be relevant. Relevant Courses Essential: Set Theory and Logic References [1] P. Aczel: Non-well-founded sets. CSLI 1988 [2] T.E. Forster: Set theory with a universal set (2nd edn ) OUP 1995 . . . and references therein. [3] Barwise and Moss: Vicious Circles, CSL

10. Fraenkel-Mostowski Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr T.E. Forster, Dr O. Caramello Fraenkel-Mostowski models for Set Theory were the technique behind the first successful partial attempt at establishing the independence of the Axiom of Choice from the other axioms of Zermelo-Fraenkel Set Theory. That the attempt was only partially successful was because the

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technique relies on the manufacture of atoms, and thereby violates extensionality (if the atoms are empty) or (in the case of a variant version due to Mendelson [3] where the atoms are objects x = {x}) violates foundation. The method rather fell out of use after Cohen’s discovery of forcing in the 1960’s (since by use of forcing one can obtain models that violate choice while satisfying both foundation and extensionality) but it has recently become of interest again. There are several reasons for this: (i) a modification due to Forster [1], which gives extensional wellfounded models that instead sacrifice some unstratified instances of replacement, (ii) The Gabbay-Pitts development of nominal Sets ([2]) and (iii) P. Freyd’s topos-theoretic methods for proving the independence of the axiom of choice ([4] and [5]). Classic expositions of FM models abound. The paper [7] surveys the different viewpoints in the subject of calculi dealing with names, including nominal sets, permutation algebras and categories of sheaves, explaining their interrelationships. An extensive treatment of Freyd’s models for the independence of the axiom of choice, explaining the connection between classical forcing and the topos-theoretic approaches to independence proofs in set theory, is carried out in [6]. Relevant Courses Useful: Topos Theory References [1] Thomas Forster. AC fails in the natural analogues of L and the cumulative hierarchy that model the stratified fragment of ZF. . . in Contemporary Mathematics 36 2004. [2] Jamie Gabbay. Foundations of nominal techniques: logic and semantics of variables in abstract syntax, Bulletin of Symbolic Logic 2011, also http://gabbay.org.uk/papers.html#fountl [3] Eliot Mendelson. The Axiom of Fundierung and the Axiom of Choice Arch Math 4 pp 65–70. [4] P. J. Freyd. The axiom of choice. J. Pure Appl. Algebra, 19 (1980), pp 103125. [5] P. J. Freyd, All topoi are localic; or, Why permutation models prevail, J. Pure Appl Alg. 46 (1987), pp 49-58. [6] A. R. Blass and A. Scedrov, Freyd’s models for the independence of the axiom of choice, Mem. Am. Math. Soc., 79 (404), 1989. [7] F. Gadducci, M. Miculan and U. Montanari, About permutation algebras, (pre)sheaves and named sets, Higher-Order and Symbolic Computation, 19 (September 2006) Issue 2-3.

11. Expansion in Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Professor B.J. Green −1 Let G be a finite group, and let S = {s1 , s−1 1 , . . . , sk , sk } be a symmetric set of generators for G. A very strong and useful property that S can have is that of being an expander. There are several equivalent definitions of this; one is that the distribution of a random product of n elements from S is very close to uniform as soon as n is around log |G|. Another is that any set A ⊆ G with |A| < |G|/2 “grows” by a factor of at least 1 + ² when you multiply it by S.

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The essay should start with a brief introduction to the expansion property and a proof that the various different definitions are equivalent. It should then give at least one proof that expanders of constant size exist in certain families of groups, for which there are various arguments of somewhat varying difficulty including the following: 1. Margulis’s argument for G = SL3 (Fp ) using the fact that SL3 (Z) has property T (which should be proven); 2. A proof for G = SL2 (Fp ) using Selberg’s theorem about the first nontrivial eigenvalue of modular surfaces; 3. The method of Bourgain and Gamburd for G = SL2 (Fp ) using additive combinatorics; 4. Kassabov’s argument for the alternating groups An . The essay might conclude with a discussion of some applications. Relevant Courses An interest in dipping into several unrelated areas (group theory, number theory, analysis...) is much more important than any serious background in any one of them. In particular, group theory beyond a basic undergraduate course is not required. Tim Gowers’s graduate course on Additive Combinatorics would help with (3). Some basic knowledge of modular forms might help with (2), but you could probably learn what you need from scratch. Some basic functional analysis and measure theory would help in the discussions of Property T necessary for (1) and (4). Reference [1] is superb, and students interested in this essay should turn there first and then come and talk to me. References [1] Alexander Lubotzky, Expanders Graphs in Pure and Applied Mathematics, notes prepared for the Colloquium Lectures at the Joint Annual Meeting of the American Mathematical Society, available online by Google Search for Alex Lubotzky Expanders.

12. Hilbert’s Fifth Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Professor B.J. Green At the start of the twentieth century David Hilbert, the foremost mathematician of the time, formulated a list of 23 problems to occupy the minds of mathematicians in the coming century. The fifth one asked “Are continuous groups automatically differential groups?” The question as posed is a little vague, but for at least one natural formulation it was solved by Gleason and Yamabe in the 1950s. This theorem states that every locally compact group is a “locally a Lie group” in a certain sense. A Lie group is a rather special type of mathematical object, whereas a priori a locally compact topological group is quite general. The theorem is therefore somewhat remarkable. The techniques they used, as well as the result itself, have very recently become important in additive combinatorics in connection with the study of so-called approximate groups.

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This essay is about Hilbert’s fifth problem, and the core of it should be an account, not necessarily with absolutely all the details, of the Gleason-Yamabe theorem. You could also include some history of the problem, and sketch links to additive combinatorics or to Gromov’s theorem on groups of polynomial growth. Relevant Courses None, though Lie Algebras might be a little bit useful. A good grounding in undergraduate analysis will be very helpful. Thankfully, Terence Tao is currently (Michaelmas 2011) giving a course on precisely this topic at UCLA; his notes [1] will be an important resource for this essay but you will need to distil them down quite a bit, and should consult other works too. Interested students should take a look at [1] and then come and talk to me. References [1] Terence Tao, Math 254A: Hilbert’s Fifth Problem, http://www.math.ucla.edu/˜tao/254a.1.11f/

13. Yau’s Solution of the Calabi Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr A.G. Kovalev The subject area of this essay is compact Kähler manifolds. Very informally, a Kähler manifold is a complex manifold admitting a metric and a symplectic form, both nicely compatible with the complex structure. The Ricci curvature of a Kähler manifold may be equivalently expressed as a differential form which is necessarily closed. Furthermore, the cohomology class defined by this form depends only of the complex manifold, but not on the choice of Kähler metric. The Calabi conjecture determines which differential forms on a compact complex manifold can be realized by Ricci forms of some Kähler metric. Substantial progress on the conjecture was made by Aubin and it was eventually proved by Yau. This result gives, among other thing, a powerful way to find many examples of Ricci-flat manifolds. The essay could discuss aspects of the proof and possibly consider some applications and examples. Relevant Courses Essential : Differential Geometry, Complex Manifolds Useful : Algebraic Topology, Elliptic Partial Differential Equations References [1] D. Joyce, Riemannian holonomy groups and calibrated geometry. OUP 2007 [2] S.-T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex MongeAmpère equation. I. Comm. Pure Appl. Math., 31 (1978), 339–411. [3] a good text on Kähler complex manifolds, e.g. D. Huybrechts, Complex geometry. An introduction. Springer 2005.

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14. Stable Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr A.G. Kovalev The term ‘stable form’ was suggested by Hitchin for an alternating form ω ∈ Λp (Rn )∗ which has an open orbit in the natural action of GL(n). Stable forms generalize the notion of nondegenerate bilinear forms to alternating forms of arbitrary degree p and non-trivial examples include stable forms of degree p = 3 in dimensions n = 6 and 7. The essay could begin by exploring the work [1] and showing how stable differential 3-forms lead to Ricci-flat metrics on 6- and 7-dimensional manifolds, as critical points of a ‘volume’ functional. In dimension 6 this gives an alternative view on Calabi–Yau complex algebraic 3-folds and in dimension 7 a special geometry associated with the Lie group G2 . The essay could investigate the significance of stable forms in one of these two geometries and its relation to metrics with special holonomy [2]. Relevant Courses Essential: Differential Geometry, Complex Manifolds Useful: Algebraic Topology, Elliptic Partial Differential Equations References [1] N. Hitchin, The geometry of three-forms in six and seven dimensions, arXiv:math.DG/0010054 or J. Differential Geom. 55 (2000), 547–576. [2] D. Joyce, Compact manifolds with special holonomy, Oxford University Press 2000. [3] a good text on Kähler complex manifolds, e.g. D. Huybrechts, Complex geometry. An introduction. Springer 2005.

15. K3 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr A.G. Kovalev A K3 surface may be defined as a simply-connected complex surface which admits a nowherevanishing differential (2,0)-form, sometimes called a holomorphic volume form (it may also be viewed as a holomorphic symplectic form). One example is a hypersurface in CP 3 defined by the equation z04 + z14 + z24 + z34 = 0. All the K3 surfaces are homeomorphic to each other, but not every K3 surface can be given algebraically, as a zero locus of polynomial(s). The essay could investigate basic topological and holomorphic invariants of K3 surfaces and discuss examples, then look into some further results and applications (e.g. Torelli theorem, Ricci-flat Kähler metrics, elliptic fibrations). Interested candidates are welcome to contact me for further details. Relevant Courses Essential: Differential Geometry, Complex Manifolds Useful: Algebraic Topology, Algebraic Geometry

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References [1] P. Griffiths and J. Harris, Principles of algebraic geometry, Wiley, 1978. [2] W. Barth, K. Hulek, C. Peters, A. van de Ven, Compact complex surfaces, Springer, 2004. Chapter VIII. [3] P.S. Aspinwall, K3 surfaces and string duality, arXiv:hep-th/9611137. Chapter 2.

16. Canonical Ramsey Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Professor I.B. Leader Canonical Ramsey theorems extend classical Ramsey theorems, which typically involve colourings involving a specified finite number of colours, to arbitrary colourings. The flavour is often different to that of the classical Ramsey theorems, although there is usually some relationship between a canonical theorem and a higher-order classical theorem. There would be two themes to the essay. One is the question of bounds for finite canonical Ramsey theorems, dealing with work of Duffus, Lefmann and Rodl and especially some remarkable bounds of Shelah that more or less completely answer the growth-speed questions. The other is the question of what the actual canonical version of a classical theorem should be, focusing on the canonical Gallai theorem due to Deuber, Graham, Promel and Voigt and the canonical Hindman theorem due to Taylor. Relevant Courses Useful: Ramsey Theory References [1] Shift graphs and lower bounds on Ramsey numbers rk (l; r), D. Duffus, H. Lefmann and V. Rodl, Discrete Math, vol 137 (1995), 177-187. [2] On Erdos-Rado numbers, H. Lefmann and V. Rodl, Combinatorica, vol 15 (1995), 85-104. [3] Finite canonization, S. Shelah, Comment. Math. Univ. Carolinae, vol 37 (1996), 445-456. [4] A canonical partition theorem for equivalence relations on Z t , W. Deuber, R. Graham, H. Promel and B. Voigt, Journal of Combinatorial Theory Series A, vol 34 (1983), 331-339. [5] A canonical partition relation for finite subsets of omega, A. Taylor, Journal of Combinatorial Theory Series A, vol 21 (1976), 137-146.

17. Pursuit on Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Professor I.B. Leader There has recently been much interest in pursuit questions on graphs. Typically, we have a number of pursuers, working as a team to catch an evader. If this takes place on a graph (so the players live on the vertices of the graph, and in each time-step they move to an adjacent vertex), how many pursuers are needed? And how does this relate to properties of the graph? Most work centres around the conjecture of Meyniel, still unproved, that the number of pursuers need be no more than about the square-root of the number of vertices.

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The essay would focus on some results for general graphs, and also on specific cases of interest like random graphs. There are some remarkable constructions (e.g. reference 4) and also some strange phenomena (e.g. reference 3). The papers below are all available online. Relevant Courses None References [1] Cops and robbers in graphs with large girth and Cayley graphs, P. Frankl [2] A new bound for the cops and robbers problem, A. Scott and B. Sudakov [3] Chasing robbers on random graphs: zigzag theorem, T. Luczak and P. Pralat [4] On a generalization of Meyniel’s conjecture on the cops and robbers game, N. Alon and A. Mehrabian.

18. Combinatorial Algorithms in the Representation Theory of Symmetric Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr S. Martin In my lecture course you will learn that many fundamental results about representations of symmetric groups [1] can be derived in a purely combinatorial manner. A good example of this phenomenon is the theorem asserting that the dimension, f λ , of the Specht module (equivalently, the degree of the ordinary irreducible character corresponding to the cycle type λ) equals the number of standard Young tableaux of type λ. The essay could deal with treatments of other algorithmic proofs of well known results for symmetric groups, for example (a) algorithms to prove the Frame-Robinson-Thrall hook formula and the Frobenius-Young determinantal formula; (b) P the Robinson-Schensted-Knuth algorithm, which provides a bijective proof that λ 2 λ`n (f ) = n!; (c) Sch¨ utzenberger’s jeu de taquin (the ‘teasing game’) which gives an alternative approach to (b) and to the so-called Knuth relations. There is a gentle account of this in Chapter 3 of Sagan’s book [3]; there are also some interesting illustrative but substantial exercises at the end of his chapter which could be incorporated into part of the body of the essay. An alternative approach is Stanley’s comprehensive tome [4] (Chapter 7). Relevant Courses Essential: Representation Theory Useful: Lie Algebras; Commutative Algebra; knowledge of ordinary character theory, e.g. from [2]. 26

References [1] G.D. James, The representation theory of the symmetric groups, SLN 682, Springer (1978). [2] W. Ledermann, Introduction to group characters, CUP (1977). [2] Bruce E. Sagan, The Symmetric group: representations, combinatorial algorithms and symmetric functions, 2nd edn. GTM 203, Springer 2001. [4] R.P. Stanley, Enumerative combinatorics, Volume 2, CUP (2001).

19. Non-Commutative Character Theory of the Symmetric Group . . . . . . . . . Dr S. Martin The lecture course gave a classical account of the linear and polynomial representation theory of symmetric groups (see also [3]) and general linear groups, with the theory based on the (commutative) C-algebra of class functions (or, with an obvious change of reference frame, of symmetric functions, as per the second essay). Following an idea of Solomon in a 1976 paper [4] in which he gives a non-commutative refinement of Mackey’s Theorem, there is a certain non-commutative superstructure which maps onto the algebra of class functions. The idea now is to transfer problems to be solved into the non-commutative setting. While the language is still combinatorial, in many ways this noncommutative world is more accessible than the classical theory, in fact more transparent and rather elementary, hence is perfect for novices to cut their teeth using the new tools offered in this approach. A splendid little book [1] gives an excellent and straightforward outline of the theory, with Parts I and II forming the main body of knowledge for the essay. Background knowledge also appears in [4]. Relevant courses Essential Representation Theory, Commutative Algebra Useful: ordinary character theory and basic group theory; Lie algebras References [1] D. Blessenohl and M. Schocker, Noncommutative character theory of the symmetric group, Imperial College Press (2005). [2] W. Fulton, Young tableaux, CUP (1997). [3] G.D. James, The representation theory of the symmetric groups, SLN 682, Springer (1978). [4] Louis Solomon, A Mackey formula in the group ring of a Coxeter group, J. Algebra 41, 255–268 (1976).

20. Symmetric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr S. Martin Results on representations of symmetric groups [1] can be proved using general facts from representation theory or we can deploy combinatorial methods as in the spirit of the first essay. 27

There is however a third way using symmetric functions, a little of which has been presented in my lecture course. At a basic level one can easily generalise the hook formula by considering a hook generating function for semistandard tableaux using a wonderful algorithm due to Hilman and Grasl. A supremely important class of symmetric function is the Schur function, with the Jacobi-Trudi determinants for the Schur functions mirroring the determinantal formula for calculating the dimension f λ . Other applications include the famous Littlewood-Richardson Rule for decomposing tensor products into irreducible summands and the Murnaghan-Nakayama Rule which gives an algorithmic way to calculate characters. The best account is in Chapter I of Macdonald [2]; there are alternative accounts in Chapter 4 of [3] and Chapter 7 of [4]. Relevant Courses Essential: Representation Theory, Commutative Algebra Useful: ordinary character theory and basic group theory References [1] G.D. James, The representation theory of the symmetric groups, SLN 682, Springer (1978). [2] I.G. Macdonald, Symmetric functions and Hall polynomials, OUP 1995 [3] Bruce E. Sagan, The Symmetric group: representations, combinatorial algorithms and symmetric functions, 2nd edn. GTM 203, Springer (2001). [4] R.P. Stanley, Enumerative combinatorics, Volume 2 CUP (2001).

21. Penrose Stability Criterion for Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr C.G. Mouhot The goal of this essay shall be to understand the mathematical proof of the famous “Penrose stability criterion” in plasma physics and galactic dynamics (1960) and its link with the linearized computation of Landau (1946) of the damping of spatial waves. The mathematical object is a partial differential equation, the linearized Vlasov-Poisson equation (when nonlinearity is neglected). Some mathematical tools from spectral theory and complex variable analysis will be used. The essay shall consist in providing a self-contained introduction to the linearized Vlasov-Poisson equation, an introduction to these linearized stability results, together with a self-contained proof following the references given. Then depending on time the student could then study nonlinear stability results in [7] related to this criterion. Relevant Courses Essential: Mathematical Topics in Kinetic Theory Useful: Perturbation and Stability Methods, Functional Analysis.

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References [1] Landau, L. On the vibration of the electronic plasma. J. Phys. USSR 10, 25 (1946). [2] Penrose, O. Electrostatic instability of a non-Maxwellian plasma. Phys. Fluids 3 (1960), 258-265. [3] Lifshitz, E. M., and Pitaevski L. P. Course of theoretical physics (“Landau-Lifshitz”) Vol. 10. [4] Maslov, V. P.; Fedoryuk, M. V. The linear theory of Landau damping. Mat. Sb. (N.S.) 127(169) (1985), no. 4, 445-559. [5] Degond, P. Spectral theory of the linearized Vlasov-Poisson equation. Trans. Amer. Math. Soc. 294 (1986), no. 2, 435-453. [6] Mouhot, C., and Villani, C. On Landau damping, arXiv:

0904.2760 (Section 3).

[7] Rein, G. Non-linear stability for the VlasovPoisson system – the energy-Casimir method, Math. Methods Appl. Sci. 17 (1994), no. 14, 1129-1140.

22. The Calder´ on Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Professor G.P. Paternain In this essay you will study the famous inverse conductivity problem due to Calderón [1]. This problem is the model for Electrical Impedance Tomography, an imaging modality proposed for use in medical and seismic imaging. In the case of anisotropic media in dimensions three and higher, the problem reduces to the question of determining a Riemannian manifold from the elliptic Dirichlet to Neumann map (or Cauchy data of harmonic functions) on its boundary. The problem is still open, but there has been some remarkable recent progress. Dos Santos Ferreira, Kenig, Salo and Uhlmann [2] opened a new direction by showing that certain smooth manifolds in a fixed conformal class are determined by the Dirichlet to Neumann map. The objective of this essay is to explain the main ideas that go into this paper and to discuss the Calderón problem in general. An excellent reference for this is the recent text [3], which also contains all the relevant background material. Relevant Courses Useful: Differential Geometry, Elliptic Partial Differential Equations References [1] A.P. Calderón, On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980) 65–73, Soc. Brasil. Mat., Rio de Janeiro, 1980. [2] D. Dos Santos Ferreira, C.E. Kenig, M. Salo, G. Uhlmann, Limiting Carleman weights and anisotropic inverse problems, Invent. Math. 178 (2009) 119–171. [3] M. Salo, The Calder´ on problem on Riemannian manifolds, available at http://www.rni.helsinki.fi/~msa/pub/index.html.

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23. Branched Double Covers of Alternating Knots . . . . . . . . . . . . . . . . . . . . . . . . . Dr J.A. Rasmussen Give a knot K in the three-sphere, we can form its double-branched cover. This is a three manifold D(K) with a natural projection D(K) → S 3 so that every point in S 3 − K has two preimages, while points in K have only one. It is not hard to find examples of different knots with the same double branched cover; e.g. any two knots which are Conway mutants of each other. (For an example of Conway mutation, take a look at the main gates of the CMS.) A recent theorem of Greene [1] provides a partial converse to this fact: if K and K 0 are alternating knots with the same double branched cover, then they are related by a sequence of Conway mutations. The aim of this essay is to understand Greene’s proof, which combines results in graph theory with an invariant of three-manifolds defined using Heegaard Floer homology. Relevant Courses Useful: Algebraic Topology, Graph Theory, Knots and 4-Manifolds References [1] J. Greene, Lattices, Graphs, and Conway Mutation; arXiv:1103.0487. [2] L. Kauffman, On Knots; Princeton University Press, 1987. [3] P. Ozsváth and Z. Szabó, On the Heegaard Floer homology of branched double covers, Adv. Math. (2005), 1-33.

24. Khovanov Homology for Tangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr J.A. Rasmussen Khovanov homology is an invariant of knots in the 3-sphere which generalizes the Jones polynomial. It is quite simple to define, but its geometric meaning is far from understood. A tangle is an embedded collection of arcs and circles in the 3-ball; two tangles are equivalent if they can be isotoped into each other while holding the end-points fixed. The aim of this essay is to understand how the definition of Khovanov homology can be generalized to tangles. To get started on the essay, you should begin by understanding Bar-Natan’s papers [1] and [2]. The second paper gives a “geometrical” definition of Khovanov homology which applies to tangles as well as knots. Next, you should learn how to rephrase this definition in terms of the Temperly-Lieb algebra. Finally, you should explain either how these ideas are generalized in Khovanov’s definition of the HOMFLY homology via the Rouquier complex [3], or how they are applied in Rozansky’s construction of a categorified Jones-Wenzl projector [4]. Relevant Courses Useful: Algebraic Topology, Knots and 4-Manifolds, Representation Theory and Invariant Theory

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References [1] D. Bar-Natan, On Khovanov’s categorification of the Jones polynomial, Algebraic and Geometric Topology 2 (2002) 337-370. arXiv:math/0201043 [2] D. Bar-Natan, Khovanov’s homology for tangles and cobordisms, Geometry and Topology 9 (2005), 1443-1499. arXiv:math/0410495 [3] M. Khovanov, Triply-graded link homology and Hochschild homology of Soergel bimodules, Int. J. Math. 18 (2007) 869-885. arXiv:math/0510265 [4] L. Rozansky, An infinite torus braid yields a categorified Jones-Wenzl projector. arXiv:1011.1958

25. K¨ ahler-Einstein Metrics on Fano Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr J.A. Ross The goal of this essay is to give an account of some of the theory behind Kähler-Einstein metrics, in particular in the case of Fano manifolds. Kähler-Einstein metrics play an important role in various branches of mathematics and mathematical physics (for example Calabi-Yau manifolds) and provide a beautiful link between Kähler and algebraic geometry. The essay should start with a review of Kähler manifolds, the definition of Kähler-Einstein metrics and a description of the three cases as separated by the first Chern class of the manifold [1]. It will then focus on the case of Fano manifolds, and in particular the sort of obstructions that are known to exist, from both an analytic and algebraic point of view (say following both [1] and [2] whilst putting in some of the background material). It should include specific examples, including a description for the case of Fano surfaces and possibly an account of the case of the Mukai threefold [Section 5, 3]. Relevant Courses Essential: Algebraic Geometry (MT), Complex Geometry (LT) References [1] G. Tian, Canonical Metrics in Khler Geometry (Lectures in Mathematics. ETH Zrich) [2] A Futaki, Asymptotic Chow Polystability in Kahler Geometry arXiv:1105.4773v2. [3] S Donaldson, Kahler geometry on toric manifolds, and some other manifolds with large symmetry arXiv:0803.0985.

26. Lagrangian Klein Bottles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr I. Smith A basic question in symplectic topology concerns the (non-)existence of Lagrangian submanifolds of Euclidean space of prescribed topology. Fairly recently, Shevchishin and Nemirovskii proved a long-standing conjecture that there are no Lagrangian Klein bottles in Euclidean space. Nemirovskii’s argument starts from a Lagrange embedding and uses surgery and information coming from classical self-linking invariants to produce an impossible symplectic manifold. The impossibility stems from a uniqueness theorem due to Eliashberg-Floer-McDuff for symplectic 31

manifolds with the standard contact sphere as boundary, which in turn relies on ideas from holomorphic curve theory. The essay will explain Nemirovskii’s argument, and then the topological aspects of the E-F-M theorem, taking an axiomatic approach to the theory of holomorphic curves and Gromov-Witten invariants underpinning the latter. Relevant Courses Essential: Algebraic Topology, Differential Geometry, Symplectic Geometry. Useful: Complex Manifolds. References [1] S. Nemirovskii. Lagrangian Klein bottles in R2n . Geom. Funct. Anal. 19 (2009), available at arXiv:0712.1760. [2] D. McDuff. Symplectic manifolds with contact type boundaries. Invent. Math. 103 (1991), 651-671.

27. Non-K¨ ahler Calabi-Yau Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr I. Smith Calabi-Yau manifolds are algebraic varieties with trivial canonical bundle. They occupy a special place in conjectural classification schemes for varieties. Recently, there has been interest in manifolds which are not algebraic but share important features with Calabi-Yau’s; either symplectic manifolds with vanishing first Chern class, or complex manifolds with holomorphically trivial canonical bundle (in each case with no compatible Kähler structure). This essay will explain a pretty construction of (simply-connected) examples of at least one and ideally both type(s) of such manifold, due to Fine and Panov, starting from hyperbolic knots in 3-space and hyperbolic structures on certain 4-manifolds or more generally 4-orbifolds. Relevant Courses Essential: Algebraic Topology, Differential Geometry, Symplectic Geometry. Useful: Complex Manifolds. References [1] J. Fine, D. Panov. Symplectic Calabi-Yau manifolds, minimal surfaces and the hyperbolic geometry of the conifold. J.Diff. Geom. 82 (2009), available at arXiv:0802.3648. [2] J. Fine and D. Panov, Hyperbolic geometry and non-K¨ ahler manifolds with trivial canonical class. Geom. Topol. 14 (2010), available at arXiv:0905.3237. [3] J. Fine and D. Panov, The diversity of symplectic Calabi-Yau six-manifolds. Preprint, available at arXiv:1108.5944.

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28. Ramsey Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Professor A.G. Thomason Ramsey long ago proved that if N is large enough then every colouring of the edges of the (s) (s) (s) complete s-uniform hypergraph KN with red and blue must include a red Km or a blue Kn . Nevertheless the value of rs (m, n), the smallest value of N for which the statement is true, remains a mystery in nearly all cases. Recently, though, there has been some progress, such as the improved bound [3] on r2 (m, n) and on r3 (m, n). Similarly, estimates for the Ramsey numbers of bounded degree graphs, classically treated by Szemerédi’s Regularity Lemma [1], have recently been greatly improved by different methods [2,4]. This essay will briefly survey the classical work and give a detailed discussion of some recent methods and results. Relevant Courses Useful: Combinatorics, Ramsey theory, Percolation on Graphs References [1] V. Chvatál, V. Rödl, E. Szemerédi and W.T. Trotter, The Ramsey number of a graph with bounded maximum degree, J. Combinatorial Theory Ser. B 34 (1983), 239–243. [2] R.L. Graham, V. Rödl and A. Ruci´ nski, On graphs with linear Ramsey numbers, J. Graph Theory 25 (2000), 176–192. [3] D. Conlon, A new upper bound for diagonal Ramsey numbers, Annals of Math. 170 (2009), 941–960. [4] D. Conlon, J. Fox and B. Sudakov, Hypergraph Ramsey numbers, J. Amer. Math. Soc. 23 (2010), 247–266.

29. Tutte’s Flow Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Professor A.G. Thomason A nowhere-zero k-flow in a graph is an assignment to each edge of a direction and an integer in {1, . . . , k − 1} so that the total flow into any vertex is zero (modulo k). Interest in such flows began with Tutte, whose conjecture that every bridgeless graph has a nowhere-zero 5-flow implies the 5-colour theorem for planar graphs. He later conjectured that every bridgeless graph without a subdivision of the Petersen graph has a 4-flow, which implies the 4-colour theorem, and that every 4-edge-connected graph has a nowhere-zero 3-flow. Jaeger, whose circular flow conjecture unifies some of the previous conjectures, also proposed weaker versions: for example, is there a k such that every k-edge-connected graph has a 3-flow? Surprisingly, this question has only just been answered. This essay will discuss some of the major proofs in this area. Relevant Courses Useful: Combinatorics, Ramsey Theory, Percolation on Graphs

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References [1] W.T. Tutte, A contribution to the theory of chromatic polynomials, Canad. J. Math. 6 (1954), 80–91. [2] P.D. Seymour, Nowhere-zero 6 flows, J. Combinatorial Theory Ser. B 30 (1981), 130–135. [3] F. Jaeger, On circular flows in graphs, Proc. Colloq. Math. János Bolyai 37 (1984), 391–402. [4] C. Thomassen, The weak 3-flow conjecture and the weak circular flow conjecture, J. Combinatorial Theory Ser. B (2011), doi:10.1016/j.jctb.2011.09.003.

30. Auslander–Gorenstein Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr S.J. Wadsley Lots of interesting rings are Auslander–Gorenstein. A ring is Auslander–Gorenstein if certain homological properties of its category of modules are well-behaved in specific ways. I imagine an essay in this area would begin by discussing the necessary properties for a ring to be Auslander–Gorenstein and some of the immediate applications of them such as the definition of a canonical dimension function. You might then continue with a discussion of some of the natural examples probably including an explanation of why a Zariski-filtered ring with Auslander–Gorenstein graded ring is itself Auslander–Gorenstein — a result that yields lots of examples. You would then probably finish by considering how the Auslander–Gorenstein condition has been used to better understand some of these specific examples. You will need to learn some non-commutative algebra to do this essay. Relevant Courses Essential: Commutative algebra References [1] J. Clark, Auslander–Gorenstein rings for beginners, International Symposium on Ring Theory (Kyongju 1999), 95–115 [2] K. Ajitabh, S. P. Smith and J. J. Zhang, Auslander–Gorenstein rings, Comm. Algebra 26 (1998), 2159–2180 [3] J. E. Björk and E. K. Ekström, Filtered Auslander–Gorenstein rings, Operator algebras, unitary representations, enveloping algebras and invariant theory (Paris, 1989), Progr. Math., vol 92, Birkhäuser, Boston, 1990, 425–448. [4] J. T. Stafford and J. J. Zhang, Homological properties of (graded) Noetherian PI rings, J. Algebra 168 (1994), 988–1026

31. Complex Moduli Spaces for Calabi–Yau Manifolds . . . . . . . . . . . . . . . . . . . . . Professor P.M.H. Wilson Calabi–Yau manifolds have played a central role in both geometry and physics over the past 25 years, in particular in complex dimension three, and the topic of their complex moduli spaces is a very wide-ranging one, where you will need to make a choice of material to cover. I think that

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any essay would probably need to cover such topics as the unobstructed theorem (Bogomolov– Tian–Todorov), the period map, the Weil–Petersson metric, and the fact the moduli spaces are special Kähler manifolds, but you will certainly wish to look at references other than those given below and probably also more recent than these. Relevant Courses Essential: Complex Manifolds, Differential Geometry Useful: Algebraic Geometry, Riemannian Geometry References [1] M. Gross: Calabi–Yau manifolds and mirror symmetry. In: Gross, Huybrechts, Joyce: Calabi–Yau manifolds and related geometries, Springer 2003, pp 71 -159. [2] C. Voisin: Mirror Symmetry, SMF/AMS Texts, 1999. [3] J. Bertin, C. Peters: Variations de structures de Hodge, variétés de Calabi–Yau et symétrie miroir. In: Bertin, Demailly, Illusie, Peters: Introduction à la théorie de Hodge, SMF 1996, pp 169-256. [4] A.N. Todorov: The Weil–Petersson geometry of the moduli space of SU (n ≤ 3) (Calabi–Yau) manifolds, I. Comm. Math. Phys. 126, 269-391 (1989). [5] C.L. Wang: On the incompleteness of the Weil–Petersson metric along degenerations of Calabi–Yau manifolds. Math. Res. Let. 4, 157-171 (1997). [6] C.L. Wang: Curvature properties of the Calabi–Yau moduli. Doc. Math. 8, 577-590 (2003). [7] A. Strominger: Special Geometry. Comm. Math. Phys. 133, 163-180 (1990). [8] D.S. Freed: Special Kähler manifolds. Comm. Math. Phys. 203, 31-52 (1999). [9] Z. Lu, X. Sun: Weil–Petersson geometry on the moduli space of polarized Calabi–Yau manifolds. J. Inst. Math. Jussieu 3, 185-229 (2004).

32. Local Langlands Correspondences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr T. Yoshida The Langlands correspondences are non-abelian generalizations of the class field theory, which amounts to the GL1 -case of the correspondence. Just like class field theory, there are global and local versions. The goal of this essay would be to state the local Langlands correspondence for GLn , and to define the objects involved in the statement (it is rather hard to define all of them). One should at least show that the local class field theory gives its GL1 -case. There is a textbook on GL2 -case ([4]), but it is still good to look at the original papers [1], [2] (and [3]) for the statement. One can elaborate on the Galois side, the representation theory side, or both; for the former, start with [5], especially the basics on Weil-Deligne representations, including their equivalence with `-adic representations of the Weil groups (`-adic monodromy theorem); for the latter, describe the classification of irreducible smooth representations of GLn over p-adic fields in terms of supercuspidals (e.g. [6]). In both sides, one can proceed to defining L and ε-factors by looking into “Tate’s thesis” and the Whittaker models; you may wish to understand its connection to automorphic representations.

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Relevant Courses Essential: Local Fields, Class Field Theory References [1] M. Harris, R. Taylor, The Geometry and Cohomology of Some Simple Shimura Varieties, Ann. of Math. Studies 151, Princeton UP, 2001. [2] G. Henniart, Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique, Invent. Math. 139 (2000), 439–455. [3] G. Henniart, Une caractérisation de la correspondance de Langlands locale pour GL(n), Bull. SMF 130-4, 2002, 587–602. [4] C.J. Bushnell, G. Henniart, The local Langlands conjecture for GL(2), Grundlehren der Math. Wiss. 335, Springer, 2006. [5] J. Tate, Number theoretic background, in: Automorphic forms, representations and Lfunctions, Proc. Symp. Pure Math., 33-2, AMS, 1979, 3–26. [6] F. Rodier, Représentations de GL(n, k) o` u k est un corps p-adique, Séminaire Bourbaki 1981/82, exp. 587, 201–218 (available at numdam.org).

33. Etale Cohomology and Galois Representations . . . . . . . . . . . . . . . . . . . . . . . . . Dr T. Yoshida Many of the recent developments in algebraic number theory came from the understanding of the absolute Galois group GQ of Q (or number fields) through its representations on finitedimensional Q` -vector spaces V (`-adic representations), and most of the interesting `-adic representations are obtained in the etale cohomology V = H i (XQ , Q` ) for varieties X over number fields. This essay should study at least one aspect of the theory of etale cohomology, with a view towards Galois representations. Here are some examples in the foundations: (a) the notion of Grothendieck topology underlying the theory; the proof of fpqc-descent and why schemes are sheaves in etale topology; (b) why V is finite-dimensional (proper base change/finiteness); (c) why V is unramified at almost all primes (smooth base change); (d) why V is acted and decomposed by algebraic correspondences (pure motives); (e) why correspondences (most notably Frobenius) act on V with Z-coefficient characteristic polynomials (Lefschetz trace formula); (f) why V has a fixed Frobenius weight i at all good primes (Weil conjectures). Each of these ingredients has its own share of interesting commutative/homological algebra in it; one can start with [1], consulting [2],[3] and the notes by J.S. Milne on his website. Grothendieck’s original SGA may not be as inaccessible as it first appears. One should keep in mind the “real-world” examples of Galois representations; either (g) Tate modules of elliptic curves and abelian varieties; CM abelian varieties give abelian Galois representations, i.e. GL1 -theory ([4]), or (h) Galois representations associated to modular forms on GL2 ([5]). Relevant Courses Essential: Commutative Algebra, Algebraic Geometry.

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References [1] P. Deligne, et al., Cohomologie étale: SGA 4-1/2, Lecture Notes in Mathematics 569, Springer, 1977. ´ [2] E. Freitag, R. Kiehl, Etale cohomology and the Weil conjecture, Ergeb. Math. Grenzgeb. (3)-13, Springer, 1988. [3] L. Fu, Etale cohomology theory, Nankai Tracts in Math. 13, World Scientific, 2011. [4] A. Weil, On the theory of complex multiplication, in: Proceedings of the international symposium on algebraic number theory (Tokyo & Nikko, 1955) (see his collected works). [5] Deligne, P., Formes modulaires et représentations `-adiques, Bourbaki Seminar 1968/69, exp. 355, 139–172 (available at numdam.org).

34. The Cosmic Censorship Conjectures in General Relativity . . . . . . . . . . . . . Professor M. Dafermos Penrose’s celebrated incompleteness theorem states that under physically plausible initial conditions, the unique classical spacetime that develops as a solution to the Einstein equations is geodesically incomplete. That is to say, there exist observers whose fate classical theory cannot predict. What happens to these poor observers? And does this have implications for other observers, in particular, those very far away? These are the questions of cosmic censorship. The weak cosmic censorship conjecture says that geodesic incompleteness is generically hidden in black hole regions. In particular, far away observers can still observe for all time without ‘seeing’ any singular behaviour. The strong cosmic censorship conjecture says that the classical spacetime determined uniquely by appropriate initial conditions is generically inextendible as a classical spacetime. Under a suitable formulation, this has the interpretation that those macroscopic observers who do not live forever are in fact destroyed. It may not be a pleasant prediction, but it makes classical theory deterministic: no observer is left unaccounted for. Note that despite the weak/strong nomenclature, these two conjectures are quite different statements and neither conjecture implies the other. The goal of this essay is to survey the general formulations of the conjectures as precise mathematical statements about the Cauchy problem in general relativity and the partial results which have been achieved under various symmetry assumptions. As the literature is now quite vast, the student may wish to concentrate more on the asymptotically flat case (gravitational collapse), where both cosmic censorship conjectures are relevant, or the cosmological case, where only strong cosmic censorship is applicable. Relevant Courses Essential: General Relativity, Differential Geometry Useful: Black Holes, PDE’s (Part II level), Cosmology

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References [1] D. Christodoulou On the global initial value problem and the issue of singularities Class. Quantum Gravity 16 (1999) A23–A35 [2] P. Chru´sciel On uniqueness in the large of solutions of Einstein’s equations (‘strong cosmic censorship’) Proceedings of the Center for Mathematical Analysis 27, Australian National University (1991) [3] M. Dafermos The evolution problem in general relativity. Current developments in mathematics, 2008, 1–66, Int. Press, Somerville, MA, 2009 [4] M. Dafermos Stability and instability of the Cauchy horizon for the spherically symmetric Einstein-Maxwell-scalar field equations Ann. of Math. 158 (2003), no. 3, 875–928 [5] M. Dafermos The interior of charged black holes and the problem of uniqueness in general relativity. Commun. Pure Appl. Math. LVIII (2005), 0445–0504 [6] M. Dafermos and A. Rendall Strong cosmic censorship for T 2 -symmetric cosmological spacetimes with collisionless matter, arXiv:gr-qc/0610075 [7] M. Dafermos and A. Rendall Strong cosmic censorship for surface-symmetric cosmological spacetimes with collisionless matter, arXiv:gr-qc/0701034v1 [8] J. Kommemi The global structure of spherically symmetric charged scalar field spacetimes, arXiv:1107.0949 [9] A. Rendall Partial differential equations in general relativity Oxford Graduate Texts in Mathematics 16 Oxford University Press, Oxford, 2008 [10] H. Ringström Strong cosmic censorship in T 3 -Gowdy spacetimes. Ann. of Math. 170 (2009), no. 3, 1181–1240

35. The Gaussian Free Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr N.E. Berestycki The (bosonic, massless) Gaussian Free Field is the canonical model of a random surface, and as such it is a central object of probability theory and statistical mechanics. Just as Brownian motion is the scaling limit of discrete random walks and one-dimensional processes, the Gaussian Free Field arises in a huge variety of questions in higher dimensions: from random matrices to domino tilings, with links to conformally invariant planar processes and in particular SchrammLoewner Evolutions. Informally, it is a collection of random variables (φx )x∈D indexed by a domain D ⊂ Rd , which is an infinite-dimensional centered Gaussian vector with zero boundary condition on ∂D and covariance structure given by E(φx φy ) = GD (x, y), where GD is the Green function. This definition is not in fact fully rigorous: φx cannot be defined pointwise, though local averages make sense. A successful essay will seek to present the definition and basic properties of the Gaussian Free Field - such as, existence as a Gaussian process in the H01 Sobolev function space, existence of local averages, domain Markov property, approximations by discrete and continuous random fields. A nice survey, written by Sheffield, already exists which lists the main properties and sketches their proofs. The goal of this essay will be to write more detailed arguments for these results (relevant sections are 2.1 to 2.7 and 4.1 to 4.2).

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Relevant Courses Useful: A familiarity with Brownian motion and with linear analysis will be useful. The course Topics on Conformal Invariance and Randomness is recommended for additional perspective on conformal invariance, though not essential. References [1] Scott Sheffield (2007), “Gaussian free fields for mathematician”, Probability Theory and Related Fields 139: 521–541. arXiv:math.PR/0312099. [2] Bertrand Duplantier and Scott Sheffield (2010). Schramm Loewner Evolution and Liouville Quantum Gravity. arXiv:1012.4800.

36. Inference about Mechanistic Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Professor C.R. Berzuini When a biologist states “the effect of gene A upon disease outcome Y depends on the allele of gene B”, he or she talks about a property of a real (biological) mechanism, within which A and B interact to produce an effect. This is an example of the general concept of mechanistic interaction, which we regard as distinct from statistical interaction, the latter being nothing more than a departure of the data from some model of independent effects and, as such, not interpretable in terms of mechanism. The essay will review possible mathematical definitions of “mechanistic”, and methods and assumptions under which such form of interaction can be detected on the basis of observational data, in the light of recent advances in statistical causal inference theory. This may possibly involve an analysis of genetic epidemiology data within a study of genetic mechanisms of disease. Among the possible methodological developments is the relaxing of certain - rather restrictive - assumptions of determinism about the dependence of the response on its causal determinants. Relevant Courses None References [1] VanderWeele, T.J., Hernandez-Diaz, S. and Hernan, M.A. (2010). Case-only geneenvironment interaction studies: when does association imply mechanistic interaction? Genetic Epidemiology, 34:327-334. [2] VanderWeele, T.J. (2010). Empirical tests for compositional epistasis. Nature Reviews Genetics, 11:166. [3] VanderWeele, T.J. (2010). Epistatic interactions. Statistical Applications in Genetics and Molecular Biology, 9, Article 1:1-22. [4] VanderWeele, T.J. and Laird, N.M. (2011). Tests for compositional epistasis under single interaction-parameter models. Annals of Human Genetics, Special Issue on Epistasis, 75:146156. [5] VanderWeele, T.J. (2009). Sufficient cause interactions and statistical interactions. Epidemiology, 20:6-13. 39

[6] Vansteelandt, S., VanderWeele, T.J., Tchetgen, E.J., Robins, J.M., (2008). Multiply robust inference for statistical interactions. Journal of the American Statistical Association, 103:16931704. [7] VanderWeele, T.J., Vansteelandt, S. and Robins, J.M. (2010). Marginal structural models for sufficient cause interactions. American Journal of Epidemiology, 171:506-514. [8] Carlo Berzuini and Philip Dawid, Mechanistic Interaction, under revision for publication on Biostatistics.

37. Causal Statistical Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Professor A.P. Dawid Statisticians have developed a variety of formal frameworks for assessing causation from experimental or observational data. These have been applied — sometimes with conflicting results — to a wide range of challenging problems, such as dealing with confounding, separating direct and indirect causation, correcting for intermediate variables, etc. This essay would survey these approaches, and ideally specialise in some area such as instrumental variable analysis, possibly in the light of real data e.g. on Mendelian randomisation. It will be important to clarify the nature and applicability of any additional assumptions that are required to support causal inferences. Relevant Courses Useful: Statistical Methods References [1] Dawid, A. P. (2007). Fundamentals of statistical causality. Research Report 279, Department of Statistical Science, University College London. 94 pp. . [2] Didelez, V. and Sheehan, N. A. (2007). Mendelian randomisation as an instrumental variable approach to causal inference. Statistical Methods in Medical Research 16, 309–30. [3] Pearl, J. (2009). Causality: Models, Reasoning and Inference (second edn). Cambridge University Press, Cambridge.

38. Forensic Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Professor A.P. Dawid The Courts often struggle with statistical issues, both simple and complex. Many of these relate to the interpretation of identification evidence, such as a DNA “match” between a suspect and a trace found at the crime scene. Interesting complications arise when the trace is a mixture of biological materials from several donors, or several traces are found on the same item, or the suspect is identified from a trawl through a DNA database. The difficulties are logical as much as mathematical, and there are unresolved differences between expert statisticians. It has been found useful to build “probabilistic expert systems” to assist with the structuring and analysis of complex cases. The essay could offer opportunities for developing these further.

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Relevant Courses Useful: Statistical Theory References [1] Dawid, A. P. (2005). Probability and statistics in the law. In Proceedings of the Tenth International Workshop on Artificial Intelligence and Statistics, January 6–8 2005, Barbados, edited by Zoubin Ghahramani and Robert G. Cowell. [2] Dawid, A. P., Mortera, J. and Vicard, P. (2007). Object-oriented Bayesian networks for complex forensic DNA profiling problems. Forensic Science International 169, 195–205. [3] Cowell, R. G., Lauritzen, S. L. and Mortera, J. (2007). A gamma model for DNA mixture analyses. Bayesian Analysis 2, 333–348.

39. Information and Decision Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Professor A.P. Dawid A statistical model is a parametrised set of probability distributions. We can consider it as a manifold, each distribution in the set constituting a point of the manifold. Using properties of probability distributions, we can then define, in a natural way, geometric properties such as Riemannian metric, affine connections, etc. This information geometry has been used to facilitate and illuminate a number of issues of statistical inference. Recently a start has been made on generalising the theory to apply to decision-theoretic problems. This essay would review the area, with the possibility of developing decision geometry further. Relevant Courses Essential: Differential Geometry Useful: Statistical Theory References [1] Amari, S., Nagaoka, H. (2000). Methods of Information Geometry. Translations of Mathematical Monographs, Vol. 191. Providence, Rhode Island: American Mathematical Society and Oxford University Press. [2] Dawid, A. P. (2007). The geometry of proper scoring rules. Annals of the Institute of Statistical Mathematics 59, 77–93. . [3] Dawid, A. P. and Lauritzen, S. L. (2006). The geometry of decision theory. In Proceedings of the Second International Symposium on Information Geometry and its Applications (12–16 December 2005). University of Tokyo, 22–28. [4] Efron, B. (1975). Defining the curvature of a statistical problem (with applications to second-order efficiency) (with Discussion). Annals of Statistics 3, 1189–1242.

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40. The Rate of Convergence of Fictitious Play . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr F. Fischer Fictitious play (FP) is a very simple learning algorithm for strategic games that proceeds in rounds. In each round, the players play a best response to the mixed strategy that is given by the empirical frequencies of actions played by the other players in previous rounds. If the empirical frequencies converge, which happens for example in (two-player) zero-sum games [4], they converge to a Nash equilibrium. A natural question concerns the rate of convergence. Shapiro [5] has shown that in zerosum games, payoffs converge to the (unique) Nash equilibrium payoffs at a rate of at most O(n−1/(r+s−2) ), where n is the number of rounds and r and s are the numbers of actions available to the two players. Karlin [2] conjectured that the correct rate of convergence is O(n−1/2 ), but no progress has been made toward this conjecture. A good way to start would be by summarizing the above results, as well as others you find yourself. The necessary background on game theory can for example be found in the textbook of Osborne and Rubinstein [3]. Although you can write a good essay by only reviewing the existing research literature, there is ample scope when working on this topic to conduct some mini-research project of your own. Possible directions include implementing FP and running simulations, comparing it to alternative procedures, or proposing a new one. You could also try to improve the asymptotic bound—this is likely a hard problem—or to derive a bound on the number of rounds required to get within a certain distance of an equilibrium. Getting close to equilibrium strategies rather than payoffs might require an exponential number of rounds, even in strict subclasses of zero-sum games [1]. The result for the most restricted class of games assumes that ties are broken in a specific way, and you could try to remove this assumption. Relevant Courses Useful: Mathematics of Operational Research. References [1] F. Brandt, F. Fischer, and P. Harrenstein. On the rate of convergence of fictitious play. In Proceedings of the 3rd International Symposium on Algorithmic Game Theory, pages 102–113. Springer, 2010. [2] S. Karlin. Mathematical Methods and Theory in Games, volume 1–2. Addison-Wesley, 1959. [3] M. J. Osborne and A. Rubinstein. A Course in Game Theory. MIT Press, 1994. [4] J. Robinson. An iterative method of solving a game. Annals of Mathematics, 54(2):296–301, 1951. [5] H. Shapiro. Note on a computation model in the theory of games. Communications on Pure and Applied Mathematics, 11:587–593, 1958.

41. Positive and Negative Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Professor G.R. Grimmett Much of mathematical analysis proceeds by first establishing inequalities between relevant quantities. A standard approach in probability theory, of particular importance in the area of in42

teracting systems, is to prove and utilise correlation inequalities. Positive association, in which random variables tend to increase or decrease together, is well understood, and there are established methods of proving positive association. The concept of negative association is much harder to formulate and explore. The proposal of this essay is to summarise the positive and negative association of random variables taking values in a partially ordered set. The essay should include accounts of the FKG, BK, and Reimer inequalities, as well as a discussion of the difficulties in setting out a satisfactory theory of negative association. Applications (actual and/or potential) to the random-cluster model or other processes should be included. Relevant Courses Useful: Percolation and Related Topics References [1] Grimmett, G. R. (1999), Percolation, Springer-Verlag, Chapter 2. [2] Grimmett, G. R. (2006), The Random-Cluster Model, Springer-Verlag, Chapter 2. [3] Pemantle, R. (2000), Towards a theory of negative dependence, Journal of Mathematical Physics 41, 1371–1390, arxiv.org/abs/math/0404095. [4] Grimmett, G. R., Winkler, S. N. (2004), Negative association in uniform forests and connected graphs, Random Structures and Algorithms 24, 444-460, arxiv.org/abs/math.PR/0302185 [5] Berg, J. van den (2011), A BK inequality for randomly drawn subsets of fixed size, arxiv.org/abs/1105.3862.

42. Lorentz/Ehrenfest Wind–Tree Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Professor G.R. Grimmett A light particle moving around a Euclidean space is subject to reflections by heavy particles. If the heavy particles are distributed at random, what can be said about the trajectory of the light particle? In particular, is it almost surely bounded? If not, is it asymptotically Gaussian? This essay will contain a summary of the existing rigorous theory and its limitations, together with a deeper study of some specific aspect such as: Harris’s theorem for two-dimensional Poissonian systems, Gaussianity for periodic systems, and Lorentzian models in the presence of extra noise Relevant Courses Useful: Percolation and Related Topics References [1] Grimmett, G. R. (1999), Percolation, Springer-Verlag, Section 13.3. [2] Spohn, H. (1991), Large Scale Dynamics of Interacting Particles, Springer-Verlag, Berlin. [3] Grimmett, G. R. (1999), Stochastic pin-ball, number 20 of www.statslab.cam.ac.uk/~grg/preprints.html. 43

43. Nonparametric Statistics on Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr R. Nickl Many high-dimensional statistical problems have a mathematical structure that can be modelled by tools from differential geometry. While such problems are classically often subsumed within the field of ‘directional statistics’ (see the monograph Mardia and Jupp (2000)), the capacity of modelling random samples on manifolds to achieve dimension-reduction has received considerable interest recently, particularly in nonparametric statistics and machine learning. See, for instance, the references [1-3]. The purpose of this essay is to branch out into one of these directions and review some of the very recent literature that has been growing on the subject. Prerequisites: Basic knowledge on statistics and differential geometry is necessary. Some knowledge of Lie group theory is also useful but not necessary, and attending the Part III course on Statistical Theory is strongly recommended. Relevant Courses Essential: Statistical Theory References [1] Giné, Koltchinskii. Empirical graph Laplacian approximation of Laplace-Beltrami operators: large sample results. In High dimensional probability IV (E. Giné et al. eds.), IMS Lecture Notes 51 (2006). [2] Huckemann, Kim, Koo, Munk. Möbius deconvolution on the hyperbolic plane with application to impedance density estimation. Ann. Statist. 38 (2010) [3] Kerkyacharian, Nickl, Picard. Concentration inequalities and confidence bands for needlet density estimators on compact homogeneous manifolds. Probability Theory and Related Fields, to appear. [4] Mardia, Jupp. Directional Statistics. Wiley, 2000.

44. Statistical Inference for L´ evy Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr R. Nickl Lévy processes are a building block in the theory of stochastic processes. They are useful to model continuous time statistical phenomena that may have jumps, particularly in the analysis of financial time series, but also to model trajectories of particles, as well as biological phenomena. A key challenge is to estimate nonparametrically the characteristics of the Lévy process from a sample of its (independent) increments. These characteristics are summarised in the Lévy-Khintchine representation: The size of the Brownian component of the process, the drift, and the jump distribution (or Lévy measure). Depending on whether low- or high-frequency observations of the increments of the Lévy process are available, different statistical procedures are necessary to make inference upon these characteristic. The purpose of this essay is to review the recent literature on the subject and to understand some of the main challenges in the field of nonparametric estimation of Lévy processes. Prerequisites: Basic knowledge of statistics and probability is required, attending the core courses on statistical theory and probability theory in Part III is also strongly recommended.

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Relevant Courses Essential: Statistical Theory References [1] Belomestny, Reiss; Spectral calibration for Lévy models. Finance and Stochastics 10 (2006). [2] Chen, Delaigle, Hall. Nonparametric estimation for a class of Lévy processes. J. Econometrics 157 (2010). [3] Comte, Genon-Catalot. Nonparametric adaptive estimation for pure jump Lévy processes. Ann. Inst. Henri Poincaré (B) 46 (2010). [4] Figueroa-Lopez, Houdre. Risk bounds for nonparametric estimation of Lévy processes. In High dimensional probability IV (E. Giné et al. eds.), IMS Lecture Notes 51 (2006). [5] Neumann, Reiss; Nonparametric estimation for Lévy processes from low-frequency observations. Bernoulli 15 (2009)

45. Multiplicative Coalescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Professor J.R. Norris The simplest multiplicative coalescent is a continuous-time Markov chain whose state is a finite set of masses (with multiplicities) and where any two masses are replaced by their sum at a rate proportional to their product. See any of the references for a more formal definition. The special structure of this process makes it amenable to investigation in many ways. In particular, there are natural links to the sparse uniform random graph process, and the largeparticle-number limit can be effectively studied using the Laplace transform. Recently, Bertoin and Normand have introduced a generalization of these processes where the mass considered is two-dimensional, and the rate is a quadratic function of the mass. Much of the basic theory carries over. There are links with sparse uniform random graphs with specified vertex degrees which are not yet fully explored. The essay will give an account of some of the theory of multiplicative coalescence and may explore new directions suggested by the work of Bertoin and Normand. Relevant Courses Useful: Advanced Probability References [1] Aldous, David. Brownian excursions, critical random graphs and the multiplicative coalescent. Ann. Probab. 25 (1997), no. 2, 812–854. [2] Bertoin, Jean. Two solvable systems of coagulation equations with limited aggregations. Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 6, 2073–2089. [3] Normand, Raoul. A model for coagulation with mating. J. Stat. Phys. 137 (2009), no. 2, 343–371. [4] Norris, J. R. Cluster coagulation. Comm. Math. Phys. 209 (2000), no. 2, 407–435. 45

46. Independent Component Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr R.J. Samworth Independent Component Analysis (ICA) is a special case of blind source separation, where the aim is to infer from a set of mixed signals both the source signals and the mixing process. A simple application is the ‘cocktail party problem’, where we try to separate underlying speech signals from the audio data of people talking simultaneously in a room. ICA is therefore related to, for example, Principal Component Analysis, and has proved an enormously popular and effective tool in statistical signal processing. In the simplest, noiseless, case of an ICA model, we observe independent and identically distributed replications of a random vector X ∈ Rd that can be expressed as X = AS, where A is a d × d non-singular matrix and S ∈ Rd is a random vector with independent components. The aim is to estimate both the mixing matrix A and the distributions of the components of S. This therefore represents a semiparametric statistical problem. There is a huge literature on ICA – starting points include the book [1] or the seminal paper [2] where the concept was introduced. However, despite the wide variety of publicly available algorithms, their statistical properties are often relatively poorly understood. This essay would describe some of the existing algorithms and a comparison of the criteria they use for model fitting. It may present what is known about the theoretical properties of these algorithms, or it may compare the performance of the algorithms on simulated and/or real data. Relevant Courses Useful: Statistical Theory, Semiparametric Statistics References [1] Hyvärinen, A., Karhunen, J. and Oja, E. (2001) Independent Component Analysis. New York: Wiley. [2] Comon, P. (1994) Independent Component Analysis: a new concept? Signal Processing, 36, 287–314.

47. Model Misspecification in Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr R.J. Samworth We are very familiar with statistical results of the following type: consider a family of densities {f (·; θ) : θ ∈ Θ}, and let X, X1 , . . . , Xn be independent observations from f (·; θ0 ) where θ0 ∈ Θ. Then for some suitably defined estimator θˆn = θˆn (X1 , . . . , Xn ) and under certain regularity p conditions, we have θˆn → θ0 . A rate of convergence may also be given. But what happens if θ0 ∈ / Θ? Results in this scenario of model misspecification are remarkably scarce, despite their obvious importance, particularly for the complex, high-dimensional or nonparametric models that are frequently encountered in modern practice. When the estimation paradigm is that of maximum likelihood, the following heuristic argument P gives some insight: since θˆn maximises `n (θ) = n−1 ni=1 log f (Xi ; θ) over θ ∈ Θ, we hope that θˆn will converge in a suitable sense to the θ∗ ∈ Θ that maximises Eθ0 {log f (X; θ)} over θ ∈ Θ. Note that under a weak condition, f (·; θ∗ ) minimises the Kullback–Leibler divergence from f (·; θ0 ), and in this sense, the maximum likelihood estimator has the desirable property of 46

converging to the closest element to the truth within the model. It requires care to make this argument rigorous, however, and early papers (e.g. [1]) often just assume that θ∗ exists and is unique. In fact, the situation is now relatively well understood for maximum likelihood in convex models ([2]), though results have only recently been obtained for non-convex models ([3],[4]), and much is still unknown. This essay would summarise the history and state of the art of the area. It may or may not focus solely on maximum likelihood as an estimation technique, but a good essay might include new examples where the properties of estimation procedures under model misspecification can be studied. Relevant Courses Useful: Statistical Theory, Semiparametric Statistics References [1] White, H. (1982) Maximum likelihood estimation of misspecified models. Econometrica, 50, 1–25. [2] Patilea, V. (2001) Convex models, MLE and misspecification. Ann. Statist., 29, 94–123. [3] Cule, M. L. and Samworth, R. J. (2010) Theoretical properties of the log-concave maximum likelihood estimator of a multidimensional density. Electron. J. Statist., 4, 254–270. [4] D¨ umbgen, L., Samworth, R. J. and Schuhmacher, D. (2011), Approximation by log-concave distributions with applications to regression. Ann. Statist., 39, 702–730.

48. Statistical Challenges with EEG Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr R.J. Samworth Electroencephalography (EEG) is a brain imaging technique that records the electrical activity in the brain using a net of (typically 128) electrodes on the scalp. It provides high temporal resolution (500Hz) data, and therefore complements other imaging techniques such as functional Magnetic Resonance Imaging (fMRI), which offers greater spatial resolution but lacks the detailed temporal information. It is a heavily used technique, for instance by experimental psychologists who are interested in studying dyslexia and other learning disabilities. Though the essay will necessarily give a short description of the biology and physics underlying the technique, the focus should be on the statistical challenges involved in extracting meaningful information from the huge data sets. It may also include the related method of Magnetoencephalography (MEG), which is often used in conjunction with EEG. Ideally, the essay would also present an analysis of a particular EEG/MEG data set, which might even use the candidate as a subject! Relevant Courses Useful: Applied Statistics

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References [1] Luck, S. J. (2005) An Introduction to the Event-Related Potential Technique. MIT Press. [2] Sanei, S. and Chambers, J. A. (2007) EEG Signal Processing. Wiley–Blackwell.

49. Dimension Reduction in Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr B. Sen Consider a random vector (Y, X) in Rd+1 having a joint density, where X = (X1 , . . . , Xd ) ∈ Rd . The main goal in regression analysis is to study the dependence of the response Y on the predictor X. The dependence can always be expressed in the form Y = g(X, U ), where g is an unknown function and U is a Uniform(0, 1) random variable independent of X. This follows −1 from the fact that we can generate Y given X and U as FY−1 |X (U |X), where FY |X (·|x) is the conditional distribution of Y given X = x. The uniform random variable U can be thought of as the error. Nonparametric regression techniques can be used to model the full conditional distribution of Y given X. But, as the dimension of X gets higher, due to the “curse of dimensionality”, most standard methods break down. In such situations, statisticians often resort to dimension reduction techniques to retrieve interesting features of such high dimensional data from lowdimensional projections. An ideal situation occurs when T Y = φ(β1T X, . . . , βK X, U ),

(1)

for unknown φ, βi ∈ Rd and K is much smaller than d (see [5], [4], [1]). This is sometimes referred to as a multiple index model; when K = 1, it is called a single-index model. Another way of looking at (1) is using the idea of sufficient dimension reduction (see [2], [3]). This essay would summarise the history and state of the art of the area. It may or may not propose a new procedure for the estimation of K and the βi ’s. The essay should also briefly summarise the large and heterogeneous collection of methods for dimension reduction in regression in which specific modelling assumptions are imposed on the regression function E(Y |X), e.g., ordinary/partial least squares, canonical correlation analysis, projection pursuit regression, etc. Relevant Courses Essential: Statistical Theory Useful: Semiparametric Statistics References [1] Chaudhuri, P., Doksum, K., and Samarov, A. (1997). On Average Derivative Quantile Regression, Ann. Statist., 25, 715–744. [2] Cook, R. (2007). Dimension Reduction in Regression. Statist. Sci., 22, 1–26. [3] Fukumizu, K., Bach, F. and Jordan, M. (2009). Kernel dimension reduction in regression. Ann. Statist., 37, 1871–1905. [4] Horowitz, J. (2009). Semiparametric and Nonparametric methods in Econometrics. Springer. [5] Li, K. C. (1991). Sliced Inverse Regression for Dimension Reduction. J. Amer. Statist. Assoc., 86, 316–342. 48

50. Nonparametric Shape Restricted Regression with Multiple Covariates . Dr B. Sen Consider a regression problem with a random vector (Y, X) ∈ Rd+1 , where X ∈ Rd (d ≥ 2) is the predictor and Y is the real valued response. We usually write Y = m(X) + ², where E(²|X) = 0 and m is the regression function, the main quantity of interest. Most nonparametric function estimation methods (e.g., kernels, splines) make smoothness assumptions on the underlying function and use local averaging techniques to construct the estimates. For univariate data these methods work well and have been applied successfully in numerous applications, but with multi-dimensional data it becomes difficult to interpret and implement these procedures – they depend crucially on tuning parameters and the choice of such parameters can be very problematic. However, in shape restricted function estimation we can completely by-pass this dependence on tuning parameters and develop estimates using the method of least squares (LS) or maximum likelihood (ML) that are fully automated in situations where prior knowledge on the shape (monotonicity, convexity, etc.) of the function is available. Such shape constraints arise naturally in numerous applications (see e.g., [4], [2] and the references therein). Recently, the nonparametric least squares estimator of a convex regression function when the predictor is multidimensional has been studied in [5]. To avoid the “curse of dimensionality” when d is large, another alternative is to consider m(x) = g(xT β) where g : R → R. If we assume that g is monotone/convex, estimation can be carried out using the method of LS or ML (see e.g., [1], [3]). The computation of these estimators is not straightforward and usually involves quadratic programming. Though the essay will necessarily give a short description of the existing literature, the focus should be on the computation of these estimators. An extensive simulation study should be undertaken to compare the performance of these estimators with other nonparametric methods (e.g., kernel smoothing). Other possible methods of extending shape restricted functions with multi-dimensional covariates could also be discussed. Ideally, the essay would also present a real data analysis. Relevant Courses Essential: Statistical Theory Useful: Mathematics of Operational Research References [1] Cosslett, S. R. (1983). Distribution-free maximum likelihood estimation of the binary response models. Econometrica, 51, 765–782. [2] Monti, M., Grant, S. and Osherson, D. (2005). A note on concave utility functions. Mind and Society: Cognitive Studies in Economics and Social Sciences, 4, 85–96. [3] Murphy, S. A., Van der Vaart, A. W. and Wellner, J. A. (1999). Current Status Regression. Math. Meth. Statist., 8, 407–425. [4] Robertson, T., Wright, F. T. and Dykstra, R. L. (1988). Order restricted statistical inference. Wiley, New York. [5] Seijo, E. and Sen, B. (2011). Nonparametric Least Squares Estimation of a Multivariate Convex Regression Function. Ann. Statist., 39, 1633–1657.

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51. Bond Market Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr M.R. Tehranchi The object of this essay is to survey the recent literature on bond market models. A bond is a contract obliging the issuer to pay the holder a fixed amount of money on a fixed future date. Let Θ ⊂ [0, ∞) be the set of maturity dates of liquidly traded bonds from a single issuer. The prices of these bonds can be modelled by a family of stochastic processes (P (t, T ) : 0 ≤ t ≤ T, T ∈ Θ) where P (t, T ) denotes the time-t price of with maturity date T . Possible topics to include in the essay are the following: • Measure-valued portfolios and generalisations. For large issuers (for instance, the US Treasury) the set Θ is very large, so it becomes mathematically convenient to identify a portfolio of bonds with a measure on Θ. The essay should include a discussion of the notion of an admissible trading strategy and market completeness. Applications to the pricing and hedging of derivative securities or optimal investment should also be addressed. Consult references [2], [3] and [4]. • Model consistency. There are several approaches to specifying the family of bond prices. One way is model the dynamics of a set of underlying economic factors (Zt )t≥0 and then assume P (t, T ) = g(Zt , T ) for some family of functions g(·, T ). Another way is to model (P (t, T ))0≤t≤T directly for each T . How are these approaches related? The essay should discuss the geometric nature of this question and how it relates to the existence of invariant manifolds for stochastic evolution equations. Consult references [1] and [5]. Relevant Courses Essential: Stochastic Calculus and Applications, Advanced Financial Models, Optimal Investment Useful: Advanced Probability References [1] Th. Björk. On the geometry of interest rate models. Paris–Princeton Lectures on Mathematical Finance 2003. Springer. pp 133–216. (2004) [2] R. Carmona and M. Tehranchi. Interest Rate Models: an infinite dimensional stochastic calculus perspective. Springer. (2006) [3] M. De Donno and M. Pratelli. A theory of stochastic integration for bond markets. Annals of Applied Probability 15(4): 2773-2791. (2005) [4] I. Ekeland and E. Taflin. A theory of bond portfolios. Annals of Applied Probability 15(2): 1260–1305. (2005) [5] D. Filipović. Consistency Problems for Heath-Jarrow-Morton Interest Rate Models. Lecture Notes in Mathematics 1760. Springer. (2001)

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52. Ranking, Reputation and Recommender Systems . . . . . . . . . . . . . . . . . . . . . . Professor R.R. Weber To quote [1], “Ranking, reputation, recommendation, and trust systems have become essential ingredients of web-based multi-agent systems. These systems aggregate agents’ reviews of one another, as well as about external events, into valuable information. Notable commercial examples include Google’s page ranking system, and Amazon and E-bay’s recommendation and reputation systems.” Your task is to read around this general field and write an interesting essay about mathematics for ranking, reputation and recommender systems. You can discuss both modelling and algorithms. You will need to be selective and focus on a few topics that you judge to be seminal (PageRank is an example), or where you find that current research is active, or where there are unsolved questions. The following references should get you started, but you can find other sources via your own explorations. I suggest you begin by reading [1] and [5]. I can provide some further references regarding the (Bayesian) recommender and ranking systems for Microsoft’s XBox Live. Relevant Courses Useful: Stochastic Networks, Mathematics of Operational Research. References [1] Andersen, A. et al., Trustbased recommendation systems: an axiomatic approach, Microsoft Research, 2008. http://research.microsoft.com/en-us/um/people/borgs/Papers/trust.pdf [2] Gyongyi, Z, et al., Combating web spam with TrustRank. Proceeding VLDB ’04 Proceedings of the Thirtieth international conference on Very large data bases - Volume 30, 2004. http://dl.acm.org/citation.cfm?id=1316740 [3] Keinberg, J. M., Authoritative sources in a hyperlinked environment, JACM 46, 1999. http://www.cs.cornell.edu/home/kleinber/auth.pdf [4] Page, L. PageRank: Bringing order to the Web. Stanford Digital Library Project, 1997. http://ilpubs.stanford.edu:8090/422/1/1999-66.pdf [5] Renick, P. and Varian, H. R., Recommender systems, http://dl.acm.org/citation.cfm?id=245121 [6] Tahajod, M. Iranmehr, A., Khozooyi, N., Trust management for semantic web, Computer and Electrical Engineering, 2009. ICCEE ’09. http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=5380157 [7] Vojnovic, M., Cruise, J., Gunawardena, D., Marbach, P., Ranking and suggesting popular items, IEEE Trans. Knowledge and Data Engineering, 12, 1133–1146, 2009. http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=4760141&tag=1

53. h → γγ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Professor B.C. Allanach The LHC is now churning out data, and has re-discovered all of the Standard Model particles barring the Higgs boson, which is becoming strongly constrained. If its mass is 115-130 GeV, 51

the loop-induced decay mode h → γγ is important for detection of the Standard Model Higgs boson. Some papers have recently claimed that all previous calculations of h → γγ are incorrect because of a deficiency in dimensional regularisation of a particular loop integral. The new papers calculate directly in four dimensions instead. The purpose of this essay is to repeat the calculations and to discover which calculation is in error. You will need to understand renormalisation in quantum field theory well, and be able to confidently calculate loop diagrams. Relevant Courses Essential: Quantum Field Theory, Standard Model, Advanced Quantum Field Theory References [1] J.R. Ellis, M.K. Gaillard and D.V. Nanopoulos, Nucl. Phys. B106 (1976) 292. [2] Gastmans, Wu, Wu, arXiv:1108.5322

54. Effective Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr D. Baumann It is a basic fact of life that Nature comes to us in many scales. Galaxies, planets, molecules, atoms and nuclei have very different sizes, and are held together by very different binding energies. However, it is another important fact of life that phenomena involving distinct scales can often be analysed by considering one relevant scale at a time. Taking advantage of scale separation in quantum field theories leads to effective field theories (EFTs). The first half of the essay should explain the basic principles of EFTs. It should show how EFTs isolate the relevant low-energy degrees of freedom, while systematically including unknown (or uncomputable) high-energy effects as corrections. The concepts that should be discussed include the integrating out heavy fields to arrive at low-energy effective actions, the importance of symmetries, power counting, renormalization and naturalness. Effective field theory is a tool, so writing a comprehensive review of its principles and applications would be like writing a treatise on the hammer. In the second half, the essay should therefore pick one concrete application of EFT and describe it in as much detail as possible. The following is an incomplete list of possible topics: • QCD Chiral Lagrangian • Landau Theory of Fermi Liquids • Standard Model as an EFT • Inflation as an EFT • General Relativity as an EFT • ... Students are most welcome to suggest their own examples of EFTs.

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Relevant Courses Essential: Quantum Field Theory Useful: Advanced Quantum Field Theory, Standard Model, Inflation References Reviews: [1] Skiba, arXiv:1006.2142. [2] Kaplan, arXiv:nucl-th/0510023. [3] Polchinski, arXiv:hep-th/9210046. [4] Burgess, arXiv:hep-th/9808176. [5] Baumann, The Physics of Inflation. Papers: To be selected in discussion with the advisor.

55. Ab initio Definite Quantities for Quantum Theory . . . . . . . . . . . . . . . . . . . . . Dr N. Bouatta, Dr J.N. Butterfield One approach to solving the measurement problem in quantum theory proposes that a certain quantity Q is ‘preferred’ in that a quantum system always has a definite value for it. So Q needs to be chosen so that: (i) its definite values appropriately explain the definiteness of the macro-realm, and this will presumably involve equations of motion for the values that mesh suitably with the quantum state’s unitary evolution; (ii) its definite values do not violate various no-go theorems such as the Kochen-Specker theorem. The best-developed example of this approach is: (a) The pilot-wave theory of de Broglie and Bohm [1, 2]. (b) But recently, several other examples, called collectively ‘modal’ interpretations, have been developed: usually choosing as Q the spectral projections of the system’s reduced density matrix [3, 4]. The purpose of the essay will be to review one (i.e. whichever the candidate chooses) of the four main aspects of this approach, that are defined by the two choices indicated above: the choice of (i) vs. (ii), and the choice of (a) vs. (b). Relevant Courses Useful: Philosophy of Classical and Quantum Mechanics, Quantum Foundations References [1] P. Holland, The Quantum Theory of Motion, C.U.P. 1993. [2] D. Bohm and B. Hiley, The Undivided Universe, Routledge 1992.

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[3] D. Dieks and P. Vermaas (eds.), The Modal Interpretation of Quantum Mechanics, C.U.P. 1998. [4] Bub, Interpreting the Quantum World, C.U.P. 1997. (copies available from JNB)

56. Information-Theoretic Aspects of Quantum Foundations . . . . . . . . . . . . . . . Dr N. Bouatta, Dr J.N. Butterfield The rise of quantum information theory has had an enormous impact on the way people address foundational and philosophical questions about quantum theory. As regards philosophy, at least three main developments can be discerned. One is the philosophical assessment [1, 2: Section 4.1.4] of recently discovered phenomena and protocols, such as teleportation [3]. A second is that interpretations of the quantum state as subjective [4, 5] or epistemic [6] have been reinvigorated; and there has of course been philosophical assessment of this [7, 8, 9]. A third development is the philosophical assessment [2: Section 4.4.2, 9] of axiomatic formulations that invoke information-theoretic principles, such as no-cloning, at their base [10, 11]. These three developments are of course inter-related. So the purpose of the essay will be to review one or two of them (i.e. according to whether the candidate wants to focus on their inter-relations). Relevant Courses Useful: Philosophy of Classical and Quantum Mechanics, Quantum Foundations, Quantum Information Theory References [1] C. Timpson, The Grammar of Teleportation, British Journal for the Philosophy of Science 57, pp. 587-621. arXiv:quant-ph/0509048 and http://philsci-archive.pitt.edu/2438/. [2] C. Timpson, Philosophical aspects of quantum information theory, in The Ashgate Companion to Contemporary Philosophy of Physics, ed. D. Rickles, Ashgate 2008; arXiv:quant-ph/0611187 and at http://users.ox.ac.uk/~bras2317/. [3] C. Bennett, G. Brassard et al, Teleporting an unknown state via dual classical and EPR channels, Physical Review Letters 70, pp. 1895-99. [4] C. Fuchs, (2002), Quantum mechanics as quantum information (and only a little more), in Khrenikov, A., editor, Quantum Theory: Reconsideration of Foundations, Vaxjo University Press. arXiv:quant-ph/0205039. [5] C. Caves, C. Fuchs and R. Schack (2007), Subjective Probability and Quantum Certainty, Studies in the History and Philosophy of Modern Physics, 38(2):255-274. [6] Spekkens, R. (2007). Evidence for the epistemic view of quantum states: A toy theory. Physical Review A, 75, 032110. arXiv:quant-ph/0401052. [7] C. Timpson, Quantum bayesianism: a study, in Studies in the History and Philosophy of Modern Physics, arXiv:quant-ph/0804.2047. [8] C. Timpson, ‘Information immaterialism instrumentalism: old and new in quantum information’, in Studies in the History and Philosophy of Modern Physics, in Foundations of Quantum Information and Entanglement, ed. A Bokulich and G Jaeger, C.U.P. 54

[9] W. Myrvold, From physics to information theory and back, in Foundations of Quantum Information and Entanglement, ed. A Bokulich and G Jaeger, C.U.P. [10] Clifton, R., Bub, J., and Halvorson, H. (2003), Characterizing quantum theory in terms of information theoretic constraints. Foundations of Physics, 33, p. 1561. arXiv:quantph/0211089. [11] G.Chiribella, G.M. D’Ariano, P. Perinotti, Informational derivation of quantum theory, Physical Review A, 84, 012311 (2011). arXiv:quant-ph/1011.6451.

57. Varieties of Locality for Quantum Fields and Strings . . . . . . . . . . . . . . . . . . . Dr N. Bouatta, Dr J.N. Butterfield Quantum field theory allows one to formulate various concepts of locality. They have usually been taken to be cornerstones of the theory; some of them are reviewed philosophically in [1]. This essay pursues the question whether one or more these concepts of locality would break down in the context of (i) black hole physics and-or (ii) string theory. As regards (i): since Hawking’s seminal work on black hole evaporation, some such as Susskind have concluded that, to preserve unitarity, a quantum theory of gravity must reject one or other of these concepts of locality [2,3]. A philosophical assessment of this debate is given by [4]. As regards (ii): formal developments in string theory and string field theory also raise concerns about some of these concepts of locality [5, 6]. Thus the purpose of the essay is to examine the prospects for various concepts of locality ideas in black hole physics and-or string theory. Relevant Courses Useful: Quantum Field Theory, Black Holes, String Theory, Philosophical Foundations of Quantum Field Theory References [1] J. Butterfield, Reconsidering relativistic causality, International Studies in the Philosophy of Science 21 (2007) 295-328. http://arxiv.org/abs/0708.2189. [2] L. Susskind, J. Lindesay, An introduction to black holes, information and the string theory revolution: The holographic universe, World Scientific 2005. [3] S. B. Giddings, “Black hole information, unitarity, and nonlocality,” Physics Review D74, (2006) 106005 [hep-th/0605196]. [4] G. Belot and J. Earman and L. Ruetsche, The Hawking information loss paradox: the anatomy of controversy, The British journal for the philosophy of science 50, (1999) 189-229. [5] N. Seiberg, in D. Gross, M. Henneaux and A. Sevrin, (eds.) The Quantum Structure of Space and Time, World Scientific, 2007. [6] T. Erler, D. Gross, Locality, causality, and an initial value formulation for open string field theory. [hep-th/0406199]. [7] S. Weinberg, The Quantum Theory of Fields I, Chapters. 3, 4, 5 and 6, Cambridge University Press, 1995.

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58. Philosophical Aspects of Spontaneous Symmetry Breaking . . . . . . . . . . . . . Dr N. Bouatta, Dr J.N. Butterfield Spontaneous symmetry breaking (SSB) within quantum field theory (QFT) has been central to our understanding of many phenomena in condensed matter physics, elementary particle physics and cosmology. Masterly introductions to the physics include [1,2]. A fine introduction to the philosophical issues is [3, 4]. The philosophical literature on SSB has emphasized the rigorous, algebraic approach to quantum field theory, as against more heuristic formalisms; for example [5,6,7]. Accordingly, the purpose of this essay is to conceptually examine SSB in QFT. Relevant Courses Useful: Quantum Field Theory, Advanced Quantum Field Theory, Philosophical Foundations of Quantum Field Theory References [1] E. Witten, From Superconductors and four-manifolds to weak interactions, BulletinAmerican Mathematical Society 44 (2007), pp. 361-391. [2] S. Weinberg. The Quantum Theory of Fields, Vol II. Cambridge University Press 1996. Chapters 19, 21. [3] J. Earman. Laws, symmetry, and symmetry breaking: Invariance, conservation principles, and objectivity. Philosophy of Science, 71 (2004), pp. 1227-1241. [4] J. Earman. Rough guide to spontaneous symmetry breaking, in K. Brading and E. Castellani (eds.) Symmetries in Physics: Philosophical Reflections, Cambridge University Press, 2003. [5] L. Ruetsche. Interpreting Quantum Theories. Oxford University Press, 2011. Chapters 12, 13 and 14. [6] C. Liu and G. Emch. Explaining quantum spontaneous symmetry breaking. Studies in History and Philosophy of Modern Physics, 36 (2005), pp. 137-163. [7] D. Baker and H. Halvorson. How is spontaneous symmetry breaking possible? Available online at: http://philsci-archive.pitt.edu/8517/

59. The Effect of Boundary Layers in Aero-Acoustics . . . . . . . . . . . . . . . . . . . . . . Dr E.J. Brambley Noise limits are a serious engineering restriction on the design of civilian aircraft engines. One popular noise reduction technique is to line the inside of the engine intake with a sound-absorbing structure. However, the simple commonly-used model of this situation has recently been shown to be incorrect for technical mathematical reasons [1]. This problem appears to be corrected by accounting for the boundary layer over the acoustic lining [2,3]. While the effect of viscosity within the boundary layer is important for stability and accuracy [4,5], it does not in itself appear to correct the problem [6]. This essay should discuss the importance of boundary layers in aeroacoustics, and some of the effects they lead to.

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Relevant Courses Essential: Perturbation and Stability Methods. Useful: Wave Propagation and Scattering. References [1] E. J. Brambley (2009), “Fundamental Problems with the Model of Uniform Flow over Acoustic Linings”, J. Sound Vib. 65, pp. 345–354. [2] E. J. Brambley (2011), “A Well-posed Boundary Condition for Acoustic Liners in Straight Ducts with Flow”, AIAA J. 49(6), pp. 1272–1282. [3] S. W. Rienstra & M. Darau (2011), “Boundary-Layer Thickness Effects of the Hydrodynamic Instability along an Impedance Wall”, J. Fluid Mech. 671, pp. 559–573. [4] Y. Aurégan, R. Starobinski & V. Pagneux (2001), “Influence of Grazing Flow and Dissipation Effects on the Acoustic Boundary Conditions at a Lined Wall”, J. Acoust. Soc. Am. 109, pp. 59– 64. [5] Y. Renou & Y. Aurégan (2011), “Failure of the Ingard–Myers Boundary Condition for a Lined Duct: An Experimental Investigation”, J. Acoust. Soc. Am. 130, pp. 52–60. [6] E. J. Brambley (2011), “Acoustic Implications of a Thin Viscous Boundary Layer over a Compliant Surface or Permeable Liner”, J. Fluid Mech. 678, pp. 348–378.

60. Numerical Techniques for Computational Aero-Acoustics . . . . . . . . . . . . . . Dr E.J. Brambley Computational Aero-Acoustics (CAA) is the computer modelling of acoustics within a fluid that is not at rest (such as being sucked through an aircraft engine). CAA is therefore a very important and widely used engineering tool in the aerospace industry. However, CAA requires a very different type of numerical approach from other types of Computational Fluid Dynamics (CFD). This is because, while most good CFD schemes are stable and damp out any small inaccuracies inherent in computer simulations, good CAA schemes are required to be neutrally-stable [1–3], as otherwise the acoustics would be artificially damped. Other interesting features peculiar to CAA include selective filtering [2], nonreflecting boundary conditions [4], and acoustically-lined boundaries [5]. This essay should review some of the numerical techniques involved in Computational AeroAcoustics, including a discussion of the relative merits of time-domain and frequency-domain simulations. It should be emphasized that this is a theoretical essay, and no computer programming is required. Relevant Courses Essential: Wave Propagation and Scattering. Useful: Numerical Solution of Differential Equations or Computational Methods in Fluid Mechanics.

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References [1] C. K. W. Tam & J. C. Webb (1993), “Dispersion-Relation-Preserving Finite Difference Schemes for Computational Acoustics”, J. Comput. Phys. 107, pp. 262–281. [2] C. Bogey & C. Bailly (2004), “A Family of Low Dispersive and Low Dissipative Explicit Schemes for Flow and Noise Computations”, J. Comput. Phys. 194, pp. 194–214. [3] J. Berland, C. Bogey & C. Bailly (2006), “Low-dissipation and Low-dispersion Fourth-order Runge–Kutta Algorithm”, Computers & Fluids 35, pp. 1459–1463. [4] F. Q. Hu (1996), “On Absorbing Boundary Conditions for Linearized Euler Equations by a Perfectly Matched Layer”, J. Comput. Phys. 129, pp. 201-219. [5] S. W. Rienstra (2006), “Impedance Models in Time Domain, including the Extended Helmholtz Resonator Model”, AIAA Paper 2006-2686.

61. Gravity Currents Flowing Over Corrugated Boundaries . . . . . . . . . . . . . . . . Dr S.B. Dalziel Gravity currents [1,2] form when horizontal density differences generate buoyancy forces directed towards an adjacent boundary. The fluid motion that ensues is predominantly horizontal, allowing substantial simplification of the equations governing the flow. The study of high Reynolds number gravity currents has mainly focused on flows in rectangular or axisymmetric geometries, with the current propagating across smooth boundaries. However, gravity currents frequently form in more complex geometries, often characterised by boundaries that are anything but flat and smooth. An essay on this topic will begin with an overview of gravity currents before undertaking a review of existing literature on the interaction of gravity currents with geometric features on a scale less than but comparable with the depth of the gravity current (e.g. [3]). From here, the essay could proceed in one of two directions to assess the impact of a corrugated boundary. Either a set of simple laboratory experiments could be conducted or a box model could be developed to determine the propagation speed and other key features of the flow. Relevant Courses Useful: Fundamentals in Fluid Mechanics of Climate, Advanced Topics in Fluid Mechanics of Climate References [1] Simpson, J.E. 1997 Gravity currents in the environment and the laboratory, 2nd Edn. Cambridge University Press. [2] Benjamin, T. B. 1968 Gravity currents and related phenomena. J. Fluid Mech. 31, 209-248. ¨ okmen, T.M., Fischer, P.F., Duan, J. & Iliescu, T 2004 Entrainment in bottom gravity [3] Ozg¨ currents over complex topography from three-dimensional nonhydrostatic simulations. Geophys. Res. Lett. 31, L13212.

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62. How Many Puffs Does it Take to Make a Jet? . . . . . . . . . . . . . . . . . . . . . . . . . Dr S.B. Dalziel At high Reynolds number, the impulsive localised injection of directed momentum through an orifice leads to the formation of a ‘vortex ring’, an inertial structure that can propagate over significant distances with only a gradual variation in size and strength. In contrast, a continual localised injection of directed momentum leads to the formation of a ‘jet’, that entrains rapidly and spreads radially as it forms an essentially conical structure. This essay will address the differences between vortex rings and jets, and seek to explore the intermediate regime where the flow is driven by a sequence of ‘puffs’. One can anticipate that a lot of puffs very close together will be indistinguishable from a continuous release, but how close must they be? Does it matter how long the puffs continue for? The essay will involve assimilating existing literature on vortex rings (e.g. [1,2]) and jets (e.g. [3,4]), and making inferences on a set of plausible behaviours. There is also scope for a set of simple laboratory experiments. Relevant Courses Useful: Advanced Topics in Fluid Mechanics of Climate References [1] Shariff, K. & Leonard, A. 1992 Vortex rings. Annual Review of Fluid Mechanics 24, 235-279. [2] Linden, P.F. & Turner, J.S. 2001 The formation of optimal vortex rings, and the efficiency of propulsion devices. Journal of Fluid Mechanics, 427 , 61-72. [3] Morton, B. R., Taylor, G. I. & Turner, J. S. 1956 Turbulent gravitational convection from maintained and instantaneous sources. Proceedings of the Royal Society of London A 234, 1-32. [4] List, E.J. 1982 Turbulent jets and plumes. Annual Review of Fluid Mechanics 14, 189-212.

63. Supersymmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Professor A.C. Davis Supersymmetry is a symmetry between bosons and fermions. Supersymmetry allows one to construct an extension to the standard model of particle physics and a supersymmetric version of QCD. One can also construct supergravity theories. Supersymmetric theories have remarkable properties in that some models could be finite. However, supersymmetry is not found in nature, so supersymmetry must be broken. It could either be broken spontaneously or dynamically. A particularly interesting case is that of dynamical breaking in supersymmetric gauge theories. The essay should include a detailed description of supersymmetry breaking. A particular model should be used to illustrate this. There should be a discussion of dynamical breaking of supersymmetry and an example given. If local supersymmetry is discussed then a detailed description of the super-Higgs effect should be given. Relevant Courses Essential: Quantum Field Theory, Supersymmetry

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References [1] The basics of supersymmetry and supersymmetry breaking can be found in the book by Bailin and Love:- ‘Supersymmetric Gauge Field Theory and String Theory’ and in Binetruy:‘Supersymmetry’. There is a review on Supersymmetry breaking [2] ‘Lectures on Supersymmetry Breaking’ by K. Intriligator and N. Seiberg Class. Quant. Grav. 24 (2007) S741-S772; hep-ph/0702069 which contains extensive references to the original literature.

64. The Two-State Vector Formalism in Quantum Mechanics . . . . . . . . . . . . . . Dr B. Groisman There is an inherent time-asymmetry in the concept of a quantum state in standard quantum theory - it is defined by the results of measurements performed in the past. The two-state vector formalism of quantum mechanics, originated from [1], is a time-symmetrized approach to standard quantum theory [2]. In this formalism a system at a given time is described by two state vectors - one evolving forwards in time (defined by the results of measurements performed in the past) and another backwards in time (defined by the results of measurements performed in the future). This approach is particularly helpful for the analysis of experiments performed on pre- and post-selected ensembles of quantum systems. There many interesting and unusual effects which naturally arise in this approach [3,4,5]. In particular, it leads to a concepts of “weak measurements” (standard measurements with weakening of the interaction) and “weak values” (the outcomes of weak measurements) [5,6]. Weak values are very unusual because they can differ considerably from the range of eigenvalues of the observable. In the recent years we have witnessed a renewed interest in the two-vector formalism [7,8]. The purpose of the essay is to provide a self-contained introduction to the topic based on the background literature and explore the latest developments in the field, such as the application of weak measurement to quantum computing [9] and the attempt to explain recent observations suggesting that neutrinos travel faster than light as an interference effect, that can be interpreted as a weak measurement [10]. Candidates will have an option to choose between giving a broader review and focusing on deeper analysis of specific issues. Relevant Courses Essential: undergraduate level Quantum Mechanics Useful: Quantum Foundations, Quantum Computation, Quantum Information Theory References [1] Y. Aharonov, P.G. Bergmann, J.L. Lebowitz, Time symmetry in the quantum process of measurement, Phys. Rev. 134, B1410 (1964). [2] Y. Aharonov and L. Vaidman, The Two-State Vector Formalism: An Updated Review, Lect. Notes Phys. 734, 399-447 (2008) [3] Y. Aharonov, D.Z. Albert, l. Vaidman, How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100, Phys. Rev. Lett. 60, 1351 (1998).

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[4] Y. Aharonov, L. Davidovich, N. Zagury, Quantum random walks, Phys. Rev. A, 48, 1687 (1993). [5] Y. Aharonov and D. Rohrlich, Quantum Paradoxes, WILEY-VCH (2005). [6] Y. Aharonov and L. Vaidman, Properties of a quantum system during the time interval between two measurements, Phys. Rev. A 41, 11-20 (1990). [7] A. Broduch and L. Vaidman, Measurements of non-local weak values, J. Phys.: Conf. Ser. 174, 012004 (2009). [8] S. Massar and S. Popescu, Estimating Pre- and Post-Selected Ensembles, preprint arXiv:1106.0405v1 [quant-ph] (http://arxiv.org/abs/1106.0405). [9] A.P. Lund, Efficient quantum computing with weak measurements, New J. Phys. 13, 053024 (2011). [10] M.V. Berry, N. Brunner, S. Popescu, P. Shukla, Can apparent superluminal neutrino speeds be explained as a quantum weak measurement? preprint arXiv:1110.2832v1 [hep-ph] (http://arxiv.org/abs/1110.2832).

65. FreeFem++ Applied to the Driven Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Professor E.J. Hinch The aim of this project is to apply to the two-dimensional driven cavity the Finite Element software FreeFem++. It will be necessary to examine the numerical performance, both accuracy and runtime, which depend on the resolution and on options in the methods available. The tested program should then be applied to study the force on the top plate as a function of the Reynolds number, and also find the critical Reynolds number for an instability. Results should be compared with data in the literature. Relevant Courses Essential: Computational Methods in Fluid Dynamics References [1] Shankar, P.N. & Deshpande, M.D. (2000) Fluid mechanics in the driven cavity Ann. Rev. Fluid Mech. 32, 93–136. [2] Benjamin, A.S. & Denny, V.E. (1979) Convergence of numerical solutions for 2-D flows in a cavity at large Re J. Comp. Phys. 33, 340–358.

66. Numerical Simulations of Incompressible Two-Dimensional Turbulence . Professor E.J. Hinch The aim of this project is to study different formulations of the nonlinear terms in collocation (pseudo-spectral) methods for the two-dimensional Navier-Stokes equation in primitive variables. Three different possibilities for the i-th term are uj ∂j ui , ∂j (uj ui ), and the anti-self-adjoint linear combination. A ‘random’ initial condition should be used to ensure that disturbances are excited on many scales. The report should discuss both aliased and de-aliased/filtered solutions for the different forms of the nonlinear terms at various Reynolds numbers.

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Relevant Courses Essential: Computational Methods in Fluid Dynamics References [1] Frisch, U. & Sulem, P.L. (1984) Numerical simulation of the inverse cascade in twodimensional turbulence Phys. Fluids 27 1921–1923. [2] Clercx, J.J.H, & van Heist, G.J.F (2009) Two-Dimensional Navier-Stokes Turbulence in Bounded Domains Appl. Mech. Rev. 62 020802.

67. The Boundary Integral Technique for Potential Flows . . . . . . . . . . . . . . . . . . Professor E.J. Hinch Consider the two-dimensional potential flow inside a boundary given by a cubic spline (x(s), y(s)), with boundary condition of given value of the potential. Represent the unknown normal gradient of the potential first with a piecewise linear function and then with cubic-Bsplines. Test your program first with a circle and then with a 3 : 1 ellipse, and with boundary data φ = rn cos(nθ), for several n, finding the order of convergence. Finally apply your code to the Hele-Shaw cell problem of withdrawing fluid through a sink which is not at the centre of an initially circle of fluid. Relevant Courses Essential: Computational Methods in Fluid Dynamics References [1] Kelly, E. & Hinch, E.J. (1998) Numerical solution of Hele-Shaw flows driven by a quadrupole E.J. Applied Maths. 8, 551–566. [2] Kelly, E. & Hinch, E.J. (1998) Numerical solutions of sink flows in the Hele Shaw cell with small surface tension E.J. Applied Maths. 8, 533–550.

68. Spontaneous Symmetry Breaking and Phase Transitions in Two Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Professor R.R. Horgan This essay is concerned with continuous phase transitions in two dimensions and the rôle of Goldstone’s theorem. The essay should discuss Goldstone’s theorem and the difficulties in its interpretation in two dimensions. The essay should then consider models in two dimensions and discuss how spontaneous symmetry-breaking is realized in these cases. The techniques of renormalization group are needed to fully explore this topic and should be briefly described. This is a technically complicated subject so do not get bogged down in trying to understand every detail especially when the calculation is complicated and not illuminating. You must make a judgement about what is useful to show you have a good grasp of the approaches and methods used. Be selective.

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You may consult the references listed below and any others you feel will be useful. Many of the references given here give good bibliographies to further reading. General reading and RG are references [1],[3]; Goldstone’s theorem is discussed in [4]–[7], and models are discussed in [8]–[12]. The original XY-transition paper [8] is somewhat hard to read and should be read together [9]. Other articles can be researched on the web at www-spires.dur.ac.uk/find/hep and xxx.soton.ac.uk. The former site goes back further in time. There are also relevant books in the DAMTP section of the Betty and Gordon Moore Library on the ground floor. Relevant Courses Essential: Statistical Field Theory, Quantum Field Theory Useful: Advanced Quantum Field Theory References [1] J. Yeomans, “Statistical Mechanics of Phase Transitions” Oxford 1992. [2] J. Cardy, “Scaling and Renormalization in Statistical Physics” Cambridge 1996. [3] M. Le Bellac, “Quantum and Statistical Field Theory” Clarendon Press 1992; particularly chapter 4. [4] J. Goldstone, “Field Theories with Superconducting Solutions”, Il Nuovo Cimento Vol XIX 154, (1961). [5] J. Goldstone et al., “Broken Symmetries”, Phys. Rev. vol 127, 965 (1962). [6] S. Coleman “There are no Goldstone Bosons ....”, Comm. Math. Phys 31, 259-264 (1973). [7] S. Coleman et al. “Spontaneous Symmetry Breaking ....”, Phys Rev D10, 2491 (1974). [8] JM Kosterlitz and DJ Thouless, “Ordering, Metastability ...”, J. Phys. C: Solid State Physics, Vol 6, 1181 (1973). [9] K. Huang and J. Polonyi, “Renormalization of the Kink Current ....”, Int. J. Mod. Phys. A, Vol 6, 409-429 (1991)† . [10] B. Nienhuis, “Critical Behaviour of Two Dimensional Spin Models....” J.Stat.Phys. Vol 34, 731,(1984)† . [11] S. Coleman, “Quantum sine-Gordon equation ...” Phys. Rev. D11, 2088 (1975). [12] J.V. Jose et al., “Renormalization, vortices ....” Phys. Rev. B16, 1217 (1977). † In Betty and Gordon Moore Library.

69. Highly Oscillatory Quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Professor A. Iserles The main concern of this essay is the computation of integrals of the form Z f (x)eiωg(x) dV, Ω

where f and g are real, smooth functions, ω À 1 and Ω is a bounded multivariate domain. Integrals of this form occur in numerous applications of mathematics, e.g. in electromagnetics, fluid dynamics, quantum physics and numerical analysis. 63

Since the integrand oscillates rapidly, standard methods of integration are useless or, at best, exceedingly expensive. In the last few years several algorithms have been proposed, all based on an asymptotic expansion in inverse powers of ω. They all share the (perhaps counterintuitive) feature that the error decays very rapidly when ω is large! Although originally proposed for univariate quadrature, these methods are currently available in a multivariate setting. They all exploit the important feature of the above integral, namely that its asymptotic behaviour is in large measure determined by its behaviour at critical points, a feature of the function f and the geometry of the underlying domain. The purpose of the essay is to review these developments. Relevant Courses Useful: Numerical Solution of Differential Equations, Perturbation and Stability Methods References [1] D. Huybrechs & S. Vandewalle, “On the evaluation of highly oscillatory integrals by analytic continuation”. SIAM J. Numer. Anal. 44, 1026–1048 (2006). [Numerical stationary phase in one variable.] [2] A. Iserles & S. Nørsett, “Efficient quadrature of highly oscillating integrals using derivatives”, Proc. Royal Soc. A 461 1383–1399 (2005), available from www.damtp.cam.ac.uk/user/na/NA_papers/NA2004_03.pdf. [Filon-type methods in one variable.] [3] A. Iserles, S. P. Nørsett & S. Olver, “Highly oscillatory quadrature: The story so far” in Numerical Mathematics and Advanced Applications (A. Berm´ udez de Castro et al., eds), Spinger-Verlag (2006), 97–118, available from www.damtp.cam.ac.uk/user/na/NA_papers/NA2005_06.pdf. [A review paper, written from a multivariate standpoint.] [4] S. Olver, “On the quadrature of multivariate highly oscillatory integrals over non-polytope domains”. Numer. Math. 103 643–665 (2006), available from www.damtp.cam.ac.uk/user/na/NA_papers/NA2005_07.pdf. [Levin-type methods.]

70. Homological Techniques for Finite Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . Professor A. Iserles Finite elements are an established and ‘mature’ methodology in the numerical solution of partial differential equations. Yet, standard finite-element spaces fall short of modelling faithfully important structural features of the underlying system and are often unstable in surprisingly ‘simple’ settings: for example, solving σ = grad u, div u = f with continuous piecewise-linear elements is unstable! A new and important approach reformulates finite elements in terms of exterior calculus and piecewise-polynomial differential forms. This allows the application of homological techniques to construct finite-element spaces which, for example, solve the above grad-div equations stably, solve Maxwell equations with the correct spectrum, and obtain a raft of new and powerful stability and well-posedness estimates and convergence results. On the face of it, this essay requires a wide range of competencies, from numerical analysis to algebraic topology. Fortunately, a very comprehensive and clearly-written review paper [1] is available, which should provide a relatively painless introduction to the subject. 64

Relevant Courses Essential: Numerical Solution of Differential Equations Useful: Differential Geometry, Algebraic Topology References [1] D.N. Arnold, R.S. Falk & R. Winther, “Finite element exterior calculus, homological techniques, and applications”, Acta Numerica 15 (2006), 1–155. [An extensive and clear survey of the subject.] [2] D.N. Arnold, R.S. Falk & R. Winther, “Finite element exterior calculus: from Hodge theory to numerical stability”, Bull. Amer. Math. Soc. 47 (2010), 281–354. [A state-of-the-art review, with an emphasis on the Hodge Laplacian.]

71. Quantum Complexity of Local Hamiltonian Problems . . . . . . . . . . . . . . . . . . Professor R. Jozsa Quantum computation is generally regarded as being more powerful than classical computation but just like the latter, it also has its limitations on what is efficiently computable. In classical complexity theory, despite the notorious openness of whether P 6= NP or not, NP-complete problems provide strong candidates for computational tasks that cannot be efficiently computed classically. In the quantum context the complexity class QMA (“Quantum Merlin Arthur”) is an appropriate and natural quantum generalisation of NP, and as in the classical scenario, QMAcomplete problems are likely to be hard even for quantum computers. In 1999 Kitaev showed that the problem of characterising the ground state energy of a local hamiltonian is complete for QMA. This seminal result lead to a flurry of research activity refining, and extending the context of, Kitaev’s result. The basic aim of the essay is to provide an exposition of the class QMA and QMA-completeness, and outline how the local hamiltonian ground state problem can be seen to be in QMA, and to be complete for it. Further (or alternative) issues that could be discussed in an essay include refinements of Kitaev’s original result (e.g. reducing the degree of locality of the hamiltonian, considering 1D rather than 2D/3D spin systems, role of translation invariance in the hamiltonian etc.) or discussing other QMA-complete problems that have subsequently appeared in the literature. Relevant Courses Essential: Quantum Computation Useful: Quantum Information Theory References [1] A. Yu. Kitaev, A. Shen, M. N. Vyali, Classical and Quantum Computation. Graduate Studies in Mathematics vol 47, American Mathematical Society, 2002. Chapter 14. [2] D. Aharonov, T. Naveh, Quantum NP – A Survey (2002). Available at http://xxx.soton.ac.uk/abs/quant-ph/0210077.

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[3] J. Kempe, A. Kitaev, O. Regev, The Complexity of the Local Hamiltonian Problem, SIAM Journal on Computing 35(5):1070-1097, 2006. Also available at http://xxx.soton.ac.uk/abs/quant-ph/0406180. [4] D. Aharonov, D. Gottesman, S. Irani, J. Kempe, The Power of Quantum Systems on a Line, Comm. Math. Physics, vol. 287, no. 1, pp. 41-65 (2009). Also available at http://xxx.soton.ac.uk/abs/0705.4077. [5] Y. K. Liu, M. Christandl, F. Verstraete, N-Representability is QMA Complete, Phys. Rev. Lett. 98, 110503 (2007). Also available at http://xxx.soton.ac.uk/abs/quant-ph/0609125.

72. Theory and Constraints on Lorentz Symmetry Violation . . . . . . . . . . . . . . . Dr E.A. Lim The purpose of this essay is to undertake a study of the theoretical landscape and observational constraints on possible violations of Lorentz Symmetry, one of the key underpinnings of modern physics. There are several overlapping frameworks for violating Lorentz Symmetry. For example, on the particle physics side, Kostelecky’s Standard Model Extension is a generic expansion of the SM Lagrangian to include all possible Lorentz symmetry violating (LV) interaction terms, where the preferred frame effects are parameterized by constant tensor fields. On the gravity side, such fields gain dynamics and also gravitate, introducing new ways where LV can manifest itself in reality – the canonical example of LV in the gravity sector is Jacobson and Mattingly’s EinsteinAether theory. Interestingly, recent work has shown that Einstein-Aether theory is equivalent in the Infra-red limit to a power counting renormalizable gravity theory called Horava-Lifshitz gravity, thus providing a surprising connection of LV to a possible quantum theory of gravity. Candidates can focus on either the theoretical aspects or observational aspects of LV. In both cases, candidate should pick a particular theoretical framework and do a detail study on it instead of providing a general overview of all possible LV mechanisms (which would be too broad). Relevant Courses Essential: General Relativity, Cosmology, Advanced Cosmology Useful: Quantum Field Theory, Advanced Quantum Field Theory, String Theory References [1] D. Colladay, V. A. Kostelecky, Phys. Rev. D58, 116002 (1998). [hep-ph/9809521]. [2] D. Mattingly, Living Rev. Rel. 8, 5 (2005). [gr-qc/0502097]. [3] T. Jacobson, S. Liberati, D. Mattingly, Annals Phys. 321, 150-196 (2006). [astro-ph/0505267]. [4] T. Jacobson, D. Mattingly, Phys. Rev. D64, 024028 (2001). [gr-qc/0007031]. [5] T. P. Sotiriou, J. Phys. Conf. Ser. 283, 012034 (2011). [arXiv:1010.3218 [hep-th]]. [6] S. Liberati, L. Maccione, Ann. Rev. Nucl. Part. Sci. 59, 245-267 (2009). [arXiv:0906.0681 [astro-ph.HE]].

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73. Fluid Flow and Elastic Flexure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr J.A. Neufeld In a number of natural and environmental settings fluids flow over elastically deforming surfaces. At the smallest length scales liquid drops placed on a floating elastic substrate deform the surface and lead to elastic wrinkling of the substrate. At very much larger length scales the characteristic size of melt water ponded on sea ice may be set by the elastic flexure of the floating ice, and the motion of the Tibetan plateau may be influenced by the flexure of the Indian plate underneath. In this essay, motivated by these settings, consider the propagation of an axisymmetric, fixed-volume, viscous gravity current over an elastic sheet floating on an inviscid ocean. Solve for the rate of propagation and the form and magnitude of the deflection of the elastic membrane (neglecting capillary effects). Importantly, consider the possibility of steady-state solutions with finite radius for a range of density contrast between current, elastic sheet and ocean. The essay should conclude with some remarks on the relevance of the problem to industrial and geophysical settings. At the discretion of the instructor, this essay may be attempted as either a theoretical or an experimental project. Relevant Courses Useful: Slow Viscous Flow References [1] Huang, J., Juszkiewicz, M., de Jeu, W. H., Cerda, E., Emrick, T., Menon, N., Russell, T. P. 2007 Capillary wrinkling of floating thin polymer films. Nature 317, 650–653. [2] Copley, A., McKenzie, D. 2007 Models of crustal flow in the India-Asia collision zone. Geophys. J. Int. 169, 683–698. [3] Kwok, R., Untersteiner, N. 2011 The thinning of Arctic sea ice. Physics Today 64, 36–41. [4] Huppert, H. E. 1982 The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface. J. Fluid Mech. 121, 43–58.

74. Eccentric Astrophysical Discs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr G.I. Ogilvie Closed Keplerian orbits around a massive body are generally non-circular. A thin Keplerian disc may be composed of nested elliptical orbits whose eccentricity e and longitude of pericentre $ vary continuously with semi-major axis a and time t. The complex eccentricity is e exp(i$) = E(a, t). When |E| and |∂E/∂ ln a| are sufficiently small, a linear evolutionary equation can be derived for the complex eccentricity, which determines how the shape of the disc propagates by means of pressure, viscosity, self-gravity and other collective effects that are weak compared to gravity. More generally, E(a, t) satisfies a nonlinear evolutionary equation and the presence of eccentricity affects the transport of mass and angular momentum in the disc. Eccentric discs are thought to exist in many astrophysical situations, including narrow planetary rings around Saturn and Uranus, protoplanetary discs around young stars, circumstellar discs around rapidly rotating Be stars, and accretion discs around compact objects in close binary systems. 67

This essay should discuss aspects of the dynamics and significance of eccentric discs in at least one of these areas of application. Apart from the derivation and interpretation of the evolutionary equation(s) for eccentric discs, theoretical topics that might be discussed include the stability of fluid flows with elliptical streamlines, the gravitational interaction of orbiting companions with a disc, and the numerical simulation of eccentric discs. A selection of starting references is provided below, and use of the NASA ADS archive is recommended. Interested candidates should contact Gordon Ogilvie for further advice. Relevant Courses Useful: Astrophysical Fluid Dynamics, Dynamics of Astrophysical Discs, Planetary System Dynamics References [1] Borderies, N., Goldreich, P. & Tremaine, S. (1983). Astron. J. 88, 1560–1568 [2] Ferreira, B. T. & Ogilvie, G. I. (2009). Mon. Not. R. Astron. Soc. 392, 428–438 [3] Goodchild, S. & Ogilvie, G. (2006). Mon. Not. R. Astron. Soc. 368, 1123–1131 [4] Kley, W. & Dirksen, G. (2006). Astron. Astrophys. 447, 369–377 [5] Lubow, S. H. (1991a). Astrophys. J. 381, 259–267 [6] Lubow, S. H. (1991b). Astrophys. J. 381, 268–277 [7] Ogilvie, G. I. (2001). Mon. Not. R. Astron. Soc. 325, 231–248 [8] Ogilvie, G. I. (2008). Mon. Not. R. Astron. Soc. 388, 1372–1380 [9] Papaloizou, J. C. B. (2005a). Astron. Astrophys. 432, 743–755 [10] Papaloizou, J. C. B. (2005b). Astron. Astrophys. 432, 757–769 [11] Tremaine, S. (2001). Astron. J. 121, 1776–1789

75. Accretion Discs and Planet Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Professor J.C.B. Papaloizou Accretion discs occur in astrophysics whenever matter is prevented from moving directly onto a central gravitating mass on account of its angular momentum content. They form during the process of star formation and can become sites of planet formation. The purpose of this essay is to give an overview of the processes associated with building up protoplanetary cores through the accumulation of solid material and their later capture of gas to form giant planets in protoplanetary accretion discs. Numerical simulations of processes such as dust sedimentation and protoplanetary core migration may be considered subsequently. Anyone considering writing this essay should consult J.C.B. Papaloizou for further advice. Relevant Courses Useful: Structure and Evolution of Stars, Astrophysical Fluid Dynamics, Dynamics of Astrophysical Discs, Planetary System Dynamics

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References [1] Lissauer, J., 1993, Annual Reviews of Astronomy and Astrophysics, 31, 129. [2] Papaloizou, John C.B.; Terquem, Caroline, 2006, Reports of Progress in Physics, 69, 119. (arXiv:astro-ph/0510487)

76. Global Modes in Shear Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Professor N. Peake The classical stability of parallel shear flows has received considerable attention, but it is only more recently that progress has been made for the weakly nonparallel and weakly nonlinear flows that are found in many practical situations (for instance, in the wake behind an airfoil). The approach to these nonparallel problems is to apply standard stability analysis at each streamwise position to determine the local stability properties. The key question then is how this local behaviour relates to the global stability or otherwise of the whole flow, and to answer this one must apply some frequency selection criterion to the local data in order to find the so-called global mode frequency for the system. Examples of the application of this idea include the prediction of the frequencies of such unsteady phenomena as vortex shedding and the breakup of a capillary jet, which can be thought of as the global modes of some initial unstable shear flow. The precise nature of the frequency selection criterion has been a matter of considerable debate, and a number of possibilities have been put forward. It is accepted, however, that absolute instability plays a key role in the development of such flows, and typically one must have a shear flow which contains a sufficiently large streamwise region of absolute instability in order to generate an unstable global mode. Work on a nonlinear selection (Pier 2001) criterion has added a further dimension to this debate. The main aim of this essay is to review previous work on the application of global-mode theories in fluid dynamics. Excellent survey articles are given below, which will provide a good starting point. Alternatively, if the student wishes, the essay could involve some research project work, in which a simple open-source Navier-Stokes solver is used to investigate the stability of an airfoil wake to vortex shedding. Appropriate support would be provided. Relevant Courses Essential: Perturbation and Stability Methods References [1] Huerre, P. & Monkewitz, P.A. 1990 Local and global instabilities in spatially developing flows. Ann. Rev. Fluid Mech. 22, 473-537. [2] Pier, B. 2001 Nonlinear self-sustained structures and fronts in spatially developing wake flows. J. Fluid Mech.435, 145 - 174. [3] J. M. Chomaz. 2005 Global instabilities in spatially developing flows: Non-normality and nonlinearity. Annual Review of Fluid Mechanics, 37, 357-392.

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77. Generalised Geometry from M-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Professor M.J. Perry The bosonic part of the supergravity limit of M-theory has a hidden geometric structure that seem to generalise what one knows of riemannian geometry. Whilst much of this structure is still unknown there is sufficient material to enable to glimpse the kind of geometry that makes its appearance in M-theory. The structure of what is possible is fairly concrete in low dimensions, specifically d=4,5,6 and 7. Your essay should include a description of what is known in these cases. The general picture can be regarded as being derived from the infinite-dimensional KacMoody-Borcherds algebra E11. A brief discussion of such considerations can also be presented in your essay. Relevant Courses Essential: General Relativity, Symmetries and Particles Useful: String Theory References [1] Generalised Geometry and M theory - D.S. Berman and M.J. Perry, arxiv 1008.1763. [2] SO(5,5) Duality in M-theory and generalized geometry, D.S.Berman, H. Godazgar and M.J. Perry, arxiv 1103.5733. [3] The local symmetries of M theory and their formulation in generalised geometry, D.S. Berman, H. Godazgar, M. Godazgar and M.J. Perry, arxiv 1110.3930. [4] Duality Invariant Action and Generalised Geometry, D.S. Berman, H. Godazgar, M.J. Perry and P. West, arxiv 1111:0459

78. Localised Convective Cells in Subcritical Convection . . . . . . . . . . . . . . . . . . . Professor M.R.E. Proctor Many types of pattern forming instability can exhibit subcritical bifurcation, so that stable patterns can co-exist with a zero (unpatterned) state. Examples include convection in binary fluids and convection in an imposed magnetic field. If such bistable states exist there exist the possibility of fronts separating the zero from the patterned state. With two such fronts localised solutions are possible in which isolated groups of convection cells can exist in a large domain with no motion otherwise. On one level these localised modes (if steady) can be thought of as homoclinic orbits of the complicated spatial equations describing steady convection, and in some cases can be looked at in terms of dynamical systems theory. Or they can be discussed as stable solutions of the time dependent convection equations. Some analysis and numerical work is possible revealing an intricate relationship between the many branches of localised modes with different numbers of convection cells. Most work has been done for two-dimensional solutions, though some interesting results for three-dimensional motion are starting to appear. The essay should give an overview of the various types of such solutions in the literature, and then focus on detailed discussion of a particular aspect.

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Relevant Courses Essential: Part II Fluid Dynamics or similar viscous fluids course Useful: Part II Dynamical Systems or similar nonlinear dynamics course, Part III Perturbation and Stability Methods (Michaelmas term) References [1] E. Knobloch. Spatially localized structures in dissipative systems: open problems. Nonlinearity. 21 (2008) T45-T60 [2] A. Bergeon and E. Knobloch. Periodic and localized states in natural doubly diffusive convection. Physica D. 237 (2008) 1139-1150 – and many other papers by Knobloch and collaborators [3] J.H.P. Dawes, The emergence of a coherent structure for coherent structures: localized states in nonlinear systems. Phil. Trans. R. Soc. A 368, 3519-3534 (2010)

79. The Backus-Gilbert Method for the Solution of Inverse Problems . . . . . . Dr O. Rath Spivack The inverse scattering problem, i.e. the problem of recovering properties of a scatterer from the knowledge of the scattered field, is of great importance in many applications, such as acoustic imaging and remote sensing, and much remains to be explored. It is an ill-posed problem in the sense that its solution does not depend continuously on the initial data, and needs to be implemented using additional techniques (regularisation) to make it stable. The Backus-Gilbert method was developed in the context of geophysics to compute various properties of the earth mantle [1,2], but is applicable to other problems in wave scattering. It is a method particularly suited to numerical solutions of the inverse scattering problem, since it regularises the problem by minimising a discretised quadratic form. It constructs approximate solutions to the moment problem Z ki (s)f (s)ds = yi , i = 1, . . . , N , where yi are given functions (the measured data) and the ki are arbitrary given function, approaching it from the point of view of finding how well all possible models can be recovered pointwise. In this essay, after introducing the ill-posed nature of inverse problems in general, and how regularisation can remedy this, the Backus-Gilbert method should be explained and its properties discussed. According to your interests and background, it is possible to develop the essay in different directions. You could concentrate on the more rigorous results and proofs of its properties [3,4,5]; alternatively, its relationship with Shannon sampling theory could be investigated and applications in signal processing explored [6,7]; finally, the Backus-Gilbert method could be explored from the point of view of Bayesian statistics, viewing the inverse problem as an inference problem [8]. Relevant Courses Essential: Knowledge of the wave equation and basic concepts in wave propagation, from any course. 71

Useful: Wave Propagation and Scattering, Topics in Analysis, knowledge of functional analysis from any course. References [1] Backus, G.E. and Gilbert, J.F. The resolving power of gross earth data Geophys, J Roy. Astron. Soc. 16, 169 (1968) [2] Backus, G.E. and Gilbert, J.F. Uniqueness in the inversion of inaccurate gross earth data Phil. Trans. Roy. Soc. London A266, 123 (1970) [3] Kirsch, A. An Introduction to the Mathematical Theory of Inverse Problems Springer (1996) (chapter 3.6) [4] Kirsch, A. Schomburg, B. and Berendt, G. The Backus-Gilbert method Inverse Problems 4, 771 (1988) [5] Schomburg, B. and Berendt, G. On the convergence of the Backus-Gilbert algorithm Inverse Problems 3, 341 (1987) [6] Caccin, B. Roberti, C. Russo, P. and Smaldone, A. The Backus-Gilbert method and the processing of sampled data IEEE Trans. Signal Proc. 40, 2823 (1992) [7] Huestis, Stephen P. The Backus-Gilbert problem for sampled band-limited functions Inverse Problems 8, 873 (1992) [8] Evans, Steven N. and Stark, Philip B. Inverse problems as statistics Inverse Problems 18, R55 (2002)

80. Sampling Methods in Inverse Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr O. Rath Spivack The inverse scattering problem, i.e. the problem of recovering properties of a scatterer from the knowledge of the scattered field, has been the object of increased interest in the past twenty years or so. The field has grown rapidly and several results and applications have been achieved, whilst much remains to be explored. The linear sampling method was first proposed by Colton and Kirsch [1]. It exploits the properties of the far field to use an ‘indicator function’ to reconstruct the support of the scatterer. This method and its variants (see, e.g., [2,3]) are usually based on a factorization of the far field operator, F : L2 (S1 ) → L2 (S1 ) defined by: Z (F g)(ˆ x) = u∞ (kˆ x, y)g(y)ds(y) , (1) S1

where S1 is the unit sphere, and u∞ (kˆ x, y) is the far field pattern of the scattered field. They do not need a priori knowledge of the scatterer and can also be used for scatterers which are not simply connected. This essay should explain the ideas on which the Linear Sampling method and variants are based [4] and how they can be implemented to obtain stable solutions, and explore the advantages and drawbacks of the method. According to interests and background, it could then be developed in different directions, for example concentrating on issues of existence and uniqueness, and convergence [5,6]; or on applications and numerical results in acoustic and electromagnetic problems [7,8].

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Relevant Courses Essential: Knowledge of the wave equation and basic concepts in wave propagation, from any course. Basic knowledge of functional analysis from any course. Useful: Wave Propagation and Scattering, Computational Methods in Fluid Dynamics References [1] Colton D and Kirsch A A simple method for solving inverse scattering problems in the resonance region. Inverse Problems 12, 383-93 (1996) [2] Potthast R A survey on sampling and probe methods for inverse problems Inverse Problems 22,R1-R47 (2006) [3] Colton D, Haddar H and Piana M The linear sampling method in inverse electromagnetic scattering theory Inverse Problems 19,S105-S137 (2003) [4] Arens T Why linear sampling works Inverse Problems 20, 163-173 (2004) [5] Colton D and Kress R Inverse Acoustic and Electromagnetic Scattering Theory Springer Verlag, 1992 [6] Arens T and Lechleiter A The linear sampling method revisited Journal of Integral Equations and Applications 21, 179-202 (2009) [7] Colton D, Giebermann K and Monk P The linear sampling method for three-dimensional inverse scattering problems ANZIAM J 42 (E), C434-C460 (2000) [8] Gebauer B, Hanke M and Schneider C Sampling methods for low-frequency electromagnetic imaging Inverse Problems 24, 015007 (2008)

81. Black Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr H.S. Reall In 4 spacetime dimensions, Hawking’s black hole topology theorem asserts that a stationary black hole must be topologically spherical. The black hole uniqueness theorems assert that a stationary black hole solution of Einstein-Maxwell theory is uniquely specified by its mass, angular momentum and electromagnetic charges. A black ring is a stationary black hole solution of the Einstein equation in d spacetime dimensions with topology S 1 × S d−3 . Explicit solutions are known only for d = 5 but approximate techniques indicate that analogous solutions exist for d > 5. The existence of black rings demonstrates that the black hole topology and uniqueness theorems do not generalize in an obvious way to d > 4 dimensions. This essay should review what is known about black rings. It should be written at a level that would be understood by another Part III student who had attended the relevant courses. Relevant Courses Essential: General Relativity, Black Holes.

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References [1] R. Emparan and H.S. Reall, “Black Rings”, Class. [arxiv:hep-th/0608012]

Quant.

Grav.

23, R169 (2006)

[2] R. Emparan and H.S. Reall, “Black holes in higher dimensions”, Living Reviews in Relativity 11, 6 (2008) [arxiv:0801.3471] [3] R. Emparan, “Blackfolds”, arXiv:1106.2021

82. Nonlinear Diffusion Equations for Image Enhancement . . . . . . . . . . . . . . . . . Dr C.-B. Sch¨ onlieb In our modern society the processing of digital images becomes more and more important, reflected in a myriad of applications: medical imaging (MRI, CT, PET), forensics, security, design, arts and many more. The most powerful image processing methods are based on mathematical principles. In the course of this essay we are interested in imaging approaches using partial differential equations (PDEs) [1-3]. The simplest PDE in this context is the heat equation. Starting with an initial degraded image this image is evolved along the heat equation in order to create a smoothed, e.g., denoised, version of it. The action of the heat equation on the given image can be understood as convolving the image with a Gaussian filter that inhibits high frequencies (oscillations, e.g., noise, in the space domain). Note however, that this smoothing is isotropic and does not depend on the given image. In particular, edges in an image are not preserved. For these reasons nonlinear PDEs have been proposed that reduce the diffusivity at those locations which have a larger likelihood to be edges. This likelihood is measured by |∇u|2 . A straightforward modification of the heat equation towards this is the Perona-Malik equation [3], which evolves an initial given image u(t = 0, x) = f (x) as ¡ ¢ ut = div g(|∇u|2 )∇u , with, e.g., g(s2 ) = 1/(1 + s2 /λ2 ), λ > 0. The aim of this essay is to study this equation and its application to image enhancement. In particular, we are interested in the properties of its solutions and how they change for different choices of the diffusivity g. How does image enhancement with Perona-Malik compare with the one using the heat equation? Relevant Courses Essential: Functional Analysis, Partial Differential Equations Useful: Numerical Analysis, Variational Calculus References [1] G. Aubert, and P. Kornprobst, Mathematical Problems in Image Processing. Partial Differential Equations and the Calculus of Variations, Springer, Applied Mathematical Sciences, Vol 147, (2006). [2] T. F. Chan, and J. J. Shen, Image Processing and Analysis - Variational, PDE, wavelet, and stochastic methods. SIAM, (2005). [3] P. Perona, and J. Malik, Scale-space and Edge Detection Using Anisotropic Diffusion, IEEE Transactions on Pattern Analysis and Machine Intelligence, 12(7), pp.629-639, July 1990.

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83. Feynman Diagrams in String Theory: the Topological Vertex . . . . . . . . . . Dr A. Sinkovics Topological string theory is a toy model of string theory where we restrict attention to certain geometric structures of the background spacetime. Besides the geometric insight it gives, this model provides powerful computational methods for certain classes of string diagrams. In particular, the topological A-model for non-compact Calabi-Yau spaces can be reformulated in terms of diagrammatic rules of a cubic field theory. This essay will study the structure of topological A-model, and the construction of its open and closed amplitudes using the topological vertex and its refined version. Relevant Courses Essential: String Theory, Quantum Field Theory Useful: Algebraic Geometry References [1] M. Aganagic, A. Klemm, M. Marino, C. Vafa, “The Topological vertex,” Commun. Math. Phys. 254, 425-478 (2005). [hep-th/0305132]. [2] A. Iqbal, C. Kozcaz, C. Vafa, “The Refined topological vertex,” JHEP 0910, 069 (2009). [hep-th/0701156]. [3] M. Marino, “Chern-Simons theory and topological strings,” Rev. Mod. Phys. 77, 675-720 (2005). [hep-th/0406005]. [4] A. Neitzke, C. Vafa, “Topological strings and their physical applications,” [hep-th/0410178].

84. Small Thermal Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr P. Skrzypczyk Down to what size can one reasonably define thermodynamic variables and thermodynamic machines? In fact it is possible to create a thermal machine by using only a pair of two-state quantum systems (commonly referred to as qubits): Each qubit is placed in contact with a separate thermal bath, both at different temperatures. An external object, if coupled correctly to the pair of qubits, can be either cooled down, heated up, or have work done upon it, therefore exhibiting the behaviour of either a refrigerator, heat pump or heat engine. This is one example of a construction of a quantum thermal machine and is one of the smallest possible constructions [1]. It is possible to understand the underlying functioning principle of such machines using only basic aspects of quantum mechanics and thermodynamics, and it shows that we can push thermodynamic concepts all the way down to the smallest of scales. This essay will explore this construction, as well as the many alternative ones, examples of which are given in [2,3,4], in a general study of quantum thermal machines. The references provided are only a starting point and there is scope to push the essay in a number of directions, depending upon the interests of the author.

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Relevant Courses Useful: Quantum Information Theory, Quantum Computation. References [1] N. Linden, S. Popescu and P. Skrzypczyk, Phys. Rev. Lett. 105, 130401 (2010); arXiv:1010.6029 (2010). [2] M. J. Henrich, M. Michel, G. Mahler, Europhys. Lett. 76, 1057 (2006); Phys. Rev. E 75, 051118 (2007). [3] A. E. Allahverdyan, K. Hovhannisyan, and G. Mahler, Phys. Rev. E 81, 051129 (2010). [4] J. P. Palao, R. Kosloff, and J. M. Gordon, Phys. Rev. E 64, 056130 (2001); E. Geva and R. Kosloff, J. Chem. Phys. 104, 7681 (1996).

85. BPS States in Supersymmetric Gauge Theories . . . . . . . . . . . . . . . . . . . . . . . . Professor D. Tong Supersymmetric theories contain special objects known as BPS states. In the classical theory, these are typically solitonic field configurations that solve the equations of motion. In the quantum theory, they are particles in the spectrum. BPS states live in short representations of the supersymmetry algebra and their mass is dictated by the charges that they carry. They provide one of the few windows into the strong coupling regime of field theories. Perhaps the most important example of a BPS state is provided by the magnetic monopole. These objects underlie our understanding of S-duality in N=4 super Yang-Mills, as well as the Seiberg-Witten solution of N=2 supersymmetric gauge theories. The purpose of this essay is to explain the meaning of BPS states, their relationship with the supersymmetry algebra, and to give some examples of their importance in understanding the dynamics of strongly coupled field theories. Relevant Courses Essential: Quantum Field Theory, Advanced Quantum Field Theory, Supersymmetry, Symmetries and Particles References [1] The importance of BPS states was first explained in a paper by Witten and Olive, Phys.Lett. B78: 97 (1978). [2] A good review of BPS magnetic monopoles and their application to N=4 super-Yang-Mills can be found in J. Harvey, “Magnetic Monopoles, Duality, and Supersymmetry”, arXiv:hep-th/9603086. [3] Many details of BPS states are covered in the lecture notes “Electromagnetic duality for children” by Jose Figueroa-O’Farrill. They can be downloaded at http://www.maths.ed.ac.uk/~jmf/Teaching/EDC.html.

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86. Global Isometric Embeddings and their Application to Black Hole Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Professor P.K. Townsend Every Riemannian manifold M can be globally and isometrically embedded in a higherdimensional Euclidean space. “Isometric” means that the Riemannian metric on M is the one induced by the higher-dimensional flat metric, and “global” means that the result holds for the whole of M. Less is known about global isometric embeddings of spacetimes, with Lorentzian-signature metrics, into higher-dimensional flat spacetimes, but many examples are known. For instance, the maximal analytic extension of the Schwarzschild spacetime can be globally isometrically embedded in a 6-dimensional Minkowski spacetime and the maximal analytic extension of the Reissner-Nordstrom spacetime can be globally isometrically embedded in a 7-dimensional flat spacetime with two time dimensions. The latter example illustrates the fact that the higher-dimensional spacetime generically must have more than one time dimension. The topic of global isometric embeddings, in particular of black-hole spacetimes, has attracted attention in recent years because of a connection to thermodynamics via a quantum field theory formula of Unruh that associates a temperature to an acceleration. A stationary observer in the black hole spacetime undergoes constant proper acceleration in the higher-dimensional flat spacetime, and the associated temperature is found to coincide with the local temperature of the black hole under the assumption of thermal equilibrium. A similar result applies to de Sitter spacetime, and various other spacetimes with event horizons, in various dimensions. Although it is not clear why the Unruh formula should apply, it does appear to connect the quantum physics of horizons with extrinsic geometric properties, whereas all classical properties are entirely intrinsic. This essay should (i) survey some of the mathematical ideas and results on global isometric embeddings, (ii) present a variety of examples of global isometric embeddings of Lorentziansignature spacetimes, with a discussion of their geometrical features, and (ii) review the connection to thermodynamics via the Unruh formula. Relevant Courses Essential: Black Holes Useful: Quantum Field Theory References [1] C. Fronsdal, Completion and embedding of the Schwarzschild solution, Phys. Rev. 116 (1959) 778. [2] J. Rosen, Embedding of various relativistic riemannian spaces in pseudo-euclidean spaces, Rev. Mod. Phys. 37 (1965) 204. [3] R. Penrose, A remarkable property of plane waves in general relativity, Rev. Mod. Phys. 37 (1965) 215. [4] C. J. S. Clarke, On the Global Isometric Embedding of Pseudo-Riemannian Manifolds, Proc. R. Soc. Lond. A 13 (1970) vol. 314 no. 1518, 417-428 [5] M. Ferraris and M. Francaviglia, An algebraic isometric embedding of Kruskal spacetime, Gen. Rel. & Grav. 10 (1979) 283.

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[6] H. Narnhofer, I. Peter and W.E. Thirring, How hot is the de Sitter space, Int. J. Mod. Phys. B10 (1996) 1507 [7] S. Deser and O. Levin, Equivalence of Hawking and Unruh temperatures through flat space embeddings, Class. Quant. Grav. 15 (1998) L85-L87 (arXiv:hep-th/9806223); Mapping Hawking into Unruh thermal properties, Phys. Rev. D59 (1999) 064004 (arXiv:hep-th/9809159). [8] L. Andrianopoli, M. Derix, G.W. Gibbons, C. Herdeiro, A Santambrogio and A. Van Proeyen, Isometric embedding of BPS branes in flat space with two times, Class. Quant. Grav. 17 (2000) 1875 (arXiv: hep-th/9912049). [9] N. L. Santos, O.J.C. Dias and J.P.S. Lemos, Global embedding of D-dimensional black holes with a cosmological constant in Minkowskian spacetimes: matching between Hawking temperature and Unruh temperature, Phys. Rev. D70 (2004) 124033 (arXiv: hep-th/0412076). [10] J.G. Russo and P.K. Townsend, Accelerating branes and brane temperature, Class. Quant. Grav. 25 (2008) 175017 (arXiv:0805.3488); Relativistic kinematics and stationary motions, J.Phys. A42 (2009) 445402 (arXiv:0902.4243). [11] E.J. Brynjolfsson and L. Thorlacius, Taking the temperature of a black hole, JHEP 09 (2008) 66 (arXiv:0805.1876). [12] R. Banerjee and B.R. Majhi, A new global embedding approach to study Hawking and Unruh effects, Phys. Lett. B690 (2010) 83 (arXiv:1002.0985).

87. Segregated Granular Collapse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr N.M. Vriend Granular materials are of tremendous importance in transport of coal and sand in industry and in natural phenomena such as snow avalanches and debris flows. Developing mathematical models for the dynamical behavior of granular materials, which exhibit both solid and fluid-like properties, remains a real challenge, but the shallow water equations (e.g. Savage & Hutter, 1989 [1]) are able to capture several aspects of shallow granular flows. An added complication in the modelling of polydisperse granular mixtures is segregation (e.g. Gray, 2010 [2]) – separations of grains leading to an inhomogeneous structure that could significantly alter its dynamical behavior. Imposing examples of segregated granular collapse are giant landslides, such as the Blackhawk Slide in California (Shreve, 1959 [3]), forming a large-scale display of granular collapse (h = 1.5 km) with long spreading paths (l = 9 km) with significant segregation in the deposit. One hypothesis is that segregation significantly enhances the run-out of these long slides, as the larger, heavier boulders can ride on top of a granulated lubrication layer. Within this essay, there is scope for a thorough literature survey on previous work on segregation effects in granular collapse. In the essay, one could either choose to proceed to extend existing models to consider segregation aspects in the shallow water equations or perform simple laboratory experiments exploring the effect of the degree of segregation of a bimodal mixture on the runout distance. Relevant Courses Essential: Geophysical and Environmental Fluid Dynamics Useful: Demonstrations in Fluid Dynamics, Fluid Dynamics of Energy 78

References [1] Savage, S. and Hutter, K. 1989, The motion of a finite mass of granular material down a rough incline. J. Fluid Mech. 199, 177215. [2] Gray J. M. N. T. 2010, Particle size segregation in granular avalanches: A brief review of recent progress. AIP Conference Proceedings. 1227, 343-362 [3] Shreve, R. L. 1959, Geology and mechanics of the Blackhawk landslide, Lucerne Valley, California. Dissertation (Ph.D.), California Institute of Technology

88. Walk, Don’t Run! – A Story of Broken Symmetry (Presented in Technicolour) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr M.B. Wingate For decades, much of particle physics research has been searching to understand the mechanism behind electroweak symmetry-breaking. The Standard Model explanation is most prominently attributed to Higgs: the W and Z bosons acquire a large mass via interactions with a scalar condensate. It is possible that this condensate is the result of some undiscovered strong dynamics. By analogy with QCD and the colour force, candidate theories of this type are called technicolour models. The essay should touch upon the following questions: Why must there be (at least) 2 scales, the technicolour and extended technicolour scales? Why cannot technicolour be simply a scaled-up version of QCD? How would a walking coupling constant generate the necessarily large ratio of these 2 scales? The essay should go into some depth regarding one aspect of technicolour. There are many papers on technicolour theories, some with pedagogical introductions emphasizing various aspects. Some searching on arXiv.org or inspirehep.net is unavoidable. A good place to start is [1] which cites many of the original papers. A comprehensive review, for its time, was given in [2]. Relevant Courses Essential: Standard Model Useful: Statistical Field Theory References [1] G Fleming, “Strong Interactions for the LHC,” arXiv:0812.2035 [2] C T Hill and E H Simmons, “Strong dynamics and electroweak symmetry breaking,” Physics Reports 381, 235 (2003).

89. Primordial Non-Gaussianity and Large-Scale Structure . . . . . . . . . . . . . . . . Professor E.P.S. Shellard, Dr X. Chen Standard slow-roll inflation generates primordial density fluctuations δ(k) with Gaussian statistics. This is arguably the most stringent prediction of standard inflation because comparing the bispectrum B(k1 , k2 , k3 ) ∝ hδ(k1 )δ(k2 δ(k3 )i (or three-point correlator) to the power spectrum 79

P (k) ∝ h|δ(k)|2 i schematically yields B/P 3/2 ≈ O(², η) P 1/2 , where ² and η are the slow-roll parameters with ², η ∼ 10−2 . Thus with P (k)1/2 ∼ 10−5 , deviations from Gaussianity are predicted to be below one part in a million! This ratio for slow-roll inflation was first calculated quantitatively by Maldacena, with the dominant “local” non-Gaussian contribution given by B(k1 , k2 , k3 ) = fNL [P (k1 )P (k2 ) + P (k2 )P (k3 ) + P (k3 )P (k1 )] , where fNL ≈ (4² − 2η) is known as the non-linearity parameter. Current CMB observations are consistent with Gaussianity and the latest WMAP constraint is −10 < fNL < 74 at 95% confidence. However, forthcoming data will test the inflationary prediction to much higher precision, notably Planck satellite results in early 2013 with ∆fNL = 5 (1-σ error). Recent work has suggested that three-dimensional data from large-scale structure, notably multi-million galaxy surveys and 21cm data, will substantially improve over CMB constraints, potentially measuring fNL ∼ O(1). Key progress on this was made by Dalal et al [3] who noticed that galaxy bias is sensitive to non-Gaussianity and can be amplified in the power spectrum. [Bias measures the degree to which galaxies are more strongly correlated than the underlying dark matter density.] This has led to some constraints on non-Gaussianity from large-scale structure (for an introduction see, for example, the reviews [1, 2]). However, the amplification of bias is also evident in the galaxy bispectrum and this should prove to be an even better measure. Studies of the large-scale bispectrum represent a formidable challenge both observationally and computationally (see e.g. [6]). This essay should provide some overview of potential sources of primordial non-Gaussianity (beyond slow-roll inflation), before discussing the basic methodology for exploiting large-scale structure to identify or constrain it. Students can also choose to invert this relative emphasis and concentrate more on the early universe and the origin of non-Gaussianity. Relevant Courses Essential: Cosmology Useful: Advanced Cosmology, General Relativity, Quantum Field Theory References [1] M. Liguori, E. Sefusatti, J.R. Fergusson, E.P.S. Shellard, “Primordial non-Gaussianity and Bispectrum Measurements in the Cosmic Microwave Background and Large-Scale Structure”, Adv. Astron. 2010: 980523 (2010). [arXiv:1001.4707]. [2] Vincent Desjacques, Uros Seljak, “Primordial non-Gaussianity in the large scale structure of the Universe”, arXiv:1006.4763. [3] X. Chen, “Primordial Non-Gaussianities from Inflation Models,” Adv. Astron. 2010, 638979 (2010), [arXiv:1002.1416 [astro-ph.CO]]. [4] J. Maldacena, “Non-Gaussian Features of Primordial Fluctuations in Single-Field Inflationary Models,” JHEP 0305, 013 (2003). [5] Neal Dalal, Olivier Doré, Dragan Huterer, Alexander Shirokov, “The imprints of primordial non-gaussianities on large-scale structure: scale dependent bias and abundance of virialized objects”, Phys. Rev. D77: 123514 (2008). [6] D.M. Regan, M.M. Schmittfull, E.P.S. Shellard, J.R. Fergusson Universal Non-Gaussian Initial Conditions for N-body Simulations, arXiv:1108.3813. 80

90. Categorical Galois Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr J. Goedecke G. Janelidze developed the concept of categorical Galois theory as a generalisation which encompasses classical Galois theory for rings, several aspects of topological descent theory, and the theory of central extensions of (for example) groups. This essay should give an outline of categorical Galois theory, and can then continue in many directions: one obvious option would be to work towards the proof of the Fundamental Theorem of Galois Theory (given in Section 11 of [1]), another option could be to compare different examples such as the classical ring theory and topological examples, or to compare different notions of central extensions arising from categorical Galois theory in different settings. There is also the classification of central extensions (see [2]). Other references than those below are available and can be discussed with me. Relevant Courses Essential: Category Theory References [1] G. Janlidze, Descent and Galois Theory, prepared for a workshop in Haute Bodeux, http://www.math.yorku.ca/ tholen/hb072.htm [2] G. Janelidze and G. M. Kelly, Galois theory and a general notion of central extension, J. Pure Appl. Algebra 97 (1994), 135-161. [3] other possible references may be discussed with me.

91. Homology in Semi-abelian Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr J. Goedecke Clasically, homological algebra is studied in the context of abelian categories. However, there are many more contexts in which homological algebra is possible and useful, such as the category of groups or Lie algebras. Semi-abelian categories give a wider setting which encompasses these examples as well as abelian categories. All diagram lemmas from homological algebra hold in this setting, but even more is possible, such as the study of semi-direct product, commutator theory and much more. There are several ways to compute homology in semi-abelian categories. One such way, which is perhaps closest to the abelian setting, is comonadic homology, which uses a functorial free presentation of an object to obtain its homology. This essay should explain the background needed to compute comonadic homology in semi-abelian categories. After that several options are available: exploring the properties of comonadic homology, or a comparison of the comonadic homology with different comonads, or a comparison with the abelian setting or perhaps even a comparison with other forms of homology in semi-abelian categories is possible. Relevant Courses Essential: Category Theory Useful: Algebraic Topology 81

References [1] F. Borceux, A survey of semi-abelian categories, in Janelidze et al., Galois theory, Hopf algebras, and semiabelian categories, Fields Institute Communications Series, vol. 43, A.M.S., 2004. [2] M. Barr and J. Beck, Homology and standard constructions, in Seminar on triples and categorical homology theory, Lecture notes in mathematics, vol. 80, Springer, 1969, pp. 245– 335. [3] Chapter 2 and 3 of my thesis: J. Goedecke, Three Viewpoints on Semi-Abelian Homology, available on my website www.dpmms.cam.ac.uk/ jg352

92. Complex Multiplication of Abelian Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . Professor A.J. Scholl An abelian variety of CM type is an abelian variety over the complex numbers of dimension g whose endomorphism ring contains an order in a number field K of degree 2g. It turns out that every abelian variety of CM type can be defined over a number field, and that there is an explicit formula for the action of Galois automorphisms on CM abelian varieties and their points of finite order. The main topic of this essay would be to explain carefully the statement, and give some details of the proof, of the “Main Theorem of Complex Multiplication”, due to Shimura and Taniyama, which describes this action for Galois automorphisms fixing the so-called reflex field of K. Time and space permitting, the essay could then go on to discuss other aspects of complex multiplication: l-adic representations and the Serre group; what happens for automorphisms which don’t fix the reflex field (theorems of Tate and Deligne); or other related topics. Relevant Courses Useful: Abelian Varieties, Class Field Theory. References [1] Shimura: “Introduction to the Arithmetic Theory of Automorphic Functions” (Princeton UP) ch.5 & 7 [2] Milne: “Notes on complex multiplication”, “Complex multiplication for pedestrians” (on www.jmilne.org)

93. p-adic Uniformisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Professor A.J. Scholl In the classical theory of Riemann surfaces one proves that every complex elliptic curve is isomorphic to the quotient of the complex plane by a lattice, and every curve of genus 2 or more can be represented as the quotient of the upper half-plane by a discrete, cocompact subgroup of P SL(2, R). In the 1950s, Tate proved a p-adic analogue of this fact for elliptic curves with non-integral j-invariant. This essay should give an account of this work, and consider a further topic: for example, Mumford’s uniformisation of curves of genus greater than one [3]; uniformisation by the Drinfeld p-adic upper half plane [2]; or abelian varieties with totally degenerate reduction. 82

Relevant Courses Useful: Elliptic Curves, Algebraic Geometry, Abelian Varieties References [1] J. Silverman: Advanced topics in the arithmetic of elliptic curves. Graduate Texts in Mathematics 151 (Springer, 1994) [2] J. Fresnel & M. van der Put: Geometrie analytique rigide et applications. (Birkhauser, 1981) [3] L. Gerritzen & M. van der Put: Schottky groups and Mumford curves. Lecture Notes in Mathematics 817 (Springer, 1980) [4] A. Robert: Elliptic curves. Lecture Notes in Mathematics 326 (Springer, 1973)

94. Smoothness Results in Deformation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . Professor N.I. Shepherd-Barron Classical deformation theory includes the construction of versal deformations of projective algebraic varieties. Often the resulting deformation space is singular, but sometimes it is smooth. For smooth Calabi-Yau varieties in characteristic zero this is due to Bogomolov, Todorov and Tian. It is analogous to the theorem of Milnor-Moore and Cartier-Shafarevich, that group schemes in characteristic zero are formally smooth, and in fact these results can be proved in the same way. An essay on this might cover Schlessinger’s deformation theory, an outline of Artin’s algebraization results and the smoothness theorems just mentioned. References [1] M. Schlessinger. Functors of Artin rings. [2] T. Ekedahl and N. Shepherd-Barron, Tangent lifting of deformations in mixed characteristic.

95. Jacobians and Prym Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Professor N.I. Shepherd-Barron The Jacobian, or Picard variety, of a curve C of genus g parametrizes the divisor classes on C. The theta divisor Θ is an effective divisor on P icg−1 that parametrizes the effective classes of C degree g − 1; the theta divisor defines a principal polarization on P ic0C . e of C of genus ge, the corresponding Prym variety is a (g −1)Given an unramified double cover C dimensional abelian subvariety of P ic0e ; it carries a natural principal polarization defined by a C

divisor Ξ on the appropriate subvariety of P icgee−1 . C

An essay on this might discuss the construction of these objects, the singularities of Θ and Ξ, and also how this can be applied to the birational geometry of conic bundles.

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References [1] Milne, Jacobian varieties, in Cornell and Silverman (and on his website) [2] Mumford, Prym varieties I [3] Mumford, Theta characteristics on algebraic curves. [4] Beauville, Variétés de Prym et jacobiennes intermédiaires.

96. Normal Subgroups of the Cremona Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Professor N.I. Shepherd-Barron The Cremona group Cr2 (k) in two variables is the group of k-birational automorphisms of the projective plane Pk2 , where k is a field. Recently Cantat and Lamy have shown that, if k is algebraically closed, then Cr2 (k) contains non-trivial normal subgroups; this solves a very old problem. Although the problem is squarely one of algebraic geometry, the ideas of the solution come more from (infinite-dimensional) hyperbolic geometry and geometric group theory. An essay on this would cover the necessary background in algebraic and hyperbolic geometry as well as the proof of this theorem. References [1] Y. Manin, Cubic Forms, North-Holland. [2] S. Cantat and S. Lamy, arXiv:1007.0895

97. Analysis of a Large and Complex Data Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr P.M.E. Altham For this Essay, the candidate is asked to carry out a detailed analysis of a complex dataset, and to write an informative and helpful report, giving i) the background of the dataset(s) chosen ii) summaries (graphical, written and tabular as appropriate) of the data iii) questions of interest posed by the datasets, and the statistical methods used to answer such questions iv) the key features of the R code used for the analysis, and the conclusions in words and graphs of such analysis v) possibilities for future research in this subject area. For a choice of dataset, candidates are advised to look at the problems in the setter’s extended R worksheets at http://www.statslab.cam.ac.uk/~pat/misc.pdf. These include interesting and topical datasets accumulated in recent years. Worksheets 18 onwards contain problems suitable for particular Part III essays. Here are some examples from these worksheets: 1. Capture-recapture Data. How many snowshoe hares are there in a given closed population? How many individuals with alcohol-related problems are there in a given closed population in a region of Northern Italy? 2. A stars plot for 48 sub-Saharan African countries (2008), and New for 2009: CO2 emissions data. 3. An Index of Child Well-being in Europe (data from Bradshaw and Richardson.) 84

4. An introduction to copulae for multivariate distributions. 5. The August 2009 Independent data on UK universities. Students may need some knowledge about multivariate analysis: this is introduced in the earlier worksheets of http://www.statslab.cam.ac.uk/~pat/misc.pdf In addition, the setter is prepared to discuss with datasets that students find of particular interest: given suitable ideas about the analysis of such a dataset, this may form a good basis for an Essay. In any case, preliminary discussions with the student on his/her choice of topic and method of analysis will be essential, and the setter will set a suitable time when available to interested students. Relevant Courses Essential: Applied Statistics References [ 1 ] W.M. Venables, B.D. Ripley, Modern Applied Statistics with S, Springer, 2002. [ 2 ] J. Maindonald, W.J. Braun, Data Analysis and Graphics Using R - An Example Based Approach, Cambridge University Press, 2007.

98. Atiyah–Singer Index Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr A.G. Kovalev The main object of study in this essay is elliptic differential operators. One well-known example of elliptic operator is the Laplacian. The elliptic property can be defined for operators acting on functions, and more generally sections of vector bundles, over smooth manifolds. If a manifold is compact then every elliptic operator over it has finite-dimensional kernel and cokernel. The difference between these two latter dimensions is called an index. A celebrated theorem due to Atiyah and Singer asserts that the index of elliptic operator, defined as an analytic quantity, can be computed entirely from topological invariants of the base manifold and vector bundles. Several different proofs of this theorem are known by now and the essay can discuss aspects of some proof and/or applications. Interested candidates could start from reading [3]. Relevant Courses Essential: Algebraic topology, Differential geometry Useful: Analysis of Operators, Elliptic partial differential equations

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References [1] J. Roe, Elliptic operators, topology and asymptotic methods, Pitman Res. Notes in Math. 179, Longman, 1988. [2] Atiyah, M.; Bott, R.; Patodi, V. K. On the heat equation and the index theorem. Invent. Math. 19 (1973), 279–330. [3] R. Mazzeo: The Atiyah–Singer Index Theorem: What it is and why you should care. http://math.stanford.edu/~mazzeo/Web/Talks/asit3.pdf

99. Algebraic Groups, Geometry of the Flag Variety, Combinatorics of the Weyl Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Professor I. Grojnowski This essay is about applying the basic algebraic geometry, algebraic topology, and representation theory (Lie algebras) you’ve learnt to study the “moment map”, or “Springer resolution” from the cotangent bundle of the flag variety to the nilpotent cone. Here is one possible list of topics; you may comfortably do much less. (You can also do more!). You’ll begin by understanding the definition of the flag variety G/B for a simple algebraic group G, that is for SLn , Sp2n , SOn , and the five exceptional cases, and by computing its singular cohomology, additively, as a consequence of Bruhat decomposition. In order to do this, you’ll need to learn the basic structure theory and representation theory of reductive algebraic groups. Then you should compute the cohomology of line bundles on the flag variety— this is the theorem of Borel-Weil-Bott, and the Weyl character formula. There are several ways of doing this. You’ll want to understand the answer both as a consequence of general theorems (Kodaira vanishing, localisation to the fixpoints), and by hand. You may also want to understand the BGG resolution of irreducible representations by Verma modules as a Grothendieck-Cousin complex here. Go back and rederive the additive structure of the cohomology of the flag variety by applying this to the de Rham complex. If you know Chern characters, or you know equivariant cohomology, compute the multiplicative structure of this cohomology (with rational coefficients). Compute the cohomology of G, and of BG. Now, define the nilpotent cone, and study the geometry of the moment map. Show the nilpotent cone is normal with rational singularities, that the fibers of the moment map are rationally connected, that the nilpotent orbits are complex symplectic varieties, that they are parameterised by homomorphisms from sl2 to the Lie algebra, and describe the dimensions of the fibers. Study the case of subregular elements, the rational double points; show the fibers are Dynkin curves. If you’re still standing, show that the cohomology of the fibers carry a representation of the Weyl group, and all irreducible representations occur in this way. Alternatively, learn D-modules and how the Beilinson-Bernstein theorem constructs all gmodules with central character.

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References Three useful textbooks: [1] N. Chriss, V. Ginzburg, Representation Theory and Complex Geometry, Birkhauser [2] R. Steinberg, Conjugacy classes in algebraic groups, Springer LNM 366 [3] J. Humphreys, Finite reflection groups.

100. Hodge Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Professor I. Grojnowski The aim of this essay is to understand the statements, and some of the proofs, of the Hodge theorem, culminating in the non-abelian Hodge theory developed by Carlos Simpson and others. You should probably begin with the abelian Hodge theorem and Lefshetz decomposition— understand the statement, some of its consequences, and perhaps one of its proofs (either the classical proof, or by reduction mod p, due to Deligne-Illusie). You can then go on and explain the statements of non-abelian Hodge theory, how they generalise the abelian case, and perhaps some of its consequences. You might want to read the characteristic p theory also. References The abelian Hodge theorem and its consequences are nicely explained in the textbook of Griffiths and Harris, but also look at the beautiful papers of Deligne: [1] P. Deligne, Travaux de Griffiths, Seminar Bourbaki 376, Lecture Notes in Math 180, Springer Verlag 1970, 213–237 [2] P. Deligne, Theorie des Hodges II, III. Inst. Hautes tudes Sci. Publ. Math. No. 40 (1971), 5–57; Inst. Hautes tudes Sci. Publ. Math. No. 44 (1974), 5–77. and for the non-abelian, some references are: [3] C. Simpson, Nonabelian Hodge theory. Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), [4] C Simpson, Higgs bundles and local systems. Inst. Hautes Etudes Sci. Publ. Math. No. 75 (1992), 5–95. [5] C. Simpson, Moduli of representations of the fundamental group of a smooth projective variety. I,II, Publ. IHES. [6] N. Hitchin, The self-duality equations on a Riemann surface. Proc. London Math. Soc. (3) 55 (1987), no. 1, 59–126.

101. Rational Homotopy Theory, and Mapping Spaces . . . . . . . . . . . . . . . . . . . . . Professor I. Grojnowski The aim of this essay is to understand approximations to the computation of the homotopy type of Maps(X,Y), for topological spaces X,Y.

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The two objectives are the computation of the mapping space for the rationally localised spaces of X and Y, and the understanding of the basic structure of the Adams-Novikov spectral sequence. Begin by understanding rational homotopy theory. Start at differential graded commutative algebras, which appear everywhere in characteristic zero geometry, topology, and algebra. For example, given a manifold, one can consider the algebra of differential forms on it; the main theorem of “rational homotopy theory” is the statement that this tells you everything about the homotopy type of the manifold if you neglect torsion. References [1] Ravenel, Douglas C. Complex cobordism and stable homotopy groups of spheres. Pure and Applied Mathematics, 121. Academic Press, Inc., Orlando, FL, [2] Bousfield, A. K.; Gugenheim, V. K. A. M., On PLPL de Rham theory and rational homotopy type., Memoirs AMS 8, 1976 [3] D. Quillen, Rational homotopy theory. Ann. of Math. 90, 1969, 205–295. [4] D. Sullivan, Infinitesimal computations in topology. Inst. Hautes Etudes Sci. Publ. Math. No. 47, (1977), 269–331.

102. Topos-Theoretic Approaches to Quantum Theory . . . . . . . . . . . . . . . . . . . . . Professor P.T. Johnstone The aim of the essay would be to develop and contrast the two approaches to topos-theoretic modelling of the foundations of quantum mechanics, developed by Doering and Isham [1] and by Heunen, Landsman and Spitters [2]. A good reference for the differences between the two approaches is the paper by Wolters [3]. Relevant Courses Useful: Category Theory, Topos Theory References [1] A. Doering and C.J. Isham, A topos foundation for theories of physics, I – IV. J. Math. Phys. 49 (2008), Issue 5, 05315–8. [2] C. Heunen, N.J. Landsman and B. Spitters, A topos for algebraic quantum theory, Commun. Math. Phys. 291 (2009), 63–110. [3] S.A.M. Wolters, A comparison of two topos-theoretic approaches to quantum theory, ArXiv Math. 1010.2031 (revised version Aug. 2011).

103. Exotic 4-Spheres and Khovanov Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr A. Juh´ asz Even though the 3-dimensional Poincaré conjecture has been settled by the work of Perelman, the 4-dimensional smooth Poincaré conjecture (SPC4) – asking whether there is a 4-manifold

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homeomorphic but not diffeomorphic to S 4 – appears completely out of reach. To make things worse, there is no consensus whether the conjecture should be true or false. Cappell-Shaneson spheres provide a family of potential counterexamples to SPC4. Freedman, Gompf, Morrison and Walker [1] proposed a simple strategy for showing that certain homotopy 4-speheres, including some Cappell-Shaneson spheres, are not diffeomorphic to S 4 . It reduces the problem to proving that a certain link in S 3 is not slice; i.e., its components do not bound smoothly embedded disks in D4 . Rasmussen’s s-invariant [2] is defined via Khovanov homology and vanishes for slice knots. Motivated by the observation that its construction is intimately tied to the geometry of S 3 , Freedman et al. have computed the s-invariant for some of the knots arising from Cappel-Shaneson spheres. Countless computer hours were spent on this project, but the s-invariant vanished in all the cases computed. This was not a coincidence, as Akbulut consequently showed in [3] using Kirby calculus that the subfamily of Cappell-Shaneson spheres tested were all standard. Recently, Kronheimer and Mrowka [4] even proved via instanton Floer homology that in general the s-invariant cannot be used to tackle SPC4 in the way proposed by [1]. In your essay, you should explain the strategy of [1] and the construction of Rasmussen’s sinvariant. After mentioning the result of Akbulut [3], you should outline the work of Kronheimer and Mrowka [4], treating instanton Floer homology as a black box. Relevant Courses Essential: Algebraic Topology, Knots and 4-Manifolds Useful: Differential Geometry References [1] M. Freedman, R. Gompf, S. Morrison, K. Walker, Man and machine thinking about the smooth 4-dimensional Poincaré conjecture, Quantum Topology 1 (2010), no. 2, 171–208. (arXiv:0906.5177) [2] J. Rasmussen, Khovanov homology and the slice genus, Invent. Math. 182 (2010), no. 2, 419–447. (arXiv:math/0402131) [3] S. Akbulut, Cappell-Shaneson homotopy spheres are standard, Ann. of Math. (2) 171 (2010), no. 3, 2171–2175. (arXiv:0907.0136) [4] P. B. Kronheimer, T. S. Mrowka, Gauge theory and Rasmussen’s invariant, Preprint. (arXiv:1110.1297)

104. Regularity of Energy Minimizing Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr N. Wickramasekera One way to produce a real valued harmonic function (i.e. a solution to the Laplace’s Equation) on a bounded domain and satisfying a constraint such as prescribed boundary values is to minimise the Dirichlet Energy functional in an appropriate class of functions satisfying the constraint. The goal of this essay is to study the problem of minimization of Dirichlet Energy subject to prescribed boundary values, but for maps which take their values in a given smooth compact Riemannian manifold. The essay should begin by establishing the existence of such minimisers, which are called energyminimising maps, in a suitable Sobolev space. The bulk of the essay should be devoted to 89

the question of regularity of energy- minimizing maps. Here the goal is to establish lowerdimensionality of the singular set which is defined as the set of points at which the map is not smooth. The essay should discuss in detail the proof of the small-energy regularity theorem, the monotonicity formula and the compactness theorem for minimizers. It should then provide at least a sketch of how to use these tools to obtain the optimal dimension estimate for the singular set in case the target space is a given arbitrary compact Riemannian manifold. A more ambitious essay can also discuss the structure of the singular set and how additional curvature or topological hypotheses on the target influences the size/structure of the singular set, and discuss known better size bounds when the target is a round sphere of a given dimension. Relevant Courses Essential: Elliptic Partial Differential Equations References [1] L. Simon, “Theorems on Regularity and Singularity of Energy Minimizing Maps”, Birkhauser, Lectures in Mathematics ETH Zurich. [2] R. Schoen, Analytic aspects of the harmonic map problem, in “Lectures on Harmonic Maps” ed. R. Schoen and S.-T. Yau, International Press. [3] F.-H. Lin and C. Wang, Stable stationary harmonic maps into spheres, Acta Math. Sinica, 22 (2006), 319–330. [4] N. Wickramasekera, On the structure of singularities of harmonic maps, minimal submanifolds and mean curvature flows. (unpublished)

105. Implied Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Professor L.C.G. Rogers The implied volatility surface is an object of central interest in derivative pricing. While the standard Black-Scholes model would say that the implied volatility surface would be constant, it never is in practice, and this has given rise to a huge number of possible explanations, including stochastic volatility models of one kind or another, log-Lévy asset dynamics, shot-noise processes for asset dynamics, hidden Markov models of various kinds, and attempts to model directly the evolution of the implied volatility surface. This essay is an opportunity to review existing methods, to comment on their good and bad points, and perhaps even to try out some ideas for improvements. Relevant Courses Essential: Advanced Financial Models, Stochastic Calculus Useful: Advanced Probability References [1] Lee, Roger W. (2004) Implied volatility: statics, dynamics and probabilistic interpretation. Recent Advances in Applied Probability, Springer, New York. [2] Gatheral, Jim (2006) The Volatility Surface. J. Wiley, Hoboken NJ. 90

106. The Potential Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Professor L.C.G. Rogers A little-explored but attractive approach to the modelling of interest rates was introduced in [1], and has since been developed and extended in various ways, see [2], [3] and references therein. This approach easily includes the modelling of FX, and of credit, and can be considered also as a candidate for modelling equities also. The essay is an invitation to understand how the approach works, how it can be applied, and perhaps also to explore some of the interest-rate models which arise from it. Relevant Courses Essential: Advanced Financial Models Useful: Advanced Probability, Stochastic Calculus References [1] Rogers, L. C. G. (1997). The potential approach to the term structure of interest rates and foreign exchange rates, Mathematical Finance 7, 157–176. [2] Kluge, T. and Rogers, L. C. G. (2008). The potential approach in practice. http://www.statslab.cam.ac.uk/ chris/papers/PFPot.pdf. [3] Rogers, L. C. G. (2006). ‘One for all: the potential approach to pricing and hedging.’ in Progress in Industrial Mathematics at ECMI 2004 eds. A. Di Bucchianico, R. M. M. Mattheij, M. A. Peletier, Springer, Berlin. (ISBN-10 3540280723).

107. General Covariance and Background Independence in Quantum Gravity Dr N. Bouatta, Dr J.N. Butterfield It is sometimes underemphasized that the theory of general relativity not only gives field equations for the geometry of spacetime, but also demands general covariance (which is closely related to background independence) from any theory built upon it. Thus when attempting to quantize general relativity, one has to decide whether to keep general covariance, and perhaps background independence, as a fundamental principle that quantum gravity also has to obey: or whether instead to treat general covariance as an emergent phenomenon. The purpose of the essay is to examine how two common approaches to quantum gravity—loop quantum gravity and string theory—handle this conceptual issue. The essay should also compare the different definitions of general covariance and background independence with each other. Relevant Courses Useful: General Relativity, Quantum Field Theory, Philosophy of Quantum Field Theory References [1] C. Rovelli - Quantum Gravity, (Cambridge Monographs on Mathematical Physics), Cambridge University Press, 2004; Chapters 1 and 2

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[2] N. Seiberg, Emergent Spacetime (Rapporteur talk at the 23rd Solvay Conference in Physics, December, 2005): http://arxiv.org/abs/hep-th/0601234 [3] M. Rozali, Comments on Background Independence and Gauge Redundancies. At: http://arxiv.org/abs/gr-qc/0809.3962v2 [4] G. Belot, Background Independence, General Relativity and Gravitation 43 (2011), pp. 2865-2884. At: http://arxiv.org/abs/1106.0920v1 [5] G.Horowitz and J. Polchinski, Gauge/gravity duality, in Approaches to Quantum Gravity, ed. D. Oriti, Cambridge University Press, 2009. At: http://arxiv.org/abs/gr-qc/0602037

108. Theories of Spiral Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Professor N.W. Evans Spiral galaxies are beautiful looking objects, but the origin of the spirality remains unclear. Galaxies rotate differentially, so if the arms were always composed of the same stars and gas, they would wind up on timescales shorter than the age of galaxies. One suggestion is that the spiral part is a manifestation of a long-lived density wave. This idea was worked out in a series of papers by C.C. Lin and collaborators. However, when it was realised that tightly wound spiral wave packets propagate radially, this suggestion was questioned. Theories in which grand design spirals are triggered by companions and ragged spirals by clumps of matter such as giant molecular clouds were then developed, associated with the work of A. Toomre. The essay should critically examine one of more of the theories of spiral structure, establishing its successes and failures. Relevant Courses Useful: Galactic Astronomy and Dynamics, Dynamics of Astrophysical Discs References [1] Athanassoula, L., 1984, The Spiral Structure of the Galaxies, Physics Reports, 114, 319 [2] Bertin G., Lin C.C., 1991, “Spiral structure in Galaxies: A Density Wave Theory”, MIT Press [3] Toomre A., 1981, In “The Structure and Evolution of Normal Galaxies”, eds S.M. Fall, D. Lynden-Bell, Cambridge University Press. [4] Toomre, A., 1977, Theories of Spiral Structure, Ann Rev Astron Astroph, 15 437

109. Instabilities in Developing Ocean Fronts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr J.R. Taylor It has been known for decades that the ocean is filled with large, energetic ‘mesoscale’ eddies on scales of O(100) km. Until recently, little was known about smaller scales O(1 − 10) km, partly due to the difficulty in sampling these scales from slow-moving research vessels. Over the past decade, new measurement techniques, high resolution satellite imagery and numerical models have uncovered a rich network of fronts and filaments on ‘sub-mesoscales’. An important outstanding question in physical oceanography is: What ultimately sets the size of sub-mesoscale fronts and filaments? A classic paper in geophysical fluid dynamics [1] demonstrated that 92

when a front is squeezed between two larger eddies, its width collapses to zero in a finite-time singularity. In reality, instabilities of the front arrest this process of ‘frontogenesis’. The goal of this project will be to examine the analytical solution presented in [1], in light of a variety of possible instability mechanisms, including Kelvin-Helmholtz instability, symmetric instability, inertial instability, etc. The findings can then be compared with very high resolution numerical simulations of an unstable, turbulent front. Relevant Courses Essential: Fundamentals in Fluid Mechanics of Climate, Advanced Topics in Fluid Mechanics of Climate References [1] Hoskins B.J. & F.P. Bretherton (1972), “Atmospheric frontogenesis models: Mathematical formulation and solution”, Journal of the Atmospheric Sciences, 29, pp. 11-37.

110. Turbulence and Plankton Ecosystem Dynamics . . . . . . . . . . . . . . . . . . . . . . . Dr J.R. Taylor Phytoplankton, free-floating algae, are responsible for about half of the photosynthesis and carbon fixation on the planet. Phytoplankton require both light and nutrients to grow, and the supply of these essential ingredients depends on fluid motion. Historically, it was believed that the primary physical control on phytoplankton growth was the depth of the ocean mixed layer. Evans and Parslow [1] introduced a three-component ecosystem model which captures many features of the ocean ecosystem, including low winter productivity, and a spring phytoplankton bloom, but the physical processes were simply represented through a prescribed mixed layer depth. Recently, it has been argued that the atmospheric forcing controls the rate of vertical turbulent mixing in the ocean, which provides a better indicator for the phytoplankton growth rate, than the mixed layer depth alone [2]. The goal of this project is to extend the model of Evans and Parslow by explicitly allowing each component of the model to depend on depth, and accounting for variable turbulent mixing using a formulation similar to that proposed in [2]. Questions that could then be addressed include: Warmer climates are likely to reduce the vertical turbulent diffusivity by enhancing stratification - how would the timing and strength of the spring and fall phytoplankton bloom respond? How does the annual primary production depend on the various parameters? Are chaotic dynamics possible in the ecosystem model? Relevant Courses Useful: Fundamentals in Fluid Mechanics of Climate, Advanced Topics in Fluid Mechanics of Climate References [1] Evans G.T. & J.S. Parslow (1985), “A model of annual plankton cycles”, Biological Oceanography, 3, 3, pp. 327-347. [2] Taylor, J.R. and R. Ferrari (2011), “Shutdown of turbulent convection as a new criterion for the onset of spring phytoplankton blooms.” Limnology and Oceanography, 56(6) pp. 2293-2307. 93

111. Statistical Methods for Cancer Genomics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Professor S. Tavar´ e Recent advances in next-generation sequencing (NGS) have made sequencing the DNA of cancer genomes a reality. The International Cancer Genome Consortium is sequencing 500 matched tumour-normal pairs (and related information such as DNA methylation and expression) from a large number of different cancer types. With the use of NGS has come a number of statistical and computational problems [2]. These include quality assessment of NGS data, identification of mutations such as single nucleotide polymorphisms and structural variants, the identification of passenger and driver mutations [3], and integrative analysis of disparate genomic, proteomic, metabolic and clinical data [2]. In this essay you are asked to outline some of the statistical issues. A good essay will include an introduction to NGS technology, and a discussion of methods used for QC, mutation identification, and integrative analysis. The papers in [2] and [3] are representative, but you are encouraged to review the literature. For a biological perspective on cancer see [1], and for an evolutionary perspective in [4]. Relevant Courses Essential: Applied Statistics Useful: Some familiarity with basic molecular biology. References [1] Hanahan D & Weinberg RA (2011) Hallmarks of cancer: the next generation. Cell, 144, 646; Hanahan D & Weinberg RA (2000) Cell, 100, 57. [2] Ding L et al. (2010) Human Molecular Genetics, 19, Review Issue 2, R188; Ovaska et al. (2010) Genome Medicine, 2, 65. [3] Youn A & Simon R (2011) Bioinformatics, 27, 175; Akavia UD et al. (2010) Cell, 143, 1005; Ding J et al. (2011) Bioinformatics, doi: 10.1093/bioinformatics/btr629 [4] Merlo LMF et al. (2006) Nat Rev Genet, 6, 924.

112. B-Spline Basis Condition Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr A. Shadrin The favourable properties of splines for numerical computations hinge on i locality and stability of the B-spline basis. This allows us to reduce linear problems to solving a system of linear equations with banded matrices, and ensures that the solution is stable with regard to round-off errors. A measure of stability is the condition number of the B-spline basis (Ni ). With k the order of B-splines, and t the underlying knot sequence, it is defined as follows P k(ai )kl∞ k bi Ni kL∞ κk,t := sup P . sup ai Ni kL∞ (bi ) k(bi )kl∞ (ai ) k It is of great importance that this quantity is bounded independently of the knot-sequence t, i.e., κk := sup κk,t < ∞, t

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This was proved by de Boor as early as in 1968, refined further [1] up to κk < k 9k , but one needs the exact behaviour of κk . It is believed that κk ∼ 2k . However, earlier hopes that the extremal knot configuration would have no interior knots (hence reducing splines to polynomials) were dashed by a counter-example [2]. Nevertheless, recently [3] it has been demonstrated that κk < k 2k , thereby proving the 2k -conjecture up to a linear factor. The essay should present an overview of the questions concerned with the notion of the B-spline basis condition number. This should include its relation to other constants in approximation theory and the proof that κk < k 2k . A related open problem (worthy of publication) Q Prove that, for any polynomial ω(x) = ki=1 (x − ti ) with ti ∈ (0, 1), there exist a function Ω such that ¯ ¯ 1) Ω(s) (x)¯ = 0, s = 0, . . . , k − 1; x=0,1

2) (−1)s sign Ω(s) (x) sign ω (k−s) (x) ≥ 0,

s = 0, . . . , k.

The latter means that, for any s, the inner zeros of Ω(s) must coincide with zeros of the polynomial ω (k−s) . Relevant Courses Useful: Approximation Theory References [1] C. de Boor, On local linear functionals which vanish at all B-splines but one, in “Theory of Approximation with Applications” (A. G. Law and B. N. Sahney, Eds.), pp.120–145, Academic Press, New York, 1976. [2] C. de Boor, The exact condition of the B-spline basis may be hard to determine, J. Approx. Theory, 60 (1990), 344–359. [3] K. Scherer, A. Yu. Shadrin, New upper bound for the B-spline basis condition number II. A proof of de Boor’s 2k -conjecture, J. Approx. Theory 99 (1999), 217–229.

113. Non-Archimedean Amoebae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr C.J.B. Brookes In [1] George Bergman studied the logarithmic limit sets of closed subvarieties X of the algebraic torus and conjectured them to be polyhedral. More recently the term ‘amoeba’ was introduced by Gel’fand, Kapranov and Zelevinsky when considering hypersurfaces in complex tori. Over an algebraically closed field endowed with a norm the amoeba of X is the closure in real space of the image of X under the map obtained by taking the logarithm of the norm of the coordinates. Bergman’s logarithmic limit set arises from the amoeba as the set of all directions of the ‘tentacles’ at infinity. Some related sets arose independently in the work of Bieri and Strebel [3] when they were considering the finite presentability of certain soluble groups. Bieri and Groves [4] later showed these sets to be polyhedral in structure. In fact they arise from amoebae over a non-archimedean field.

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I suggest you start by looking at [6] which is the draft of a book - the topic is related to section 1.6. You might write about it from the computational aspect in the context of amoebae and tropical geometry, but it’d also be possible to concentrate on the group theory. The article [5] explains how the amoebae in the non-archimedean case are related to tropical varieties and you will also find a lot of references to them in the literature of tropical geometry - they sometimes appear there as Bergman sets and fans. Relevant Courses Essential: Commutative Algebra References [1] G.M. Bergman, The logarithmic limit-set of an algebraic variety, Trans. Amer. Math.Soc 157 (1971), 459-469. [2] I.M. Gel’fand, M.M. Kapranov, A.V. Zelevinsky, Discriminants, resultants and multidimensional determinants (Birkhauser, 1994). [3] R. Bieri, R. Strebel, A geometric invariant for modules over an abelian group, J. reine angew. Math. 322 (1981), 170-189. [4] R. Bieri and J.R.J. Groves, The geometry of the set of characters induced by valuations, J. Reine Angew. Math. 347(1984), 168-195. [5] M. Einsiedler, M. Kapranov, D. Lind, Non-archimedean amoebas and tropical varieties, J. Reine Angew. Math. 601 (2006), 139-157. http://arxiv.org/abs/math/0408311 [6] D. Maclagan, B. Sturmfels, Introduction to tropical geometry www.warwick.ac.uk/staff/D.Maclagan/papers/TropicalBook.pdf

114. The Multiplication Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Professor B.J. Green Write out, in the way you did at primary school, the n × n multiplication table. How many different elements does it contain? This is a surprisingly tricky question, and recently Kevin Ford showed that the answer is bounded above and below by (different) constants times n2 log 2 1− 1+log log 2

(log n)

(log log n)3/2

.

The proof involves an extremely penetrating study of the structure of the set of divisors of a typical integer. The aim of this essay would be to explore this area of number theory. Relevant Courses None, but you should have the desire to take a highly technical argument, to try to extract the important underlying ideas from it, and then to present them in an entertaining form.

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References Please note that both of these papers are available on the author’s webpage. [1] K. Ford, The distribution of integers with a divisor in a given interval, to appear in Annals of Math. [2] K. Ford, Integers with a divisor in (y, 2y], in Anatomy of integers (ed. de Koninck, Granville, Luca), CRM Proc. and Lect. Notes 46, AMS 2008.

115. Entanglement and Entropies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dr N. Datta Entanglement plays a fundamental role in quantum information theory. Communication protocols such as teleportation of quantum states and superdense coding showed how entanglement can be seen as a valuable resource. Many quantum information-processing tasks rely on the use of entanglement. These include tasks as diverse as quantum error correction, state merging, channel simulation, entanglement-assisted information transmission, and quantum cryptography. Hence the characterization of the entanglement of a quantum state is an important problem in quantum information theory. There are various criteria for establishing whether a given bipartite quantum state is entangled. These include Peres’ criterion and entanglement witnesses. Entanglement monotones are used to quantify the amount of entanglement in a quantum state. Entanglement monotones do not increase under local operations and classical communication, and they vanish on separable states. Many different entanglement monotones have been defined for bipartite quantum states. Most of these are given in terms of entropic quantities. Some of these entanglement monotones are particularly interesting because they have the additional property of having an operational interpretation in quantum information theory. This essay will focus on the study of various characterizations of the entanglement of bipartite states, the connection between entanglement monotones and entropies, and the operational interpretations of certain entanglement monotones. Relevant Courses Useful: Quantum Information Theory References [1] R. Horodecki, P. Horodecki, M. Horodecki and K. Horodecki, Quantum entanglement; Rev.Mod.Phys.81:865-942,2009 [2] M. Horodecki, P. Horodecki, R. Horodecki, Separability of Mixed States: Necessary and Sufficient Conditions; arXiv:quant-ph/9605038. [3] M. Christandl and A. Winter, “Squashed Entanglement” - An Additive Entanglement Measure ; J. Math. Phys. 45, No 3, pp. 829-840 (2004); arXiv:quant-ph/0308088 [4] P. Hayden, M. Horodecki, B. Terhal, The asymptotic entanglement cost of preparing a quantum state: J. Phys. A: Math. Gen. 34(35):6891-6898, 2001. [5] N. Datta, Max- Relative Entropy of Entanglement, alias Log Robustness; International Journal of Quantum Information, 7, no.2, 475-491, 2009. 97

116. Instabilities in Dilute Suspension of Self-Propelled Microscopic Swimmers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Professor R.E. Goldstein The swimming mechanisms of microorganisms such as bacteria or algae are quite complex. Being immersed in a slow viscous fluid (Stokes flow), they operate in the low Reynolds number regime where damping is very strong. Therefore, unlike human swimmers, microorganisms cannot impart inertia to the fluid and instead have to use non time-reversible swimming mechanisms to achieve locomotion (Purcell’s or Scallop theorem). Based on a modern model for a system of dilute suspension of self-propelled particles (Saintillan and Shelley 2008), an understanding of a promising fluid dynamical modelling approach to microscopic swimmers is to be gained. Additionally, the results of linear stability analyses for both aligned and isotropic suspensions are to be understood. Relevant Courses Useful: Slow Viscous Flow References [1] Saintillan D, Shelley MJ (2008) Instabilities, pattern formation and mixing in active suspensions. Phys Fluids 20:12330 [2] Pedley, T. J. & Kessler, J. O. 1990 A new continuum model for suspensions of gyrotactic micro-organisms. J. Fluid Mech. 212, 155-182. [3] Batchelor, G.K.: Introduction to Fluid Dynamics, Cambridge University Press 1967 [4] Happel, J. and Brenner, H.: Low Reynolds Number Hydrodynamics, Kluwer 1965

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Faculty of Mathematics Part III Essays: 2011 - 2012

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