Story of Maths - Part 4.2: Topology - Graph theory (Euler and Poincarré)

13:56 - 23:54

The 7 bridges of Konigsberg (Topology / Graph theory) - Euler Today the city of Konigsberg is known as Kaliningrad, a little outpost of Russia on the Baltic Sea surrounded by Poland and Lithuania. It started as an 18th-century puzzle.

Is there a route around the city which crosses each of these seven bridges only once? Finding the solution is much more difficult than it looks. It was eventually solved by the great mathematician Leonhard Euler, who in 1735 proved that it wasn't possible. There could not be a route that didn't cross at least one bridge twice. He solved the problem by making a conceptual leap. He realised, you don't really care what the distances are between the bridges. What really matters is how the bridges are connected together. This is a problem of a new sort of geometry of position - a problem of topology. Many of us use topology every day. Virtually all metro maps the world over are drawn on topological principles. You don't care how far the stations are from each other but how they are connected.

Topology - Poincaré Although topology had its origins in the bridges of Konigsberg, it was in the hands of Poincare that the subject evolved into a powerful new way of looking at shape. Some people refer to topology as bendy geometry because in topology, two shapes are the same if you can bend or morph one into another without cutting it. So for example if I take a football or rugby ball, topologically they are the same because one can be morphed into the other. Similarly a bagel and a tea-cup are the same because one can be morphed into the other. Even very complicated shapes can be unwrapped to become much simpler from a topological point of view. But there is no way to continuously deform a bagel to morph it into a ball. The hole in the middle makes these shapes topologically different. Poincare knew all the possible two-dimensional topological surfaces. But in 1904 he came up with a topological problem he just couldn't solve. If you've got a flat two-dimensional universe then Poincare worked out all the possible shapes he could wrap that universe up into. It could be a ball or a bagel with one hole, two holes or more holes in. But we live in a three-dimensional universe so what are the possible shapes that our universe can be? That question became known as the Poincare Conjecture. It was finally solved in 2002 in St Petersburg by a Russian mathematician called Grisha Perelman. His proof is very difficult to understand, even for mathematicians. Perelman solved the problem by linking it to a completely different area of mathematics. To understand the shapes, he looked instead at the dynamics of the way things can flow over the shape which led to a description of all the possible ways that three dimensional space can be wrapped up in higher dimensions.

Mathematics were for Hilbert a universal language. He believed that this language was powerful enough to unlock all the truths of mathematics, a belief he expounded in a radio interview he gave on the future of mathematics on the 8th September 1930.In it, he had no doubt that all his 23 problems would soon be solved and that mathematics would finally be put on unshakeable logical foundations. There are absolutely no unsolvable problems, he declared, a belief that's been held by mathematicians since the time of the Ancient Greeks. He ended with this clarion call, "We must know, we will know" - "'Wir mussen wissen, wir werden wissen." Unfortunately for him, the very day before, in a scientific lecture that was not considered worthy of broadcast, another mathematician would shatter Hilbert's dream and put uncertainty at the heart of mathematics.