Pauli Lectures 2018

After completing his PhD at Cambridge in 1990, Gowers held a faculty position at University College London before returning to Cambridge as Rouse Ball Professor of Mathematics in 1998. Gowers received the Fields Medal at the International Congress of Mathematicians in Berlin in 1998 for his remarkable solutions of outstanding problems in the theory of Banach spaces (some posed by Stefan Banach himself in the 1930s). Gowers has also done profound work in other areas: the ideas in his new proof of Endre Szeméredi’s theorem on arithmetic progressions have played an important role in the development of additive combinatorics. Gowers has been a fellow of the Royal Society since 1999 and was knighted in 2012 for his services to mathematics.

In recent years, Gowers has expressed influential ideas on many aspects of the practice of mathematics. He has suggested the possibility that large informal collaborations of mathematicians might lead to significant new results. Some of these Polymath projects have been very successful.

It is known that the problem of determining whether a given mathematical statement has a proof is computationally hard, in the sense that there is no algorithm that can do it in general. Even if one restricts attention to proofs of at most a given length, it is widely believed that there is no algorithm that can do it in a reasonable time. And yet human beings somehow manage to find proofs that can be long and complex. I shall argue that this seeming paradox can be resolved without appealing to some mysterious property of human brains that computers could never hope to emulate.

A typical theorem in Ramsey theory states that if some mathematical structure is “coloured” with a small number of colours, then there must be a substructure that is monochromatic — that is, only one colour is used for the substructure. There are many beautiful results of this kind, and also many fascinating open problems. Often the best known results have quite simple proofs, and yet improving these results, even slightly, is extremely challenging. I shall illustrate these points with several examples.

A Latin square is an n x n grid filled up with n labels in such a way that the same label never appears twice in the same row or column. It satisfies the quadrangle condition if whenever one rectangle is labelled abcd and another is labelled abcd’ we must have that d=d’. It is a simple exercise to show that a Latin square satisfies the quadrangle condition if and only if it is isomorphic to the multiplication table of a group. But what happens if we weaken the quadrangle condition and now assume only that of all pairs of rectangles that share three labels, a positive proportion share the fourth as well? This kind of weakening is common in additive combinatorics, and one often finds that the conclusion one can draw is surprisingly strong. In this case, one can show that a large portion of the Latin square is derived from a "group-like object”, in a sense that I shall make precise. This is joint work with Jason Long.