Proving Theorems with Computers
Kevin Buzzard · Notices of the American Mathematical Society · 2020
Superhuman MathematicsHow is breakthrough mathematics achieved?Here is one example, from algebraic number theory.In his 1987 paper "Deforming Galois representations," Barry Mazur observes that the geometric concept of a smoothly varying complex family of representations of a group has an arithmetic analogue.He sets up deformation theory in the non-geometric setting of mod 𝑝 and 𝑝-adic Galois representations, and makes some interesting observations about the relationship between arithmetic deformation rings and Galois cohomology.By 1990, Mazur and Tilouine have raised a profound question about whether a certain universal deformation ring coming out of this theory is isomorphic to one of Hida's Hecke algebras.In 1993 Wiles uses new techniques in commutative algebra to reduce a variant of this question to a numerical criterion, and a year later, aided by Taylor, he has pushed the strategy through.The semistable Shimura-Taniyama conjecture (at that time often called the semistable Shimura-Taniyama-Weil conjecture) follows, and hence, by earlier work of Ribet, Fermat's Last Theorem.This is but one of very many examples where crossfertilization has occurred in mathematics.The breadth of Mazur's mathematical knowledge (he was initially a topologist) played a key role here.In a 2014 article [Maz14] for the Math.Intelligencer, Mazur writes: "Reasoning by analogy is the keystone: it is present in much (perhaps all) daily mathematical thought, and is also often the inspiration behind some of the major long-range projects in mathematics."