The Shaw Prize 2021 in the Mathematical Sciences goes to Jean-Michel Bismut and Jeff Cheeger for their remarkable insights that have transformed, and continue to transform, modern geometry.
Bismut […] imported ideas from probability into index theory, reproving all the main theorems and vastly extending them, which enabled him to link index theory to other parts of mathematics. This has led to many applications in areas as far afield as Arakelov geometry, which is used in number theory to study high-dimensional Diophantine equations, and physics, where the tools developed by Bismut have been used to compute the genus 1 Gromov–Witten invariant. In recent years, his work has been changing the way we think about the Selberg trace formula, a fundamental tool in representation theory and modern number theory.
A major theme of modern geometry, to which Cheeger has made profound contributions, is to understand the impact of curvature conditions on the structure of manifolds. His work in this area has had a huge impact — for example, Perelman made essential use of it in his solution of the Poincaré conjecture. He is also a household name in combinatorics and theoretical computer science owing to his introduction of what we now call the Cheeger constant. This is the smallest area of a hypersurface that divides a manifold into two parts, which Cheeger related to the first non-trivial eigenvalue of the Laplace–Beltrami operator on that manifold.
Bismut and Cheeger have also worked together, and are particularly celebrated for their extension of a famous invariant, the so-called eta invariant, from manifolds to families of manifolds, which allowed them to compute explicitly the limit of the eta invariant along a collapsing sequence of spaces.
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