Geometry is one of the oldest branches of mathematics, going back to the Greeks and beyond. A famous problem left open by the Greeks and not resolved until the 19th century was whether the parallel postulate, which states that given a line in the plane and a point not on that line, there is exactly one line through the point that does not meet the first line, could be deduced from Euclid’s other axioms. It was shown by Gauss, Bolyai and Lobachevsky that the answer was no, and that there are different, mathematically consistent geometries in which the notions of Euclidean geometry such as points and lines have natural interpretations, and in these geometries the other axioms hold but the parallel postulate does not. This demonstrates that the parallel postulate cannot be a consequence of the other axioms. Moreover, these non-Euclidean geometries, far from being mere curiosities, are fundamental to modern mathematics.
From these ideas, thanks in particular to the work of Riemann, the concept of a manifold became central to geometry. A manifold can be thought of as a higher-dimensional generalization of the notion of a surface in three-dimensional space, though a manifold is often better thought of “intrinsically” rather than with reference to a larger space in which it lives. Manifolds are ubiquitous in mathematics and physics ― for example, they are needed to make sense of the notion of curved spacetime, which is essential to Einstein’s theory of general relativity ― and their study has led to remarkable developments and many fascinating open problems.
One of these developments is the realization that global topological quantities of a manifold can often be computed using local tools. For example, a famous theorem of Gauss and Bonnet shows that the number of “holes” a surface has (where, for instance, a torus has one hole, the surface of a figure-of-8-shaped pretzel has two, and so on), can be obtained by integrating a local quantity, the curvature, over the surface. This idea has subsequently been vastly generalized, a particular highlight being the famous Atiyah–Singer index theorem from 1963. This theorem led to an entire subfield of mathematics devoted to index theory.
Bismut has played a central role in this subfield. In the early part of his career, he made profound contributions to probability theory that have had a major impact on the theory of mathematical finance. Later, he 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 common feature of all his works is that using index theory he is able to prove explicit formulas for quantities that people would previously never have dared to try to compute.
A major theme of modern geometry, to which Cheeger has made profound contributions, is to understand the impact of curvature conditions, such as assuming that the curvature is everywhere non-negative, 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. Cheeger 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. A discrete analogue of this result for graphs has played an extremely important role in the study of random walks on graphs, which in turn has led to the development of important algorithms for random sampling, integration in high dimensions, and many other applications.
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.
More generally, over the last few decades, including right up to the present day, Bismut and Cheeger, as well as solving long-standing open problems, have introduced important new ideas and built tools that have greatly extended the range of what is possible in modern geometry, and as a result have transformed the subject.
28 October 2021 Hong Kong