The Shaw Prize Lecture in Mathematical Sciences 2021

About the Laureate

Professor Jean-Michel Bismut

Professor Jean-Michel BISMUT is currently Emeritus Professor of the Mathematics Department at Université Paris Sud, Orsay, France. He has made profound lifetime contributions to probability theory, representation theory as well as index theory. His findings have had major impacts in various areas, from mathematical finance to the Selberg trace formula to as far as Arakelov geometry.

Born in 1948 in Lisbon, he graduated from École Polytechnique, France and earned his PhD in Mathematics in 1973 from the Université Paris VI, France. He was an Engineer of the Corps des Mines in France and a Lecturer at École Polytechnique before joining the Université Paris Sud as an Associate Professor. He is a member of the French Academy of Sciences.

Lecture Abstract

In this lecture, Professor BISMUT will review elementary properties of the hypoelliptic Laplacian. If  is a compact Riemannian manifold, the hypoelliptic Laplacian is a family of differential operators  acting on , the total space of the tangent bundle, that interpolates between , the Laplacian of , and the vector field of  on , the generator of the geodesic flow. The dynamical system counterpart is an interpolation between Brownian motion and the geodesic flow via the Langevin dynamics. This interpolation may preserve the full spectrum of the original Laplacian. The case of the circle will be described in detail, while connections with Selberg’s trace formula and applications to complex Hermitian geometry will also be explained.

About the Laureate

Professor Jeff Cheegar

Professor Jeff CHEEGER is currently Silver Professor of Mathematics at the Courant Institute of Mathematical Sciences. A major theme of modern geometry, to which Professor Cheeger has made profound contributions, is to understand the impact of curvature conditions on the structure of manifolds. He is also a household name in combinatorics and theoretical computer science, owing to his introduction of the Cheeger constant and Cheeger's inequality, which has had a vast number of descendants. 

He was born in 1943 in New York, USA and obtained his Bachelor’s degree from Harvard University, USA before receiving his MS and PhD in Mathematics from Princeton University in 1966 and 1967 respectively. Between the late 1960s and 1992, he was a National Science Foundation Postdoctoral Fellow at the University of California, Berkeley; he worked at University of Michigan as well as the State University of New York, Stony Brook. He has been a Professor at New York University since 1989 and is a member of the US National Academy of Sciences and the American Academy of Arts and Sciences.

Lecture Abstract

Professor CHEEGER will discuss the simplest instance of a notion called quantitative differentiation. More sophisticated instances have played a key role in contemporary advances in the analysis of singularities of solutions to nonlinear partial differential equations such as the minimal surface equation. The simplest case pertains to differentiable functions,  , where the only information available is a bound on the derivative, for instance ​​. Although the discussion requires only elementary calculus, it provides a framework for more advanced generalisations. Nonetheless, it is not covered in standard courses.