I was born on December 1, 1943, at the Brooklyn Jewish Hospital. I had a normal childhood, engaging in the usual games and sports. My father introduced me to mathematics at the level of elementary algebra when I was seven. Intermittently, he would teach me more. Soon I was hooked.
In the seventh grade, I made a great new friend, Mel Hochster, a fellow math enthusiast, later my college roommate, now, an eminent mathematician. I attended Erasmus Hall, a large public high school with many famous alumni. There were some very bright students and the honours classes were at a good level. Eventually, I became captain of the math team.
At Harvard, my teachers Shlomo Sternberg and Raul Bott were charismatic and encouraging. As a junior, with no practice, I tied for 21st in the country on the Putnam exam. This relatively modest accomplishment meant a lot to me. As a senior, I took a graduate course in PDE from a young Assistant Professor named Jim Simons.
In graduate school at Princeton, after deciding to study differential geometry, I consulted Jim, who was a specialist in that area. Coincidentally, he had just moved to Princeton and was working as a code breaker at the Institute for Defense Analyses. My advisor was the legendary Salamon Bochner, but my teacher was Jim. For a year, he told me what to read and patiently answered all my questions. Then he suggested a thesis problem. After I solved it, it morphed into something very different, a finiteness theorem for manifolds of a given dimension admitting a Riemannian metric with bounds on curvature and diameter and a lower bound on volume. This needed a corresponding lower bound for the injectivity radius, which I think of as my first real theorem. The finiteness theorem brought a certain change in perspective to Riemannian geometry, now subsumed under Cheeger–Gromov compactness.
The major part of my career has been spent at Stony Brook (1969–1989) and the Courant Institute (1989–). First, I spent an exciting year at Berkeley and another at Michigan. Significant stays in Brazil, Finland, IHES in France and IAS in Princeton were enormously fruitful. I have had exceptionally brilliant collaborators and some great students. Several collaborators are mentioned below. Unfortunately, space constraints forced the omission of many others.
When I started doing research, my viewpoint was geometric and topological. As I learned more analysis, my work evolved into a mixture of all three fields. Several times, I noticed things which were hiding in plain sight, but which proved to have far reaching consequences. In retrospect, a significant part of my work involved finding structure in contexts which might initially have seemed too naive or too rough. Occasionally, a specific problem led to new developments that went far beyond what was needed for the original application.
With strong mutual connections and the mention of a few highlights, my work could be summarized as follows. (1) Curvature and geometric analysis; see below. (2) A lower bound for the first nonzero eigenvalue of the Laplacian, which has had a vast, varied and seemingly endless number of descendants. (3) Analysis on singular spaces: The precursor was my proof of the Ray–Singer conjecture on the equality of Ray–Singer torsion, an analytic invariant and Reidemeister torsion, a topological invariant. Simultaneously, Werner Muller gave a different proof. Independently, I discovered Poincaré duality for singular spaces, in the guise of L2-cohomology. Later, I showed it was equivalent to the contemporaneously defined intersection homology theory of Goresky–MacPherson. I pioneered index theory and spectral theory on piecewise constant curvature pseudomanifolds. Applications included a local combinatorial formula for the signature. Adiabatic limits of η-invariants and local families index for manifolds with boundary were joint with Jean–Michel Bismut. (4) Metric measure spaces: I showed that properly formulated, all of first order differential calculus is valid for metric measure spaces whenever the measure is doubling and a Poincaré inequality holds in Heinonen-Koskela’s sense. Examples include non-selfsimilar fractals with dimension any real number. Related work with Bruce Kleiner and Assaf Naor had applications to theoretical computer science.
Curvature. My thesis (1967) and my first paper with Detlef Gromoll (1969) on the soul theorem for complete manifolds of nonnegative curvature were purely geometric. In 1971, we proved the fundamental splitting theorem for complete manifolds of nonnegative Ricci curvature. The statement was geometric but the proof involved partial differential equations (PDE). Both works with Detlef were early examples of rigidity theorems. Here is the principle. When geometric hypotheses are sufficiently in tension, they can mutually coexist only in highly non-generic situations where specific special structure is present. Similarly, Misha Gromov and I characterized collapse with bounded curvature in terms of generalized circular symmetry (1980–1992) in the end joining forces with Kenji Fukaya (1992). Work with Toby Colding (1995–2000) on Ricci curvature was a mixture of geometry and PDE. We proved quantitative versions of rigidity theorems which, together with scaling, vastly increased their range of applicability. Specifically, if the hypotheses of rigidity theorems fail to hold by only a sufficiently small amount, then the conclusions hold up to an arbitrarily small error. Quantitative rigidity theorems were the basis of our structure theory for weak geometric limits (Gromov–Hausdorff limits) of sequences of smooth Riemannian manifolds with Ricci bounded below. These geometric objects play the role that distributions play in analysis. In particular, limit spaces can have singularities living on lower dimensional subsets and we proved a sharp bound on their dimension. Aaron Naber and I gave the first quantitative theory of such singular sets (2011–2021). Beyond bounding their dimension, we bounded their size. Our flexible techniques were rapidly applied to numerous nonlinear elliptic and parabolic geometric PDE’s. In 2015, we proved a longstanding conjecture on noncollapsed Gromov–Hausdorff limits of sequences of n-dimensional Einstein manifolds: Singular sets have dimension at most n – 4 .
28 October 2021 Hong Kong
