Chair, Department of Mathematics
Andrew MacLeish Distinguished Professor of Mathematics
Most of my research is concerned with understanding things geometrically or understanding geometric things. The main directions are:
- Topology of (mainly high dimensional) manifolds.
- Global analysis (e.g. \(L^2\) cohomology and index theory) on noncompact manifolds and its coarse nature.
These topics are somewhat related to the Novikov conjecture (although that is only one important aspect). Jonathan Rosenberg maintains a web page of developments related to this problem (and to the Borel and Baum-Connes conjectures). In general, one often connects the fundamental group to invariants of manifolds with that fundamental group.
- Singularities, e.g. orbifolds, but much more serious as well. The main reference is probably to my book "The topological classification of stratified spaces" (but that is somewhat out of date.)
- Applications of logical and computer scientific ideas to variational problems and the large scale geometry of certain moduli spaces. Again I have written a book on this ("Computers, Rigidity, and Moduli: The Large-Scale Fractal Geometry of Riemannian Moduli Space"): this is mainly joint work with Alex Nabutovsky.
- Quantitative topology: i.e. studying the precise nature of solutions to problems that are produced existentially by algebraic topology. This is a vague theme that includes a number of points of contact with the previous topics. But, you can do well to look at the papers in Gromov's bibliography that allegedly refer to this to capture a good deal of the scope.
I have also been involved recently in applications of algebraic topology to the analysis of large data sets, to robotics, and to economics. Some of this is the focus of a program at MSRI in Fall 2006, and a conference in Zurich in July 2006. See this article about the last one.