Shmuel Weinberger

Andrew MacLeish Distinguished Professor of Mathematics

Shmuel Weinberger
Eckhart 403
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Most of my research is concerned with understanding things geometrically or understanding geometric things. The main directions are:

  1. Topology of (mainly high dimensional) manifolds.
  2. 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.

  1. 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 The main reference is probably to my book "The topological classification of stratified spaces" (but that is somewhat out of date.)
  2. 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.
  3. 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.