Singularities in Geometry and Topology

New York University, New York NY

Investigators

Abstract

Abstract Award: DMS-0706968 Principal Investigator: Jeff Cheeger, Shmuel Weinberger This conference will explore differing approaches to non-smooth spaces that arise in modern geometry and topology with an eye towards cross-fertilization among the various groups of researchers. Two of the central thrusts are topological methods based on the ideas of a stratification, and metric measure spaces, and their analysis, especially their differential calculus. Others are based on cohomology with growth conditions, or resolution of singularities, intersection homology, sheaf theory etc. Moreover the applications of these studies are diverse, from the Novikov conjecture, knot theory, and transformation groups to Gromov-Hasudorff convergence techniques in Riemannian geometry and geometric group theory to problems of interest to theoretical computer scientists (embedding finite metric spaces in various Banach spaces). It is anticipated that as a result of this meeting, progress can be expected in the directions of producing analytic geometric structures on singular spaces, the quasi-isometric theory of groups, and characteristic classes. Many of the basic ideas of analysis and geometry are based on the idea of linear approximation. The first example of this is calculus itself where linear approximation defines the basic notion of the derivative. A more modern example is the idea of a manifold in topology, which is a space that is, in small regions, well approximated by Euclidean space. Countless further developments have shown the need for more general tools to deal with situations where such approximations are not feasible or may not exist. By bringing together workers who have dealt with these issues in varied contexts, this conference will spur the development of new syntheses and bring to the fore common issues that arise in completely different settings. The conference web site is http://www.math.uchicago.edu/~shmuel/Cappelliday.