Conference on Topology, Geometry, and Physics; May 2006; New York, NY

Columbia University, New York NY

Investigators

Abstract

Abstract Award: DMS-0540236 Principal Investigator: Robert D. Friedman, Peter S. Ozsvath This is a proposal for a four-day-long, broad conference at Columbia University on "Topology, Geometry, and Physics.'' The conference will focus on the following major themes, and their interconnections: hyperbolic manifolds and geometrization gauge theory, three- and four-manifolds, Hodge theory, and interactions between mathematics and modern physics. Such a conference is particularly timely, in view of the many breakthroughs in each of these subjects. In particular, the recent spectacular work of Perelman on the Poincare conjecture and Thurston's geometrization conjecture, as well as new developments in hyperbolic geometry, make this an especially opportune moment to survey the new landscape of three-manifold theory. At the same time, new techniques of gauge theory and new combinatorial methods have deepened the current understanding of three- and four-manifolds and it is time to take stock of these developments and their relation to other work in the field. Many of the new results in geometric topology have been based upon ideas arising in mathematical physics and geometric analysis, and these kind of interactions will serve as a unifying theme for the conference. A fundamental problem in mathematics is to describe all possible shapes. In the case of surfaces (two dimensional objects), possible shapes include a sphere (the surface of the earth, for example), a torus (the surface of a tire) and more complicated generalizations. A great deal of research has been concerned with dimension three, the dimension of space, and dimension four, the dimension of space-time. Because of its relevance to understanding our physical world, understanding all possible shapes in these dimensions is particularly important. Paradoxically, these cases are much harder to understand and to classify than higher-dimensional objects (which in turn are much harder to visualize in any meaningful way). The study of dimensions three and four has drawn on a rich variety of mathematical and physical ideas. Recent work of a Russian mathematician, G. Perelman, seems to confirm one of the famous outstanding conjectures of topology, the Poincare conjecture (which gives a complete characterization of the three dimensional analogue of a sphere), as well as a profound generalization of this conjecture, the geometrization conjecture of Thurston, which gives in principle a scheme whereby one could describe all possible three-dimensional shapes. A major goal of this conference is to understand these new ideas, as well as other recent work in dimensions three and four, and to evaluate our new understanding of geometry and topology in these dimensions.