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

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AbstractAward: DMS-0540236Principal Investigator: Robert D. Friedman, Peter S. OzsvathThis is a proposal for a four-day-long, broad conference atColumbia University on "Topology, Geometry, and Physics.'' Theconference will focus on the following major themes, and theirinterconnections: hyperbolic manifolds and geometrization gaugetheory, three- and four-manifolds, Hodge theory, and interactionsbetween mathematics and modern physics. Such a conference isparticularly timely, in view of the many breakthroughs in each ofthese subjects. In particular, the recent spectacular work ofPerelman on the Poincare conjecture and Thurston's geometrizationconjecture, as well as new developments in hyperbolic geometry,make this an especially opportune moment to survey the newlandscape of three-manifold theory. At the same time, newtechniques of gauge theory and new combinatorial methods havedeepened the current understanding of three- and four-manifoldsand it is time to take stock of these developments and theirrelation to other work in the field. Many of the new results ingeometric topology have been based upon ideas arising inmathematical physics and geometric analysis, and these kind ofinteractions will serve as a unifying theme for the conference.A fundamental problem in mathematics is to describe all possibleshapes. In the case of surfaces (two dimensional objects),possible shapes include a sphere (the surface of the earth, forexample), a torus (the surface of a tire) and more complicatedgeneralizations. A great deal of research has been concerned withdimension three, the dimension of space, and dimension four, thedimension of space-time. Because of its relevance tounderstanding our physical world, understanding all possibleshapes in these dimensions is particularlyimportant. Paradoxically, these cases are much harder tounderstand and to classify than higher-dimensional objects (whichin turn are much harder to visualize in any meaningful way). Thestudy of dimensions three and four has drawn on a rich variety ofmathematical and physical ideas. Recent work of a Russianmathematician, G. Perelman, seems to confirm one of the famousoutstanding conjectures of topology, the Poincare conjecture(which gives a complete characterization of the three dimensionalanalogue of a sphere), as well as a profound generalization ofthis conjecture, the geometrization conjecture of Thurston, whichgives in principle a scheme whereby one could describe allpossible three-dimensional shapes. A major goal of thisconference is to understand these new ideas, as well as otherrecent work in dimensions three and four, and to evaluate our newunderstanding of geometry and topology in these dimensions.