Topology of Manifolds and Algebraic Varieties

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This proposal concerns work at the interface of topology and geometry. Perelman has made great advances in applying Ricci flow to the topology of 3-manifolds. There remain open questions about using his ideas to completely classify closed 3-manifolds (i.e., establish Thurston's Geometrization Conjecture). These questions revolve around the theory of sufficiently collapsed 3-manifolds and their relationship to Alexandrov spaces of dimension at most 2. Sorting these issues out will give a complete, detailed argument proving the Geometrization Conjecture. There should be an analogous theory of sufficiently collapsed 4-manifolds and the relation of these objects to Alexandrov spaces of dimension at most 3. Developing such a theory for 4-manifolds will be important in its own right, but may also be crucial in applying Ricci flow to prove topological results in dimension 4. The other area of study is mirror symmetry for Calabi-Yau 3-folds in toric varieties.Here the idea is to establish mirror symmetry for these examples at the fairly delicate level of variations of Hodge structure and monodromy representations.There have been exciting developments in the subject of topology in the last few years which this proposal will explore. These involve the interplay between topology and other, more geometric and analytic, areas of mathematics. Perelman's application of a heat-type flow equation and techniques from geometry to establish the Poincare Conjecture, the oldest and most fundamental of all topological questions, is a major watershed moment in the subject. This is one of the deepest and most beautiful applications of partial differential equations to a purely topological problem ever. There remain many questions of how to apply the same ideas to all 3-dimensional spaces, not just simplest, which is the 3-dimensional sphere. More speculatively, there is the question of how to extend these ideas to provide new insights into 4-dimensional spaces, about which we know little, except that the possibilities are much wider than in dimension 3. Another main theme in topology and geometry over the past few years is to make sense of the mirror symmetry. This principle is derived by non-rigorous physics arguments and is unlike anything normally seen in `classical mathematics.' It is some version of `quantum mathematics.' This notion has led to an incredibly rich flourishing of mathematics on the interface of topology and geometry as mathematicians attempt to understand special cases of this notion, to formulate this notion precisely, and to establish it mathematically.