Foundations of the theory of J-holomorphic curves

Barnard College, New York NY

Investigators

Abstract

This project seeks to rework some basic foundational constructions in symplectic geometry. Symplectic invariants (usually counts of curves of various kinds) are defined using solutions to perturbed systems of equations, and the issues involved in finding coherent perturbations are not yet fully worked out. A main part of this proposal is to continue McDuff?s project with Wehrheim that reworks the traditional approach (via finite dimensional reduction) to this problem. So far, they have resolved the main topological issues, but questions concerning isotropy and how to deal with the lack of smoothness caused by gluing have yet to be worked out in detail, even in the simplest case of closed curves. Once this most basic problem has been dealt with, many important variants of the construction also need reworking, for example what happens when there is a circle symmetry. McDuff also proposes to study a variety of more geometric questions. For example, how does the existence of a very non generic holomorphic curve in a symplectic four manifold affect the other curves in the manifold? How "bendable" are symplectic structures: is there a circle action with isolated fixed pointswhose moment map defines a (singular) fibration over a circle? Symplectic geometry has grown into a very important tool for understanding the structure of manifolds. These are spaces which, like the space-time of physics, locally look like familiar Euclidean space but might have global twisting. As in the recent solution to the Poincare conjecture concerning three dimensional manifolds, it has turned out that instead of looking at plain vanilla manifolds, one should give them extra structure: a metric (which is a way of measuring distance) or perhaps a complex or symplectic structure. The latter two involve making two-dimensional measurements (symplectic structures measure area), and are related by a mysterious mirror symmetry that was first suggested by physicists and recently given various mathematical interpretations. In its eagerness to work with these exciting new ideas from physics, the mathematical community has been using various foundational tools without setting them up with sufficient rigor. This has become a serious problem. Mathematics develops via intuition and imagination, but, because results cannot be verified by experimentation, without careful and valid proofs one is left with mere speculation. This project is largely motivated by the desire to remedy this. It proposes a detailed reworking of basic constructions (from topology and analysis) that allow one to count objects in a consistent way. Once completed, symplectic geometers will be able to move ahead with a sure way to test the correctness of their arguments.