Pseudoholomorphic Curves in Topology and Symplectic Geometry

点击翻译按钮获取中文摘要

This research project addresses several areas within symplectic topology and its interaction with gauge theory, complex geometry, and low-dimensional topology. Lagrangian morphisms in the symplectic category are viewed as morphisms between manifolds that are not necessarily symplectomorphic, but their geometric composition is generically singular. A previous project partially resolves this problem, with consequences including the construction of invariants for three-manifolds and knots; the next goal is to fit these invariants into a framework of higher topological quantum field theories, in particular four-manifold invariants. Another goal is to extend the existing framework to a refined Fukaya category, leading to a tool for mirror symmetry proofs. Moreover, the proposal seeks to solidify and provide accessible expositions of the differential-topological foundations for regularizations of moduli spaces of elliptic PDEs.The research projects described in this proposal concern symplectic geometry, the geometric structure that lies behind the Hamiltonian formulation of mechanics. Recent applications of ideas from symplectic geometry include invariants that help to distinguish one (three or) four-dimensional space from another. Broader impacts of this project concentrate on promoting women in mathematics, e.g. by continuing a lecture series by accomplished researchers and expositors, named after Dorothy Weeks, the first female graduate from the MIT math department.