Pseudoholomorphic Curves in Topology and Symplectic Geometry

本研究计画探讨辛拓扑学的几个领域，以及它与规范理论、复杂几何学和低维拓扑学的相互作用。辛范畴中的拉格朗日态射被视为流形之间的态射，这些流形不一定是辛态的，但它们的几何组成是一般奇异的。以前的一个项目部分解决了这个问题，其结果包括三流形和结的不变量的构建;下一个目标是将这些不变量融入更高拓扑量子场论的框架，特别是四流形不变量。另一个目标是将现有的框架扩展到一个精炼的福谷类，从而得到一个镜像对称证明工具。此外，该提案旨在巩固和提供可访问的expositions的微分拓扑基础的正则化模空间的椭圆PDEs.The研究项目中描述的建议关注辛几何，几何结构，背后的哈密顿制定的力学。辛几何思想的最新应用包括有助于区分一个（三维或）四维空间的不变量。该项目的更广泛影响集中在促进妇女参与数学，例如，继续举办以麻省理工学院数学系第一位女毕业生多萝西·威克斯命名的有成就的研究人员和学者的系列讲座。

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.