Pseudoholomorphic Curves in Topology and Symplectic Geometry

这项研究项目涉及辛拓扑中的几个领域及其与规范理论、复杂几何和低维拓扑的相互作用。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同构的，但它们的几何组成是一般奇异的。之前的一个项目部分解决了这个问题，结果包括构造三维流形和纽结的不变量；下一个目标是将这些不变量适合于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Fukaya范畴，从而产生一个用于镜像对称性证明的工具。此外，该提案寻求巩固和提供椭圆偏微分方程模空间的正则化的微分拓扑基础的易懂的论述。在该提案中描述的研究项目涉及辛几何，即隐藏在力学的哈密顿公式背后的几何结构。辛几何思想的最新应用包括帮助区分一个(三维或)四维空间与另一个空间的不变量。该项目的更广泛影响集中在促进女性在数学方面，例如，通过继续由有成就的研究人员和说明者举办的一系列讲座，以麻省理工学院数学系第一位女性毕业生多萝西·威克斯的名字命名。

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.