CAREER: The symplectic category, Floer field theory, and relations to gauge theory and topology

奖项：dms -0844188首席研究员：Katrin wehrheim该奖项由2009年美国复苏与再投资法案（公法111-5）资助。本研究项目涉及辛拓扑及其与规范理论、复杂几何和低维拓扑的相互作用的几个领域。辛范畴中的拉格朗日态射被认为是流形之间的态射，不一定是辛纯的，但它们的几何组成是一般奇异的。先前的一个项目部分地解决了这个问题，其结果包括构造三流形和节的不变量；下一个目标是将这些不变量放入拓扑量子场论的框架中，另一个目标是将现有的框架扩展到一个精炼的Fukaya类别，从而导致镜像对称性证明的工具。正在进行的关于atiyah - floer猜想的项目将继续进行，更多的推测性工作将从无限维辛的角度考虑PDE流中的非挤压效应。本提案中描述的研究项目涉及辛几何，即力学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空间和另一个三维空间的不变量。该项目的广泛影响集中在促进女性在数学领域的发展，并在麻省理工学院举办了一场针对初级研究人员的女性数学会议，以及女孩之角（Girls’Angle），这是一个基于与数学领域女性导师密切非正式互动的女孩数学俱乐部。更多信息请参见http://www.girlsangle.org/。

AbstractAward: DMS-0844188Principal Investigator: Katrin WehrheimThis award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).This research project addresses several areas within symplectictopology and its interaction with gauge theory, complex geometry,and low-dimensional topology. Lagrangian morphisms in thesymplectic category are viewed as morphisms between manifoldsthat are not necessarily symplectomorphic, but their geometriccomposition is generically singular. A previous projectpartially resolves this problem, with consequences including theconstruction of invariants for three-manifolds and knots; thenext goal is to fit these invariants into a framework oftopological quantum field theory, and another goal is to extendthe existing framework to a refined Fukaya category, leading to atool for mirror symmetry proofs. An ongoing project on theAtiyah-Floer conjecture will continue, and more speculative workwill consider non-squeezing effects in PDE flows from aninfinite-dimensional symplectic point of view.The research projects described in this proposal concernsymplectic geometry, the geometric structure that lies behind theHamiltonian formulation of mechanics. Recent applications ofideas from symplectic geometry include invariants that help todistinguish one three-dimensional space from another. Broaderimpacts of this project concentrate on promoting women inmathematics, and feature both a conference at MIT on women inmathematics that is aimed at junior researchers and Girls' Angle,a math club for girls that is based on close informal interactionwith female mentors from mathematical areas. See http://www.girlsangle.org/ for more information.