Structure of Low-Dimensional Floer Homologies

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。花同调理论是一种利用偏微分方程研究光滑/辛拓扑问题的技术，于20世纪80年代首次提出。从那时起，这些理论导致了许多戏剧性的发现，包括辛几何中的阿诺德猜想和戈登关于透镜空间手术的猜想的解决，但它们的结构仍然有些神秘。这个项目的重点是更好地理解某些花同源理论的结构。一个特别的目标是继续发展Lipshitz-Ozsvath-Thurston的“边界花同调”理论，这是通过一种二阶退化来公理化Heegaard花同调的一种努力。另一个目标是构造拉格朗日交Floer同调的“Floer同伦型”，这一思想由Cohen-Jones-Segal提出，并由Manolescu、Kronheimer-Manolescu和Sarkar在其他情况下进行。第三个目标是将结花同调上的映射与结配合关联起来，希望能得到关于3维和4维heegard花理论的深层结构结果。大多数数学可以分为两类。一类研究连续问题，如流体流动或最小表面（气泡）形成，通常通过偏微分方程。另一个研究更严格的代数问题，如现实世界物体（如马赛克或粒子物理理论）或数学对象（如群和场）的隐藏对称性。一些最引人注目的数学在这两种类型的交叉点上，使用代数方法来研究难以定量处理的连续系统的定性性质。这种方法的一个强有力的例子是Floer同调，它使用链配合物、Hochschild同调和拓扑场理论等代数结构来研究来自物理学的某些偏微分方程族。花理论然后重新解释结果来回答拓扑（大规模或定性几何）中的问题。本项目将通过考虑更深入的代数和拓扑结构来扩展几种Floer同调理论，以获得更强的几何结果。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). Floer homology theory, a technique using partial differential equations to study problems in smooth / symplectic topology, were first introduced in the 1980's. Since then, these theories have led to many dramatic discoveries, including the resolution of the Arnold conjecture in symplectic geometry and Gordon's conjecture on lens space surgeries, but their structure remains somewhat mysterious. The focus of this project is on better understanding the structure of certain Floer homology theories. One particular goal is to continue to develop Lipshitz-Ozsvath-Thurston's theory of "bordered Floer homology," an effort to axiomatize Heegaard Floer homology by a type of second-order degeneration. Another goal is to construct a "Floer homotopy type" for Lagrangian intersection Floer homology, an idea suggested by Cohen-Jones-Segal and carried out in other contexts by Manolescu, Kronheimer-Manolescu, and Sarkar. A third goal is to associate maps on knot Floer homology to knot cobordisms, hopefully leading to deep structural results about 3- and 4-dimensional Heegaard Floer theory.Most of mathematics falls into one of two categories. One category studies continuous problems, like fluid flow or minimal surface (bubble) formation, often through partial differential equations. Another studies more rigid, algebraic problems like hidden symmetries of real world objects (e.g., mosaics or particle physics theories) or mathematical objects (like groups and fields). Some of the most striking mathematics sits at the intersection of the two types, using algebraic methods to study qualitative properties of continuous systems which are quantitatively intractable. One powerful example of this method is Floer homology, which uses algebraic structures like chain complexes, Hochschild homology, and topological field theories to study certain families of partial differential equations coming from physics. Floer theory then re-interprets the results to answer questions in topology (large-scale or qualitative geometry). This project will extend several Floer homology theories by considering even deeper algebraic and topological structure to obtain stronger geometric results.