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。弗洛尔同调理论，一种使用偏微分方程来研究光滑/辛拓扑问题的技术，最早是在20世纪80年代引入的。 从那时起，这些理论导致了许多戏剧性的发现，包括辛几何中阿诺德猜想的解决和关于透镜空间手术的戈登猜想，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弗洛尔同源理论的结构。 一个特别的目标是继续发展Lipshitz-Ozsvath-Thurston的“边界Floer同源性”理论，这是一种通过二级退化将Heegaard Floer同源性公理化的努力。 另一个目标是为拉格朗日交Floer同调构造一个“Floer同伦类型”，这是科恩-琼斯-西格尔提出的一个想法，并由马诺列斯库、克朗海默-马诺列斯库和萨卡尔在其他上下文中进行。 第三个目标是将纽结弗洛尔同调映射与纽结配边映射联系起来，希望能得到关于3维和4维Heegaard Floer理论的深层结构结果。 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，如真实的世界物体的隐藏对称性（例如，马赛克或粒子物理理论）或数学对象（如组和场）。 一些最引人注目的数学位于这两种类型的交叉点，使用代数方法来研究连续系统的定性性质，而这些性质在数量上是难以解决的。 这种方法的一个强有力的例子是弗洛尔同调，它使用代数结构，如链复形，Hochschild同调和拓扑场论来研究来自物理学的某些偏微分方程族。 弗洛尔理论然后重新解释的结果，以回答问题的拓扑结构（大规模或定性几何）。 这个项目将通过考虑更深层次的代数和拓扑结构来扩展几个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.