CAREER: Equivariant Floer Theory and Low-dimensional Topology

拓扑学研究的是不同空间的形状。低维拓扑是对三维和四维空间的研究，这是基本上最不了解的维度。一维和二维空间足够“小”，不会发生任何有趣的事情；五维空间足够“大”，让有趣的东西有变得无趣的空间。低维拓扑中的一个主要问题是同调协群的结构，同调协群是具有许多与三维球面相同的代数特征的三维空间群。这一群体与四维拓扑和光滑范畴之间的本质区别以及高维拓扑中的结构问题有着深刻的联系。在过去的三十年里，利用规范理论（处理物理中的偏微分方程的解）和花理论（处理空间中具有面积概念的刚性曲线）中的不变量，在这个问题和其他中心拓扑问题上取得了实质性进展。本项目将利用Floer理论的工具来研究同调协群等拓扑问题，并开展新的Floer理论工作，为低维拓扑研究提供有用的工具。在开展研究工作的同时，该项目还包括进一步推进PI的指导和推广工作的计划，重点是提高早期阶段数学的可及性，以及培养年轻研究人员的教学和指导技能。这些计划包括扩展密歇根州立大学现有的本科生研究项目，为中学生举办数学日营，以及安排建立学术沟通技巧的讲习班。这个项目的工具是弗洛尔理论中不变量的等变版本。项目的第一部分使用了由首席研究员和C. Manolescu构造的三流形不变量Heegaard flower同调的等变版本，给出了同调配的新不变量。项目负责人计划对这一理论进行改进，类似于平行规范论不变量的工作，并利用它来解决同调协群中元素的扭转和不可分问题。项目的第二部分集中在拉格朗日花同调，基础的辛几何构造Heegaard花同调和许多其他拓扑不变量。包含Z/ 2z对称信息的拉格朗日花上同调的等变版本在过去六年中得到了广泛而富有成果的发展，包括首席研究员。然而，文献中缺乏类似的Z/ pz对称理论。首席研究员计划与R. Lipshitz和S. Sarkar一起构建一个，并用它来研究具有自然对称性的低维拓扑中的许多情况。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Topology is the study of the shapes of different spaces. Low-dimensional topology is the study of three- and four-dimensional spaces, which are the dimensions that are substantially least understood. One-and two-dimensional spaces are "small" enough that nothing interesting can happen; five-dimensional spaces are "large" enough that interesting things have room to become uninteresting. A major question in low-dimensional topology is the structure of the homology cobordism groups, groups of three-dimensional spaces with many algebraic features in common with the three-dimensional sphere. This group has deep connections to the substantial difference between the topological and smooth categories in four-dimensions, and to structural issues in higher dimensional topology. In the past thirty years, substantial progress on this and other central topological questions has been made using invariants from gauge theory (which deals with solutions of partial differential equations from physics) and Floer theory (which deals with rigid curves in spaces with a notion of area). This project will use tools from Floer theory to study the homology cobordism groups and other topological questions, and to undertake new theoretical work in Floer theory that will produce useful tools for low-dimensional topology. In parallel to the research component, the project includes plans to further the PI's mentoring and outreach efforts, with a focus on increasing the accessibility of mathematics at early stages and on building pedagogical and mentorship skills in young researchers. These plans include an extending Michigan State University's existing undergraduate research program, running mathematics day camps for middle school students, and arranging for workshops for building academic communication skills. The tools of this project are equivariant versions of invariants from Floer theory. The first part of the project uses an equivariant version of the three-manifold invariant Heegaard Floer homology constructed by the Principal Investigator and C. Manolescu, which gives new invariants of homology cobordism. The Principal Investigator plans to construct a refinement of this theory, in analogy with work done in parallel gauge-theoretic invariants, and use it to address questions of torsion and indivisibility of elements in the homology cobordism group. The second part of the project focuses on Lagrangian Floer homology, the symplectic geometry construction underlying Heegaard Floer homology and many other topological invariants. Equivariant versions of Lagrangian Floer cohomology that incorporate the information of a Z/2Z-symmetry have been extensively and fruitfully developed in the past six years, including by the Principal Investigator. However, the literature lacks an analogous theory for Z/pZ-symmetries. With R. Lipshitz and S. Sarkar, the Principal Investigator plans to construct one, and to use it to study many situations in low-dimensional topology that possess natural symmetries.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.