CAREER: Equivariant Floer Theory and Low-dimensional Topology

拓扑学是研究不同空间的形状。低维拓扑学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”，没有什么有趣的事情会发生;五维空间足够“大”，有趣的事情有空间变得无趣。低维拓扑学中的一个主要问题是同调配边群的结构，这些群是三维空间的群，具有与三维球面相同的许多代数特征。这群人与四维空间中拓扑范畴和光滑范畴之间的实质性差异，以及高维空间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（处理物理学中偏微分方程的解）和弗洛尔理论（处理具有面积概念的空间中的刚性曲线）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。）。本项目将使用Floer理论的工具来研究同调配边群和其他拓扑问题，并在Floer理论中进行新的理论工作，这将为低维拓扑提供有用的工具。在研究部分的同时，该项目还包括计划进一步推动PI的辅导和外联工作，重点是增加早期阶段数学的可及性，并培养年轻研究人员的教学和辅导技能。 这些计划包括扩大密歇根州立大学现有的本科研究计划，为中学生举办数学日营，并安排研讨会，以培养学术沟通技能。这个项目的工具是来自Floer理论的不变量的等变版本。该项目的第一部分使用了由首席研究员和C。Manolescu，它给出了新的同调配边不变量。首席研究员计划构建一个完善的这个理论，在类比平行规范理论不变量所做的工作，并用它来解决问题的扭转和不可分割的元素在同调配边组。该项目的第二部分侧重于拉格朗日弗洛尔同调，辛几何基础Heegaard弗洛尔同调和许多其他拓扑不变量的建设。包含Z/2 Z对称信息的拉格朗日弗洛尔上同调的等变版本在过去六年中得到了广泛而富有成效的发展，包括首席研究员。然而，文献缺乏Z/pZ对称性的类似理论。与R. Lipshitz和S. Sarkar，首席研究员计划构建一个，并使用它来研究低维拓扑中具有自然对称性的许多情况。该奖项反映了NSF的法定使命，并已被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.