CAREER: Equivariant Floer Theory and Low-dimensional Topology

拓扑学是研究不同空间形状的学科。低维拓扑学是对三维和四维空间的研究，这些空间是人们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”，有趣的事情有空间变得不有趣。低维拓扑中的一个主要问题是同调余边群的结构，同调余边群是具有许多与三维球面相同的代数特征的三维空间的群。这一组与四维拓扑范畴和光滑范畴之间的本质差异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，使用规范理论(处理物理学中的偏微分方程解)和Floer理论(处理空间中带有面积概念的刚性曲线)中的不变量，在这个和其他中心拓扑问题上取得了实质性的进展。这个项目将使用Floer理论中的工具来研究同调余边群和其他拓扑问题，并在Floer理论中进行新的理论工作，这将产生有用的工具来研究低维拓扑。除研究部分外，该项目还包括进一步推动国际数学协会的指导和推广工作的计划，重点是在早期阶段提高数学的可及性，以及培养年轻研究人员的教学和指导技能。这些计划包括扩大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，并安排研讨会培养学术交流技能。这个项目的工具是Floer理论中的不变量的等变版本。项目的第一部分使用了由首席调查者和C.Manolescu构造的三维流形不变量Heegaard Floer同调的等变版本，它给出了新的同调协边不变量。首席调查者计划构建这一理论的改进，类似于在平行规范理论不变量中所做的工作，并用它来解决同调余边群中元素的扭转和不可分割性问题。项目的第二部分集中在拉格朗日Floer同调，Heegaard Floer同调背后的辛几何构造和许多其他拓扑不变量。包含Z/2Z对称性信息的拉格朗日Floer上同调的等变形式在过去的六年中得到了广泛而卓有成效的发展，包括首席研究员。然而，文献中缺乏关于Z/PZ对称性的类似理论。首席调查员计划与R.Lipshitz和S.Sarkar一起建造一个，并利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NSF的法定使命，并通过使用基金会的智力优势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.