CAREER: Equivariant Floer Theory and Low-dimensional Topology

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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.