Instantons, Lagrangians, and Low Dimensional Topology

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Low dimensional topology is an area of mathematics that studies qualities of three- and four-dimensional spaces which are insensitive to continuous deformations like stretching and bending. These spaces model real world objects, and low dimensional topology is highly relevant to other scientific disciplines. For instance, several homology theories, that are fundamental constructions in topology, have been used to analyze datasets in the field of data analysis. In addition, topology plays an essential role in formulating modern theories in physics. Perhaps more surprisingly, tools from modern physics, more specifically quantum filed theory, have yielded significant progresses in low dimensional topology. This NSF award provides support for projects that promote systematic application of ideas in physics to topology and vice versa. The PI aims to investigate applications of the Yang-Mills theory of high energy physics in the topological properties of three-and four-dimensional objects. As a step toward this goal, the PI will further develop homology theories that he constructed together with his collaborators. The research also partly focuses on foundational questions in symplectic geometry, a field with close ties with physics. Last but not the least, the project will support training of graduate students and early career researchers in the field of topology through mentoring, conference and workshop organization, and dissemination of expository materials. The PI will also engage in undergraduate and K-12 outreach. The project aims to study equivariant instanton homology theories, which provide invariants of knots and three-manifolds and are developed by the PI in collaboration with Chris Scaduto and Michael Miller Eismeier. The PI will apply different versions of instanton Floer homology to the study of problems in low dimensional topology and group theory. Another focus of the project is the Atiyah-Floer conjecture. This conjecture states that one can apply methods from symplectic geometry to define three-manifold invariants. Furthermore, the resulting invariant, often called symplectic instanton Floer homology, is isomorphic to instanton Floer homology. The PI and his collaborators will use a geometric partial differential equation, called the mixed equation, to address various versions of the Atiyah-Floer conjecture.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.