Low-Dimensional Topology, Floer Homology, and Categorification

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This project investigates a variety of questions in low-dimensional topology, the study of the global shapes of 3- and 4-dimensional spaces and of knots and surfaces contained within them. This subject lies at the crossroads of many disparate areas of mathematics, and it has a wide variety of applications ranging from cosmology (the shape of the universe) to biochemistry (the knotting of DNA molecules) to mathematical physics. Surprisingly, many problems in low dimensions are usually more difficult than their analogues in higher dimensions and require the use of invariants that go beyond traditional algebraic topology. The PI's particular area of expertise is in Heegaard Floer homology, a collection of invariants for 3- and 4-dimensional manifolds, which has been one of the most fruitful areas of research in low-dimensional topology since the early 2000s. These tools bring together several different fields of mathematics, including representation theory, differential geometry, and analysis, and the PI hopes to elucidate the connections between these different areas and expand the discourse among researchers in these fields.The specific goals of the project are (1) to make progress on a variety of concrete problems in 4-manifold topology, including knot concordance, exotic smooth structures, and embeddings of non-orientable surfaces; (2) to understand the relationship between the Heegaard Floer homology of a 3-manifold and topological properties such as the existence of incompressible surfaces, taut foliations, and left-orderings on the fundamental group; (3) to establish relationships between the knot invariants arising from gauge theory and symplectic geometry and those coming from representation theory and quantum algebra.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.