Four-Manifolds and Categorification

Duke University, Durham NC

Investigators

Abstract

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 more difficult than their analogues in higher dimensions and require the use of invariants that go beyond traditional algebraic topology. The PI’s primary tools come from Heegaard Floer homology and Khovanov homology, two important packages of invariants developed in 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. In addition, the PI is deeply committed to integrating his research with a passion for education at a variety of levels and mathematical outreach. He has numerous projects in mind for both undergraduate and graduate students in the coming years, and he is deeply invested in expanding the pipeline of women and underrepresented minorities in the field. The specific research goals of the project are organized around two main areas. (1) The PI has made numerous contributions regarding knot concordance, the study of which knots in 3-dimensional space bound smoothly embedded disks in 4 dimensions, along with various other problems concerning smoothly embedded surfaces in 4-dimensional manifolds. These questions are deeply tied to the fundamental strangeness of 4-dimensional topology, as compared with higher dimensions. The PI plans to investigate a number of open questions in this area, including homology slice knots, piecewise-linear concordance of knots in homology 3-spheres, ribbon concordance, new constructions of exotic 4-manifolds, and invariants for knotted 2-spheres in 4-dimensional space. (2) The PI also works on questions that are more internal to the structures of Heegaard Floer homology and Khovanov homology and the relationship between them. While rather technical, these problems may have more topological applications down the road. In particular, the PI plans to investigate some technical issues that arise in the behavior of Heegaard Floer homology under surgery and to continue a long-standing effort to construct a spectral sequence between Khovanov homology and knot Floer homology in characteristic 2. 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.