New Invariants of Knots and 3-Manifolds

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This award provides funding for a project in topology, a central field in modern mathematics. The main goal is to use ideas from both algebra and theoretical particle physics (gauge theory) to develop new tools for studying geometric shapes. The PI plans to apply these tools to study the problem of which high dimensional shapes can be triangulated, that is, decomposed into simple pieces (similar to a decomposition of a surface into triangles). Triangulations have applications beyond pure mathematics, for example in computer graphics. The project will support graduate education and help disseminate mathematical ideas to the general public.In more technical terms, the project concerns some of the homological invariants that are associated to knots and three-manifolds. The PI will investigate extensions of Khovanov homology to knots in manifolds other than the three-sphere, and study their properties with regard to cobordisms and surfaces in various four-manifolds. In a different direction, the PI will use Pin(2)-equivariant Seiberg-Witten Floer stable homotopy and involutive Heegaard Floer homology to understand more about the structure of the three-dimensional homology cobordism group, and thus shed light on the classification of triangulations of high-dimensional manifolds. Additionally, the PI aims to construct knot Floer stable homotopy types and Heegaard Floer stable homotopy types.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.