Floer Homology and Low-Dimensional Topology

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Low-dimensional topology is the study of spaces of dimensions three and four and their qualitative geometric properties. The classification of these spaces remains a fundamental problem today. The main theme of this project is to study topological properties of spaces using certain algebraic invariants, called Floer homology and Khovanov homology. These invariants have become central tools in modern topology and have connections to fields ranging from symplectic geometry to quantum physics to biology. In addition to its research component, the project includes plans for mentoring and outreach efforts, with a focus on increasing the accessibility of mathematics to groups underrepresented in the mathematical sciences. The project is devoted to studying three- and four-dimensional manifolds by further developing techniques in Floer homology and Khovanov homology. The first part of the project is to study homology cobordism and knot concordance, including constructing new concordance homomorphisms for knots in homology spheres. The second part of the project concerns properties of the monodromy of open book decompositions and Stein fillability of contact three-manifolds. The third part is to study properties of a link invariant called symplectic sl(n) homology and its connection to Khovanov-Rozansky homology.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.