The proposed research focuses on applying Floer-theoretic invariants to study objects in low-dimensional topology and contact geometry. The first project develops combinatorial approaches to symplectic field theory. The foundation for this study is a combinatorial formulation of Legendrian contact homology for Seifert fibered spaces which will be used to explore Legendrian knot theory in rational homology three-spheres. The second project studies generalizations of the notion of knot genus. We will develop techniques for studying non-orientable surfaces in link complements, adapting existing Heegaard Floer invariants to detect minimal-complexity non-orientable surfaces.
Without the sensory apparatus to detect quantum phenomena, our experience of the universe is as a three-dimensional physical space or as a four-dimensional spacetime; this project studies the properties and interactions of three-, and four-dimensional mathematical objects called manifolds. The last decade has seen success in studying these objects using techniques from Floer theory, a powerful yet technically difficult approach. One of the primary goals of the proposed research is developing versions of Floer-theoretic tools which are easier to use. This will facilitate the study of low-dimensional manifolds and shed new light on the original constructions.
