The Geometry/Topology RTG

I study homotopy theory and homotopical algebra in the sense of Quillen. Specifically, I'm interested in algebraic structures, especially monoidal structures, on (∞, n)-categories and the algebraic structure of the category of (∞, n)-categories itself. I am also interested in stable homotopy theory and the application of stable homotopy-theoretic methods to questions in algebra. My research statement is available here.

My current research involves investigating the geometry of gerbes. More generally, I am interested in n-torsors, which are analogues of principal bundles and gerbes in the world of higher stacks, i.e. sheaves of n-categories. This is related to describing a sheafified Dold-Kan correspondence in various models of infinity-categories, thereby giving geometric realizations of cohomology classes and generalizing results about ordinary principal bundles.

My research is in the field of symplectic geometry and topology. I am interested in invariants of manifolds defined by counting pseudoholomorphic curves, such as Floer homology and Fukaya categories, and the relationship between these invariants and algebraic geometry, in light of the homological mirror symmetry conjecture. For example, some of my current work concerns aspects of this conjecture for certain Stein manifold. A particularly important invariant in this case are the /symplectic homology groups/. In certain cases, these groups contain the global regular functions on the mirror variety, and so can be used to understand its geometry. Moreover, these groups come with a "canonical basis" that is evident from their geometric origin, giving rise to generalized "theta functions" on the mirror variety (Gross-Hacking-Keel).

My research mostly consists of analyzing objects from low-dimensional topology using various Floer theories, which are invariants constructed from either gauge theory or symplectic geometry. One mysterious question is to find the relationship between these Floer theories and the fundamental groups of three-manifolds. I am currently studying a conjectural relationship between a certain Floer homology group and the existence of left-invariant orders on the fundamental group. This is also related to the codimension 1 foliations that can occur on a three-manifold. This motivates the study of the behavior of all of these objects under natural topological operations, such as covering maps, Dehn surgery, and JSJ decompositions.