The Geometry Research Group

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.

Eric Katz studies a tropical geometry which is a way of turning algebraic problems into combinatorial ones involving polyhedral geometry. He has been doing work on the foundations of tropical geometry and in applying tropical geometry to other areas of mathematics.

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.