Georgia Institute of Technology College of Sciences

I'll talk about joint work with Sam Taylor. We characterize convex cocompact subgroups of mapping class groups that arise as subgroups of specially embedded right-angled Artin groups. We use this to construct convex cocompact subgroups of Mod(S) whose orbit maps into the curve complex have small Lipschitz constants.

In this talk I will discuss bounds on the slice genus of aknot coming from it's representation as a braid closure, starting withthe slice-Bennequin inequality. From there I will use surfacebraiding techniques of Rudolph and Kamada to exhibit a new lower boundon the ribbon genus of a knot, given some knowledge about what slicesurfaces it bounds.

We discuss a simple construction a finite dimensional algebra("bipartite algebra") to a bipartite oriented graph, and explain how thestudy of the representation theory of these algebras produces acategorification of the cut and flow lattices of graphs. I'll also mentionwhy we suspect that bipartite algebras should arise naturally in severalother contexts. This is joint work with Anthony Licata.

For a group G, stable G-equivariant homotopy theory studies (the stabilizations of) topological spaces with a G-action up to G-homotopy. For a field k, stable motivic homotopy theory studies varieties over k up to (a stable notion of) homotopy where the affine line plays the role of the unit interval. When L/k is a finite Galois extension with Galois group G, there is a functor F from the G-equivariant stable homotopy category to the stable motivic homotopy category of k. If k is the complex numbers (or any algebraically closed characteristic 0 field) and L=k (so G is trivial), then Marc Levine has shown that F is full and faithful. If k is the real numbers (or any real closed field) and L=k[i], we show that F is again full and faithful, i.e., that there is a "copy" of stable C_2-equivariant homotopy theory inside of the stable motivic homotopy category of R. We will explore computational implications of this theorem.This is a report on joint work with Jeremiah Heller.