Essential embedded surfaces in an irreducible 3-manifold correspond to non-trivial splittings of its fundamental group. In joint work with Khánh Lê, we give some conditions on the fundamental group of a Haken hyperbolic 3-manifold which guarantee that any other 3-manifold group with the same set of finite quotients must have a non-trivial splitting. Using one of these conditions, we show that every finite regular cover of an aspherical integer homology three sphere with positive first Betti number will have first Betti number at least four and this is optimal. We will also discuss examples of Haken 3-manifolds to which the theorems in this work apply. 

Abstract: In this talk we will review past and recent results pertaining to the emerging field of integrable space-time nonlocal nonlinear evolution equations. In particular, we will discuss blow-up in finite time of soliton solutions as well as the physical derivations of many integrable nonlocal systems.

Abstract: I will explain the combinatorics behind the cluster algebra structure on braid and Richardson varieties. This generalizes the previously known constructions for positroid varieties and double Bruhat cells. The cluster algebra is described in terms of a 3-dimensional generalization of Postnikov’s plabic graphs, and the underlying quiver is induced by the intersection form of the Goncharov–Kenyon conjugate surface associated to a 3D plabic graph. Joint work with T. Lam, M. Sherman-Bennett, and D. Speyer