In 2002, Farb and Mosher introduced the notion of convex cocompactness in the mapping class group to capture coarse geometric information of the associated surface group extensions. Convex cocompact subgroups are necessarily finitely generated and purely pseudo-Anosov, but it is an open question whether the converse is true. Several partial results are known in certain settings, however. For example, work of Dowdall, Kent, Leininger, Russell, and Schleimer give a positive answer for subgroups of fibered 3-manifold groups (aka surface-by-cyclic extensions) naturally embedded in punctured mapping class groups via the Birman exact sequence. We present a generalization of this in the setting of surface-by-abelian extensions.

We discuss recent work that pinpoints the spectral decomposition for the Anderson tight-binding model with an unbounded random potential on the Bethe lattice of sufficiently large degree. We prove that there exist a finite number of mobility edges separating intervals of pure-point spectrum from intervals of absolutely continuous spectrum, confirming a prediction of Abou-Chacra, Thouless, and Anderson. A central component of the proof is a monotonicity result for the leading eigenvalue of a certain transfer operator, which governs the decay rate of fractional moments for the tight-binding model’s off-diagonal resolvent entries. This is joint work with Amol Aggarwal. We will also draw connections to the problem of establishing a mobility edge in the spectrum of heavy-tailed random matrices, which was considered in recent work with Amol Aggarwal and Charles Bordenave.

We have impromptu and (sometimes) scheduled talks, on topics in probability, combinatorics, geometry, and dynamics.

Everyone is welcome!