For discrete subgroups of semisimple Lie groups, the limit cone serves as a key invariant in studying asymptotic properties. Anosov subgroups, well-known for their rich geometric and dynamical features, form an important class of these discrete subgroups. A natural question arises: does the limit cone vary continuously along deformations of Anosov subgroups? In this talk, we discuss a sufficient condition for such continuity, along with applications like the continuity of the growth indicators along deformations. Based on joint work with Hee Oh.

Let F be a pseudo-Anosov homeomorphism of a hyperbolic surface S. In this talk, we’ll describe joint work with Tarik Aougab and Dave Futer that predicts the number of fixed points of F, up to constants that depend only on the surface S. If F satisfies a mild condition called “strongly irreducible,” then the logarithm of the number of fixed points of F is coarsely equal to its translation length on the Teichmuller space of S. Without this condition, there is still a coarse formula involving subsurface projections of F’s invariant laminations.

Please join us for pre-seminar Tea

The Bernoulli free boundary problem is a classical model associated with certain interface problems from applications (capillarity, jet flows), and also arising in shape-optimization problems for Dirichlet eigenfunctions.  I will explain a result on the large-scale regularity theory of minimizing (and almost minimizing) one-phase free boundaries in periodic media, and a corresponding Liouville theorem for global minimizing solutions.  In a forthcoming work with Farhan Abedin (Lafayette College) we have also obtained analogous results in the two-phase case.  If times permits I will also discuss an application to quantitative homogenization in a shape-optimization problem for the principal Dirichlet eigenvalue.