In 3-dimension topology, the study of foliations, flows and \pi_1-actions on 1-manifolds are closely related. Given a 3-manifold M, one can construct a \pi_1(M)-action on a circle from either a taut foliation or a pseudo-Anosov flow in M by works of Thurston, Calegari-Dunfield and Fenley. When the foliation is depth-one and the pseudo-Anosov flow is transverse to the foliation, we show (with some additional assumptions) that the circle actions from both settings are topologically conjugate. Moreover, the two circles admit extra structures that are compatible in the most natural sense.

The Langlands program predicts some surprising connections between number theory and representation theory, most of which are still conjectural. In recent years, powerful new tools in non-archimedean geometry have led to dramatic progress on these questions. One fascinating aspect of these new tools is that in a certain precise sense, they allow us to treat a fixed prime number p as a variable, which opens the door to importing ideas from the geometric Langlands program into more arithmetic settings. I will give a historically motivated introduction to this circle of ideas, and give a big-picture view of some of the most exciting recent and ongoing developments.

A relaxed-pace seminar on impromptu subjects related to the interests of the audience. Everyone is welcome. The subjects are geometry, probability, combinatorics, dynamics, and more!