I will discuss recent work on quantitative estimates for the passive scalar transport equation when the advecting velocity field is below the Lipschitz regularity class. In this setting, the classical Cauchy-Lipschitz theory fails to give control on the rate at which the solution may grow, usually measured by Sobolev norms. Despite this, well-posedness of the transport equation has been known when the vector field lies in W^{1,1} since 1989, and for vector fields in BV since 2004, under the additional assumption that the divergence remains bounded. It was conjectured by Bressan in 2003 that the same quantitative “mixing estimates” given by the Cauchy-Lipschitz theory should hold also in this setting. Since then, much progress has been made in recovering these estimates when the vector field lies in W^{1,p} with p>1, however the endpoint case p=1, and in particular BV, has remained a challenging open problem. In this talk I will discuss the first quantitative mixing and stability estimates for a passive scalar transported  by W^{1,1} and BV autonomous vector fields with zero divergence. The proof involves quantitative weak harmonic estimates in BV, and a double pigeonhole principle to quantify the weak-* compactness of the solution space.

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

Everyone is welcome! 

Global class field theory provides a universal abelian reciprocity laws for 1-dimensional global fields. It has been generalized in a number of different directions. In the geometric setting, it manifests as a duality between (different kinds of) sheaves on the moduli spaces of line bundles and 1-dimensional representations of the fundamental group of a curve. In a different direction, Kato and Saito gave a generalization to higher dimensional varieties. I will describe a categorification of the Kato-Saito higher class field theory, in which the moduli space of line bundles is generalized to the moduli space of zero cycles. This is joint work in progress with Elmanto.

The ends of finite-volume hyperbolic (n+1)-manifolds are cusps of the form B x R+ for some compact, flat n-manifold B.  In 2009, McReynolds built on work on Long and Reid to prove that every such n-manifold arises as the cusp cross section of some hyperbolic (n+1)-manifold using an arithmetic construction.  A natural further question to ask is under what conditions each cross-section can arise.  In this talk, we give an algebraic condition that describes exactly when a given flat manifold arises (as a cusp cross-section) in a commensurability class of cusped arithmetic hyperbolic manifolds.  Time permitting, we will also discuss some applications, including a potential new way to prove a hyperbolic manifold is non-arithmetic.  This is joint work with Duncan McCoy.

These lectures will describe connections between dynamical systems and topics including algebraic geometry, number theory,
unipotent flows and fractal Jordan curves.

All talks will be for a general audience, and no talk is a prerequisite for any other.

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

Everyone is welcome!