Zimmer proposed the study of higher-rank semisimple group
actions in the 1980's after showing certain rigidity results,
especially about associated cocycles in the measurable category. In
this talk, I will describe recent progress towards...
The geodesic flow (for the hyperbolic metric) of an infinite
Riemann surface is ergodic if and only if Brownian motion is
recurrent, which is also equivalent to the divergence of the
Poincaré series. Surfaces with ergodic geodesic flows are
most...
In this talk I will describe new results establishing the
existence of a strong spectral gap for geometrically finite,
n dimensional, hyperbolic manifold with critical exponent greater
than (n−1)/2. This settles a conjecture of Mohammadi and Oh...
Let Γ be a countable group with a Cayley graph G. Suppose we are
running an independent identical probabilistic algorithm on each
vertex (or each edge) and vertices are only allowed to communicate
with their neighbors. The goal of this distributed...
Given a one-parameter degenerating family of rational maps on
the projective line, it is possible to construct a non-archimedean
limit which captures how this family degenerates. Recently,
Luo used ultrafilters to construct limits for an arbitrary...
Motivated by the study of billiards in polyhedra, we study
linear flows in a family of singular flat 3-manifolds which we call
translation prisms. Using ideas of Furstenberg, and Veech, we
connect results about weak mixing properties of flows on...
The theory of hierarchically hyperbolic groups, due to
Behrstock, Hagen, and Sisto, was developed by abstracting work of
Masur and Minsky on mapping class groups. Study of the large scale
geometry of the outer automorphism group Out(Fn)
Thurston gave an explicit construction of pseudo-Anosov
elements
in the mapping class group of a compact surface, using Dehn
twists in pairs of filling multicurves. We show that the
probability that a random walk of length n on the mapping
class...
Consider a closed surface M of genus greater than or equal to 2.
For negatively curved metrics on M and their corresponding geodesic
flow, we can study the topological entropy, the Liouville entropy,
and the mean root curvature. In 2004, Manning...
