MSCS Seminars Today
Calendar for Saturday November 7, 2009
Saturday November 7, 2009
Midwest Dynamical SystemsEquidistribution and counting points on orbits of geometrically finite hyperbolic groups
Nimish Shah (Ohio State University)
9:00 AM in SEO 636
In this joint work with Hee Oh, we consider various sphere packing configurations,
which happen to be invariant under actions of geometrically finite hyperbolic groups,
and estimate the cardinality of spheres of curvature (with respect to euclidean,
or spherical, or hyperbolic metric) at most T for some large T.
This sphere counting problem is studied by formulating and proving certain ``weighted equidistribution''
results related to the geodesic flow on the unit tangent bundle of a hyperbolic $n$-manifold $H^n/\Gamma$,
where $\Gamma$ is a geometrically finite discrete group of isometries of $H^n$.
Midwest Dynamical SystemsEntropy in measurable dynamics
Lewis Bowen (Texas A & M University)
10:30 AM in SEO 636
In 1958, Kolmogorov defined the entropy of a probability measure preserving transformation.
Entropy has since been central to the classification theory of measurable dynamics.
In the 70's and 80's researchers extended entropy theory to measure preserving actions of amenable groups (Kieffer, Ornstein-Weiss).
My recent work generalizes the entropy concept to actions of sofic groups; a class of groups that contains for example, all subgroups of GL(n,C).
Applications include the classification of Bernoulli shifts over a free group. This answers a question of Ornstein and Weiss.
Midwest Dynamical SystemsLimit shapes in groups
Moon Duchin (University of Michigan)
11:30 AM in SEO 636
Consider larger and larger metric spheres in a group. Under nice circumstances, these converge to a definite "limit shape" as the radius goes to infinity. For instance in finitely generated nilpotent groups one may use the rescaling homothety in the ambient Lie group to shrink down large spheres, and by work of Pansu (extended by Breuillard) this gives a well-defined limit. For a simple example, in the free abelian group $Z^2$, if we take the standard generating set, the limit shape is a diamond (and the limiting metric, for which this is the unit sphere, is the $L^1$ metric on the plane). It is natural to ask whether the counting measure on the discrete spheres converges to a measure on the limit shape. I'll discuss our work on this question, and give some ergodic applications and some averaging applications for limit shapes. Parts of this project are joint work with Samuel Lelievre, Christopher Mooney, and Ralf Spatzier.
Midwest Dynamical SystemsFinite approximation, large conjugacy classes, and dynamics of automorphism groups
Christian Rosendal (UIC)
2:30 PM in SEO 636
Automorphism groups of first order structures exhibit a number of
interesting dynamical phenomena not easily encountered in other
topological groups. We shall focus on the existence of large, i.e., dense
or even comeagre, conjugacy classes in automorphism groups and see how the
existence of these relates to strengthened versions of residual finiteness
for countable groups. Moreover, we shall indicate how the existence of
comeagre conjugacy classes strongly determines the structure of a
topological group.
Midwest Dynamical SystemsFast growth in the Folner function for Thompson's group F
Justin Moore (Cornell University)
4:00 PM in SEO 636
While it is not known whether Thompson's group $F$ is amenable, I will
establish a lower bound on the F\"{o}lner function for $F$. In particular,
I will demonstrate the following: For each generating set, there is a
constant $C > 1$ such that if $A$ is a $C^{-n}$-F\"{o}lner set in $F$,
then $A$ contains at least $H(n)$ elements, where $H(0) = 0$ and $H(n+1)
= 2^{H(n)}$.