Mathematics

Choquet-Deny groups and the Infinite conjugacy class property

Josh Frisch (Caltech)

Abstract: The Poisson Boundary of a random walk on a group is a measure space that corresponds to the space of different asymptotic trajectories that the random walk might take. Given a group $G$ and a probability measure $\mu$ on $G$ the Poisson boundary is trivial (i.e. has no non-trivial events) if and only if $G$ supports a bounded $mu$-harmonic function. In this talk I will give an introduction to the notion of the poisson boundary and discuss a recent characterization of exactly which countable groups $G$ have a trivial Poisson Boundary for every measure $\mu$ (the so called Choquet-Deny groups). This characterization will immediately imply that, for finitely generated groups, these groups are exactly those of polynomial growth. This answers a question of Kaimanovich and Vershik. Surprisingly, the proof does not rely at all on growth estimates for a group, and instead relies on the algebraic infinite conjugacy class property. This is joint work with Yair Hartman, Omer Tamuz, and Pooya Vahidi Ferdowski.

Asymptotic dimension and coarse embeddings in the quantum setting

Alejandro Chavez-Dominguez (University of Oklahoma)

Abstract: We generalize the notions of asymptotic dimension and coarse embeddings, from metric spaces to quantum metric spaces in the sense of Kuperberg and Weaver. We show that the quantum asymptotic dimension behaves well with respect to several natural operations, and in particular with respect to quantum coarse embeddings. Moreover, in analogy with the classical case, we prove that a quantum metric space that equi-coarsely contains a sequence of quantum expanders must have infinite asymptotic dimension. This is done by proving a vertex-isoperimetric inequality for quantum expanders, based upon a previously known edge-isoperimetric one. Joint work with Andrew Swift.

Distribution of eigenvalues of random matrices (part I)

Peixue Wu (UIUC Math)

Abstract: Last time we proved a famous semicircular law for the limit distribution of the empirical measure of the eigenvalues of Wigner's matrix (i.i.d. under the symmetry restriction). When we go over the proof in detail, we find two essential ingredients to the proof: 1. Stochastic independence of the entries. 2. Most matrix entries are centered and have the same variance. Using the similar idea (methods of moments) we will show that semicircular law holds for a much larger class of random matrices. We will also talk about the joint distribution for the eigenvalues of the Gaussian Orthogonal (Unitary) Ensembles (GOE or GUE).