Friday, November 15th, 2019

Time: 10:30 a.m Place: Jeffery Hall 319

Speaker: Elizabeth Field (University of Illionois Urbana-Champagn)

Title: Trees, dendrites, and the Cannon-Thurston map.

Abstract: When 1 -> H -> G -> Q -> 1 is a short exact sequence of three word-hyperbolic groups, Mahan Mitra (Mj) has shown that the inclusion map from H to G extends continuously to a map between the Gromov boundaries of H and G. This boundary map is known as the Cannon-Thurston map. In this context, Mitra associates to every point z in the Gromov boundary of Q an ``ending lamination'' on H which consists of pairs of distinct points in the boundary of H. We prove that for each such z, the quotient of the Gromov boundary of H by the equivalence relation generated by this ending lamination is a dendrite, that is, a tree-like topological space. This result generalizes the work of Kapovich-Lustig and Dowdall-Kapovich-Taylor, who prove that in the case where H is a free group and Q is a convex cocompact purely atoroidal subgroup of Out(F_n), one can identify the resultant quotient space with a certain R-tree in the boundary of Culler-Vogtmann's Outer space.

Thursday, October 17th, 2019

Time: 3:30 p.m Place: Jeffery Hall 319

Speaker: Heejoung Kim (University of Illinois, Urbana-Champaign)

Title: Algorithms detecting stability and Morseness for finitely generated groups.

Abstract: For a word-hyperbolic group G, the notion of quasiconvexity is independent on the choice of a generating set and a quasiconvex subgroup of G is quasi-isometrically embedded in G. Kapovich provided a partial algorithm which, on input a finite set S of G, halts if S generates a quasiconvex subgroup of G and runs forever otherwise. However, beyond word-hyperbolic groups, the notion of quasiconveixty is not as useful. For a finitely generated group, there are two recent generalizations of the notion of a quasiconvex subgroup of a word-hyperbolic group, a ``stable'' subgroup and a ``Morse'' subgroup. In this talk, we will discuss various detection and decidability algorithms for stability and Morseness of a finitely generated subgroup of mapping class groups, right-angled Artin groups, toral relatively hyperbolic groups.

Friday, September 13th, 2019

Time: 10:30 a.m Place: Jeffery Hall 319

Speaker: James Mingo (Queen's University)

Title: Can a free group have a fractional number of generators? In the free world it can.

Abstract: Starting with a countable discrete group G we complete the group ring in a suitable topology to get L(G), the group von Neumann algebra. We let L(n) be the group von Neumann algebra for the free group on n ≥ 2 generators. Using random matrix theory Voiculescu showed that if we tensor L(m) with the k x k matrices we get an algebra isomorphic to L(n). The relation between k, m, and n is the same as in Schreier's index theorem for subgroups of free groups. With this we can define L(t) for any real t > 1 as the group algebra of the free group with t generators. I will explain the main ideas in the proof. No prior knowledge of free probability will be assumed.

Friday, August 23rd, 2019

Time: 10:30 a.m Place: Jeffery Hall 319

Speaker: Marco Lenci (Universita di Bologna)

Title: Infinite-volume mixing and the case of one-dimensional maps with an indifferent fixed point.

Abstract: I will first discuss the question of mixing in infinite ergodic theory, which will serve as a motivation for the introduction of the notions of "infinite-volume mixing". Then I will focus on a prototypical class of infinite-measure-preserving dynamical systems: non-uniformly expanding maps of the unit interval with an indifferent fixed point. I will show how the definitions of infinite-volume mixing play out in this case. As it turns out, the most significant property, and the hardest to verify, is the so-called global-local mixing, corresponding to the decorrelation in time between global and local observables. I will present sufficient conditions for global-local mixing, which will cover the most popular examples of maps with an indifferent fixed point (Pomeau-Manneville and Liverani-Saussol-Vaienti). If time permits, I will also present some peculiar limit theorems that can be derived for these systems out of the property of global-local mixing.

Friday, April 12th, 2019

Time: 2:30 p.m Place: Jeffery Hall 422

Speaker: Reda Chhaibi (Toulouse)

Title: Quantum SL_2, infinite curvature and Pitman's 2M-X Theorem.

Abstract: Pitman's theorem (1975) is an aesthetic theorem from probability theory, with geometry and representation theory related to SL_2, in disguise. Many proofs do exist, and the goal of this talk is to present a unified point of view regarding two proofs. - A proof by Bougerol and Jeulin - which generalizes to all semi-simple groups. They consider a Brownian motion on the symmetric space $H^3 = SL_2(\mathbb{C})/SU_2$, with varying curvature r and then take the limit $r \rightarrow \infty$. - Biane defined and studied quantum random walks on the enveloping algebra of SL_2, in the 90s. Then in the years 2000, he made the connection to the representation theory of the Jimbo-Drinfeld quantum group $\mathcal{U}_q(sl_2)$, in the crystal regime, i.e $q \rightarrow 0$.

Why should the crystal regime $q=0$ for quantum groups be related infinite curvature in symmetric spaces? The goal of this talk is to convince the audience that the parameter q, from the point of view of quantization and Kirillov's orbit method, is not a quantum parameter but indeed a curvature parameter. The simple relationship is q=e^{-r}, after a modification of the classical definition of quantum groups. I shall only mention the rank 1 group SL_2, and assume no knowledge of quantum groups, since they will have to be (re)defined anyway.

Joint work with F. Chapon.