Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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Dynamics, Geometry, & Groups Seminar

Dynamics, Geometry, & Groups - Heejoung Kim (UIUC)

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.

Dynamics, Geometry, & Groups - Ian Frankel (Queen's University)

Friday, September 20th, 2019

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

Speaker: Ian Frankel (Queen's University)

Title: Local geometry of Teichmüller space: flat and quasiconformal.

Abstract: The Teichmüller distance between two homeomorphic Riemann surfaces X and Y is a number that quantifies the following question: Given a homeomorphism from X to Y, how non-conformal does the map have to be?

The optimal quasiconformal maps, i.e. those with smallest quasiconformal constant, are characterized by choices of special singular flat metrics on X and Y, and in fact fit into a large familes of maps, and the dynamics of SL(2,R) acting on this family have been the subject of many celebrated results in the past decade.

Now, suppose we are given X and Y but with singular flat metrics that are not related to the optimal map. We will describe how we can still estimate the Teichmüller distance from X to Y.

Dynamics, Geometry, & Groups - James Mingo (Queen's University)

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.

Dynamics, Geometry, & Groups - Catherine Pfaff (Queen's University)

Tuesday, August 27th, 2019

Time: 1:30 p.m Place: Jeffery Hall 319

Speaker: Catherine Pfaff (Queen's University)

Title: Typical Trees: An Out(F_r) Excursion.

Abstract: Random walks are not new to geometric group theory (see, for example, work of Furstenberg, Kaimonovich, Masur). However, following independent proofs by Maher and Rivin that pseudo-Anosovs are generic within mapping class groups, and then new techniques developed by Maher-Tiozzo, Sisto, and others, the field has seen in the past decade a veritable explosion of results. In a 2 paper series, we answer with fine detail a question posed by Handel-Mosher asking about invariants of generic outer automorphisms of free groups and then a question posed by Bestvina as to properties of R-trees of full measure in the boundary of Culler-Vogtmann outer space. This is joint work with Ilya Kapovich, Joseph Maher, and Samuel J. Taylor.

Dynamics, Geometry, & Groups - Marco Lenci (Universita di Bologna)

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.

Dynamics, Geometry, & Groups - Catherine Pfaff (Queen's University)

Wednesday, May 15th, 2019

Time: 2:00 p.m Place: Jeffery Hall 319

Speaker: Catherine Pfaff (Queen's University)

Title: Counting Conjugacy Classes: Groups Rebelling Against Dynamics.

Abstract: This talk will start with a gentle introduction to ways of viewing outer automorphisms of free groups and then will discuss joint work with Ilya Kapovich regarding counting conjugacy classes of these outer automorphisms. That is, inspired by results of Eskin and Mirzakhani counting closed geodesics of bounded length in the moduli space of a fixed closed surface, we consider a similar question in the Out(F_r) setting and discover bounds revealing behavior not present in the surface setting or in classical hyperbolic dynamical systems.

Dynamics, Geometry, & Groups - Reda Chhaibi (Toulouse)

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.

Dynamics, Geometry, & Groups - Alessandro Portaluri (U of Torino)

Friday, March 22nd, 2019

Time: 10:30 a.m Place: Jeffery Hall 102

Speaker: Alessandro Portaluri (University of Torino, Italy)

Title: Visiting Kepler with a couple of symplectic friends.

Abstract: Starting from the classical planar Kepler problem, by using the conservation law of the angular momentum, we reduce the problem to a one degree of freedom singular problem. Thanks to this reduction and after a suitable time scaling we show that, for negative energy, the orbit is an ellipse. Finally, by using a refined version of the Conley-Zehnder intersection index , we give a homotopic classification of all bounded motions.

Dynamics, Geometry, & Groups - Francesco Cellarosi (Queen's)

Friday, October 5th, 2018

Time: 10:30 a.m Place: Jeffery Hall 422

Speaker: Francesco Cellarosi (Queen's University)

Title: Central Limit Theorem via spectral method

Abstract: I will explain the Nagaev-Guivarc'h method to obtain a Central Limit Theorem for sequences of random variables coming from a large class of 1-dimensional dynamical systems, namely uniformly expanding maps of the interval. The idea is work in a suitable Banach space to establish a spectral gap for the transfer operator, and then use a perturbative argument. This talk is based on a paper by Sébastien Gouëzel.