Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
Subscribe to RSS - Dynamics

Dynamics, Geometry, & Groups Seminar

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 (POSTPONED)

Thursday, August 15th, 2019 (POSTPONED to a later date)

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

Speaker: Catherine Pfaff (Queen's University)

Title: Random free group automorphisms & the boundary of outer space.

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, 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 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 - 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.

Dynamics, Geometry, & Groups - Jacob Russell (CUNY)

Friday, September 14th, 2018

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

Speaker: Jacob Russell (CUNY)

Title: The geometry of groups via their boundaries

Abstract: Gromov revolutionized the study of finitely generated groups by purposing the study of groups as geometric objects. The success of this geometric viewpoint has inspired a whole program of classifying groups geometrically. From the geometry of the group, one can construct various boundaries; topological spaces which record the geometry of the group "at infinity". One of these boundaries, the Morse boundary, is particularly nice as the group has a natural action on it by homeomorphisms. The Morse boundary can also be equipped with a natural cross-ratio and we will discuss how the topology of the boundary coupled with this cross-ratio is actually sufficient to encode the entire geometry of the group.

Dynamics, Geometry, & Groups - Camille Horbez

Friday, September 7th, 2018

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

Speaker: Camille Horbez (Laboratoire de Mathématiques d’Orsay)

Title: Growth under automorphisms of hyperbolic groups

Abstract: Let G be a finitely generated group, let S be a finite generating set of G, and let f be an automorphism of G. A natural question is the following: what are the possible asymptotic behaviors for the length of f^n(g), written as a word in the generating set S, as n goes to infinity, and as g varies in the group G?

We investigate this question in the case where G is a torsion-free Gromov hyperbolic group. Growth was completely described by Thurston when G is the fundamental group of a hyperbolic surface, and can be understood from Bestvina-Handel’s work on train-tracks when G is a free group. We address the case of a general torsion-free hyperbolic group. We show in particular that every element g has a well-defined exponential growth rate under iteration of f, and that only finitely many exponential growth rates arise as g varies in G.

This is a joint work with Rémi Coulon, Arnaud Hilion and Gilbert Levitt.

Dynamics, Geometry, & Groups - Derrick Wigglesworth (Fields Inst.)

Friday, August 10th, 2018

Time: 10:30 am Place: Jeffery Hall 422

Speaker: Derrick Wigglesworth (Fields Institute)

Title: Groups acting on trees

Abstract: I'll discuss several of the ways one can learn about groups via their actions on trees.  There will be many examples and pictures.  Then, we'll briefly discuss folding paths; a tool for understanding complicated actions. Finally, I'll mention some applications of folding paths.