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 - 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.