Graduate Groups and Dynamics Seminar : Past Events
Past Seminars
The following seminars have already happened, you may instead view upcoming seminars in this series.
Sept. 4, 2018
Alex Furman :
3 p.m. in 1227 SEO
Abstract
Organizational meeting for this Graduate Groups and Dynamics Seminar.
Sept. 11, 2018
Samuel Dodds :
3 p.m. in 1227 SEO
Sept. 18, 2018
Samuel Dodds :
3 p.m. in 1227 SEO
Abstract
The Ruziewicz (a.k.a Banach-Ruziewicz) problem on the $n$-sphere $S^n$ is the question whether the
Lebesgue measure is the <i>unique</i> rotation invariant normalized <i>mean</i> on the Lebesgue
sigma-algebra of the sphere. Here a <i>mean</i> is a finitely additive probability measure.
For $n=1,2$ the answer is negative; for $n\ge 5$ it is positive as was proved independently by
Margulis and Sullivan. For $n=2,3$ the positive answer was proven by Driendfeld.
In the talk the connection between the problem and an existence of a group $G<SO(n)$
acting with a spectral gap on $L^2_0(S^n)$ will be explained, and proven for $n\ge 5$.
Sept. 25, 2018
Wouter van Limbeek :
3 p.m. in 1227 SEO
Abstract
We say a group of matrices in a compact Lie group has spectral gap if the associated averaging operator has eigenvalues bounded away from 1. This property is a geometric analogue of the algebraic criterion of expansion in Cayley graphs, and is intimately connected with a number of interesting problems such as construction of expanders and behavior of random walks. In this talk, we discuss a result of Benoist-De Saxcé connecting the spectral gap property to diophantine properties of matrices, and establishing spectral gap for groups with algebraic entries. This is the first of a series of talks on this result.
Oct. 2, 2018
Robert Kozma :
3 p.m. in 1227 SEO
Abstract
The basis for the Hausdorff-Banach-Tarski "paradox"
is the fact that the free group $F_2$, and therefore any group containing $F_2$,
admits a paradoxical decomposition.
Amenable groups have no paradoxical decompositions.
While not all n-n-amenable groups contain $F_2$, Tarski proved that
every non-amenable group admits a paradoxical decomposition.
The talk presents a proof of this theorem.
Oct. 16, 2018
Alex Furman :
3 p.m. in 1227 SEO
Oct. 23, 2018
Alex Furman :
3 p.m. in 1227 SEO
Oct. 30, 2018
Alex Furman :
3 p.m. in 1227 SEO
Nov. 6, 2018
Wouter van Limbeek :
3 p.m. in 1227 SEO
Abstract
In 1965, Selberg proved that for every congruence cover of the principal modular curve, the spectrum of the Laplacian is uniformly bounded below by 3/16. This implies that the Cayley graphs for SL(2,F_p) form an expander. We will explain the connection with expanders, and discuss recent arguments by Sarnak-Xue (simplified further by Gamburd and Tao) that establish a weak version of Selberg's theorem with smaller gap, but which are more amenable to generalization.
Nov. 13, 2018
Sebastian Hurtado :
3 p.m. in 1227 SEO
Nov. 20, 2018
Wouter van Limbeek :
3 p.m. in 1227 SEO
Abstract
Previously we have established a spectral gap for weakly diophantine groups (first part of the work of Benoist-De Saxce). Today we'll cover the second part: We show that groups with algebraic entries are weakly diophantine.
Jan. 23, 2019
Alex Furman :
4 p.m. in 612 SEO
Abstract
This semester we plan to discuss the recent work of Abert-Bergeron-Biringer-Gelander-Nikolov-Raimbault-Samet) and to
(hereafter 7s) that, among other things, discusses the asymptotic growth of the Betti numbers of compact locally symmetric manifolds
$b_i(M)/vol(M)$ as $vol(M)\to \infty$, where $M=\Gamma\backslash X$
are quotients of a fixed (higher rank) irreducible symmetric space.
To understand these results (and to put them in perspective) we need to discuss a variety of important and cool math topics:
- L^2-Betti numbers
- Luck's Approximation Theorem
- Benjaminy-Schramm convergence
- Invariant Random Subgroups
- Stuck-Zimmer theorem
- and more...
There is some topology, geometry, dynamics, and number theory in this all.
In this first talk we will give a general overview, and discuss soem organizational topics.
Feb. 6, 2019
Wouter van Limbeek :
4 p.m. in 612 SEO
Abstract
We define L^2 (co)homology of a space using L^2 (co)chains or forms and associate a dimension to these (Hilbert) spaces.
This gives rise to L^2 Betti numbers, which possess a number of marvelous properties their classical cousins lack (e.g. multiplicativity under taking covers).
We then turn to a proof of the main theorem connecting the classical and the L^2, namely
Lueck's approximation theorem (realizing L^2 Betti numbers as an appropriate limit of ordinary Betti numbers along a tower of covers).
Feb. 13, 2019
Wouter van Limbeek :
4 p.m. in 612 SEO
Abstract
Last time we introduced L^2 homology and their Betti numbers. This time we prove the main theorem connecting
the classical and the L^2, namely Lueck's approximation theorem (realizing L^2 Betti numbers as an appropriate
limit of ordinary Betti numbers along a tower of covers).
Feb. 20, 2019
Samuel Dodds :
4 p.m. in 612 SEO
Abstract
An Invariant Random Subgroup (IRS for short) for a group $G$ is a probability measure
on the space ${\rm Sub}_G$ of all closed subgroups of $G$, invariant under the natural $G$-action by conjugation.
This concept is a simultaneous generalization of closed normal subgroups and of lattices.
In the lecture we will discuss the compact space ${\rm Sub}_G$, and IRS and give some basic properties of those.
March 13, 2019
Mikolaj Fraczyk :
4 p.m. in 612 SEO
Abstract
I will explain the relation between Benjamini-Schramm convergence of sequences of locally symmetric spaces and certain bounds on the geometric side of the Arthur-Selberg trace formula. After giving a brief description of arithmetic congruence lattices I will sketch a proof of the Benjamini-Schammm convergence for sequences of congruence arithmetic hyperbolic 2 and 3 orbifolds. We will treat the number theoretic estimates as a black box and focus on the “ergodic” part of the proof, which is a nice application of the Borel density theorem for IRS’ses (joint work with Jean Raimbault).
March 20, 2019
Alex Furman :
4 p.m. in 612 SEO
Abstract
Margulis's Normal Subgroup Theorem states that for an irreducible lattice $\Gamma$ in a higher rank center-free
semi-simple Lie group $G$ does not have normal subgroups of infinite index.
Stuck and Zimmer have proved that every non-transitive p.m.p. ergodic action of $G$ is essentially free.
This strengthens Margulis' NST.
More recently, Peterson proved character super-rigidity for higher rank lattices,
strengthening the result of Stuck-Zimmer.
In the talk, I will discuss the relations between these statements and will sketch some ideas of the proofs.
April 3, 2019
Alex Furman :
4 p.m. in 612 SEO
April 17, 2019
Sebastian Hurtado :
4 p.m. in 612 SEO
Aug. 27, 2019
Alex Furman :
4 p.m. in TBD
Abstract
In this meeting we plan to discuss the topics for the semester, decide on the near future plans. My suggestions starts from
Patterson-Sullivan theory, and several topics in manifolds of negative curvature. I will give some overview of some results that I propose to study in the seminar.
Sept. 3, 2019
Samuel Dodds :
4 p.m. in 612 SEO
Abstract
This the first in a sequence of talks on Patterson-Sullivan theory that concerns discrete subgroups of isometries of a hyperbolic space, or more generally rank-one symmetric space.
Sept. 10, 2019
Samual Dodds :
4 p.m. in 612 SEO
Sept. 17, 2019
Samual Dodds :
4 p.m. in 612 SEO
Sept. 24, 2019
Alex Furman :
4 p.m. in 612 SEO
Oct. 1, 2019
Yanlong Hao :
4 p.m. in 612 SEO
Oct. 8, 2019
Yanlong Hao :
4 p.m. in 612 SEO
Oct. 15, 2019
Alex Furman :
4 p.m. in 612 SEO
Oct. 22, 2019
Wouter van Limbeek :
4 p.m. in 612 SEO
Abstract
For negatively curved surfaces, we study visual metrics on the boundary at infinity of their universal covers.
We prove that the visual metric is classified up to bi-Lipschitz equivalence by its Hausdorff dimension, and
we use this to construct many non-isometric negatively curved surfaces whose universal covers are almost-isometric (= quasi-isometric with multiplicative constant 1).
As an application, we answer a question Alex posed in the first seminar this semester (due to him? Or Hamenstaedt?):
Are there negatively curved metrics on a surface such that they are not isometric in any finite cover but almost-isometric on the universal cover?
Joint work with Jean-François Lafont and Ben Schmidt.
Oct. 29, 2019
Subhadip Dey :
4 p.m. in 612 SEO
Nov. 5, 2019
Sebastian Hurtado :
4 p.m. in 612 SEO
Jan. 21, 2020
Alex Furman :
4 p.m. in 612 SEO
Jan. 28, 2020
Alex Furman :
4 p.m. in 612 SEO
Abstract
2nd talk on the paper by Bader, Fisher, Miller, Stover.
Feb. 4, 2020
Yanlong Hao :
4 p.m. in 612 SEO
Feb. 11, 2020
Alex Furman :
4 p.m. in 612 SEO
Abstract
In this talk (a prequel to last week's talk by Yanlong Hao) we will discuss the result of Navas, and possibly a theorem of Thurston that is used there.
Feb. 18, 2020
Ethan Fricker :
4 p.m. in 612 SEO
Feb. 25, 2020
Alex Furman :
4 p.m. in 612 SEO
April 21, 2020
Anders Karlsson :
4 p.m. in 612 SEO
Abstract
TBA
Sept. 8, 2020
Alex Furman :
5 p.m. in Zoom
Abstract
If you are interested in participating, but have not registered yet,
please fill out this form https://forms.gle/BsGdGzmeVyiB7fFP7,
or contact Alex directly
Sept. 15, 2020
Alex Furman :
5 p.m. in Zoom
Abstract
This is part 1.5 of the series of talks on a theorem of Benoist about Cartan and Jordan
projections of Zariski dense semi-groups.
Sept. 22, 2020
Alex Furman / Homin Lee :
5 p.m. in Zoom
Oct. 6, 2020
Homin Lee and Wouter van Limbeek :
5 p.m. in Zoom
Aug. 26, 2025
Alex Furman :
3 p.m. in 427 SEO