"Groups, Dynamics & Topology" Seminar

The seminar will take place in-person on Thursdays 4PM (CET).  The topics covered include: locally symmetric spaces,  homogeneous dynamics, representation theory, geometric and measured group theory. 

Upcoming meetings


Past meetings

April 11, 16:00 (Faculty of Mathematics and Computer Science, room 0009). 

Title:  Measured property (T) and approximability"

Abstract: The aim of this talk is to present the following characterization: a p.m.p. countable borel equivalence relation (cber) has measured property (T) if and only if every ergodic extension of it is non-approximable. The starting point of the talk will be a dynamical characterization of measured property (T) à la Connes-Weiss, developed in joint work with Łukasz Grabowski and Sam Mellick. We will then study the dynamics induced by approximations of an equivalence relation and use our observations to prove the announced characterization. The remainder of the talk will be devoted to motivate said result through the discussion of some illustrative examples. Among these, we will see a brief proof of why graphings with planar connected components cannot have measured property (T).

April 4, 16:00 (Faculty of Mathematics and Computer Science, room 0009). 

Title:  Doubling and Brunn—Minkowski in compact Lie groups

Abstract: Given a subset A of a locally compact group, the doubling constant is the ratio of the measure m(A^2) of the set A^2 of products of two elements of A to the measure m(A) of A. This constant is a central object in additive combinatorics, in the study of random walks on groups, in geometric analysis and in many other fields. 

In Euclidean spaces, doubling is now particularly well understood. Beyond that, the situation is far more mysterious. A conjecture of Breuillard and Green predicts that in a compact simple group this constant must be at least 2 to the power of the minimal co-dimension of a proper subgroup. 

In this talk, I will discuss the proof of this conjecture. I'll also explain how the tools employed open the door to other results, such as a Brunn-Minkowski inequality or a stability result.


March 21, 16:00 (Faculty of Mathematics and Computer Science, room 0009). 

Title: On the General Rankin-Selberg Problem

Abstract:  We will see the general Rankin-Selberg problem. In a joint work, with Kummari Mallesham, Prof. Ritabrata Munshi, Saurabh Kumar Singh, we break the general Rankin-Selberg's bound (also the well known Rankin-Selberg's bound) on the error term. Our work will also generalize Bingrong Huang's result ``On the Rankin-Selberg problem" (Math. Ann. https://doi.org/10.1007/s00208-021-02186-7) and gives a better bound. This is an upcoming work.

March 7, 16:00 (Faculty of Mathematics and Computer Science, room 0009). 


Title: Quantum ergodicity on the Bruhat-Tits building for PGL(3) in the Benjamini-Schramm limit


Abstract: Originally, quantum ergodicity concerned equidistribution properties of Laplacian eigenfunctions with large eigenvalue on manifolds for which the geodesic flow is ergodic. More recently, several authors have investigated quantum ergodicity for sequences of spaces which "converge" to their common universal cover and when one restricts to eigenfunctions with eigenvalues in a fixed range. Previous authors have considered this type of quantum ergodicity in the settings of regular graphs, rank one symmetric spaces, and some higher rank symmetric spaces. We prove analogous results in the case when the underlying common universal cover is the Bruhat-Tits building associated to PGL(3, F) where F is a non-archimedean local field. This may be seen as both a higher rank analogue of the regular graphs setting as well as a non-archimedean analogue of the symmetric space setting.


February 29, 16:00 (Faculty of Mathematics and Computer Science, room 0009). 

Title: Topological tilings, mean dimension, and shift embeddability.

Abstract: We introduce the question of cubic shift embeddability for topological dynamical systems and explain the previously known results. We state our two main theorems:

1. systems with Uniform Rokhlin Property and comparison satisfy sharp shift embeddability and

2. for a large class of amenable groups, having a technical condition which we call property FCSB implies that the system has the Uniform Rokhlin Property and comparison. 

If time permits, we will attempt to sketch the proof of the first result.

December 15, 14:00 (Faculty of Mathematics and Computer Science, room 0009). (note the new time)

Title: Torsion subgroups of small cancellation groups

Abstract: We prove that torsion subgroups of groups defined by, C(6), C(4)-T(4) or C(3)-T(6) small cancellation presentations are finite. 

This follows from more general results about locally elliptic action on small cancellation complexes.


November 24 , 14:00 (Faculty of Mathematics and Computer Science, room 0009).  (note the new time)


Title: Torsion homology growth and cheap rebuilding of inner-amenable groups

Abstract: In the first half, I will give a short introduction on inner-amenable groups and (torsion) homology growth invariants. One tool for showing vanishing of these invariants is the cheap rebuilding property, which was recently introduced by Mikołaj Fraczyk together with Abért, Bergeron and Gaboriau. We prove that certain inner-amenable groups have this property, thus extending vanishing results that were already known for amenable groups.

November 9, 16:00 (Faculty of Mathematics and Computer Science, room 0009).
Title: Koopman representations for positive definite functions. 


Abstract: It is a classical result of Raikov and Gelfand that if G is a locally compact second countable group and \phi:G-> C is a continuous positive definite function, then there exists a representation U of G on Hilbert space H and a cyclic vector f\in H for which \phi(g)=<U(g)f,f>. Through the use of the Gaussian Measure Space Construction (GMSC), the previous result can be refined by taking H=L^2(X,\mu) and letting U be the Koopman representation of a measure preserving action of G. Our first main result is to further refine the latter result by showing that the measure preserving action of G can be assumed to be ergodic. Our second main result is when G is abelian, in which case we refine the result of the GMSC in a direction by showing that f\in L^2(X,\mu) can be taken to satisfy |f|=1 a.e. We will also review a classical result of Foias and Stratila that shows that we cannot always take the system in the previous result to be ergodic. If time permits, we will discuss connections to the study of van der Corput sets. 


Slides 

October 26, 16:00 (Faculty of Mathematics and Computer Science, room 0009). 

Title: Quasi-arithmeticity and commensurability

Abstract: In their construction of nonarithmetic real hyperbolic lattices in each dimension, Gromov and Piatetski-Shapiro exploited the fact that two arithmetic lattices coinciding on a Zariski-dense subgroup must be commensurable. We discuss a rephrasing of the latter that holds more generally for lattices that are quasi-arithmetic in the sense of Vinberg, and how this can be used to demonstrate the abundance of compact hyperbolic Coxeter polyhedra in dimension 4, as well as of high-dimensional closed hyperbolic manifolds with equal small systole. This is largely based on joint work with Nikolay Bogachev and Jean Raimbault.

October 19, 16:00 (Faculty of Mathematics and Computer Science, room 0009). 

Title: Application of extended quotients to the Langlands program

Abstract: We will first introduce the notion of extended quotient, which originates in noncommutative geometry. Next, we will explain how it naturally occurs in both the representation theory of p-adic reductive groups and the study of enhanced versions of local Langlands parameters. Finally, we will show how to interpret the local Langlands correspondence into a correspondence between extended quotients, and provide applications

The seminar will be preceded by an introductory lecture on representations of p-adic groups by Anna Szumowicz (IMPAN) 11:00-12:00 in the room 0146. 

September 18, 16:00 (Faculty of Mathematics and Computer Science, room 0009). 

Title: On the  Rankin-Selberg bounds

Abstract: