August 18th – Kyle Austin

Nuclear Dimension of C*-Algebras with Applications in Dynamical Systems and Coarse Geometry

Nuclear dimension was created by Zacharias and Winter as an extension of the covering dimension of locally compact Hausdorff spaces. It also has deep connections in coarse geometry as the asymptotic dimension of a bounded geometry metric space bounds the nuclear dimension of its reduced uniform Roe algebra. In my talk, I plan on introducing nuclear dimension for C* algebras and to briefly explain its connections with covering dimension and with asymptotic dimension.

Depending on time, I would like to discuss current work (mine included) on the nuclear dimension of crossed product and groupoid C*-algebras. These include the Rokhlin dimension created by Winter, Hirshberg, and Zacharias and the dynamical asymptotic dimension of Guentner, Willett, and Yu.

We meet at 10.15 in room 403.

June 23rd – Assaf Bar-Natan

A k-intersecting family of arcs on a surface with boundary S is a collection of essential arcs between the boundary components, such that no two are homotopic, and such that the arcs pairwise intersect at most k times. We can also look at closed curves in place of arcs when the surface is closed. In this expository talk, I will talk about some elementary results in the field, and then discuss some cardinality results of P. Przytycki and C. Smith about the maximal cardinality of 1- and 2-intersecting families of arcs, and show how these can give interesting upper bounds on 1-intersecting families of curves on closed surfaces. If time permits, I will also show some ideas about finding the precise maximal number of 1-intersecting curves on the genus g surface.

We meet at 10.15 in room 403.

June 7th – Tom Church

On Tuesday, June 7th at 14.15 Tom Church will give a talk entitled Applications of representation stability.

Abstract:

I will give a gentle survey of representation stability, viewed through the lens of its applications in topology and elsewhere. Representation stability is an asymptotic version of representation theory; for example, it captures the behavior of families of representations of permutation groups S_n as n goes to infinity.
I will discuss applications to configuration spaces of manifolds; homology of arithmetic groups; the stable mod-p Langlands correspondence; uniform generating sets for congruence subgroups and “congruence” subgroups; and distributional stability for random squarefree polynomials over finite fields. Based on joint work with Ellenberg, Farb, Putman, and Nagpal.
The talk will take place in IMPAN, room 403.

June 2nd – Damian Sawicki

Admitting a coarse embedding is not preserved under group extensions

I will present the recent preprint of Arzhantseva and Tessera. Apart from confirming the title, the obtained group generalises from box spaces to groups another result of the authors – that there exist spaces (‘relative expanders’) non-embeddable into the Hilbert space yet without weakly/coarsely embedded expanders. The construction is a wreath product of $\mathbb{Z}_2$ and a monster group constructed by Osajda.
We meet at 10.15 in room 403.

May 19th – Jarek Kędra (Aberdeen)

Jednostajnie ograniczone grupy

Grupa G nazywa sie ograniczona jeśli każda niezmiennicza na sprzężenia
norma na G ma skończoną średnicę. Przykładami ograniczonych grup są grupy
dyfeomorfizmów niektórych rozmaitości, niektóre niekozwarte kraty (np
SL(n,Z) dla n>2), komutant grupy Thompsona F, przykłady z nowej pracy Gala
i Gismatullina. Nie wiadomo czy kraty w półprostych grupach Liego wyższej
rangi są ograniczone.

Wprowadzę definicję jednostajnie ograniczonej grupy i przedystkutuję
cztery przyklady: SO(n), SL(n,R), SO(n,Z[1/5]), SL(n,Z) oraz jako
zastosowanie podam metryczny dowód twierdzenia Delzanta o tym, że
niezwarte proste grupy Liego nie działają hamiltonowsko na zamkniętych
rozmaitościach symplektycznych.

Jednostajna ograniczoność jest silniejsza od ograniczoności oraz uogólnia
jednostajną prostotę.

Praca wspólna z Assafem Libmanem.

We meet at 10.15 in room 403

April 28th – Tomasz Odrzygóźdź

Bent walls in random groups.

I will present new method of finding actions of random groups on CAT(0) cube complexes. Its well known idea to find an action of such groups on spacrs with walls. This structure of space with walls is usually provided by the system of hypergraphs in the Cayley complex. However This method often does not work in some range of parameters of random groups. I will show how to extend this method to these cases.

We meet at 10.15 in room 403.

April 21st – Marek Kaluba

Non-injectivity via persistence topology

Consider cloud of points in \mathbb{R}^n. We would like to know if the points has been sampled from a (local) patch of a manifold. Such sampling usually occurs with some addition of Noise, so we may think of points as values of a (stochastic) “function”: Y = F(X,N), where N = \text{Noise} is assumed to be

  1. uni-modal, and
  2. (strongly) independent of X.

The basic problem approached can be summarized as follows. Given a set of points \{(X_i,Y_i)\}_i \subset \mathbb{R}^n = \mathbb{R}^k \times \mathbb{R}^{n-k}, determine if there is a function f\colon \mathbb{R}^k \to \mathbb{R}^{n-k} or g\colon \mathbb{R}^{n-k} \to \mathbb{R}^k of which a “noised sample” F(X,N) represents the data.

I will present easy 2-dimensional case where the persistence homology of appropriately thresholded Delaunay complex can supply computationally cheap hints to the true answer. Higher dimensional cases are also tractable with Rips complex. These considerations have some applications to the causality problem, where the task to infer the underlying casual structure of two random variables X, Y when only the set of points is given. Surprisingly though the presented approach works for non-functional relations (e.g. in the presence of con-founder).

We meet at 10.15 in room 403.

April 7th – Izhar Oppenheim

Angles between groups in Banach spaces and Banach property (T)

The idea of using angle between subspaces to deduce property (T) and vanishing of group cohomology with unitary representations was introduced by Dymara and Januszkiewicz in their paper from 2002 dealing with groups acting on Buildings. Since then this idea was developed and refined by several authors, perhaps culminating in the work of Ershov, Jaikin-Zapirain and Kassabov on property (T) for groups graded by root systems.

In my talk I will discuss a generalization of this idea in the Banach setting, i.e., how to define angles between groups acting on a Banach space and use this notion of angle to deduce (a strengthened version of) Banach property (T) and vanishing of group cohomology with Banach representations.

We meet at 10.15 in room 403