An epsilon-representation of a discrete group G is a map from G to the unitary group U(n) that is epsilon-multiplicative in norm uniformly across the group. In the 1980s, Kazhdan showed that surface groups of genus at least 2 are not uniform-to-local stable in the sense that they admit epsilon-representations that cannot be perturbed, even locally (on the generators), to genuine representations.
 

In this talk, Marius Dadarlat of Purdue University will discuss the role of bounded 2-cohomology in Kazhdan's construction and explain why many rank-one lattices in semisimple Lie groups are not uniform-to-local stable, using certain K-theory properties reminiscent of bounded cohomology.

Given a von Neumann algebra M, we can consider the set of values of p such that Lp(M) has the approximation property: the identity on it is a limit of finite rank operators for a suitable topology. Apart from the case when p is infinite, which has been the subject of a lot of work initiated by Haagerup in the late 70s, this invariant has not been very much exploited so far. But ancient works in collaboration with Vincent Lafforgue and Tim de Laat suggest that, maybe, it can distinguish the factors of SL(n,Z) for different values of n. I will explain something that I realized only recently, and that explains why this is a difficult question: it implies some form of the classical Kakeya conjecture, which predicts the shape of sets in the Euclidean space in which a needle can be turned upside down. This talk from Mikael de la Salle will be an opportunity to discuss other connections between classical Fourier analysis and analysis in group von Neumann algebras, including in collaboration with Javier Parcet and Eduardo Tablate

Join us for the inaugural session of Mathematrix book club! Have you heard that office workplaces often have the thermostat set at a temperature that is too cold for women to work comfortably? This month we will be discussing the academic articles behind concepts that often come up in conversations about gender inequality in the workplace. The goal of book club is to develop an evidence-based understanding of diversity in mathematics and academia. 

Join us for an initial welcome pizza lunch to start the academic year to learn about what's happening in Mathematrix in 2025/26! Meet other students who are from underrepresented groups in mathematics and allies :) 

Please RSVP here to confirm your spot: https://form.jotform.com/252814345456864

The typical energy estimate for the Navier-Stokes equations provides a bound for the gradient of the velocity; energy-stable numerical methods that preserve this estimate preserve this bound. However, the bound scales with the Reynolds number (Re) causing solutions to be numerically unstable (i.e. exhibit spurious oscillations) on under-resolved meshes. The dissipation of enstrophy on the other hand provides, in the transient 2D case, a bound for the gradient that is independent of Re.

We propose a finite-element integrator for the Navier-Stokes equations that preserves the evolution of both the energy and enstrophy, implying gradient bounds that are, in the 2D case, independent of Re. Our scheme is a mixed velocity-vorticity discretisation, making use of a discrete Stokes complex. While we introduce an auxiliary vorticity in the discretisation, the energy- and enstrophy-stability results both hold on the primal variable, the velocity; our scheme thus exhibits greater numerical stability at large Re than traditional methods.

We conclude with a demonstration of numerical results, and a discussion of the existence and uniqueness of solutions.