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.

Let \( G = \exp(\mathfrak{g}) \) be a connected, simply connected nilpotent Lie group with Lie algebra \( \mathfrak{g} \), and let \( H = \exp(\mathfrak{h}) \) be a closed subgroup with Lie algebra \( \mathfrak{h} \). Consider a unitary character \( \chi \) of \( H \), given by \(\chi(\exp X) = \chi_{f}(\exp X) = e^{i f(X)}, \  X \in \mathfrak{h}, \) for some \( f \in \mathfrak{g}^{\ast} \). Let \( \tau = \operatorname{Ind}_{H}^{G} \chi \) denote the monomial representation of \( G \) induced from \( \chi \).

The object of interest is the algebra \( D_{\tau}(G/H) \) of \( G \)-invariant differential operators acting on the homogeneous line bundle associated with the data \( (G, H, \chi) \). Under the assumption that \( \tau \) has finite multiplicities, it is known that \( D_{\tau}(G/H) \) is commutative.

In this talk, I will discuss the Polynomial Conjecture for the representation \( \tau \), which asserts that the algebra \( D_{\tau}(G/H) \) is isomorphic to  
\(\mathbb{C}[\Gamma_{\tau}]^{H}\),  the algebra of \( H \)-invariant polynomial functions on \( \Gamma_{\tau} \). Here, \( \Gamma_{\tau} = f + \mathfrak{h}^{\perp} \) denotes the affine subspace of \( \mathfrak{g}^{\ast} \).

I will present recent advances toward proving this conjecture, with a particular emphasis on Duflo's Polynomial Conjecture concerning the Poisson center of \( \Gamma_{\tau} \). Furthermore, I will discuss the case where \( \tau \) has discrete-type multiplicities in the exponential setting, shedding light on a counterexample to Duflo's conjecture.