University of Cambridge

In the context of categorical Langlands, there are many ways in which one could define the notion of n-finitely presented smooth representation. We will explore and compare two different definitions, relating them with the notion of n-coherence for the corresponding Iwasawa ring.

There has been considerable interest in the problem of whether every metric space of bounded geometry coarsely embeds into a uniformly convex Banach space due to the work of Kasparov and Yu that established a connection between such embeddings and the Novikov conjecture. Brown and Guentner were able to prove that a metric space with bounded geometry coarsely embeds into a reflexive Banach space. Kalton significantly extended this result to stable metric spaces and asked whether these classes are coarsely equivalent, i.e. whether every reflexive Banach space coarsely embeds into a stable metric space. Baudier introduced the notion of upper stability, a relaxation of stability, for metric spaces as a new invariant to distinguish reflexive spaces from stable metric spaces. In this talk, we show that in fact, every reflexive space is upper stable and also establish a connection of upper stability to the asymptotic structure of Banach spaces. This is joint work with F. Baudier and Th. Schlumprecht.

A factorisation $x=u_1u_2\cdots$ of an infinite word $x$ on alphabet $X$ is called ‘super-monochromatic’, for a given colouring of the finite words $X^{\ast}$ on alphabet $X$, if each word $u_{k_1}u_{k_2}\cdots u_{k_n}$, where $k_1<\cdots<k_n$, is the same colour. A direct application of Hindman’s theorem shows that if $x$ is eventually periodic, then for every finite colouring of $X^{\ast}$, there exist a suffix of $x$ that admits a super-monochromatic factorisation. What about the converse?

In this talk we show that the converse does indeed hold: thus a word $x$ is eventually periodic if and only if for every finite colouring of $X^{\ast}$ there is a suffix of $x$ having a super-monochromatic factorisation. This has been a conjecture in the community for some time. Our main tool is a Ramsey result about alternating sums. This provides a strong link between Ramsey theory and the combinatorics of infinite words.

Joint work with Imre Leader and Luca Q. Zamboni

We formulate and solve a generalized inverse Navier–Stokes boundary value problem for velocity field reconstruction and simultaneous boundary segmentation of noisy Flow-MRI velocity images. We use a Bayesian framework that combines CFD, Gaussian processes, adjoint methods, and shape optimization in a unified and rigorous manner.
With this framework, we find the velocity field and flow boundaries (i.e. the digital twin) that are most likely to have produced a given noisy image. We also calculate the posterior covariances of the unknown parameters and thereby deduce the uncertainty in the reconstructed flow. First, we verify this method on synthetic noisy images of flows. Then we apply it to experimental phase contrast magnetic resonance (PC-MRI) images of an axisymmetric flow at low and high SNRs. We show that this method successfully reconstructs and segments the low SNR images, producing noiseless velocity fields that match the high SNR images, using 30 times less data.
This framework also provides additional flow information, such as the pressure field and wall shear stress, accurately and with known error bounds. We demonstrate this further on a 3-D in-vitro flow through a 3D-printed aorta and 3-D in-vivo flow through a carotid artery.

Stable commutator length (scl) is a measure of homological complexity in groups that has attracted attention for its various connections with geometric topology and group theory. In this talk, I will introduce scl and discuss the (hard) problem of computing scl in surface groups. I will present some results concerning isometric embeddings of free groups for scl, and how they generalise to surface groups for the relative Gromov seminorm.

The character table of GL(2,Fq), for a prime power q, was constructed over a century ago. Many of these characters were determined via the explicit construction of a corresponding representation, but purely character-theoretic techniques were first used to compute the so-called discrete series characters. It was not until the 1970s that Drinfeld was able to explicitly construct the corresponding discrete series representations via l-adic étale cohomology groups. This work was later generalised by Deligne and Lusztig to all finite groups of Lie type, giving rise to Deligne-Lusztig theory.

In a similar vein, we would like to construct the representations affording the (smooth) characters of compact groups like GL(2,Zp), where Zp is the ring of p-adic integers. Deligne-Lusztig theory suggests hunting for these representations inside certain cohomology groups. In this talk, I will consider one such approach using a non-archimedean analogue of de Rham cohomology.

Fluids sculpt many of the shapes we see in the world around us. We present a new mathematical model describing the shape evolution of a body that dissolves or melts under gravitationally stable buoyancy-driven convection, driven by thermal or solutal transfer at the solid-fluid interface. For high Schmidt number, the system is reduced to a single integro-differential equation for the shape evolution. Focusing on the particular case of a cone, we derive complete predictions for the underlying self-similar shapes, intrinsic scales and descent rates. We will present the results of new laboratory experiments, which show an excellent match to the theory. By analysing all initial power-law shapes, we uncover a surprising result that the tips of melting or dissolving bodies can either sharpen or blunt with time subject to a critical condition.

I will explain how various results in arithmetic statistics by Bhargava, Gross, Shankar and others on 2-Selmer groups of Jacobians of (hyper)elliptic curves can be organised and reproved using the theory of graded Lie algebras, following earlier work of Thorne. This gives a uniform proof of these results and yields new theorems for certain families of non-hyperelliptic curves. I will also mention some applications to rational points on certain families of curves.

After a general introduction to the study of random walks on groups, we discuss the relationship between limit theorems for random walks on Lie groups and Diophantine properties of the underlying distribution. Indeed, we will discuss the classical abelian case and more recent results by Bourgain-Gamburd for compact simple Lie groups such as SO(3). If time permits, we discuss some new results for non-compact simple Lie groups such as SL_2(R). We hope to touch on the relevant methods from harmonic analysis, number theory and additive combinatorics. The talk is aimed at a general audience. 

I will review the series of recent results with Merle (IHES), Rodnianski (Princeton) and Szeftel (Paris Sorbonne) concerning the description of implosion mechanisms for viscous three dimensional compressible fluids. I will explain how the problem is connected to the description of blow up mechanisms for classical super critical defocusing models.