The classical sphere packing problem asks: what is the densest possible arrangement of identical, non-overlapping spheres in $\mathbb{R}^d$?
I will discuss a recent proof that there exists a sphere packing with density at least
\[
(1-o(1))\frac{d \log d}{2^{d+1}}.
\]
This improves upon previous bounds by a factor of order $\log d$ and is the first improvement by more than a constant to Rogers' bound from 1947.
This is joint work with Marcelo Campos, Marcus Michelen and Julian Sahasrabudhe.

The self-organization of turbulence is a remarkable property of flows with two sign-definite conserved quantities. When such flows are forced at small scales, a coherent flow called a condensate emerges, and is sustained by turbulence. The organizational principle for the condensate is that it should occupy the entire domain, respect its symmetries and be independent of small-scale details. One class of flows where condensation occurs is a rapidly rotating shallow fluid layer under the influence of gravity. This family of two-dimensional flows is characterized by a single parameter, the Rossby deformation radius R, which determines the range of influence of a flow perturbation. When R is much larger than the domain size, the flow reduces to two-dimensional Navier-Stokes. In the opposite limit of vanishing R, a regime termed LQG, interactions between fluid elements become strictly local. We uncover an unexpected organizational principle in the latter: the condensate area is determined by the ratio between the forcing scale and the UV cutoff. In particular, the large-scale flow can take different configurations depending on this ratio, including regions of bi-stability of configurations and spontaneous symmetry breaking in the thermodynamic limit (increasing system size). We explain how this behavior arises from the spatial distribution of fluxes of the conserved quantities in the system.

The Wiener-Hopf factorisation of a Lévy process has two forms. The first describes how the process makes new maxima and minima, by decomposing it into two so-called 'ladder processes'. The second expresses its characteristic exponent as the product of two functions related to the ladder processes. Since the latter is analytic in nature, the question naturally arises: is such a decomposition unique? The answer has been known for killed Lévy processes since at least Rogozin's work in 1966, but appears to have remained open in general. We show that, indeed, uniqueness holds in all cases. This gives a solid foundation to the 'theory of friendship', which allows one to construct a Lévy process with known Wiener-Hopf factorisation. The results also hold for random walks. Joint work with Leif Döring (Mannheim), Mladen Savov (Sofia) and Lukas Trottner (Aarhus).

Motivated by various contraction conjectures, categorical statements, and classification theorems, and also by the seemingly insatiable urge to rewrite all of mathematics using only the letter C, I will describe the full A_infty structure associated to a general (-3,1)-curve inside a smooth CY 3-fold. This sounds complicated, but it turns out to be combinatorial and easy. Of course, most of the talk will be about background, and the motivation for considering these questions, including the analytic classification of 3-fold flops using noncommutative data. This is all joint work with Gavin Brown.

Onwards from the mid-twentieth century, the stochastic filtering problem has caught the attention of thousands of mathematicians, engineers, statisticians, and computer scientists. Its applications span the whole spectrum of human endeavour, including satellite tracking, credit risk estimation, human genome analysis, and speech recognition. Stochastic filtering has engendered a surprising number of mathematical techniques for its treatment and has played an important role in the development of new research areas, including stochastic partial differential equations, stochastic geometry, rough paths theory, and Malliavin calculus. It also spearheaded research in areas of classical mathematics, such as Lie algebras, control theory, and information theory. The aim of this paper is to give a brief historical account of the subject followed by a recent filtering application to data assimilation for geophysical fluid dynamics models.  

We discuss necessary and sufficient criteria for certain Fourier multiplication operators to satisfy the Liouville property (bounded harmonic functions are a.s. constant) and the local continuation property (bounded functions, that are harmonic and identically zero on a domain, are a.s. zero on the whole space). Since the operators generate stochastic processes, there is also a probabilistic interpretation of these findings.

Kernel Stein discrepancies (KSDs) measure the quality of a
distributional approximation and can be computed even when the target
density has an intractable normalizing constant. Notable applications
include the diagnosis of approximate MCMC samplers and goodness-of-fit
tests for unnormalized statistical models. The present work analyzes
the convergence control properties of KSDs. We first show that
standard KSDs used for weak convergence control fail to control moment
convergence. To address this limitation, we next provide sufficient
conditions under which alternative diffusion KSDs control both moment
and weak convergence. As an immediate consequence we develop, for each
q>0, the first KSDs known to exactly characterize q-Wasserstein
convergence.

The Widom conjecture concerns the asymptotic spectral density of Toeplitz operators of the form $\Pi_U F \Pi_V F^* \Pi_U$, where $\Pi_U$ is the operator of multiplication by the indicator of an open set $U$ and $F$ is the Fourier transform, in a semclassical limit where the size of $U$ and/or $V$ tends to infinity. This conjecture was proved by Widom himself in the 80's and by A. Sobolev and his collaborators a decade ago.

Widom's initial motivation was to prove an analogue of a theorem by Basor on large Toeplitz matrices with indicator symbols, and in both cases one can translate the spectral asymptotics into probabilistic quantities for natural point process models -- for instance, Basor's result describes the number of eigenvalues of a random large unitary matrix which lie inside an interval of the unit circle.

In turns, this interpretation prompts potential generalisations of the Widom conjecture to operators built with other kinds of projectors, such as general spectral projectors for quantum hamiltonians. In this talk, I will present an overview of the Widom conjecture, the probabilistic interpretation, and my joint work with Gaultier Lambert (some of it in progress) towards the generalised Widom conjecture.