University of Cambridge

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. 

The statistical analysis of data which lies in a non-Euclidean space has become increasingly common over the last decade, starting from the point of view of shape analysis, but also being driven by a number of novel application areas. However, while there are a number of interesting avenues this analysis has taken, particularly around positive definite matrix data and data which lies in function spaces, it has increasingly raised more questions than answers. In this talk, I'll introduce some non-Euclidean data from applications in brain imaging and in linguistics, but spend considerable time asking questions, where I hope the interaction of statistics and topological data analysis (understood broadly) could potentially start to bring understanding into the applications themselves.

We consider, in a framework of iterated random conformal maps, a two-parameteraggregation model of Hastings-Levitov type, in which the size and intensity of new particles are each chosen to vary as a power of the density of harmonic measure. Then we consider a limit
in which the overall intensity of particles become large, while the particles themselves become
small. For a certain `sub-critical' range of parameter values, we can show a law of large numbers and fluctuation central limit theorem. The admissible range of parameters includes an off-lattice version of the Eden model, for which we can show that disk-shaped clusters are stable.
Many open problem remain, not least because the limit PDE does not yet have a satisfactory mathematical theory.

This is joint work with Vittoria Silvestri and Amanda Turner.

The dynamics of quantum many-body systems is intricately related to random matrix theory (RMT), to such a degree that quantum chaos is even defined through random matrix level statistics. However, exact results on this connection are typically precluded by the exponentially large Hilbert space. After a short introduction to the role of RMT in many-body dynamics, I will show how dual-unitary circuits present a minimal model of quantum chaos where this connection can be made rigorous. This will be illustrated using a new kind of emergent random matrix behaviour following a quantum quench: starting from a time-evolved state, an ensemble of pure states supported on a small subsystem can be generated by performing projective measurements on the remainder of the system, leading to a projected ensemble. In chaotic quantum systems it was conjectured that such projected ensembles become indistinguishable from the uniform Haar-random ensemble and lead to a quantum state design, which can be shown to hold exactly in dual-unitary circuit dynamics.

We derive quantitatively the weak and strong Harnack inequality for kinetic Fokker--Planck type equations with a non-local diffusion operator for the full range of the non-locality exponents in (0,1).  This implies Hölder continuity.  We give novel proofs on the boundedness of the bilinear form associated to the non-local operator and on the construction of a geometric covering accounting for the non-locality to obtain the Harnack inequalities.  Our results apply to the inhomogeneous Boltzmann equation in the non-cutoff case.

Previous joint work with Caroline Terry had identified model-theoretic stability as a sufficient condition for the existence of strong arithmetic regularity decompositions in finite abelian groups, pioneered by Ben Green around 2003. 
Higher-order arithmetic regularity decompositions, based on Tim Gowers’s groundbreaking work on Szemerédi’s theorem in the late 90s, are an essential part of today's arithmetic combinatorics toolkit.
In this talk, I will describe recent joint work with Caroline Terry in which we define a natural higher-order generalisation of stability and prove that it implies the existence of particularly efficient higher-order arithmetic regularity decompositions in the setting of finite elementary abelian groups. If time permits, I will briefly outline some analogous results we obtain in the context of hypergraph regularity decompositions.