University of Bath

Recently, Terence Tao used a new quantitative approach to infer that certain ‘slightly supercritical’ quantities for the Navier–Stokes equations must become unbounded near a potential blow-up time. In this talk I’ll discuss a new strategy for proving quantitative bounds for the Navier–Stokes equations, as well as applications to behaviours of potentially singular solutions. This talk is based upon joint work with Christophe Prange (CNRS, Cergy Paris Université).

Climate- and weather prediction centres such as the Met Office rely on efficient numerical methods for simulating large scale atmospheric flow. One computational bottleneck in many models is the repeated solution of a large sparse system of linear equations. Preconditioning this system is particularly challenging for state-of-the-art discretisations, such as (mimetic) finite elements or Discontinuous Galerkin (DG) methods. In this talk I will present recent work on developing efficient multigrid preconditioners for practically relevant modelling codes. As reported in a REF2021 Industrial Impact Case Study, multigrid has already led to runtime savings of around 10%-15% for operational global forecasts with the Unified Model. Multigrid also shows superior performance in the Met Office next-generation LFRic model, which is based on a non-trivial finite element discretisation.

The recent, rapid advances in modern biology and data science have opened up a whole range of challenging mathematical problems. In this talk I will discuss a class of interacting particle models with anisotropic repulsive-attractive interaction forces. These models are motivated by the simulation of fingerprint databases, which are required in forensic science and biometric applications. In existing models, the forces are isotropic and particle models lead to non-local aggregation PDEs with radially symmetric potentials. The central novelty in the models I consider is an anisotropy induced by an underlying tensor field. This innovation does not only lead to the ability to describe real-world phenomena more accurately, but also renders their analysis significantly harder compared to their isotropic counterparts. I will discuss the role of anisotropic interaction in these models, present a stability analysis of line patterns, and show numerical results for the simulation of fingerprints. I will also outline how very similar models can be used in data classification, where it is desirable to assign labels to points in a point cloud, given that a certain number of points is already correctly labeled.

Gibbs measures on spaces of functions or distributions play an important role in various contexts in mathematical physics.  They can, for example, be viewed as continuous counterparts of classical spin models such as the Ising model, they are an important stepping stone in the rigorous construction of Quantum Field Theories, and they are invariant under the 
flow of certain dispersive PDEs, permitting to develop a solution theory with random initial data, well below the deterministic regularity threshold. 

These measures have been constructed and studied, at least since the 60s, but over the last few years there has been renewed interest, partially due to new methods in stochastic analysis, including Hairer’s theory of regularity structures and Gubinelli-Imkeller-Perkowski’s theory of paracontrolled distributions. 

In this talk I will present two independent but complementary results that can be obtained with these new techniques. I will first show how to obtain estimates on samples from of the Euclidean $\phi^4_3$ measure, based on SPDE methods. In the second part, I will discuss a method to show the emergence of phase transitions in the $\phi^4_3$ theory.

Quantum limits are objects describing the limit of quadratic quantities (Af_n,f_n) where (f_n) is a sequence of unit vectors in a Hilbert space and A ranges over an algebra of bounded operators. We will discuss the motivation underlying this notion with some important examples from quantum mechanics and from analysis.

Nonlocal minimal surfaces are hypersurfaces of Euclidean space that minimize the fractional perimeter, a geometric functional introduced in 2010 by Caffarelli, Roquejoffre, and Savin in connection with phase transition problems displaying long-range interactions.

In this talk, I will introduce these objects, describe the most important progresses made so far in their analysis, and discuss the most challenging open questions.

I will then focus on the particular case of nonlocal minimal graphs and present some recent results obtained on their regularity and classification in collaboration with X. Cabre, A. Farina, and L. Lombardini.

I will describe new solutions to the stationary Ginzburg-Landau equation in 3 dimensions with vortex lines given by interacting helices, with degree one around each filament and total degree an arbitrary positive integer. I will also present results on the asymptotic behavior of vortices in the entire plane for a dissipative Ginzburg-Landau equation. This is work in collaboration with Manuel del Pino, Remy Rodiac, Maria Medina, Monica Musso and Juncheng Wei.

Description:In his seminal work from the 1950s, William Feller classified all one-dimensional diffusions in terms of their ability to access the boundary (Feller's test for explosions) and to enter the interior from the boundary. Feller's technique is restricted to diffusion processes as the corresponding differential generators allow explicit computations and the use of Hille-Yosida theory. In the present article we study exit and entrance from infinity for jump diffusions driven by a stable process.Many results have been proved for jump diffusions, employing a variety of techniques developed after Feller's work but exit and entrance from infinite boundaries has long remained open. We show that the these processes have features not observes in the diffusion setting. We derive necessary and sufficient conditions on σ so that (i) non-exploding solutions exist and (ii) the corresponding transition semigroup extends to an entrance point at `infinity'. Our proofs are based on very recent developments for path transformations of stable processes via the Lamperti-Kiu representation and new Wiener-Hopf factorisations for Lévy processes that lie therein. The arguments draw together original and intricate applications of results using the Riesz-Bogdan--Żak transformation, entrance laws for self-similar Markov processes, perpetual integrals of Lévy processes and fluctuation theory, which have not been used before in the SDE setting, thereby allowing us to employ classical theory such as Hunt-Nagasawa duality and Getoor's characterisation of transience and recurrence.

For a finite subgroup $G$ of $SL(2,C)$ and for $n \geq 1$,  the Hilbert scheme $X=Hilb^{[n]}(S)$ of $n$ points on the minimal resolution $S$ of the Kleinian singularity $C^2/G$ provides a crepant resolution of the symplectic quotient $C^{2n}/G_n$, where $G_n$ is the wreath product of $G$ with $S_n$. I'll explain why every projective, crepant resolution of $C^{2n}/G_n$ is a quiver variety, and why the movable cone of $X$ can be described in terms of an extended Catalan hyperplane arrangement of the root system associated to $G$ by John McKay. These results extend the algebro-geometric aspects of Kronheimer's hyperkahler description of $S$ to higher dimensions. This is joint work with Gwyn Bellamy.