Analysis

I will present an application of optimal transport theory to simplified models of large-scale rotational flows (weather).

The semi-geostrophic equation is used by researchers at the Met Office to diagnose problems in simulations of more complicated weather models. It has also attracted a lot of attention in the applied analysis community, e.g., Alessio Figalli's work on the semi-geostrophic equation is listed in his Fields Medal citation. In this talk I will discuss the semi-geostrophic equation in geostrophic coordinates (SG), which is a nonlocal transport equation, where the transport velocity is defined via an optimal transport problem. Using recent results from semi-discrete optimal transport theory, we give a new proof of the existence of weak solutions of the SG equations. The proof is constructive and leads to an efficient numerical method. I will conclude talk by showing some simulations of weather fronts.

This is joint work with Charlie Egan, Théo Lavier and Beatrice Pelloni (Heriot-Watt University and the Maxwell Institute for Mathematical Sciences), Mark Wilkinson (Nottingham Trent University), Steven Roper (University of Glasgow), Colin Cotter (Imperial College London) and Mike Cullen (Met Office - retired).

The Landau-Lifshitz-Gilbert equation (LLG) is a continuum model describing the dynamics for the spin in ferromagnetic materials. In the first part of this talk we describe the properties and dynamical behaviour of the self-similar solutions of this model in one dimension.  Time permitting, and motivated by the properties of these solutions, we consider the Cauchy problem for the LLG-equation and provide a global well-posedness result provided that the BMO norm of the initial data is small. 

The model of N hard spheres has been studied intensively in applied mathematics and physics due to its ability to provide a relatively-simple description of matter on atomic length scales. In the analysis community, hard spheres have been of particular interest in recent years within the context of Hilbert’s Sixth Problem, wherein one tries to establish that solutions of the Liouville equation for hard sphere dynamics converge to solutions of the Boltzmann equation. The study of hard sphere dynamics and the associated Liouville equation is surprisingly rich, and it has beautiful connections with symplectic geometry as well as algebraic geometry. In this talk, we present new results on the existence and properties of invariant measures for hard sphere dynamics. We achieve these by establishing new representation formulae for flow maps on the tangent bundle of the hard sphere table, some of which arise from algebraic structures in Lie algebra theory. 

We discuss the typical behavior of two important quantities on compact Riemannian manifolds: the number of primitive closed geodesics of a certain length and the error in the Weyl law. For Baire generic metrics, the qualitative behavior of both of these quantities has been well understood since the 1970's and 1980's. Nevertheless, their quantitative behavior for typical manifolds has remained mysterious. In fact, only recently, Contreras proved an exponential lower bound for the number of closed geodesics on a Baire generic manifold. Until now, this was the only quantitative estimate on the number of geodesics for typical metrics, and no such estimate existed for the remainder in the Weyl law. In this talk, we give stretched exponential upper bounds on the number of primitive closed geodesics for typical metrics. Furthermore, using recent results on the remainder in the Weyl law, we will use our dynamical estimates to show that logarithmic improvements in the remainder in the Weyl law hold for typical manifolds. The notion of typicality used in this talk is a new analog of full Lebesgue measure in infinite dimensions called predominance. Based on joint work with Y. Canzani. 

A nodal domain refers to a connected region where the eigenfunction is nonzero. The simplest topological invariant of nodal domains is the nodal count. In this talk, we review the celebrated Courant and Pleijel nodal domain theorems and some of the recent developments in this direction. The main focus of the talk will be on the Pleijel type nodal domain theorem for the Robin problem without restriction on the sign of the Robin parameter. This is joint work with David Sher. 

The restriction conjecture, one of the most central problems in harmonic analysis, studies the Fourier transform of functions defined on curved surfaces; specifically, it claims that the level sets of such Fourier transforms are relatively small. The Mizohata-Takeuchi conjecture further studies the shape of these level sets, and in particular the extent to which they can avoid clustering on lines. In this talk we will present some small recent progress on the Mizohata-Takeuchi conjecture. This is partially joint work with Anthony Carbery and Hong Wang.