In this talk we prove an analogue of the Brakke's $\epsilon$-regularity theorem for the parabolic Allen--Cahn equation. In particular, we show uniform $C^{2,\alpha}$ regularity for the transition layers converging to smooth mean curvature flows as $\epsilon\rightarrow 0$. A corresponding gap theorem for entire eternal solutions of the parabolic Allen--Cahn is also obtained. As an application of the regularity theorem, we give an affirmative answer to a question of Ilmanen that there is no cancellation in BV convergence in the mean convex setting.

An $r$-uniform tight cycle of length $k>r$ is a hypergraph with vertices $v_1,\ldots,v_k$ and edges $\{v_i,v_{i+1},…,v_{i+r-1}\}$ (for all $i$), with the indices taken modulo $k$. Sós, and independently Verstraëte, asked the following question: how many edges can there be in an $n$-vertex $r$-uniform hypergraph if it contains no tight cycles of any length? In this talk I will review some known results, and present recent progress on this problem.

Random walks are a common model for the exploration and discovery of complex networks. While numerous algorithms have been proposed to map out an unknown network, a complementary question arises: in a known network, which nodes and edges are most likely to be discovered by a random walker in finite time? In this talk we introduce exposure theory, a statistical mechanics framework that predicts the learning of nodes and edges across several types of networks, including weighted and temporal, and show that edge learning follows a universal trajectory. While the learning of individual nodes and edges is noisy, exposure theory produces a highly accurate prediction of aggregate exploration statistics. As a specific application, we extend exposure theory to better understand human learning with its typical mental errors, and thus account for distortions of learned networks.

This talk is based on https://arxiv.org/abs/2202.11262

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.

Mass transfer in multicomponent systems occurs through convection and diffusion. For a viscous Newtonian flow, convection may be modelled using the Navier–Stokes equations, whereas the diffusion of multiple species within a common phase may be described by the generalised Onsager–Stefan–Maxwell equations. In this talk we present a novel finite element formulation which fully couples convection and diffusion with these equations. In the regime of vanishing Reynolds number, we use the principles of linear irreversible dynamics to formulate a saddle point system which leads to a stable formulation and a convergent discretisation. The wide scope of applications for this novel numerical method is illustrated by considering transport of oxygen through the lungs, gas separation processes, mixing of water and methanol and salt transport in electrolytes.

Stably density stratified shear flows arise widely in geophysical settings. Instabilities of these flows occur on scales that are too small to be directly resolved in numerical simulations, e.g., of the oceans and atmosphere, yet drive diabatic mixing events that often exert a controlling influence on much larger-scale processes. In the limit of strong stratification, the flows are characterized by the emergence of highly anisotropic layer-like structures with much larger horizontal than vertical scales. Owing to their relative horizontal motion, these structures are susceptible to stratified shear instabilities that drive spectrally non-local energy transfers. To efficiently describe the dynamics of this ``layered anisotropic stratified turbulence'' regime, a multiple-scales asymptotic analysis of the non-rotating Boussinesq equations is performed. The resulting asymptotically-reduced equations are shown to have a generalized quasi-linear (GQL) form that captures the essential physics of strongly stratified shear turbulence. The model is used to investigate the mixing efficiency of certain exact coherent states (ECS) arising in strongly stratified Kolmogorov flow. The ECS are computed using a new methodology for numerically integrating slow--fast GQL systems that obviates the need to explicitly resolve the fast dynamics associated with the stratified shear instabilities by exploiting an emergent marginal stability constraint.

Throughout a product development project, many decisions must be made. These include whether to start, stop, continue, or re-direct a project based on the learnings of the project team. Some of these decisions are related to the risk of achieving certain product performance attributes and they are often based on experimental observations in the laboratory or in field applications of early prototypes. Sometimes, these observations provide sufficient insight but often a significant uncertainty remains. Mathematical simulation can provide deeper insight into the mechanisms, may indicate limiting parameters and transport steps, and allows exploration of novel prototypes without actually making them. This talk will illustrate how Mathematics have been used to inform project development projects and their guiding decisions at WL Gore by describing examples from three very different applications.