We model a tokamak fusion reaction, combining Maxwell's equations with the Navier-Stokes equations, the heat equation and the Seebeck effect giving a model of thermoelectric magnetohydrodynamics (TEMHD). At leading order, we showed that the free surface must be flat, that the pressure is constant, and that the temperature decouples from the governing equations relating the fluid velocity and magnetic field. We also find that the fluid flow is driven entirely by the temperature gradient normal to the free surface. Using singular perturbation methods we obtained velocity profiles which exhibit so-called Hartmann layers and thicker side layers. The role of the aspect ratio has been seldom considered in classical MHD duct flow literature as a varying parameter. Here, we show it's importance and derive a relationship between the aspect ratio and Hartmann number that maximises flow rate of fluid down the duct.

Shortwave radiation penetrating beneath an ice-sheet surface can cause internal melting and the formation of a near-surface porous layer known as the weathering crust, a dynamic hydrological system that provides home to impurities and microbial life. We develop a mathematical model, incorporating thermodynamics and population dynamics, for the evolution of such layers. The model accounts for conservation of mass and energy, for internal and surface-absorbed radiation, and for logistic growth of a microbial species mediated by nutrients that are sourced from the melting ice. I will discuss one-dimensional steadily melting solutions of the model, which suggest a range of changes in behaviour of the weathering crust and its microbial community in response to climate change. In addition, time-dependent solutions of the model give insight into the formation and removal of the weathering crust in frequently changing weather conditions.

PDEs frequently exhibit certain physical structures that guide their behaviour, e.g. energy/helicity dissipation, Hamiltonians, and material conservation. Preserving these structures during numerical discretisation is essential.

Although the finite-element method has proven powerful in constructing such models, incorporating inhomogeneous(/non-zero) boundary conditions has been a significant challenge. We propose a technique that addresses this issue, deriving structure-preserving models for diverse inhomogeneous problems.

Moreover, this technique enables the derivation of novel structure-preserving timesteppers for time-dependent problems. Analogies can be drawn with the other workhorse of modern structure-preserving methods: symplectic integrators.

In this talk I will discuss Onsager's conjecture for energy conservation. Moreover, in 1949 Onsager conjectured that weak solutions to the incompressible Euler equations, that were Hölder continuous with Hölder exponent greater than 1/3, conserved kinetic energy. Onsager also conjectured that there were weak solutions that were Hölder continuous with Hölder exponent less than 1/3 that didn't conserve kinetic energy. I will discuss the results regarding the former, focusing mainly on the case where the spacial domain is bounded with C^2 boundary, as proved by Bardos and Titi.

I will describe algorithms for regularizing and training deep neural networks. Soft constraints, which add a penalty term to the loss, are typically used as a form ofexplicit regularization for neural network training. In this talk I describe a method for efficiently incorporating constraints into a stochastic gradient Langevin framework for the training of deep neural networks. In contrast to soft constraints, our constraints offer direct control of the parameter space, which allows us to study their effect on generalization. In the second part of the talk, I illustrate the presence of latent multiple time scales in deep learning applications.

Different features present in the data can be learned by training a neural network on different time scales simultaneously. By choosing appropriate partitionings of the network parameters into fast and slow parts I show that our multirate techniques can be used to train deep neural networks for transfer learning applications in vision and natural language processing in half the time, without reducing the generalization performance of the model.

I will discuss the difficult problem of proving reasonable bounds in the multidimensional generalization of Szemerédi's theorem and describe a proof of such bounds for sets lacking nontrivial configurations of the form (x,y), (x,y+z), (x,y+2z), (x+z,y) in the finite field model setting.

In this talk, I will review some recent progress on random graph matching problems, that is, to recover the vertex correspondence between a pair of correlated random graphs from the observation of two unlabelled graphs. In this talk, I will touch issues of information threshold, efficient algorithms as well as complexity theory. This is based on joint works with Hang Du, Shuyang Gong and Zhangsong Li.