L3

In this talk I will present a few topics of recent interest that centre around the theme of “driven interfacial hydrodynamics”: fluid mechanical systems in which droplets and particles are self-propelled through interaction with the environment. I will also present some very recent work on using differentiable physics (a branch of physics-informed machine learning) to determine constitutive relations for highly plasticised metals.

This talk will contain elements of fluid dynamics, experimental mechanics, dynamical systems, statistical physics, and machine learning.

You will likely be familiar with the notion of a hydrogen atom, having seen something about its discrete energy levels and orbitals at some point or another. This is an example of a quantum system. In this talk, we explore what transpires when taking a quantum system and placing it into a three-dimensional container having some prescribed geometry. In the limit where the container is large (relative to the natural lengthscale of the quantum system), its influence over the quantum system is negligible; yet, as the container is made small (comparable to the aforementioned lengthscale), geometric information intrinsic to the container plays an important role in determining the energy and orbital structure of the system. We describe how to do (numerically-assisted) perturbation theory in this small-container limit and then match it to the large-box regime, using a combination of these asymptotics and direct simulations to tell the story of geometrically confined quantum systems. Much of our focus will be on linear Schrödinger equations governing single-particle quantum systems; however, time permitting, we will briefly discuss how to do similar things to study geometrically confined nonlinear Schrödinger equations, with geometric confinement of Bose-Einstein condensates being a primary motivation. Geometric confinement of an attractive Bose-Einstein condensate can, for instance, modify the collapse threshold and enhance stability, with the particular choice of confining geometry shifting the boundary of instability, staving off the collapse which is prevalent in three-dimensional attractive condensates.

We all sleep. But what determines when and for how long? In this talk I’ll describe some of the fundamental mechanisms that regulate sleep. I’ll introduce the nonsmooth coupled oscillator systems that form the basis of current mathematical models of sleep-wake regulation and discuss their dynamical behaviour. I will describe how we are using models to unravel environmental, societal and physiological factors that determine sleep timing and outline how constructing digital-twins could enable us to create personalised light interventions for sleep timing disorders.

I will present an overview of a range of interdisciplinary modelling challenges that I have been working on in collaboration with experimentalists and external partners. I will begin with mathematical modelling of calcium signalling in In-Vitro fertilization (IVF) and embryogenesis, illustrating how multiscale approaches can link molecular dynamics to cellular and developmental outcomes. I will then discuss our ongoing work on modelling viral transmission in indoor environments, carried out in collaboration with architects and policymakers, with the aim of informing evidence-based policy decisions for future epidemics.

Gaussian fields arise in a variety of contexts in both pure and applied mathematics. While their geometric properties are well understood, their topological features pose deeper mathematical challenges. In this talk, I will begin by highlighting some motivating examples from different domains. I will then outline the classical theory that describes the geometric behaviour of Gaussian fields, before turning to more recent developments aimed at understanding their topology using the Wiener chaos expansion.

We present an adjoint method for optimization of the spatially inhomogeneous Boltzmann equation for rarefied gas dynamics. The adjoint method is derived using a "discretize then optimize" approach. Discretization (in time and velocity) is via the Direct Simulation Monte Carlo (DSMC) method, and adjoint equations are derived from an augmented Lagrangian.  The boundary conditions that are included in this analysis include spectral reflection, thermal reflection, and inflow boundary conditions. For thermal reflection, a "score function" is included as a statistical regularization. This is joint work with Yunan Yang (Cornell). This special seminar is jointly held with the Keble Complexity Research Cluster.

Fundamentally motivated by the two opposing phenomena of fragmentation and coalescence, we introduce a new stochastic object which is both a process and a geometry. The Brownian marble is built from coalescing Brownian motions on the real line, with further coalescing Brownian motions introduced through time in the gaps between yet to coalesce Brownian paths. The instantaneous rate at which we introduce more Brownian paths is given by λ/g^2  where g is the gap between two adjacent existing Brownian paths. We show that the process "comes down from infinity" when 0<λ<6  and the resulting space-time graph of the process is a strict subset of the Brownian Web on R×[0,∞) . When λ≥6 , the resulting process "does not come down from infinity" and the resulting range of the process agrees with the Brownian Web.