Mathematical Institute (University of Oxford)

When you ask someone what maths is, their answer will massively depend on how they’ve been exposed to maths up until that point. From a 10-year-old who will tell you it’s adding up numbers, to a Fields medalist who may say to you about the idea of abstraction of logical ideas, there is no clear consensus as to the “right” answer to this question. Our individual journeys as mathematicians give us a clear idea about what it means to us, and this affects how we then communicate our ideas to an audience of other mathematicians and the general public. However, a pitfall that we easily fall into as a result is forgetting that others can understand maths in a different way to ourselves, and by only offering our preferred perspective, we are missing out on the chance to effectively communicate our ideas.

In this talk, I will explore how our individual understanding of what mathematics is can shape our methods of communication. I will review which methods of communication mathematicians utilise, and show examples where each method does well, and not so well.  Examples of communication methods include writing equations, plotting graphs, creating diagrams and storytelling. Given this, I will cover how by using a collection of these different methods, you can increase the impact of your research by engaging with the various different mindsets your audience may have on what mathematics is.

The Onsager framework for linear irreversible thermodynamics provides a thermodynamically consistent model of mass transport in a phase consisting of multiple species, via the Stefan–Maxwell equations, but a complete description of the overall transport problem necessitates also solving the momentum equations for the flow velocity of the medium. We derive a novel nonlinear variational formulation of this coupling, called the (Navier–)Stokes–Onsager–Stefan–Maxwell system, which governs molecular diffusion and convection within a non-ideal, single-phase fluid composed of multiple species, in the regime of low Reynolds number in the steady state. We propose an appropriate Picard linearisation posed in a novel Sobolev space relating to the diffusional driving forces, and prove convergence of a structure-preserving finite element discretisation. The broad applicability of our theory is illustrated with simulations of the centrifugal separation of noble gases and the microfluidic mixing of hydrocarbons.

Statics and dynamics of droplets on lubricated surfaces

Zhaohe Dai

The abstract is "Slippery liquid infused porous surfaces are formed by coating surface with a thin layer of oil lubricant. This thin layer prevents other droplets from reaching the solid surface and allows such deposited droplets to move with ultra-low friction, leading to a range of applications. In this talk, we will discuss the static and dynamic behaviour of droplets placed on lubricated surfaces. We will show that the layer thickness and the size of the substrate are key parameters in determining the final equilibrium. However, the evolution towards the equilibrium is extremely slow (on the order of days for typical experimental parameter values). As a result, we suggest that most previous experiments with oil films lubricating smooth substrates are likely to have been in an evolving, albeit slowly evolving, transient state.

Modeling and Design Optimization for Pleated Membrane Filters

Yixuan Sun

Membrane filtration is widely used in many applications, ranging from industrial processes to everyday living activities. With growing interest from both industrial and academic sectors in understanding the various types of filtration processes in use, and in improving filter performance, the past few decades have seen significant research activity in this area. Experimental studies can be very valuable, but are expensive and time-consuming, therefore theoretical studies offer potential as a cost-effective and predictive way to improve on current filter designs. In this work, mathematical models, derived from first principles and simplified using asymptotic analysis, are proposed for pleated membrane filters, where the macroscale flow problem of Darcy flow through a pleated porous medium is coupled to the microscale fouling problem of particle transport and deposition within individual pores of the membrane. Asymptotically-simplified models are used to describe and evaluate the membrane performance numerically and filter design optimization problems are formulated and solved for industrially-relevant scenarios. This study demonstrates the potential of such modeling to guide industrial membrane filter design for a range of applications involving purification and separation.

In 2020, cirrhosis and other liver diseases were among the top five causes of death for
individuals aged 35-65 in Scotland, England and Wales. At present, the only curative
treatment for end-stage liver disease is through transplant which is unsustainable.
Stem cell therapies could provide an alternative. By encapsulating the stem cells we
can modulate the shear stress imposed on each cell to promote integrin expression
and improve the probability of engraftment. We model an individual, hydrogel-coated
stem cell moving along a fluid-filled channel due to a Stokes flow. The stem cell is
treated as a Newtonian fluid and the coating is treated as a poroelastic material with
finite thickness. In the limit of a stiff coating, a semi-analytical approach is developed
which exploits a decoupling of the fluids and solid problems. This enables the tractions
and pore pressures within the coating to be obtained, which then feed directly into a
purely solid mechanics problem for the coating deformation. We conduct a parametric
study to elucidate how the properties of the coating can be tuned to alter the stress
experienced by the cell.

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.

We present models of a vapour bubble produced during ureteroscopy and laser lithotripsy treatment of kidney stones. This common treatment for kidney stones involves passing a flexible ureteroscope containing a laser fibre via the ureter and bladder into the kidney, where the fibre is placed in contact with the stone. Laser pulses are fired to fragment the stone into pieces small enough to pass through an outflow channel. Laser energy is also transferred to the surrounding fluid, resulting in vapourisation and the production of a cavitation bubble.

While in some cases, bubbles have undesirable effects – for example, causing retropulsion of the kidney stone – it is possible to exploit bubbles to make stone fragmentation more efficient. One laser manufacturer employs a method of firing laser pulses in quick succession; the latter pulses pass through the bubble created by the first pulse, which, due to the low absorption rate of vapour in comparison to liquid, increases the laser energy reaching the stone.

As is common in bubble dynamics, we couple the Rayleigh-Plesset equation to an energy conservation equation at the vapour-liquid boundary, and an advection-diffusion equation for the surrounding liquid temperature.1 However, this present work is novel in considering the laser, not only as the cause of nucleation, but as a spatiotemporal source of heat energy during the expansion and collapse of a vapour bubble.
 

Numerical and analytical methods are employed alongside experimental work to understand the effect of laser power, pulse duration and pulse pattern. Mathematically predicting the size, shape and duration of a bubble reduces the necessary experimental work and widens the possible parameter space to inform the design and usage of lasers clinically.

Optimal execution of large positions over a given trading period is a fundamental decision-making problem for financial services. In this talk we explore reinforcement learning methods, in particular policy gradient methods, for finding the optimal policy in the optimal liquidation problem. We show results for the case where we assume a linear quadratic regulator (LQR) model for the underlying dynamics and where we apply the method to the data directly. The empirical evidence suggests that the policy gradient method can learn the global optimal solution for a larger class of stochastic systems containing the LQR framework, and that it is more robust with respect to model misspecification when compared to a model-based approach.

TBA.

Many applications involve non-Euclidean data, where exploiting Riemannian geometry can deliver algorithms that are computationally superior to standard nonlinear programming approaches. This observation has resulted in an increasing interest in Riemannian methods in the optimization and machine learning community. In this talk, we consider the problem of optimizing a function on a Riemannian manifold subject to convex constraints. We introduce Riemannian Frank-Wolfe (RFW) methods, a class of projection-free algorithms for constrained geodesically convex optimization. To understand the algorithm’s efficiency, we discuss (1) its iteration complexity, (2) the complexity of computing the Riemannian gradient and (3) the complexity of the Riemannian “linear” oracle (RLO), a crucial subroutine at the heart of the algorithm. We complement our theoretical results with an empirical comparison of RFW against state-of-the-art Riemannian optimization methods. Joint work with Suvrit Sra (MIT).

There's an idea going back at least to Kirchberger in 1902 that since the only operations we can ultimately compute are +, -, *, and /, all of numerical computation must reduce to rational functions.  I've been looking into this idea and it has led in some interesting directions.