C5

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.

Understanding the impact of societal and economic change on the labour market is important for many causes, such as automation or the post-carbon transition. Occupational mobility plays a role in how these changes impact the labour market because of indirect effects, brought on by the different levels of direct impact felt by individual occupations. We develop an agent-based model which uses a network representation of the labour market to understand these impacts. This network connects occupations that workers have transitioned between in the past, and captures the complex structure of relationships between occupations within the labour market. We develop these networks in both space and time using rich survey data to compare occupational mobility across the United States and through economic upturns and downturns to start understanding the factors that influence differences in occupational mobility.

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.

Michael Negus

Modelling high-speed droplet impact onto an elastic membrane

The impact of a high-speed droplet onto an elastic membrane is a highly nonlinear process and poses a formidable modelling challenge due to both the multi-scale nature of the flow and the fluid-structure interaction between the droplet and the membrane. We present two modelling approaches for droplet impact onto elastic membranes: matched asymptotics and direct numerical simulations (DNS). Inviscid Wagner theory is used in the former to derive analytical expressions which approximate the behaviour of the droplet during the early stages of impact, while the DNS builds on the open-source volume-of-fluid code Basilisk. We demonstrate the strong influence that the thickness, tension and stiffness of the membrane have on the dynamics of the droplet and the membrane. We also quantitatively show that the speed the droplet spreads across the substrate is notably decreased when the membrane is more compliant, which is consistent with experimental findings that splashing can be inhibited by impacting onto a soft substrate. We conclude by showing how these methods are complementary, as a combination of both can lead to a thorough understanding of the droplet impact across timescales.

Helen Saville

Lubrication model of a valve-controlled, gravity-driven bioreactor

Hospitals sometimes experience shortages of donor blood platelet supplies, motivating research into in vitro production of platelets. We model a novel platelet bioreactor described in Shepherd et al. [1]. The bioreactor consists of an upper channel, a lower channel, and a cell-seeded porous collagen scaffold situated between the two. Flow is driven by gravity, and controlled by valves on the four inlets and outlets. The bioreactor is long relative to its width, a feature which we exploit to derive a lubrication reduction of Navier-Stokes flow coupled to Darcy. Models for two cases are considered: small amplitude valve oscillations, and order one amplitude valve oscillations. The former model is a systematic reduction; the latter incorporates a phenomenological approximation for the cross-sectional flow profile. As the shear stress experienced by cells influences platelet production, we use our model to quantify the effect of valve dynamics on shear stress.

1: Shepherd, J.H., Howard, D., Waller, A.K., Foster, H.R., Mueller, A., Moreau, T., Evans, A.L., Arumugam, M., Chalon, G.B., Vriend, E. and Davidenko, N., Biomaterials, 182, pp.135-144. (2018)

Here we present the findings of our summer research project: an 8-week investigation of C*-Algebras. Our aim was to explore when a group is uniquely determined by its reduced group C*-algebra; i.e. when the group is C*-rigid. In particular, we applied techniques from topology, algebra, and analysis to prove C*-rigidity for a certain class of Bieberbach groups.

I will report on recent progress concerning eigenvalues of Schrödinger operators with complex potentials. We are interested in the magnitude and distribution of eigenvalues, and we seek bounds that only depend on an L^p norm of the potential.

These questions are well understood for real potentials, but completely new phenomena arise for complex potentials. I will explain how techniques from harmonic analysis, particularly those related to Fourier restriction theory, can be used to prove upper and lower bounds. We will also discuss some open problems. The talk is based on recent joint work with Sabine Bögli (Durham).

Group theoretic Dehn filling, motivated by Dehn filling in the theory of 3- manifolds, is a process of constructing quotients of a given group. This technique is usually applied to groups with certain negative curvature feature, for example word-hyperbolic groups, to construct exotic and useful examples of groups. In this talk, I will start by recalling the notion of word-hyperbolic groups, and then show that how group theoretic Dehn filling can be used to answer the Burnside Problem and questions about mapping class groups of surfaces.

A big mapping class group is the mapping class group (MCG) of a surface of infinite type. Although several aspects of big MCGs remain mysterious, their geometric definition allows some simple, interesting arguments. In this talk, we will use big MCGs as an excuse to survey some (more or less) classical results in geometric group theory: we will present a quick introduction to infinite type surfaces, highlight differences between standard and large MCGs, and use Higman’s embedding theorem to deduce that there exists a big MCG that contains every finitely presented group as a subgroup.

TBA