Forthcoming events in this series


Tue, 13 Jun 2023
12:30
C3

Hydrocephalus shunt simulations

Lizzi Hayman
Abstract

Hydrocephalus is a serious medical condition which causes an excess of cerebrospinal fluid (CSF) to build up within the brain. A common treatment for congenital hydrocephalus is to implant a permanent drainage shunt, removing excess CSF to the stomach where it can be safely cleared. However, this treatment carries the risk of vascular brain tissues such as the Choroid Plexus (CP) being dragged into the shunt during drainage, causing it to block, and also preventing the shunt from being easily replaced. In this talk I present results from our fluid-structure interaction model which simulates the deflection of the CP during the operation of the hydrocephalus shunt. We seek to improve the shunt component by optimising the geometry with respect to CP deflection.

Tue, 30 May 2023
12:30
C2

Compromised clearance and cognitive decline

Georgia Brennan
Abstract

We describe a network model for the progression of Alzheimer's disease based on the underlying relationship to toxic proteins. From human patient data we construct a network of a typical brain, and simulate the concentration and build-up of toxic proteins, as well as the clearance, using reaction--diffusion equations. Our results suggest clearance plays an important role in delaying the onset of Alzheimer's disease, and provide a theoretical framework for the growing body of clinical results.

Tue, 16 May 2023
12:30
C3

Structure-Preserving Finite-Element Methods for Inhomogeneous and Time-Dependent PDEs

Boris Andrews
Abstract

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.

Tue, 02 May 2023
12:30
C2

An Introduction to Holography

Alice Luscher
Abstract

Holography, which reveals a specific correspondence between gravitational and quantum theories, is an ongoing area of research that has known a lot of interest these past decades. The duality of holography has many applications: it provides an interpretation for black hole entropy in terms of microstates, it helps our understanding of solid state properties such as superconductivity and strongly coupled quantum systems, and it even offers insight into the mysterious realm of quantum gravity. 

In this talk, I will first introduce the concept of holography and some of its applications. I will then discuss some notions of string theory and geometry that are commonly used in holography. Finally, if time permits, I will present some of our latest results, where we match the energy of membranes in supergravity to properties of the dual quantum models.

Tue, 07 Mar 2023
12:30
C3

Mathematical modelling of liquid lithium inside a tokamak fusion reactor

Oliver Bond
Abstract

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.

Tue, 21 Feb 2023
12:30
C3

Modelling the weathering crust and microbial activity on an ice-sheet surface

Tilly Woods
Abstract

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.

Tue, 07 Feb 2023
12:30
C3

Studying occupational mobility using online resume data

Rohit Sahasrabuddhe
Abstract

Data sets of self-reported online resumes are a valuable tool to understand workers' career trajectories and how workers may adapt to the changing demands of employers. However, the sample of workers that choose to upload their resumes online may not be representative of a nation's workforce. To understand the advantages and limitations of these datasets, we analyze a data set of more than 1 Million online resumes and compare the findings with a administrative data from the Current Population Survey (CPS).
 

Tue, 24 Jan 2023
12:30
C3

Onsager's conjecture for energy conservation

Samuel Charles
Abstract

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.

Tue, 29 Nov 2022

12:30 - 13:00
C3

Spatial analysis to investigate the emergent dynamics of a cellular automaton model of tumour-immune interactions.

Roisin Stephens
Abstract

Baseline T cell infiltration and the spatial distribution of T cells within a tumour has been found to be a significant indicator of patient outcomes. This observation, coupled with the increasing availability of spatially-resolved imaging data of individual cells within the tumour tissue, motivates the development of mathematical models which capture the spatial dynamics of T cells. Agent-based models allow the simulation of complex biological systems with detailed spatial resolution, and generate rich spatio-temporal datasets. In order to fully leverage the information contained within these simulated datasets, spatial statistics provide methods of analysis and insight into the biological system modelled, by quantifying inherent spatial heterogeneity within the system. We present a cellular automaton model of interactions between tumour cells and cytotoxic T cells, and an analysis of the model dynamics, considering both the temporal and spatial evolution of the system. We use the model to investigate some of the standard assumptions made in these models, to assess the suitability of the models to accurately describe tumour-immune dynamics.

Tue, 15 Nov 2022

12:30 - 13:00
C3

A Hele-Shaw Newton's cradle and Reciprocity in Fluids

Daniel Booth and Matthew Cotton
Abstract

A Hele-Shaw Newton's cradle: Circular bubbles in a Hele-Shaw channel. (Daniel Booth)

We present a model for the motion of approximately circular bubbles in a Hele-Shaw cell. The bubble velocity is determined by a balance between the hydrodynamic pressures from the external flow and the drag due to the thin films above and below the bubble. We find that the qualitative behaviour depends on a dimensionless parameter and is found to agree well with experimental observations.  Furthermore, we show how the effects of interaction with cell boundaries and/or other bubbles also depend on the value of this dimensionless parameter For example, in a train of three identical bubbles travelling along the centre line, the middle bubble either catches up with the one in front or is caught by the one behind, forming what we term a Hele-Shaw Newton's cradle.
 

Reciprocity in Fluids (Matthew Cotton)

Reciprocity is a useful, and often underused, way to calculate integrated quantities when a to solution to a related problem is known. In the remaining time, I will overview these ideas and give some example use cases

Tue, 01 Nov 2022

12:30 - 13:00
C3

Asymptotic Analysis of Deep Residual Networks

Alain Rossier
Abstract

Residual networks (ResNets) have displayed impressive results in pattern recognition and, recently, have garnered considerable theoretical interest due to a perceived link with neural ordinary differential equations (neural ODEs). This link relies on the convergence of network weights to a smooth function as the number of layers increases. We investigate the properties of weights trained by stochastic gradient descent and their scaling with network depth through detailed numerical experiments. We observe the existence of scaling regimes markedly different from those assumed in neural ODE literature. Depending on certain features of the network architecture, such as the smoothness of the activation function, one may obtain an alternative ODE limit, a stochastic differential equation (SDE) or neither of these. Furthermore, we are able to formally prove the linear convergence of gradient descent to a global optimum for the training of deep residual networks with constant layer width and smooth activation function. We further prove that if the trained weights, as a function of the layer index, admit a scaling limit as the depth increases, then the limit has finite 2-variation.

Tue, 18 Oct 2022
12:30
C3

Recovering scattering distributions from covariance-map images of product distributions

Brady Metherall
Abstract

Molecules can be broken apart with a high-powered laser or an electron beam. The position of charged fragments can then be detected on a screen. From the mass to charge ratio, the identity of the fragments can be determined. The covariance of two fragments then gives us the projection of a distribution related to the initial scattering distribution. We formulate the mathematical transformation from the scattering distribution to the covariance distribution obtained from experiments. We expand the scattering distribution in terms of basis functions to obtain a linear system for the coefficients, which we use to solve the inverse problem. Finally, we show the result of our method on three examples of test data, and also with experimental data.

Tue, 31 May 2022

12:30 - 13:30
C5

Communicating Mathematics: From equations to storytelling

Michael Negus
(Mathematical Institute (University of Oxford))
Abstract

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.

 

Tue, 17 May 2022

12:30 - 13:30
C5

Finite element methods for the Stokes–Onsager–Stefan–Maxwell equations of multicomponent flow

Francis Aznaran
(Mathematical Institute (University of Oxford))
Abstract

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.

Tue, 03 May 2022

12:30 - 13:30
C5

A model of internal stresses within hydrogel-coated stem cells in transit to the liver

Simon Finney
(Mathematical Institute (University of Oxford))
Abstract

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.

Tue, 08 Mar 2022

12:30 - 13:30
C5

Modelling the labour market: Occupational mobility during the pandemic in the U.S.

Anna Berryman
(University of Oxford)
Abstract

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.

Tue, 22 Feb 2022

12:30 - 13:15
C5

Modelling laser-induced vapour bubbles in the treatment of kidney stones

Sophie Abrahams
(Mathematical Institute (University of Oxford))
Abstract

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.

Tue, 08 Feb 2022

12:30 - 13:30
C5

Reinforcement Learning for Optimal Execution

Huining Yang
(Mathematical Institute (University of Oxford))
Abstract

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.

Tue, 30 Nov 2021
12:30
C5

Modelling high-speed droplet impact onto an elastic membrane (Negus). Lubrication model of a valve-controlled, gravity-driven bioreactor (Saville)

Michael Negus & Helen Saville
(Mathematical Institute (University of Oxford))
Abstract

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)

Tue, 16 Nov 2021

12:30 - 13:30
C5

Contact problems in glaciology

Gonzalo Gonzalez De Diego
(Mathematical Institute (University of Oxford))
Abstract

Several problems of great importance in the study of glaciers and ice sheets involve processes of attachment and reattachment of the ice from the bedrock. Consider, for example, an ice sheet sliding from the continent into the ocean, where it goes afloat. Another example is that of subglacial cavitation, a fundamental mechanism in glacial sliding where the ice detaches from the bedrock along the downstream area of an obstacle. Such problems are generally modelled as a viscous Stokes flow with a free boundary and contact boundary conditions. In this talk, I will present a framework for solving such problems numerically. I will start by introducing the mathematical formulation of these viscous contact problems and the challenges that arise when trying to approximate them numerically. I will then show how, given a stable scheme for the free boundary equation, one can build a penalty formulation for the viscous contact problem in such a way that the resulting algorithm remains stable and robust.

Tue, 02 Nov 2021

12:30 - 13:00
C5

A homogenisation approach to mass transport models for organoid culture

Meredith Ellis
(Mathematical Institute (University of Oxford))
Abstract

Organoids are three–dimensional multicellular tissue constructs. When cultured in vitro, they recapitulate the structure, heterogeneity, and function of their in vivo counterparts. As awareness of the multiple uses of organoids has grown, e.g. in drug discovery and personalised medicine, demand has increased for low–cost and efficient methods of producing them in a reproducible manner and at scale. We are working in collaboration with the biotechnology company Cellesce, who develop bioprocessing systems for the expansion of organoids at scale. Part of their technology includes a bioreactor, which utilises flow of culture media to enhance nutrient delivery to the organoids and facilitate the removal of waste metabolites. A key priority is ensuring uniformity in organoid size and reproducibility; qualities that depends on the bioreactor design and operating conditions. A complete understanding of the system requires knowledge of the spatial and temporal information regarding flow and the resulting oxygen and metabolite concentrations throughout the bioreactor. However, it is impractical to obtain this data empirically, due to the highly–controlled environment of the bioreactor posing difficulties for online real–time monitoring of the system. Thus, we exploit a mathematical modelling approach, to provide spatial as well as temporal information.

In the bioreactor, organoids are seeded as single cells in a layer of hydrogel. We present a general model for the nutrient and waste metabolite concentrations in the hydrogel and organoid regions of the bioreactor. Resolving for the millions of organoids within the hydrogel is computationally expensive and infeasible. Hence, we take a mathematical homogenisation approach to understand how the behaviour of the organoids on the microscale influences the macroscale behaviour in the hydrogel layer. We consider the case of growing organoids, with a temporally and spatially dependent radii, and exploit the separation of scales to systematically derive an effective macroscale model for metabolite transport. We explore some canonical problems to understand our homogenised system.