These seminars (unless otherwise stated) will take place on Tuesdays at 1pm on an almost weekly basis.

## 5 October 2023 in 25 Gordon Street - Room 416 - 2pm-3pm

**Speaker: David Angdinata **

Supervisor: Prof V Dokchitser

##### TITLE: AN ELEMENTARY PROOF OF THE GROUP LAW OF ELLIPTIC CURVES

**Abstract: **

Elliptic curves are fundamental objects in number theory whose points form a group, but proving associativity typically involves advanced algebraic geometry or very tedious computation. I will give an elementary yet conceptual proof of the group law, which only uses linear algebra and ring theory. This has been formalised in Lean, and is joint work with Junyan Xu.

## 12 October 2023 in Bentham House - Room G20 - 2pm-3pm

**Speaker 1: Anushka Herale**

Supervisor: Dr P Pearce

##### TITLE: A MINIMAL CONTINUUM MODEL OF CLOGGING IN SPATIO-TEMPORALLY VARYING CHANNELS

**Abstract: **

Particle suspensions in confined geometries exhibit rich dynamics, including flowing, jamming, and clogging. It has been observed that jamming and clogging in particular are promoted by variations in channel geometry or fluid material properties - such variations are often present in industrial systems (e.g. local confinements) and biological systems (e.g. stiffening of red blood cells in deoxygenated conditions in sickle cell disease). The aim of this talk is to shed light on the macroscopic dynamics of particulate suspensions in these conditions. To this end, we present a continuum two-phase model of particle suspensions that accounts for spatio-temporally varying material properties or channel geometries. The model comprises a continuous particle phase which advects with flow and has material properties dependent on the particle volume fraction, and a suspending fluid which flows through the particle phase obeying Darcy’s law. We are able to show the emergence of high and low particle density regions and complete clogging of the particle phase in both pressure-driven flows. These results clarify how spatial variation in material and channel properties can contribute to clogging of particle suspensions.

**Speaker 2: Emily Cook**

Supervisor: Dr D Hewitt

##### TITLE: APPLICATIONS AND LIMITATIONS IN MODELLING PURELY THIXOTROPIC FLUID DYNAMICS

**Abstract: **

‘Thixotropy’ is a form of non-Newtonian fluid behaviour that is characterised by variation in viscosity over time due to the break down or build-up of internal structure over time and in response to stress. This evolution in structure can result in surprising and extreme changes in behaviour, such as avalanching, ageing, shear banding, and time-dependent yield behaviour. Such phenomena exist in a wide variety of complex fluids with both biological and industrial applications, including toothpaste, ketchup, clay-rich muds and drilling fluids.

It is widely acknowledged that many thixotropic fluids also exhibit other complex behaviours (take elasticity, for example) which has motivated the development of various comprehensive and somewhat complicated rheological models. Such models can be difficult to analyse and fully interpret.

Here we will instead focus on analysing, characterising and interpreting the possible behaviour that can be predicted by basic but general mathematical models of purely thixotropic fluids. This is done by considering only the requirements of a ‘generalised Newtonian’ constitutive relation, to capture a wide variety of behaviours associated with thixotropy (yielding, avalanching, time dependent yield stresses etc.). The aim here is to understand the range of rheological behaviour that can be predicted by such thixotropic models, the conditions under which different qualitative behaviour applies, and, importantly, the limitations of this modelling framework for capturing some observed rheological behaviour. The results of this analysis and the implications for more general rheological modelling will be discussed.

## 19 October 2023 in Medical Sciences G46 H O Schild Phamacology LT - 2pm-3pm

**Speaker: Jiajie Tao **

Supervisor: Prof H Ni

##### TITLE: A PDE APPROACH FOR SOLVING THE CHARACTERISTIC FUNCTION OF THE GENERALISED-SIGNATURE PROCESS

**Abstract: **

The signature of a path, as a fundamental object in Rough path theory, serves as non-communicative monomials on the path space. It transforms the path into a grouplike element in the tensor algebra space, summarizing the path faithfully up to a negligible equivalence class. Our work concerns the characteristic function of the signature of stochastic processes. In contrast to the expected signature, it determines the law on the random signatures without any regularity condition. The computation of the characteristic function of the random signature offers potential applications in stochastic analysis and machine learning, where the expected signature plays an important role. We focus on a time-homogeneous Ito diffusion process, adopting a PDE approach to derive the characteristic function of its signature defined at any fixed time horizon.

## 26 October 2023 in 25 Gordon Street - Room 416 - 2pm-3pm

**Speaker: Techheang Meng **

Supervisor: Prof R Halburd

##### TITLE: NATURAL BOUNDARY OF ANALYTIC CONTINUATION OF THE LAURENT SERIES SOLUTIONS OF CERTAIN FUNCTIONAL DIFFERENTIAL EQUATIONS (FDE) AROUND AN ATTRACTING FIXED POINT

**Abstract: **

In this talk, I would like to present some connections between Mahler’s work towards Mahler’s equation (y(z)^2-y(z^2)+c=0) and Marshall-Van Brunt-Wake’s work towards the Pantograph equation (y’’(z)+ay’(z)+by(z)-cy(z^2)=0) where they show analytic continuation of Laurent and respectively Taylor series solutions around $0$ lead to interesting features such as branch points or natural boundary. If there is some time left, I will mention briefly on my work towards their approach.

## 2 November 2023 - No seminar

## 16 November 2023 in 25 Gordon Street - Room 416 - 2pm-3pm

**Speaker: Michael Nguyen**

Supervisor: Prof ER Johnson

##### TITLE: HOW DOES A RIVER PROPAGATE INTO AN OCEAN (USING A FLUID DYNAMICS MODEL)?

**Abstract: **

This talk will focus on how a river outflow spreads into an ocean. Normally we may think water releases from a river will spread out equally in all directions, but because the Earth is rotating, water tends to veer towards the right (in the Northern Hemisphere). But there has been outflows that also veer to the left. We would like to find out why using Geophysical fluid dynamics!

We begin by deriving a simplified equation (the “QG equation”) from the Navier-Stokes equation, which governs most of all fluids. We then analyse the boundary of the current that spreads into the ocean, which we will call a “coastal front” using asymptotic analysis, particularly going into the field of dispersive equations (equations involving 3 derivatives) where we can still use maths to usefully analyse what’s happening. Then we will compare these with numerically derived results and see how far our theoretical predictions align with the full computations.

## 23 November 2023 in 25 Gordon Street - Room 416 - 2pm-3pm

**Speaker: Dr Giulia Celora**

##### TITLE: SELF-ORGANIZED PATTERNING IN COMPLEX FLUIDS

**Abstract: **

Understanding how multicomponent systems self-organise to control their emergent dynamics across spatial and temporal scales is a fundamental problem with important applications in many areas; from the design of soft materials to the study of developmental biology. In this talk, I will discuss how we can use mathematical modelling to understand the role of microscale physical interactions in the self-organisation of complex fluids. I will illustrate this by presenting two examples. Firstly, I will discuss self-organization in stimuli-responsive polyelectrolyte gels surrounded by an ionic solution; secondly, I will discuss self-organization during collective migration of multicellular communities. Our results reveal hidden connections between these two initially disconnected applications hinting at the existence of general principles controlling self-organisation of both inanimate and living matter.

## 30 November 2023 in 25 Gordon Street - Room 416 - 2pm-3pm

**Speaker: Xintong Ji**

Supervisor: Prof H Wilson

##### TITLE: THEORETICAL ANALYSIS OF VISCOUS AND VISCOELASTIC FLOW THROUGH A CROSS-SLOT

**Abstract: **

A cross-slot (or cross-channel) is a flow geometry with four ‘arms’. The inflow comes from two opposite (horizontal as default) ‘arms’ and the outflow goes away from the other two opposite (vertical as default) ‘arms’. A slow flow through this geometry produces a pattern with reflectional symmetry and produces a good approximation to a pure planar extension near the stagnation point at the centre.

For viscoelastic flow through the cross-slot, there is substantial evidence of a symmetry-breaking event as the flow rate is increased. A body of experimental evidence follows from the original work of Arratia et al. (2006) and numerical evidence (e.g. Poole et al. (2007)) to show an initial bifurcation to a steady asymmetric flow that results from purely elastic effects (generating) and inertial effects (stabilising); however, as yet we have no analytical insight into this complex fluid behaviour.

In this work we begin by studying Stokes flow in the cross-slot geometry, in order to provide a theoretical foundation on which to base an analysis of viscoelastic flow. We use a conformal mapping to transform one quarter of the cross-slot geometry (an L-shape strip, relatively complex geometry) into an infinite strip with unit height (relatively simple geometry). The velocity and the stream function in the L-shape strip (and therefore, in the whole cross-slot) can be calculated and the curved streamlines can be drawn. We will build on this solution to address linear instability and the effects of viscoelasticity.

## 7 December 2023 in 25 Gordon Street - Room 416 - 2pm-3pm

**Speaker: Jun Cheng**

Supervisor: Dr FA Diaz de la O

##### TITLE: INTRODUCTION TO SEQUENTIAL MONTE CARLO METHODS AND THEIR APPLICATIONS

**Abstract: **

Bayesian methods are well-known for their wide applications in diverse fields in which statistics is applied, such as economics, signal processing and epidemiology. Bayesian framework allows one to update prior distributions about unknown quantities of interest in models with likelihood information obtained from their relevant observations. The resulting posterior distributions play a crucial role in guiding the subsequent stages of statistical inference.

Many problems, such as the estimation of hidden states in hidden Markov models or the real-time calibration of expensive computer models, are characterized by observations that arrive sequentially. For such problems, particularly if the models are non-linear or non-Gaussian, it is wasteful to recalculate the target from scratch when new observations arrive.

Sequential Monte Carlo (SMC) methods combine importance sampling with Monte Carlo methods and could be used to provide approximations for these sequential Bayesian inference tasks as they could calculate the new target by using information from previous posterior distributions when new observations come. Also, they circumvent the need for linearity or Gaussianity assumptions and can be effective even when the chosen importance proposal cannot handle non-linearity and non-Gaussianity, as these characteristics can be contained within the likelihood function.

## 14 December 2023 in 25 Gordon Street - Room 416 - 2pm-3pm

**Speaker: Venkata Dhruva Pamulaparthy**

Supervisor: Dr R Harris

##### TITLE: TOWARDS MACHINES FOR NON-EQUILIBRIUM FLUCTUATION ANALYSIS

**Abstract: **

This talk is a gentle invitation to the broad area of nonequilibrium fluctuation analysis. We focus here on an approach based on the theory of large deviations .

Large deviation theory was pioneered by Donsker, Varadhan and others in the 70's to study 'rare events' and tails of distributions. In the last thirty years or so it has emerged as a useful framework for nonequilibrium fluctuation analysis. Apart from delivering fundamental theoretical insights (ex. the Gallavoti-Cöhen symmetry), the large deviations approach also provides guiding principles to develop numerical schemes for measuring rare events (i.e., extreme fluctuations) when analytical calculations are not possible. At its core is the Gärtner-Ellis relation, a free-energy/entropy type connection that allows the development of variational methods for fluctuation analysis, including machine learning! We shall discuss the specific challenges in developing such variational methods and how a machine for measuring nonequilibrium fluctuations/rare-events may look like. We shall be sticking to Markovian (memoryless) examples and touch upon non-Markovian systems (and my research topic) if time permits.