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

## Autumn 2022

### Talks are being given by 2nd and 3rd year Mathematics PhD students for PhD students.Seminars this term will be in room G01 in 222 Euston Road unless otherwise stated.

## 04 October 2022

**Speaker: Phil (Mingfei) Lu**

Supervisor: Prof E Burman

##### Title: Stabilized finite element method for unique continuation problem with respect to Schrödinger equation

**Abstract:**

The main purpose in this talk is to present a stabilized finite element methods solving the ill-posed unique continuation problem for Schrödinger equation, i.e. -\laplace u+Vu = f. We will build conditional stability estimate for Schrödinger equation and introduce PDE constrained optimisation, and construct a stabilized finite element method with H^1 conforming elements to reach an optimal convergence approximation. Stability of the scheme and data perturbation will also be considered along with numerical examples.

## 11 October 2022

**Speaker: Srinath Kailasa**

Supervisor: Prof T Betcke

##### Title: Rust for Computational Science

**Abstract:**

In this talk we introduce Rust as a tool for high-performance scientific software development, and some of our recent work in applying Rust to computational electromagnetics. Traditionally HPC tools have been written in C or Fortran, and more recently in C++. However, contributing to these tools required significant software expertise, and building from source for use on desktops or HPC systems with diverse hardware and software architectures, is often a headache for those who are solely interested in science. This is the main motivation for using Rust. Its build system and package manager (Cargo) make conflict-free builds significantly simpler to manage than in C or C++, and can often be compressed to a single command. Its basic syntax will be familiar to programmers of all abilities, and its modern multi paradigm style with zero cost abstractions is expressive and highly-readable, like an interpreted language such as Python. Furthermore, with tools such as Maturin it’s simple to develop interfaces to other languages such as Python or C/C++, and take advantage of existing software tools if required. Although many packages required for scientific simulation don’t yet exist in pure Rust, we believe that its advantages will make it a major future language for scientific software.

## 18 October 2022

**Speaker: Vivienne Leech**

Supervisor: Dr A Manhart

##### Title: A modelling framework to explore the alignment of keloid fibroblasts in fibrotic tissue

**Abstract:**

Fibroblasts are cells that help to maintain the structural framework of tissue. They are the main cells involved in the formation of keloid scars. Through mathematical modelling, we hope to better understand the mechanisms behind the long-range alignment of fibroblasts in keloid scar tissue, as a pathway to better understanding the causes of tissue fibrosis. Keloid fibroblasts have been observed to align over a longer length-scale and shorter timescale compared to normal fibroblasts, but the mechanism behind this long-range alignment is unclear. Models so far have explored the nature of this alignment, but not whether the alignment can arise purely from force-based interactions. We start by deriving and simulating an agent-based model that incorporates physical, force-based interactions between cells to see whether this is enough to account for the long-range alignment that is observed experimentally. Data fitting techniques are used to compare simulations to experimental data and assess the validity of the model. We find that while this leads to some alignment, it is not enough alignment to account for what we observe experimentally. The model can then be extended accordingly, incorporating further mechanisms such as cell-cell junctions or alignment feedback with the extra cellular matrix.

## 25 October 2022

**Speaker: Yohance Osborne**

Supervisor: Dr I Smears

##### Title: Analysis and Numerical Approximation of Stationary Second-order Mean Field Game Partial Differential Inclusions

**Abstract:**

The formulation of Mean Field Games (MFG) via partial differential equations typically requires continuous differentiability of the Hamiltonian in order to determine the advective term in the Kolmogorov--Fokker--Planck equation for the density of players. However, in many practical applications the underlying optimal control problem may exhibit bang-bang controls, which typically lead to nondifferentiable Hamiltonians. In this talk we will present results on the analysis and numerical approximation of stationary second-order MFG systems for the general case of convex, Lipschitz, but possibly nondifferentiable Hamiltonians. In particular, we will propose a generalization of the MFG system as a Partial Differential Inclusion (PDI) based on interpreting the derivative of the Hamiltonian in terms of subdifferentials of convex functions. We present results that guarantee the existence of unique weak solutions to the stationary MFG PDI under a monotonicity condition similar to one that has been considered previously by Lasry and Lions. Moreover, we will propose a monotone finite element discretization of the weak formulation of the MFG PDI, and present results that confirm the strong H^1-norm convergence of the approximations to the value function and strong L^q-norm convergence of the approximations to the density function. The performance of the numerical method will be illustrated in experiments featuring nonsmooth solutions. This talk is based on joint work with my supervisor Iain Smears.

## 01 November 2022 at 12pm in B03 Ricardo LT, Drayton House

**Speaker: Jamie Bell**

### Supervisor: Prof V Dokchitser

##### Title: Ranks of Elliptic Curves

**Abstract:**

An elliptic curve is one given by an equation of the form y^2=x^3+ax+b for some fixed a and b. We are particularly interested in the rational points on this curve. Unlike degree 1 and 2 equations, it is not easy to find all solutions. There are however many applications to solving them, for instance the Ancient Greek congruent number problem: which integers are the areas of right-angled triangles with rational side lengths?

The modern study of elliptic curves began in 1922 when Mordell proved that their points form a finitely generated abelian group. This means that the structure of these points can be specified by two pieces of data: its torsion (which is easy to find) and its rank (which isn’t). This is the subject of the Birch–Swinnerton-Dyer conjecture, a Millenium Prize problem. In my talk I will introduce elliptic curves and explain why we should care about them, and talk about some open problems related to my research. No number theory experience necessary!

## 08 November 2022

**NO SEMINAR**

## 15 November 2022

**NO SEMINAR**

## 22 November 2022 at 12pm in Room 944, 20 Bedford Way (IoE)

**Speaker: **Antonio D’Alfonso Del Sordo

Supervisor: TBC

##### Title: The Cosmic Latte: modelling the universe as a perfect fluid

**Abstract:**

Between 1907 and 1915, Einstein developed the general theory of relativity. In the first part of the talk, I will give an overview of the main mathematical ideas underpinning general relativity, starting from the question, ‘How do we measure distances?’

General relativity is based on a set of assumptions that we can try to dissect, break, and extend. This is where modified theories of gravity come into play. I will give a layman introduction to these theories and cosmology, which is the study of the universe as a whole using general relativity. The universe is modelled as a perfect fluid, which, for an Italian, is of course Coffee! I will finish the talk by introducing my own research and how it connects to these ideas.

## 29 November 2022 at 12pm in Room 944, 20 Bedford Way (IoE)

**Speaker: Gareth Jenkins**

Supervisor: Prof HJ Wilson

##### Title: A model for the evidence dynamics of forensic trace materials

**Abstract:**

Microscopic particles of contraband materials can be deposited onto surfaces via fingerprints, and these trace evidence samples can then be analysed to infer key details of suspected criminal activities. A forensic reconstruction such as this will be more robust if we can develop an understanding of how these materials behave.

Suppose, for example, that an investigation is carried out around a crime scene involving the handling of a particular contraband material, and a series of prints are found and analysed with varying quantities of the material identified in each. The investigation then wants to know the order in which these prints were made, but there are uncertainties in determining this. Can we reliably assume that the amount of transferred material will decrease monotonically in each print? To what extent do the physical properties of the contraband and the imprinted surfaces affect this reliability?

We are working towards a model which can replicate the transfer patterns found in experimental data of crystalline explosive particles. We use a coarse-grained model for the crystalline particles, approximating them as aggregates of elastic spheres with breakable bonds.

In this talk we cover the methodology of this model, discuss possible reductions of the parameter space, and show off simulations which demonstrate various desired behaviours of the evidence particles.

## 06 December 2022 at 12pm in Room 944, 20 Bedford Way (IoE)

**Speaker: Techheang Mang**

Supervisor: Prof R Halburd

##### Title: Singularity behaviour of meromorphic solutions of the Lotka-Volterra system in 3D space

**Abstract:**

Lotka-Volterra (L-V) system appears in many applications from chemical reaction model to indicate prey-predator behaviour. Local existence of solutions is guaranteed by theorems from ODE, but we are interested in finding global solutions. Extension of local solutions could obtain various types of singularities such as branches, logarithmic and essential singularities to which sometimes we cannot detect them in prior. To help ease the case, we only prefer to look for meromorphic solutions in 3-dimensional space Lotka-Volterra system.

Local series expansion has been a useful tool to characterise behaviour about singularities and we will go through it during the talk. The analysis of local series expansion will give necessary information such as leading order terms, positive resonance, and resonance conditions and these will provide further argument on the existence of meromorphic solutions of the L-V system.

## 13 December 2022 at 12pm in Room 944, 20 Bedford Way (IoE)

**Speaker: Marios Voskou**

Supervisor: Prof Y Petridis

##### Title: Hyperbolic Counting Problems

**Abstract:**

The Gauss' Circle Problem is concerned with estimating the number of lattice points inside a Euclidean circle. In this talk we will discuss various analogue problems in the hyperbolic space. We will explain why the geometric methods used by Gauss fail, and we will investigate the much more efficient approach of Spectral Theory. Essentially, we will see how some counting problems can be interpreted as problems in the Spectral Theory of Automorphic Forms, demonstrating one of the most essential aspects of Analytic Number Theory: the unlucky duality between the discrete and the continuous.