These seminars (unless otherwise stated) will take place on Tuesdays at 12pm-1pm on an almost weekly basis.

## Spring 2023

### Talks are being given by 2nd and 3rd year Mathematics PhD students for PhD students.

## 17 January 2023 in Bedford Way (26) Room G03

**Speaker: Ignacia Fierro Piccardo**

Supervisor: Prof T Betcke

##### TITLE: AN OSRC PRECONDITIONER FOR THE EFIE

**Abstract: **

The Electric Field Integral Equation (EFIE) is a boundary integral equation used to solve high-frequency electromagnetic scattering problems. However, the matrices derived from BEM (Boundary Elements Method) are densely populated, so the system of equations cannot be solved using direct methods (complexity order of O(N^3)). Indirect methods like the GMRES (Generalised Minimal Residual method) can be applied instead, but the EFIE matrix is also ill-conditioned, so a regulariser (preconditioner) must be applied to preserve the accuracy of the solution and reduce the number of iterations of the GMRES. The most common preconditioner for this equation is the so-called Calderón preconditioner, but as we will see, its application comes with a cost that we would like to avoid.

Here we present an alternative operator as a preconditioner: a local surface approximation of the so-called Magnetic-to-Electric operator for time-harmonic Maxwell’s equations can be efficiently evaluated through the solution of sparse linear systems and it has the properties of a potential preconditioner. In this presentation we discuss such properties, we show its construction and numerical experiments performed on both open and closed geometries that pose different challenges due to the nature of the functional spaces on which this formulation relies.

## 24 January 2023 in Bedford Way (26) Room G03

**Speaker: Helena Coggan**

Supervisor: Prof KM Page

##### TITLE: Agent-based modelling of lung cancer organoids sheds light on how mutant cells interact

**Abstract: **

Mutations in the epidermal growth factor receptor (EGFR) are common in non-small cell lung cancer (NSCLC), particularly in never-smoker patients. However, these mutations are not always carcinogenic and have been found in normal tissue from healthy patients. Why these mutations lead to cancer in some people and not others is unknown. Experiments reveal that one of these mutations induces a characteristic `bubbling' structure in cells grown in a 3D hydrogel. On-lattice simulation suggests that this can be explained by an increase in reproductive fitness combined with inhibitory neighbour-neighbour signalling, prohibiting division amongst surrounded cells whilst massively boosting the fitness of cells on the surface of the organoid. Analysis of the experimentally observed growth rates of the clusters also suggested that their overall fitnesses follow a skewed-normal distribution which is largely unaffected by mutation, and that initially-weak clusters tend to lose fitness much faster than initially-fit ones. We hypothesise that the effect of this mutation is to confer an evolutionary strategy which prioritises reproduction amongst edge-cells at the expense of surrounded ones, which renders it highly invasive but does not much change its overall growth rate. We further hypothesise that the ability of mutant cells to suppress the division their neighbours should give them a significant advantage over wild-type cells. We suggest that the likelihood of L858R-fuelled tumorigenesis is affected not just by random fluctuations in cell fitness but by whether the mutation arises in a spatial environment that allows cells to reproduce without being forced to encounter each other. This has important possible implications for cancer prevention strategies and for understanding NSCLC progression.

## 31 January 2023 in Harrie Massey LT, 25 Gordon Street

**Speaker: King Ming Lam**

Supervisor: Prof M Hadzic

##### TITLE: Nonradial stability of self-similarly expanding goldreich-weber stars

**Abstract: **

In simple terms, we established non-linear stability of a class of known spherically symmetric expanding solutions to a PDE that model the dynamics of a star. In doing so, we have proven global-in-time existence for a larger class of expanding solutions around the known class that's not spherically symmetric but behaves similarly.

More precisely, we have the following. Goldreich-Weber solutions constitute a finite-parameter of expanding and collapsing solutions to the mass-critical Euler-Poisson system. A subclass of this family corresponds to compactly supported density profiles suitably modulated by the dynamic radius of the star that expands at the self-similar rate λ(t)∼t^{2/3} as t→∞. We prove that any given self-similarly expanding Goldreich-Weber star is codimension-4 stable in the class of irrotational perturbations. The codimension-4 condition is optimal and reflects the presence of 4 unstable directions of the linearised operator in self-similar coordinates, which are induced by the conservation of the energy and the momentum. This result can be viewed as the codimension-1 nonlinear stability of the "manifold" of self-similarly expanding GW-stars in the class of irrotational perturbations.

## 7 February 2023 in Bedford Way (26) G03

**Speaker: Kyriakos Katsamaktsis**

Supervisor: Dr S Letzer

##### TITLE: The Absorption method for the travelling salesman problem

**Abstract: **

The Travelling Salesman Problem asks whether given a collection of cities and road connections between them it is possible to drive through all cities by visiting each one precisely once. If this is the case, we say that the graph (of the cities and road connections between them) is Hamiltonian. A major open problem is whether there exists an efficient algorithm for determining the Hamiltonicity or not of an arbitrary graph. However, if a graph satisfies certain simple conditions, one can prove it is Hamiltonian, and many variants of this problem are well studied. I will describe a general proof strategy, the absorption method, that has been used successfully over the last 15 years to tackle such problems.

## 14 February 2023 in Bedford Way (26) G03

**Speaker: Nicholas Jelicic**

Supervisor: Dr S Cooper

##### TITLE: incorporating patient flow into infection control decisions

**Abstract: **

Healthcare-associated infections (HCAIs) pose a significant risk to both staff and patients; they can complicate patients' treatments and lead to serious illness. As a consequence, infection prevention control (IPC) measures are often put in place to reduce the transmission within the healthcare system. However, these measures can have a negative impact on patient flow through the system. The aim of this research is to evaluate what the impact of IPC decisions is on patient flow, and to provide a tool for decision makers to help them better consider the trade-offs between HCAIs and patient flow.

Here we describe a general framework which can be used to deal with specific instances which might occur. Further, we present two models to solve the general problem - a Continuous Time Markov Chain (CTMC) model, and a Discrete Event Simulation (DES) model. We then formed a classification problem to determine the circumstances in which a particular IPC decision should be taken.

## 21 February 2023 on Zoom at 10am (https://ucl.zoom.us/j/99040075995#success)

**Speaker: Daniel Bussell**

Supervisor: Dr CA Garcia Trillos

##### TITLE: multistep deep learning methods for nonlinear pdes and bsdes

**Abstract: **

In this talk we introduce a deep learning neural network based method for the resolution of moderate to high dimensional nonlinear parabolic PDEs and their corresponding Backward Stochastic Differential Equations (BSDEs).

We rely on a probabilistic framework relying on discretising the BSDE and incorporating multiple time step learning for error reduction compared to classical one step methods. An error approximation result is proved using a specific class of differentiable shallow neural networks and numerical examples for PDEs of up to dimension 100 are presented.

## 28 February 2023 in Bedford Way (26) G03

**Speaker: Mads Christensen**

Supervisor: Dr L Garcia Martinez

##### TITLE: Some Topological invariants in hyperbolic space

**Abstract: **

In complex analysis, the winding number of a closed loop in the (complex) plane counts how many times the loop winds around a fixed point. This is an example of a topological invariant. Another example of topological invariants are linking numbers, which count how many times that two loops in 3-dimensional space “link” together. But instead of placing our loops inside the familiar Euclidean space we can also try to use other spaces such as hyperbolic space. I will give an overview of some known and some conjectural results in this direction.

## 7 March 2023 in Bedford Way (26) G03

**Speaker: Myles Workman**

Supervisor: Prof C Bellenttini

##### TITLE: calculus

**Abstract: **

Interesting behaviour of functions tend to happen near critical points. We can identify and classify these points by calculus (looking at the functions first and second derivatives). Let us explore how we can push this idea to answer questions in geometry and PDE theory.

## 14 March 2023 in Bentham House SB31 Denys Holland Lecture Theatre

**Speaker: Dr Jakob Stein**

Supervisor: Dr L Foscolo

##### TITLE: Non-autonomous ode systems in gauge theory

**Abstract: **

In this talk, we will discuss a neat example of solving a non-autonomous (i.e. time-dependant) ODE system that arises from gauge theory. We will see how qualitative features of solutions relate to the geometry, and use this example to talk about gauge theory on spaces called G2-manifolds.

## 21 March 2023 in Bentham House SB31 Denys Holland Lecture Theatre

**Speaker: Luke Debono**

Supervisor: Prof HJ Wilson

##### TITLE: An introduction to the molecular origins of viscosity with an application to particle-based fluid simulations

**Abstract: **

In this introductory-level talk, I will first discuss the concept of viscosity and relate observed deformation properties (the rheology) to a molecular length scale. These observations will then be quantified through a linear relation known as Newton’s law of viscosity. This discussion poses two questions: how exactly might one measure viscosity, and in the context of molecular dynamics, how might one simulate a particle-based system to measure viscosity? To answer this question, a simple experimental approach is presented; followed by an introduction to reverse-nonequilibrium methods and the Müller-Plathe Algorithm as a particle-based simulation technique for measuring viscosity. This talk will conclude with a discussion of the utility of particle-based methods in fluid simulations.