# Abstracts

## Speakers

### Brigitte Stenhouse, The Open University

## “I thought it unjust that women should have been given a desire for knowledge if it were wrong to acquire it”: Rediscovering mathematics through the Personal Recollections of Mary Somerville

Mary Somerville (1780-1872) was a mathematician and scientist, who during her lifetime was widely known as the only person who understood the physical astronomy of Pierre-Simon Laplace; that is, Laplace’s mathematical treatment of the forms and motions of the planets, moons, and comets. Despite her intelligence and ambition, as a woman she was ineligible to attend university or to hold memberships of learned societies. Seven years after her death, a college for the higher education of women was founded in Oxford. In honour of Somerville, as an intellectual with liberal politics, the founders named their new institution Somerville Hall, later Somerville College. It was as an undergraduate at this college, in 2015, that I first came across her autobiographical Personal Recollections (first published in 1873). Having struggled since arriving at university with imposter syndrome, and in the process of disentangling myself from an abusive relationship, at this time I had somewhat fallen out of love with maths. Reading about Somerville’s own passion for the subject, how it was a way for her to understand the world around her, helped me to see maths as a collaborative, human activity that can be for fun, not just for exams. It was also the first stage of a 9 year (and counting) research journey to understand what it really meant for Somerville – as a 19th-century woman – to be a mathematician, and how she and many other women overcame gendered barriers that continue to affect the mathematical community today.

In this talk, I will briefly trace my own career trajectory from an undergraduate maths student to a Lecturer in the history of mathematics at the Open University (and explain what this job entails!). I will highlight a couple of areas of my research – from Somerville sending her husband off to the Royal Society with a to-do list to enable her research, to her studies of quaternions at 90 years old – and reflect on how this research informs my own work to mitigate gender gaps in mathematics.

### Mura Yakerson, The University of Oxford

## Don Juan, or the Love of Algebraic Geometry

In this talk, I will share some personal experiences and struggles on my way into math, explain the celebrated Grothendieck-Riemann-Roch theorem which brought me into algebraic geometry, and say few words about its relation to my research. The talk hopes to be accessible to a general math audience!

### Tacey O'Neil, The Open University

## Hiding out in mathematics (living in the shadows of analysis)

I shall talk both about how I eventually came out along with some of the fractal problems that interest me as a mathematician – I'm not entirely sure these two things are connected! I intend that the talk be accessible to a general mathematical audience.

### Melanie Rupflin, The University of Oxford

## (Almost) Optimal geometric objects

Many interesting geometric objects are characterised as minimisers of natural geometric quantities such as the length of a curve, the area of a surface or the energy of a map, and one can think of objects which minimise such a quantity as “optimal states”. When studying a minimisation problem, it is natural to not only consider the properties of exact minimisers, but to also ask whether objects that nearly minimise the given quantity must essentially look like a minimiser. We will discuss this for the model problem of the Dirichlet energy of maps between spheres, where it has long been known that the minimisers are given by Moebius Transforms and other meromorphic functions that you might know from your undergraduate course in Complex Analysis, but where this simple question remained unanswered until very recently.

## Christl Donnelly, The University of Oxford

## A statistical epidemiologist's life on the edge (of the science-policy interface)

Ebola, MERS, pandemic influenza, SARS and Zika have all posed serious threats to our health and economic wellbeing in recent years. In each of these cases, statistical (and more broadly mathematical) epidemiologists contributed to top-level policy discussions of diseases control policy development, implementation and contingency planning. The methods build upon foundations of epidemiological modelling and analysis of both human and animal diseases (HIV/AIDS, BSE, vCJD, bovine TB and foot-and-mouth disease, among others). The potential impact of such analyses is enormous, but it can be challenging to provide robust answers to key scientific and policy questions. In the midst of an epidemic response effort, it really does feel like living on the edge.

## Early-Career Speakers

### Pure

### Shaked Bader, The University of Oxford

## CAT(0) polygonal complexes are 2-median

A (1-)median space is a space in which for every three points the intersection of the three intervals between them is a unique point. Having this in mind, in the talk I will define a 2-median space which is a 2 dimensional variation of the median space, explain the title and present the idea of the proof of the theorem in the title. This talk is based on my Master Thesis done under the supervision of Nir Lazarovich.

### Alexandra Embleton, Royal Holloway, University of London

## Cohomological invariants of Bredon modules

Given the orbit category O_{F}G, for any given family F of subgroups, we define a Bredon module to be a functor M(−) : O_{F}G → Ab. From this one can define the notion of free, projective, injective and Bredon Gorenstein projective modules and related cohomological invariants. Subsequently some analogous results, to those know for Z-modules, can be proved relating invariants such as silp_{F}G, spli_{F}G, cd_{F}G, gldim_{F}G, fin.dim_{F}G and Gcd_{F}G.

### Sofía Marlasca Aparicio, The University of Oxford

## Ultrasolid geometry and deformation theory

We will talk about the category of ultrasolid modules, a new framework first proposed by Dustin Clausen. This generalises the already existing solid modules over F_p or Q of Clausen and Scholze. We will discuss this category as well as some of the applications to derived and formal geometry.

### Isabel Rendell, King's College London

## Rational points on curves

If we are given a curve, then we would like to be able to determine points that lie on it which have rational co-ordinates - these are called its rational points. This turns out to be an extremely non-trivial question and is in fact a very active area of research in number theory. The question can be divided into three situations, determined by the genus of the curve. In this talk I will give an overview of this problem, particularly focusing on the case where the genus is at least 2.

### Applied

### Anna Berryman, The University of Oxford

## Region and occupation bottlenecks of the net zero transition

Countries around the world require economy-wide transitions to their reach net zero targets. While much modelling has been carried out to understand the economic and environmental implications of different transition scenarios, most models that include labour market impact do not account for mobility frictions and the speed of the transition that determine the outcomes for workers. We combine a data-driven occupational mobility network with an agent-based labour market model to account for labour mobility and frictions that might enable, or slow down, the net zero transition. Our results indicate that labour market frictions can slow down the Brazilian transition scenarios we analyse in this paper and that policy makers need to address both supply and demand of workers to enable a smooth transition. The proposed modelling approach could also be used to understand the labour market impacts of other net zero transition policies, as well as other economic transitions, such as automation.

### Otilia Casuneanu, The University of Oxford

## Mathematical modelling of carnivorous pitcher plants and chemotactic attraction of foraging ants

Nepenthes pitcher plants are carnivorous plants. They attract prey to a rim, called the peristome, which becomes very slippery, causing insects to fall into the trap, where they cannot escape and are thus digested, providing vital nutrients to the plant. The geometry of the peristomes widely diverges, raising evolutionary questions about these variations. In this talk I will present a mathematical model of this phenomena, as well as a mathematical description of the random walk of an insect on the surface of the peristome.

### TD Dang, University College London

## Three-dimensional melting of ice in shear flow

This study is motivated by a few industrial applications. Ice formation on various surfaces have been known to pose efficiency issues and safety hazards, particularly in aeroplanes where ice growth on wings which deform the overall wing shape or within engines, can pose a significant risk to the aircraft. Another relevant application of this work may be the ice formation inside domestic or commercial pipes, presenting plumbing issues in buildings. The majority of icing literature present equivalent two-dimensional models however there is clearly scope for improvement as more complex three-dimensional models present a much more realistic scenario. We present a novel approach to modelling a three-dimensional ice lump of finite extent melting on a flat wall, with the oncoming fluid surrounding the ice, near the wall, being warmer than the ice. Further away from the wall, there may be a heat transfer effect that increases or decreases the temperature of the water at distances normal to the wall. The wall temperature is assumed to be the same as the near wall water, except for the area directly underneath the ice lump, which is the same temperature as the ice. The flow behaviour is studied both analytically and computationally and in-flow predictions of the ice shape are presented as well as the induced pressure gradient caused by the flow.

### Gillian Grindstaff, The University of Oxford

## Statistics in the space of phylogenetic trees

Phylogenetic trees represent the historical branching relationship between the products of a divergent evolutionary process, such as the "tree of life" relating all living species with their ancestral counterparts. Typically, phylogenetic trees are inferred from genetic or phenotypic data, but their complex shape presents unique challenges for statistical analysis - what is the average of a set of potential trees? A confidence interval? In my work, I study the topology and geometry of moduli spaces of phylogenetic trees, in order to develop statistical tools that respect the algebraic constraints of tree-shaped data.

### Anushka Herale, University College London

## A minimal continuum model of clogging in spatio-temporally varying channels

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. stiƯening 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 based on granular rheology that accounts forspatio-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 solve the system using a finite-volume method and simulate the evolution of an initially uniform particle density. We find that varying material properties and varying geometry can induce heterogeneity in particle volume fraction. We are able to show the emergence of high and low particle density regions in volume-driven flows. These results clarify how spatial variation in material and channel properties can contribute to clogging of particle suspensions.

## Poster Presentations

### Zaineb Bel-Afia, The University of Oxford

## Chebyshev Varieties

Chebyshev varieties are algebraic varieties parametrized by Chebyshev polynomials. They arise when solving polynomial equations expressed in the Chebyshev basis. Chebyshev polynomials are widely used in optimization and numerical analysis, and their analytic aspects have been studied intensively. On the other hand, their algebraic properties remain relatively unexplored. This work aims to bridge this gap. We define Chebyshev varieties and discuss their defining equations, degrees and dimensions. Through examples, we motivate their use in effective computations. This is joint work with Chiara Meroni (Harvard University) and Simon Telen (MPI MiS).

### Maria Tasca, The University of Oxford

## Parameter identifiability analysis of spatiotemporal models of cell invasion

Identifiability analysis questions if the parameters of a mathematical model can be uniquely identified, or not, given the system observations. Its importance is given by the need of having confidence in the predictions from mathematical and computation models that rely on parameter values estimated experimentally. The use of mathematical models to quantify their behaviour through parameters has increased significantly in delicate areas such as drug development. However, currently there are no tools for identifiability analysis for models that incorporate space. The novelty of our work is the analysis on PDE reaction diffusion models, which, to the best of our knowledge, hasn’t been developed properly yet. We expect that our framework is applicable to a broad range of linear reaction-diffusion equation models

### Tara Trauthwein, The University of Oxford

## Normal Approximation of Poisson Functionals via Malliavin-Stein Method

When considering a random network and a cost function on it, you might ask yourself what happens at very large scales. We present a method allowing to show convergence to a Gaussian distribution of functions of certain random point collections when the underlying collection is large. Applications include central limit theorems for graphs, among them a graph that was developed to model the behaviour of the internet.

### Sara Veneziale, Imperial College London

## Machine learning detects terminal singularities

This poster describes recent work in the application of machine learning to explore questions in algebraic geometry, specifically in the context of the study of Q-Fano varieties. These are Q-factorial terminal Fano varieties, and they are the key players in the Minimal Model Program. In this work, we ask and answer if machine learning can determine if a toric Fano variety has terminal singularities. We build a high-accuracy neural network that detects this, which has two consequences. Firstly, it inspires the formulation and proof of a new global, combinatorial criterion to determine if a toric variety of Picard rank two has terminal singularities. Secondly, the machine learning model is used directly to give the first sketch of the landscape of Q-Fano varieties in dimension eight. This is joint work with Tom Coates and Al Kasprzyk.

### Phoebe Valentine, The University of Warwick

## Connected Tangents & 1-Dimensional Trickery

We all have an intuitive notion of dimension: a point is 0 dimensional, a line is 1 dimensional and so on. In some way, our idea of 'dimension' is actually measuring complexity. So, can we construct something made of points that is so complicated we end up being 1 dimensional, even though it contains no lines? Unfortunately for our intuition, the answer is yes! In geometric measure theory, we call these unlikely objects 'unrectifiable sets'. It turns out, every set is either rectifiable or unrectifiable - there is no in-between and they have overwhelmingly opposite geometric properties. This poster examines whether the existence of tangents (in a very weak sense) is enough to characterise 1-dimensional rectifiability. Based on joint work with David Bate.