# Welcome and Information

Welcome to the fifth annual SAMBa Summer Conference, taking place 5th to 7th of July. For a photo without the words overlay see here

The SAMBa conference is an opportunity for students to showcase their work to members of the department, outside the department and at other Universities in a supportive environment. The work of SAMBa students covers the entire spectrum of statistical applied mathematics: including projects in statistics, probability, analysis, numerical analysis, mathematical biology, fluid dynamics, machine learning and high-performance computing. The conference is organized by students and contains talks by SAMBa students, external speakers, and students from other departments and institutions.

## Important Documents

Important information to the conference will be located here as they become available.

• All attendees are expected to follow the code of conduct.
• If you would like to upload a selfie or photo of yourself to be added to the conference photo you can upload them here.
• Welcome Slides can be found here

# Registration

Public registration for the conference has now closed. This is in order to ensure numbers at social events and in person attendance. Anyone who wishes to register for virtual attendance should email one of the conference organisers as soon as possible, using the contact details at the bottom of this page.

# Schedule

Below is the schedule for the conference. The list of speakers can be found below the timetable with their titles and abstracts updated when they become available. Each session of talks is linked nominally by theme, given to provide an idea of focus for the session.

 Time Monday (5th July) Tuesday (6th July) Wednesday (7th July) 13:00 Opening Remarks Industrial and Applied Mathematics Chair: Katie Speakers: Piotr Morawiecki Katerina Kaouri Gianluca Audone Mathematical Biology and Collective Behaviour Chair: Shahzeb Speakers: Weini Huang Yvonne Krumbeck Jeremy Worsfold 13:15 Discrete Probability Chair: Chris Speakers: Edward Crane Carlo Scali Thomas Bartos 13:30 13:45 14:00 14:15 14:30 14:45 15:00 Poster Session and break Break 15:15 Break and Conference Photo 15:30 Probability and Analysis Chair: Yi Sheng Speakers: Paolo Grazieschi Stefano Bruno Zsofia Talyigas 15:45 Statistics and Machine Learning Chair: Chris Speakers: Allen Hart and Kevin Olding Eleanor Barry Margaret Duff 16:00 16:15 16:30 Lightning Talks 16:45 17:00 17:15 Closing Remarks 17:30 End End End (Evening) Virtual Conference Event In-Person Conference Dinner

## Conference Social Event

We will be holding a virtual social event on the Evening of July 5th and a socially distanced in person social event on Campus in the evening of July 6th. Conference Attendees will be contacted with more information of these events.

# Speakers

Find below the full list of speakers, with titles and abstracts added as they become available.

## Guest Speakers

### Edward Crane

University of Bristol

Title: Self-organized criticality in the mean-field forest fire model

Abstract: Various complex systems in nature are observed to have characteristics of criticality. This means they are qualitatively similar to simple physical systems that are tuned to a point of phase transition in their parameter space. For example, power law decay is empirically observed in the size distributions of avalanches, earthquakes, neuronal firing cascades in the brain, forest fires, and even financial crashes. An explanation is that these systems all have negative feedback mechanisms that drive them into a critical state: they show self-organized criticality. A famous mathematical model for self-organized criticality is the abelian sandpile model. This talk will be about another abstract mathematical model which has been rigorously proven to have self-organized criticality. This is the mean field forest fire model introduced in 2009 by Rath and Toth. It is a stochastic process of random graphs, closely related to the Erdos-Renyi random graph model. It has been analysed in two quite different ways, using a controlled inviscid Burgers equation and using critical multitype branching processes. (Joint work with Nic Freeman, Balint Toth, Balazs Rath and Dominic Yeo.)

### Weini Huang

Queen Mary University of London

Title: Mathematical models in evolutionary medicine — example of modeling ecDNA dynamics.

Abstract: Evolutionary medicine is an interdisciplinary field applying methods from evolutionary biology, relevant mathematical models such as differential equations and stochastic process in understanding disease progress. Together with experimental biologists and clinicians, we work on quantifying the underlying stochastic process and properties of various biological complex systems. Our projects range from classical coevolutionary system as host-virus, host-parasite, predator-prey system to healthy liver tissue, brain/colon cancer through genomic data. Here, I will present one of our most recent projects on modelling the stochastic process of Extra-chromosomal DNA (ecDNA) and application on glioblastoma data. Glioblastoma is an aggressive type of brain cancer, where ecDNA can be a driving force of the tumour progress and treatment resistance. Typically thought to be produced during the cell replication, ecDNA is circular DNA found in cells outside of the chromosomes. Recently large-scale studies showed that ecDNA is highly prevalent in many types of cancer, with the highest prevalence in glioblastoma. Each cell can carry from zero to a large number of ecDNA copies. Because the lack of centromeres, ecDNA copies are segregated randomly into daughter cells, rather than evenly separated as standard chromosomal DNA. This stochastic process leads to an interesting ecDNA distribution among cells, which can be analysed mathematically. With collaboration with experimental groups, we developed a stochastic model and first confirmed random ecDNA segregation in experiments. Our theoretical results on the ecDNA distribution leads to testable predictions on how cells carrying ecDNA are positively selected and spread in the tumour population. We validate our predictions in various glioblastoma data, for example the ecDNA dynamics in GBM39 cell line experiments suggesting a 300% cell proliferation increase due to ecDNA. Our work estimates quantified properties of those ecDNA elements in individual tumours, which otherwise are hard to measure directly in clinical/experimental setups but important to adjust treatment strategies.

### Katerina Kaouri

Cardiff University

Title: Modelling the transmission of COVID-19 in indoor spaces

In Collaboration: Zechariah Lau1,2, Ian Griffiths1, Aaron English3, Raquel González Fariña1, Alexander Ramage1, Katerina Kaouri1
1 School of Mathematics, Cardiff University
2 Mathematical Sciences Institute, University of Oxford
3 Department of Mechanical, Aerospace and Civil Engineering, The University of Manchester

Abstract: As the pandemic continues to rage we need to urgently find ways to safely resume economic and social activities while containing the virus. The transmission of COVID-19 in indoor spaces has been intensively researched since the beginning of the pandemic, especially in highly frequented spaces such as healthcare clinics, schools, nursing homes, supermarkets and public transport. We have, thus, developed an efficient model for determining the spatiotemporally varying risk of infection from COVID-19 in indoor spaces. The airborne infectious particles are emitted by an infected person, advected by airflow, diffused due to turbulence, and removed due to the room ventilation, biological inactivation of the virus and gravitational settling. We model these processes using an advection-diffusion-reaction equation and determine the concentration of the particles in the room, leveraging the semi-analytic solution to run very fast simulations. Also, as there are many particle sizes involved in viral transmission (due to breathing, talking, coughing etc.) we have extended the modelling framework to account for realistic particle size distributions. We find that the one-particle-size assumption used in many models can lead to wrong predictions of the infection risk in a room. Finally, we determine the Time to Probable Infection (TTPI), paving the way for formulating policy recommendations. Good agreement with CFD models and existing data has been obtained. For more information: https://arxiv.org/abs/2012.12267

## Student Speakers

### Gianluca Audone

SAMBa Aligned Cohort 6

Title: Listening to the Ocean through time series

Abstract: The Earth is covered with large oceans, making up close to two thirds of its surface. Marine ecosystems are extremely reliant on sounds, for navigation, communication and predation. Underwater soundscapes are increasingly affected by human activities (like shipping) and climate change (warming oceans). In this talk, I will dive into the world of time series, in particular those collected worldwide by the international Comprehensive Test Ban Treaty Organization. The starting point of our journey will be ergodic stationary processes through which we will give a rigorous definition of time series and overview some analysis methods. All the presented notions will be applied to real world data showing how we can use them as a preliminary analysis tool.

### Eleanor Barry

SAMBa Cohort 4

Title: The inclusion of prevalent cohort data to understand the possible increased risk of cancer in patients with systemic sclerosis

Abstract: In epidemiological studies, we are often interested in what effect an exposure (or disease or treatment) has on an outcome of interest. In the case of my research, we are interested in patients with systemic sclerosis (SSc), a rare autoimmune disease, and the outcome of interest is the incidence of cancer in this patient group. This study may lead us to identify a possible link between SSc and cancer. Within this research, the baseline group of patients are those who have or develop SSc, but the disease ‘of interest’, within this patient group, is cancer.
Epidemiological survival data are now commonly collected in the UK. This data may be from an ‘incident cohort’ (in our case, this is the cohort of patients who had SSc at the start of the study period or who develop SSc through the time period of the study – these patients may or may not develop cancer). Alternatively, the data may be from a ‘prevalent disease’ cohort (in our case, this is patients who had an SSc diagnosis prior to the study but who did not have a cancer diagnosis at that time – they may have subsequently developed cancer).
The dataset I am working with has the benefit of enabling us to study both incident and prevalent cohorts. Prevalent cohorts are often biased due to being left truncated, which leads to an oversampling of patients with longer disease duration. In my talk I will discuss prevalent cohort bias, and common methods to adjust for this. The disease SSc has the added complication of many patients being ‘censored’ due to an increased death risk, either from SSc itself or from other causes, meaning that we do not observe our event of interest (which in our case is cancer incidence). We term this complication a competing event. I will discuss this interaction between the issues of biasing and competing events, the implications of these within a causal framework, and the impact of these on our study results.

### Thomas Bartos

SAMBa Cohort 5

Title: Accessibility percolation in a dynamic fitness landscape

Abstract: The concept of a fitness landscape was originally introduced to model the evolution of a biological population. The landscape can be modelled as a finite graph with each vertex representing a genotype and each edge representing a possible mutational step between genotypes. Recently, progress has been made analysing a model known as ‘accessibility percolation’ in which random fitness values are assigned to vertices of a directed landscape and a path between vertices is called ‘accessible’ if the fitness values increase monotonically along the path. Here we introduce a dynamical version of accessibility percolation on a n-ary tree, which for the static model is known to exhibit a phase transition depending on the height of the tree. Using tools from the dynamical percolation and noise sensitivity literature, we investigate the probability that so-called ‘exceptional times’ of accessibility exist under different parameter regimes, our main result being that at the critical parameter value such times exist with high probability, in contrast to the static case.

### Stefano Bruno

SAMBa Cohort 4

Title: Regularity in time and space for stochastic porous media equations.

Abstract: Stochastic porous media equations (SPME) are well-studied models describing non-linear diffusion dynamics perturbed by noise. In this talk, I will discuss the spatial and time regularity for solutions to these stochastic PDEs in Sobolev spaces. I will introduce the notion of kinetic formulation for conservation laws which allows to transform SPME into “linear” stochastic PDEs and use this formulation in the derivation of the regularity estimates. At the end of the talk, I will compare our regularity results with the optimal regularity estimates obtained for the deterministic porous medium equation. This is joint work with Benjamin Gess and Hendrik Weber.

### Margaret Duff

SAMBa Cohort 5

Title: Inverse problems and generative models

Abstract: Solving an inverse problem is the process of calculating an unknown quantity from observed, potentially noisy, measurements. Inverse problems are everywhere. In just a few examples: smart phone cameras are denoising, deblurring, sharpening and correcting images; modern medicine uses MRI, CT and x-ray; and geophysics uses seismic imaging to explore the centre of the earth. Often the data is not sufficient to give a unique or stable solution to the inverse problem, and more information must be added. In a brain MRI scan we could say that the image must be sparse in some basis, must have a limited number of points with large gradients or must overall have small magnitude. In this work, what we would really like to do is tell the mathematics that we are looking at an MRI scan and moreover we are looking at a brain! To do this, we use the technology that brought us the deep fake image of the queen in the Christmas message, that brought us apps that made us look younger or older (and may or may not steal our data) and could be used to make celebrities say incriminating things in videos. Generative models are designed to generate data similar to some predefined training set. Perhaps instead of celebrities we can train our models to generate MRI images of brains, and then given some observed data, we choose the best image that matches the data? We explore this question in the talk, introducing a two common generative models, variational autoencoders and generative adversarial networks, and then considering a number of ways they can be applied to inverse problems. The talk will try and avoid too much mention of neural networks and will hopefully contain some cat images!

### Paolo Grazieschi

SAMBa Aligned Cohort 5

Title: Mesoscopic interaction, scaling limits and SPDEs

Abstract: The Ising-Kac model is an interacting particle system, where the interaction is of mesoscopic type: as such, each particle interacts with an infinite amount of other particles, yet a infinitesimal amount in the limit with respect to the total number of particles in the system. We show how to properly rescale this model and we analyse the limit, thereby introducing new recent results for the solution of Stochastic Partial Differential Equations.

### Allen Hart & Kevin Olding

Both SAMBa Cohort 4

Title: Using Echo State Networks to Approximate Value Functions for Control

Abstract: An Echo State Network (ESN) is a type of single-layer recurrent neural network with randomly-chosen internal weights and a trainable output layer. We prove under mild conditions that a sufficiently large Echo State Network can approximate the value function of a broad class of stochastic and deterministic control problems. Such control problems are generally non-Markovian. We describe how the ESN can form the basis for novel and computationally efficient reinforcement learning algorithms in a non-Markovian framework. We demonstrate this theory with two examples. In the first, we use an ESN to solve a deterministic, partially observed, control problem which is a simple game we call Bee World’. In the second example, we consider a stochastic control problem inspired by a market making problem in mathematical finance. In both cases we can compare the dynamics of the algorithms with analytic solutions to show that even after only a single reinforcement policy iteration the algorithms arrive at a good policy.

### Yvonne Krumbeck

SAMBa Aligned Cohort 5

Title: Fluctuation spectra of large random dynamical systems reveal hidden structure in ecological networks

Abstract: Understanding the relationship between complexity and stability in large dynamical systems – such as ecosystems – remains a key open question in complexity theory which has inspired a rich body of work developed over more than fifty years. The vast majority of this theory addresses asymptotic linear stability around equilibrium points, but the idea of ‘stability’ in fact has other uses in the empirical ecological literature. The important notion of temporal stability’ describes the character of fluctuations in population dynamics, driven by intrinsic or extrinsic noise. Here we apply tools from random matrix theory to the problem of temporal stability, deriving analytical predictions for the fluctuation spectra of complex ecological networks. We show that different network structures leave distinct signatures in the spectrum of fluctuations, and demonstrate the application of our theory to the analysis of ecological timeseries data of plankton abundances.

### Piotr Morawiecki

SAMBa Cohort 6

Title: From testing chemicals to foreseeing riots: Why Study Groups with Industry are fun!

Abstract: : Study Groups with Industry originated over 50 years ago in UK, which spread all over the world. These week-long workshops are great opportunity for the PhD students to get experience in using mathematical methods for solving real-world problems in collaboration with industry. During this talk I will promote these workshops by giving the audience a taste of a variety of research topics that participants can work on. Three mini case studies will cover topics ranging from probability (optimising chemical testing process), through mathematical modelling (estimating conductance of heterogenous materials) to statistics (detecting abnormal behaviour in crowds), so people from all backgrounds may find something interesting.

### Carlo Scali

SAMBa Cohort 6

Title: Scaling limits for sub-ballistic random walks in random environment

Abstract: Random walks in random environment were introduced by Solomon in 1975. This first model was defined by assigning independent and identically distributed jump probabilities on the 1-dimensional lattice Z with nearest neighbours’ links. One of the many reasons why this basic model is very interesting is that the transient version showcases two different regimes. I will present the results in the literature that describe the sub-ballistic regime (i.e. the one for which the walk is transient but atypically slow due to the appearance of trapping phenomena), linking this to my current research that investigates a similar model: the random walk on random conductances. In particular, I will try to convince you that, even if the trapping mechanisms behave quite differently, the limiting behaviour of the two models is very much comparable.

### Zsofia Talyigas

SAMBa Cohort 5

Title: Genealogies in a branching random walk with selection

Abstract: The N-particle branching random walk (N-BRW) is a particle system, which can be viewed as a toy model of an evolving population affected by natural selection. In the N-BRW at each time step, N particles have locations on the real line. Each of the N particles has two offspring, which have a random displacement from the location of their parent according to some fixed jump distribution. Then among the 2N offspring particles, only the N rightmost particles survive to form the next generation. The long term behaviour of the N-BRW heavily depends on the tail of the jump distribution. In this talk I will explain our result about the genealogies of the surviving particles in a late generation, in the case when the jump distribution has polynomial tails.

### Jeremy Worsfold

SAMBa Cohort 6

Title: Stochastic Kinematic Flows: Why do Fish Swarm and Traffic Jams form?

Abstract: Organisms make decision based on individual information and goals yet they often form patterns or clusters due to these decisions. Both fluid models and interacting particles systems are used to try to understand this collective behaviour as it arises in many real-world scenarios from fish and locusts to humans in crowds or driving cars. In this talk I will present a general system of interacting Brownian particles which we will call agents. I will show that, for large numbers of particles, this is equivalent to some “kinematic flows”. When we do not have a very large number of agents (as is often the case) we will also see that the random decisions made by agents can significantly change the overall behaviour. To see why this is useful I will then apply this to examples such as vehicle traffic and swarming fish.

## Posters

We are fortunate to have a selection of SAMBa students also presenting posters. These will be on display in the atrium during the conference, as well as held virtually in our online environment. For the full list of posters with their abstracts please look here.

# Coordinators

The conference is being organised by 4 second year SAMBa PhD students. If you have any questions please feel free to contact any of us using the information below.

Katie PhillipsShahzeb Raja NoureenChris DeanYi Sheng Lim
Fluid dynamics:
lubrication theory and interfacial waves
Stochastic modeling of melanoblast neural crest cellsPolya urn processes with infinite initial conditionsSpectral theory of random operators
kap39 at bath.ac.uksrn32 at bath.ac.ukcbcd20 at bath.ac.ukysl64 at bath.ac.uk