The PSS is a weekly seminar for maths postgraduates (MSc or PhD) to give talks on mathematical topics which interest them, to an audience of other maths postgrads. It is an excellent opportunity to give a talk in a friendly enviroment to an audience which spans the department. People often speak about topics relating to their research but many also take the chance to look at something new to them and then present it to the rest of us.

More importantly, after each and every talk has been finished there is FREE CAKE !

## When, Where and What's On?

For up to date information look at Dan Green's (2016-17's organiser) webpage.

### PSS Archive

By popular demand I have compiled a history of the talks given over the last few years which can be found via the links: 2012-13, 2013-14 and 2014-15. The seminars I organised can be found below.

## 2015-16 Semester 1

### Title and Abstract

1 Oct Kieran Jarrett What is Ergodic Theory?

... is the question which often follows when people ask what I research. Now I am going to attempt to answer it! I'll explain what the theory is about, and use some simple accessible examples to motivate the approach. Then I will briefly describe how the subject is formulated, and move on to give some of the fundamental questions the theory tries to answer. If we have time, perhaps I'll explain some of answers which don't require you to have done a course in the subject.

For those of you who want a small hint at the answer, in his book 'Ergodic Theory' Karl Peterson's first sentence reads 'Ergodic theory is the mathematical study of the long-term behaviour of (dynamical) systems'.

8 Oct Katy Gaythorpe Antimicrobial resistance in modelling

With the growing threat of antibiotic resistant bacteria at the forefront of everyones minds (possibly), how is mathematics being used to help. I detail a model of some of the effects of antibiotic use in food animals. I will also give some insight into similar models of resistant bacteria transmission.

15 Oct Alge Wallis An Introduction to Special Holonomy

The aim of this talk is simply to explain how and why humanity has ended up funding me to do my research. We will take a whistle stop, one-slide tour from the roots of geometry, to (hopefully) looking at impacts of special holonomy in our understanding of the natural world. . A note of caution: Historical, and scientific accuracy may have been compromised in the writing of this talk.

22 Oct Pite Satitkanitkul Lévy processes and self-similar Markov processes

In this talk, I will start by defining the Lévy processes and how they could naturally fit in various models. Then, I will explain how are they are related to the positive-valued self-similar Markov processes (pssMp) and how I would extend it to my research on Markov Additive processes and real-valued self-similar processes...

29 Oct George Frost The Geometry of Surfaces in $$\mathbb{R}^3$$

Many surfaces in $$\mathbb{R}^3$$, such as the sphere or cylinder, have an obvious notion of "curvature": it's tempting to say that the sphere has "positive curvature" at every point and in every direction, while the cylinder has "positive curvature" around its circumference and "zero curvature" along its axis. I will attempt to formalise these vague notions by introducing the classical theory of surfaces as developed by Gauss et al.

In case geometry isn't your cup of tea, you can look forward to my differential geometric variant of "Who's that Pokémon?" (https://youtu.be/Akt-0oKULAc or https://youtu.be/IfQumd_o0Gk in case your childhood was rubbish), which I will shoehorn in at some point.

5 Nov Dan Green How do you solve a problem like a Lyapunov Equation?

In this talk I'll attempt to answer:

How do you solve a problem like a Lyapunov Equation?
How do you choose a method and pin it down?
How do you find a use for a Lyapunov Equation?
A flibbertigibbet! A will-o'-the wisp! A clown!

12 Nov James Roberts Harmonic maps with a 1D domain

In Euclidean space, a curve has zero acceleration if and only if it is a straight line. A straight line between two points is the shortest path connecting these points. More generally, we may equip a Riemannian manifold with a 'natural' notion of acceleration and then ask 'for which curves is the acceleration zero?'. Such curves are called (Riemannian) Geodesics (or Harmonic maps with a 1D domain!) and turn out to be critical points of a Dirichlet energy. Do these curves also minimise the distance between points? I will introduce all the concepts required to permit the discussion of Geodesics, give some examples and discuss some of their properties.

19 Nov Thomas Burnett Designing adaptive enrichment trials for the pharmaceutical industry

When developing a new treatment a company may be able to identify several sub-populations they expect to receive a benefit. The true benefits for these populations are usually unknown, making it difficult to choose an appropriate population for a confirmatory trial. Adaptive enrichment designs allow this decision to be delayed, changing the trial population after a proportion of the recruitment is complete. I will discuss my work on adaptive enrichment designs and how working with Roche Products Limited has influenced the project.

26 Nov Will Saunders An introduction to Molecular Dynamics and cell based methods for short range interactions

This talk will be similar to the NA one on the 20th. If you do not want the Dave rerun I suggest you just come for the cake.

Molecular Dynamics (MD) has become a major tool in condensed matter physics and chemistry. However useful results often require substantial computational resource to produce. A major proportion of the University of Bath HPC facility is spent on calculations on or related to MD. I will introduce the concepts behind a MD simulation and provide motivation as to why a MD simulation is worthwhile. Furthermore I will demonstrate how a naive O(N^2) approach to computing the short range interactions can be improved to O(N) using cell based algorithms.

3 Dec Matthew Thomas
Not a visitor from Cardiff! But he is Welsh.
Global modelling of air pollution using multiple data sources

Air pollution is an important determinant of health and poses a significant threat globally. The World Health Organisation (WHO) are at the forefront of health modelling and policy development worldwide and must ensure that this is based on accurate and convincing evidence. A coherent framework for integrating data from various sources is required that provides accurate and effective analysis and yield exposure estimates with associated uncertainty. These estimates should be consistent with raw data and provide a means for explanation when there are discrepancies. I will explore the current methodology used within WHO to estimate air pollution levels and how changes in framework can significantly improve model predictions. I will also explain how Bayesian melding can be used to match the requirements for air pollution modelling within WHO and the associated challenges that arise from using this technique.

15 Dec Ray Fernandes PSS Xmaths Special

Come along to the PSS Xmaths special, with plenty of prizes to be won!* As usual there will be minimal maths and even less talk. If all else fails just turn up to laugh when things inevitably go tits up.

*total prize fund may consist of a box of tic-tacs

## 2015-16 Semester 2

### Title and Abstract

4 Feb Marcus Kaiser The Large Deviation Principle

We know from introductory courses to probability the Law of Large Numbers (LLN) and the Central Limit Theorem (CLT). For a sequence of IID random variables, the LLN yields us convergence of the average of these random variables to the mean. The CLT on the other hand identifies the Gaussian nature of small fluctuations (=deviations) around the mean.

In this talk we will present some ideas related to the Large Deviation Principle (LDP) which can be seen as a refinement of the (weak) LLN and the CLT that also yields information on deviations 'far' away from typical events.

11 Feb Leonard Hardiman Some Title Containing the Phrase 'Lie Algebra', e.g. this one

Why does anyone care about Lie algebras? What is a root system? Why do certain strange people sidle up to each other in the dead of night and whisper 'A ¬ D ¬ E' over and over again, then grin maniacally like they're in on some secret cosmic joke? My PSS will expose all!

18 Feb Aoibheann Brady Bayesian hierarchical modelling for quantifying uncertainty in extreme climate indices

I'm going to talk about some data (that I don't entirely have) and some methodology (that I haven't yet implemented) to solve a problem (that I don't yet know).

(I'll discuss the problem of modelling extreme climate indices and quantifying the uncertainty in calculating them. I'll talk about past research in the field, what's needed from new research and how/why I'll implement a Bayesian hierarchical model to look at this. And, to give a SAMBa/applied student perspective, I'll tell you who might care about this research and why!).

25 Feb Ben Robinson Sea Ice: Data, modelling and uncertainty

How can we model the concentrations of sea ice in the oceans around both the Arctic and Antarctic? And why would we want to do this?
Understanding the past climate is a major area of research for the Met Office, with implications for policy making and planning for future climate events. As well as being a quantity of interest itself, sea ice forms a boundary for ocean and atmospheric models, so a better understanding of this would be very valuable.
I will present some initial ideas for a probabilistic model, inspired by the physical properties of the problem. I will talk about how to combine data sources which vary substantially in completeness and accuracy, and how we might account for seasonal variation, possible long term trends, and spatial correlation.

(Pictures of penguins may be included.)

3 Mar Joel Cawte Elasticity and sulcus formation

Most people's knowledge of Elastic Materials only stretches (pun intended) to Hooke's Law, and the Young's Modulus of a material, if you did A level physics. There's only a very simple one-dimensional mathematical equation that arises from this, so where do we go from here? How can we make a 3D version, and does this actually work for all materials?
From a simplified conclusion of what is essentially the Elasticity lecture course available this semester, I'll begin to try and describe more complicated deformations that occur in the real world, such as crease formation (or as worded by my supervisor, 'sulcus' formation), and how this relates to my thesis.

10 Mar Federico Cornalba The introduction of stochastic analysis in the study of the "spinodal decomposition" phenomena: a concise history.

We first describe the physical phenomenon called "spinodal decomposition". This phenomenon occurs, for example, when a binary alloy, which is initially very hot, is abruptly cooled down. Throughout this cooling process, the alloy follows various specific stages during which one can observe distinctive geometric patterns. These configurations are approximated by the solution of the Cahn-Hilliard partial differential equation (CH).
We then provide a concise history of the mathematical results associated with the stochastic analysis of the (CH) equation, starting from its very formulation (1958) and arriving at the contemporary state of the art.

17 Mar Matt Durey Droplets on a vibrating bath: an analogue to quantum mechanics.

A small droplet of silicon oil may bounce on a vertically vibrated bath of the same fluid, creating waves on the bath surface. As the vibration acceleration is increased, the bouncing is destabilised. A new stable regime forms in which the droplet 'walks' across the surface of the bath. The wave-droplet coupling gives rise to analogues of many phenomena previously only seen in the quantum world!
From the linearised Navier-Stokes equations, we derive (without excessive detail) a simplified model for the fluid system. This transforms the complicated PDEs to a system of homogeneous ODEs with jump conditions. Through mathematical analysis, we show that this model still captures many of the complicated dynamics observed, despite its simplicity! Results are presented through many pretty graphs.

Please note: this will be a different talk from my NA seminar and will be free of all numerical analysis.

7 Apr Claudio Onorati How to take a quotient?

In this talk I want to show a (not so) new way to think about manifolds and varieties in geometry which was developed in the last decades. Since the theory is very technical and cumbersome, I will spend all the talk on a simple example, the quotient of a set by a finite group, and I will try to give at least one concrete example in which this construction in very useful.

14 Apr Alice Davis Using Cumulative Hazards for Modelling Survival Data

I shall first introduce what survival analysis is, discussing common techniques for modelling survival data. I will then propose a new mathematical structure for cumulative hazard functions that allows flexible regression models for survival data. This structure includes commonly used models such as the proportional hazards and accelerated failure time models. I will illustrate the methodology using survival data of liver transplants in the UK.

21 Apr Xavier Pellet Different points of view on the famous Heat Equation

If you are a probabilist, analyst, or statistician, you have heard about it. But why do we use so many different formalisms to speak about it? How should we connect these notions? Why do all traders on the financial markets use it without knowing where it comes from? This is a short introduction on the different aspects needed to understand this PDE.
If you are an algebraist, you have to come, for your general culture, to see at least what the heat equation is.

Keywords : Fun, Martingale, Stochastic Process, Gradient Flow, Optimal transport, PDE.

28 Apr James Green What's the difference between a fish and a piano?

Tuning a piano is not as simple as it seems. There have been many different types of tunings throughout history, so what are the advantages and disadvantages of modern tuning: equal temperament? I will attempt to give a brief explanation of various tuning methods: harmonic, Pythagorean, (quarter-comma) meantone and equal.