### Postgraduate Seminar Series (PSS) 2015/16

## The Jarrett Jurisdiction

This is the compilation of the talks given in the 2015/16 Postgraduate Seminar Series which was organised by Kieran Jarrett.

This is the compilation of the talks given in the 2015/16 Postgraduate Seminar Series which was organised by Kieran Jarrett.

1 Oct 2015, **Kieran Jarrett**

... 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 2015, **Katy Gaythorpe**

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 2015, **Alge Wallis **

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 2015, **Pite Satitkanitkul **

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 2015, **George Frost **

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 2015, **Dan Green **

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 2015, **James Roberts **

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 2015, **Thomas Burnett **

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 2015, **Will Saunders **

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 2015, **Matthew Thomas**

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 2015, **Ray Fernandes**

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

4 Feb 2015, **Marcus Kaiser**

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 2015, **Leonard Hardiman**

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 2015, **Aoibheann Brady**

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!).

~~18 Feb 2015, ~~**Aoibheann Brady**

~~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 will briefly discuss an education dataset with two different cohorts and
different means of communicating the course material. The aim is to use multi-level
models (via STAN) to investigate patterns of engagement. This could potentially be used
to suggest mid-course interventions where students aren't engaging, and drive future
means of communicating a course. All opinions and suggestions are welcome!)

25 Feb 2015, **Ben Robinson**

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 2015, **Joel Cawte**

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 2015, **Federico Cornalba**

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 2015, **Matt Durey**

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 2015, **Claudio Onorati**

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 2015, **Alice Davis**

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 2015, **Xavier Pellet**

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 2015, **James Green**

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.