## The Epoch of Roberts

This is the compilation of the talks given in the 2014/15 Postgraduate Seminar Series which was organised by James Roberts.

# Semester 1 2014/15

2 Oct 2014, James Roberts

### Fractional Harmonic Maps

Fractional Harmonic Maps in $$m$$ dimensional space can be understood by considering a PDE in an $$m +1$$ dimensional half space, together with an appropriate boundary condition. I will discuss solutions to the PDE and how they relate to the fractional Laplace operator as well as some theory from the Calculus of Variations.

9 Oct 2014, Horacio G. Duhart

### Hipster maths: Counting with complex numbers (because natural numbers are too mainstream)

This talk shall be considered as the ultimate beginners' introduction to Analytic Combinatorics. I will attempt to explain what this concept means and how techniques from Complex Analysis are involved in such discrete and finite objects. If you know a little about this subject, then it'll be great to have you there so that you may answer my questions. However, if you know nothing about this, come and learn before it is cool.

16 Oct 2014, Katy Gaythorpe

### Natural disasters and disease

In a crisis situation, what is the best way to use limited time, money and resources to minimise the spread of infectious diseases? Consider the threat of environmentally transmitted diseases such as Cholera to a developing world city after severe flooding- how should we control them? I shall talk through abstracting this problem into a mathematical model in which epidemiological dynamics occur through a heterogeneous metapopulation. We will examine different analyses and use different techniques to highlight the most effective control strategies . I shall show that the outcome depends on the nature of the heterogeneities in the system and the target criteria. These insights provide a framework in which to make better informed decisions about the most efficient deployment of limited resources to contain infectious outbreaks in the aftermath of a natural disaster.

23 Oct 2014, George Frost

### Quaternionic Manifolds; or, Complex Geometry isn't Complex Enough

Frobenius' theorem tells us that the only associative normed division algebras over R are R, C and H, so it's natural to study geometries which have the structure of the quaternions. I will give a gentle introduction to the quaternions and to manifolds, before defining quaternionic manifolds and giving some examples. I will finish by answering the question "why on earth would you want to do this!?" with some applications in Physics and Pure Mathematics.

30 Oct 2014, Carlos Galeano Rios

### Bouncing droplets: A problem with many unsuspected repercussions

As early as in 2005 it was already shown by Yves Couder et al (Paris 7 Denis Diderot) that a liquid droplet can be kept in vertical periodic motion as it bounces on a bath of the same fluid, provided the bath is shaken vertically. The bath then acts as trampoline that keeps shooting the drop upwards and also propagates waves on its surface as a natural consequence of the impacts. Hence the droplet bounces on its own wave field creating what can be said to be the first known macroscopic object to present wave and particle behavior simultaneously. I will give a short overview of the problem and comment on some mathematical approaches for it, some of which are being developed here, at the University of Bath.

6 Nov 2014, Nathan Prabhu-Naik

### Life on the projective plane

I will give various basic constructions of the projective plane and explain why we may want to use it. Most of my talk will involve me drawing lots of pictures and waving my hands, avoiding all technicalities.

13 Nov 2014, Sam Gamlin

### An electric talk on graphs

In this talk I will show how electrical networks are related to random walks and spanning trees.
I will explain probabilistic interpretations for current and voltage, before culminating in the transfer-current theorem, which tells us about the probability of a set of edges being in a given spanning tree.

20 Nov 2014, Matt Pressland

### Non-Commutative Geometry and Quivers

A standard approach in geometry is to study a space by instead studying the ring of functions on it, replacing geometric problems by algebraic ones. These rings of functions are always commutative (meaning $$fg=gf$$ for any two functions $$f$$ and $$g$$) but it is sometimes helpful to consider non-commutative rings as if they were rings of functions on some "non-commutative space". I will explain some applications of this point of view to resolutions of singularities. This lofty ambition will mainly serve as a pretext for introducing a few more straightforward geometric and algebraic constructions (modules, quivers and simple GIT quotients) to non-experts.

27 Nov 2014, Thomas Burnett

### Adaptive Enrichment Designs for Clinical Trials

Clinical trials are conducted by recruiting from a pre-determined study population. It may be possible to split this population into sub-populations. The challenge becomes knowing whether to focus our trial on one of these sub-populations. Adaptive Enrichment designs offer flexibility in this choice, allowing for adjustment of the study population during the trial. We will discuss how an adaptive enrichment trial may be conducted, in particular how to maintain control of the type 1 error rate.

04/12/2014 2014, Ben Boyle

### Will no one rid me of this turbulent nonlinear PDE? - An overview of the Navier-Stokes Equations

The Clay Institute's Millenium Problems include one of the most enduring mysteries in all of analysis - given any reasonable initial data in $$\mathbb{R}^3$$, can we be assured that the solution of the Navier-Stokes equations exists and is smooth into perpetuity? In this talk we will fill in some of the history of the problem, establish exactly why it has thwarted the best and brightest for so long, and conclude with an overview of some of the progress that has been made, hopefully without getting bogged down in details.

11 Dec 2014, Euan Spence

### PSS XMaths special

In many ways, the PSS XMaths specials are a bit like supermarket Christmas adverts: to some they represent a cheesy, but heartwarming experience, to others they represent all that is wrong with our celebration of the festive season. Come and find out which side of the fence you're on next Thursday! Note that, unlike Christmas adverts, at the end of the XMaths PSS you get to eat the cake (and not just look at it!).

# Semester 2 2014/15

5 Feb 2015, Horacio G. Duhart

### The Marvels of Gamma - Convergence

What is Gamma - convergence? What is it for? Is this a probability or an analysis talk? What is homogenisation? Why is it that in any talk I give I have to talk about large deviations? And how are these related in any way? As a talk that seems to have more questions than answers approaches, assemble to the first talk of 2015 to be introduced to and marvelled by a bunch of interesting concepts... who knows you might even find them useful!

12 Feb 2015, Jack Blake

### Domain Decomposition: what is it and what can it do for you?

In this talk we will be considering domain decomposition methods for solving the neutron transport equation: a linear integro-pde of interest within the nuclear industry. After motivating why this is a good idea, we will outline two possible approaches to building a simple domain decomposition method and will talk about the advantages and disadvantages of each. If this all sounds too mathsy for you, then I invite you to attend for the brightly coloured diagrams and not-so-subtle film references instead.

19 Feb 2015, Sandra Palau Calderon

### Branching processes

Branching processes are mathematical representations of the development of a population whose members reproduce and die, subject to laws of chance. In this talk I will discuss some properties of branching processes. I will start with the simplest case, the Galton-Watson process, and I will finish with continuous state branching processes in a random environment.

26 Feb 2015, James Roberts

### The singular set of Harmonic maps

Given a solution of a PDE, the next thing to ascertain is how regular this solution is. More regular solutions are more nicely behaved. Some PDEs will admit completely smooth solutions but sometimes solutions are discontinuous everywhere! Restricting to a subclass of solutions can yield better results; Harmonic maps from an open subset  of $$\mathbb{R}^m$$ to a compact subset of  $$\mathbb{R}^n$$ have this property. Harmonic maps in this setting satisfy the Euler-Lagrange equation for the energy functional and may be discontinous everywhere. If we require that a harmonic map satisfies an additional stationarity condition then, at worst, the map will be smooth away from a 'small' singular set. If we have an energy minimiser it is even possible to reduce the potential size of the singular set. I will give an introduction to harmonic maps and discuss the ideas used to reduce the size of the singular set.

5 Mar 2015, Amy Spicer

### Liquid crystals

Liquid crystals are a phase of matter between a solid and a liquid. The molecules in a liquid crystal sample prefer to align in the same direction, this results in interesting optical properties. I will introduce the mathematical modelling of liquid crystals and show how we map transitions between liquid and liquid crystals states.

12 Mar 2015, George Frost

### Projective differential geometry

We all know the shortest distance between two points in euclidean space is the straight line between them. However, on the sphere the shortest distance is an arc of a great (i.e. equatorial) circle. In the context of riemannian geometry, the notion of distance-minimising curves are encapsulated in the geodesics of the metric.
Projective differential geometry is a classical subject concerning riemannian metrics which have the same geodesics. In this talk I will give an introduction to manifolds and riemannian geometry, before discussing some of the key questions (and their answers) in projective differential geometry. Expect no formal arguments and copious amounts of hand-waving.

19 Mar 2015, James Green

### Resolving the Kleinian surface singularities

In 1884 Felix Klein classified the finite subgroups of $$SL(2,\mathbb{C})$$ (complex 2x2 matrices with determinant 1). We can use these groups to create surfaces in $$\mathbb{C}^3$$. Each surface contains a singularity, which has a resolution graph identical to an ADE Dynkin diagram (special connected graph).
I will attempt to give a whistle-stop tour of this result, which begins with Platonic solids and ends with blowing stuff up. In continuing the proud tradition of PSS, I will be waving my hands a fair amount.

26 Mar 2015, Jennifer Jones

### Modelling the Formation of a Buprenorphine Reservoir in the Stratum Corneum

It has been shown that prolonged presence of a drug in the body can cause a build-up of that drug in the skin. This ‘reservoir’ of drug, if understood, could provide information about the recent drug taking history of the patient.
In this talk I will discuss the structure of the model I have created to describe this system, present some results and discuss the implications of the modelling and results for drug monitoring

16 Apr 2015, Elizabeth Arter

### Low-rank approximations of matrices arising in problems involving the Laplace operator

The solution of many Boundary Value Problems for the Laplace equation involve evaluating matrix-vector products where the matrix arises from sampling the fundamental solution on a grid of points. In this talk I will show these matrices can be approximated by low-rank matrices, a property which can be used to speed up the evaluation of the matrix-vector products. Since it's just been Easter I may add 5 minutes of bonus material on Christianity/Mathematics.  Since it's the PSS there is likely be some handwaving.

1 May 2015, William Holderbaum

### Immersion in Reductive Homogeneous Space with Application to 3 Dimensional Homogeneous Manifold

This talk will aim at demonstrating the existence of an immersion into a reductive homogeneous space. This generalization will give an extension to the Bonnet theorem with a focus on isometric immersions into 3-dimensional homogeneous manifolds. In particular we will deduce a necessary and sufficient conditions for isometric immersions into 3-dimensional homogeneous manifold. The main ingredient is the interpretation of the Lie group acting on the ambient space.