## The Age of Pressland

This is the compilation of the talks given in the 2012/13 Postgraduate Seminar Series which was organised by Matthew Pressland.

# Semester 1 2012/13

2 Oct 2012, Matthew Pressland

### The Fundamental Group and Triangulations

The fundamental group is an important homotopy invariant of topological spaces, essentially describing the ways in which loops can be mapped into them. This talk will (loosely) explain the definition of the fundamental group and why it is a group, and then explain how to calculate it using triangulations, with some examples from $$3$$-manifold topology.

9 Oct 2012, Horacio González

### Wazzup with the WASEP?

This talk will revise the physical motivation of interacting particle systems. From there we will define what is the Simple Exclusion Process and all its variations suchas the symmetric, totally asymmetric and weakly asymmetric. Finally we will talk about the hydrodynamic limits of these processes. Our approach will be very intuitive and no knowledge of statistical physics or stochastic processes is required.

16 Oct 2012, Andrew Bate

### Predators, Prey and Prevalence

It's October. Autumn is falling upon us; the mixture of rain and fallen leaves become a slipping hazard; squirrels are hoarding for the winter and the freshers' flu epidemic hits its peak. I sometimes wonder: do squirrels get freshers' flu too? I do not know the answer, but I could consider the consequences.

For this talk, I will give an introduction to a simple disease model and a simple ecological model (predator-prey) before combining the two models to get some interesting results.

23 Oct 2012, Alex Watson

### Discrete and Continuous Family Trees

I will showcase some nice objects and results from the theory of random processes: some discrete objects, which date back to a letter in the Times from 1873, and some continuous ones, which are at the cutting edge of current research.

Galton-Watson processes are random processes which count a population, while Galton-Watson trees are randomly evolving discrete trees which show the genealogical record of a population. Galton-Watson trees are characterised in the set of (discrete) random trees by a certain regenerative property. A recent paper proves that the same is true in a continuous context: here, we deal with continuous-state branching processes, which correspond to Levy trees, which are then neatly characterised among the set of all real trees. Since these objects are all entertainingly weird, I will hand-wave my way through some vague definitions and draw some pictures in order to very approximately demonstrate how all of this works.

30 Oct 2012, Alex Collins

### The Classical McKay Correspondence

The McKay correspondence describes a mysterious relationship between the representation theory of certain finite groups and the structure of the singular surface that arises from their action on the plane. I will tell you about it.

6 Nov 2012, Elvijs Sarkans

### Time Invariance for Dynamical Systems

In classical mathematics time invariance for dynamical systems revolves around autonomous differential equations and for good reason: many areas of physics fit this framework very nicely. However there are some drawbacks of this approach in control and signals processing. In 1980s Jan C. Willems proposed an alternative first principles approach to studying time invariant systems and this theory features some beautiful yet reasonably simple mathematics. I will try to present some of these ideas for linear discrete time systems.

13 Nov 2012, Anthony Masters

### Measure Resolutions and Rearrangements: Utilising an Old Idea

Measure resolutions are a rarely-used concept, but are related to the non-atomic properties of a measure space. I will show that measure resolutions may be utilised to remove a standard assumption on the underlying measure space for results involving rearrangements.

20 Nov 2012, Curdin Ott

### Russian options, American options with floating strike or Bottleneck options

I'll try to explain in a very informal way and with the help of pictures what these fancy names stand for and what their connection to the area of "optimal stopping" (a specific area in probability theory) is.

27 Nov 2012, Matthew Dawes

### Moduli of Irreducible Symplectic Manifolds and K3 Surfaces

The theory of K3 surfaces is vast and fascinating and has origins in classical algebraic geometry. In higher dimensions, one may generalise K3 surfaces in two directions: one may study Calabi-Yau varieties or one may study Irreducible Symplectic Manifolds (also known as hyperkähler manifolds). I am interested in the latter. Over the last sixty years, many important questions about K3 surfaces have been answered but very much less is known about hyperkähler manifolds in general. In this talk, we will mostly be interested in discussing their moduli. I will start by showing how questions about the birational geometry of the moduli of K3 surfaces can be understood by studying modular forms and then move on to discussing how the K3 theory generalises to the hyperkähler case. I will also mention some open problems. There will be no proofs but there will be some pictures and there should be enough background to satisfy a general audience.

4 Dec 2012, Jack Blake and Steve Cook

### The Mathematics of Starcraft

We will introduce the computer game 'Starcraft 2' and give some examples of how simple mathematical models can help understand different strategies of play, as well as mentioning some basic maths that can be applied whilst playing. We will also talk about how user input can show that the assumptions required for our models are wrong. This will be a very informal talk, aiming at giving an overview of the game and an insight into its depth of strategy. No knowledge of Starcraft 2 is required.

10 Dec 2012, Ray Fernandes

### XMaths Special

In true Xmaths PSS style, expect much craziness, but not much in the way of maths or talk. What it's on is always kept a secret (often because I don't know myself), but anyone that's been to a previous Xmaths special will know to expect the unexpected. If things go badly at least there will be cake.

# Semester 2 2012/13

5 Feb 2013, Matthew Pressland

### Rings and Varieties: An Introduction to Algebraic Geometry

This talk will be a gentle introduction to algebriac geometry, using commutative algebra. I will define varieties and ideals, and briefly discuss Hilbert's Nullstellensatz, the Zariski topology, and coordinate rings. The punchline will be to use the algebra to explain why degree $$n$$ polynomials over $$\mathbb{C}$$ have exactly $$n$$ roots, even though some of them clearly don't; in other words, I will explain why some roots are counted more than once.

12 Feb 2013, Maren Eckhoff

### Solving Partial Differential Equations with a Cloud of Smoke

We discuss two or three stochastic processes and associated parabolic differential equations. The close connection between probability and PDE theory provides intuition about the coefficients of these deterministic PDEs in terms of properties of random processes.

19 Feb 2013, George Frost

### A Brief History of Geometry

In this non-technical talk, I will give a brief history of the study of geometry. Geometry has evolved somewhat from Euclid's study of triangles and circles; ultimately, the modern generalisation of what features a "geometry" should have is encapsulated in a so-called "Cartan geometry". There are two possible routes to this generalisation from familiar Euclidean geometry: Riemannian geometry and Kleinian geometry. I will describe in basic terms each of these four approaches, and give some justification for why Euclidean geometry is not enough for modern geometers.

The talk will be entirely non-technical, assuming no prior (non-Euclidean!) geometrical knowledge.

26 Feb 2013, Jenny Jones

### Reverse Iontophoresis as a Technique for Drug Monitoring

Reverse iontophoresis is a relatively new non-invasive drug monitoring technique in which a small current is passed across the skin causing an ion flow to the surface where it is collected, allowing the amount of drug to be measured and hence an accurate estimate of the blood drug concentration.

Initial reverse iontophoresis readings are unusually high and unrepresentative of the blood drug concentration suggesting that ions are at first being collected from somewhere other than blood; one such candidate is the dead cells that make up the surface layer of the skin (stratum corneum).

I will give a little more detail about the above and an explanation of what this could mean for drug monitoring in reality. I will then go into how I am approaching the modelling of this and probably explain "Michaelis-Menten kinetics" and anything else that seems relevant at the time.

5 Mar 2013, Sam Gamlin

### An Introduction to the Abelian Sandpile Model and the Connection to Uniform Spanning Trees

The Abelian Sandpile model is a probabilistic model defined on a graph, which was originally proposed as an example of self-organised criticality. In this talk I will define the model and then show how it is related to Uniform Spanning Trees. No prior knowledge will be assumed.

12 Mar 2013, Aretha Teckentrup

### Numerical Methods for Ordinary and Stochastic Differential Equations

I will give a simple derivation of Euler's method for ordinary differential equations, and show how the same approach can be used to derive the Euler-Maruyama method for stochastic differential equations (I will explain what these are). I will then show how the same approach can in fact be generalised to derive numerical methods of arbitrarily high order, and point out some of the difficulties in going from ordinary to stochastic differential equations.

19 Mar 2013, Zheyuan Li

### A Family Tree of Statistics and Practical Statistical Modelling

Statistical modelling of data has never been more popular than it is today. However, when given a data set, we are likely to feel lost in the ocean of existing models. Sometimes we are even not so clear why a specific kind of model is as it is. This talk is aimed to depict you a picture of this, by drawing an interesting family tree in world of statistics. In particular, by looking at the black smoke data from a long established monitoring network in UK from 1966 to 1996, we will together examine a branch of this tree: "statistical inference $$\to$$ parametric statistical inference $$\to$$ frequentist approach $$\to$$ linear models $$\to$$ nonparametric regreesion $$\to$$ additive models $$\to$$ additive mixed models". The talk is designed to be introductory, and fun.

9 Apr 2013, Ben Boyle

### Similarity Solutions of PDEs

Similarity methods are some of the most powerful available for learning about solutions to PDEs, especially nonlinear ones. In this talk I will (very) briefly introduce concepts from Dimensional Analysis and show how they can be applied to to find 'self-similar' solutions to certain PDEs.

16 Apr 2013, James Clarke

### Designing Optimal Controls for Networked Disease Systems: Chlamydia Gets the Treatment

Everybody is worried about chlamydia. I hope to enhance everyone's calm by describing a method whereby deterministic models are used to determine optimal controls for stochastic networks. By extracting specific information from networks it is possible to parametrise a certain simple model. This can then give you the means to force a network to do whatever you want it to. No knowledge of maths will be assumed.

23 Apr 2013, James Roberts

### The Direct Method in the Calculus of Variations

In the calculus of variations, the direct method provides a way of constructing minimizers of sufficiently 'nice' integral functionals. (I will explain what these things are). Motivated by the classical Weierstrass theorem on extrema for continuous functions we will first collect some topological and functional analitic results. In the right setting (which will be provided) we may combine these results to give an existence theorem for minimizers of a certain class of functional.

30 Apr 2013, Katy Gaythorpe and Amy Spicer

### Fun with Plagues

An informal introduction to mathematical epidemiology.