# Postgraduate Seminar 2011-12

### Term 3 2011/12 - The seminars are held on Wednesdays 12:00 in MS.04 - Mathematics Institute

**Organiser: Robert Tang**

- Wednesday 9 May 2012

**David McCormick**

*(Magnetic) Fluid Dynamics from a PDE-Theoretical Point of View*

The Navier-Stokes equations, a system of nonlinear partial differential equations (PDEs), are probably the simplest possible model that accurately captures the complex behaviour of fluid flow. However, it remains one of the most important open questions in mathematics whether the Navier-Stokes equations in three dimensions have a unique solution for all time, given any initial conditions. In this talk, I will give a gentle introduction to the theory and language that has been built up over the last 100 years to formulate and solve such problems of existence, uniqueness and regularity, and give some idea of the key difficulties in proving existence and uniqueness in three dimensions. I will then describe the related models of magnetohydrodynamics (MHD), which concern a fluid with a magnetic field. If there's time I hope to explain an "analogy", originally suggested by Moffatt (1985), linking the Euler equations and MHD in an unexpected way.

- Wednesday 30 May 2012

**Gareth Speight**

*Porosity and Differentiability*

A set is porous if each point of the set sees nearby holes of radius proportional to their distance away. Porous sets are in some sense small - they are nowhere dense and have Lebesgue measure zero in Euclidean spaces. Surprisingly, in infinite dimensional Banach spaces the size of porous sets has applications to Frechet differentiability of Lipschitz maps.

We give precise definitions, state several generalizations of Rademacher's theorem on Lipschitz maps, discuss intersections of surfaces with porous sets then state a recent theorem on Frechet differentiability involving the size of porous sets.

- Wednesday 13 June 2012

**Janosch Ortmann**

*The KPZ universality class*

A large class of processes arising in various branches of the sciences can be described by Gaussian statistics in the large-scale limit. Because of the central limit theorem one expects this kind of behaviour to happen whenever some independence assumption can be made,

However, there are also systems where one does not have Gaussian-like scaling limits. Recently, a different class of models, called the KPZ universality class, have received much attention from probabilists and mathematical physicists. I will describe some of the scaling behaviour that is shared by models in this class. Most of the talk will be focused on examples, ranging from interacting particle systems, surface growth and polymers to random permutations.

A remarkable role in this story is played by a distribution that occurs in random matrix theory.

### Term 2 2011/12

#### Organiser: Robert Tang

- Wednesday 11 January 2012

**Jenny Cooley**

*Generating points on cubic surfaces*

The seminar is about a recent theorem of Siksek that shows a special case where all the rational points on a cubic surface can be generated from one point. I will define a cubic surface and talk a little about some interesting properties of cubic surfaces, introduce the idea of rational points and how one generates points, talk about the four lemmas that make up the proof of the theorem and, if there is time, prove the first lemma to give a flavour of how these sort of proofs work.

- Wednesday 18 January 2012

**Mark Bell**

*The thin parts of Teichmuller space*

A surface is a metrizable topological space and its Teichmuller space parametrizes the different hyperbolic metrics on it. In the first half of this talk we will discuss some of the properties of Teichmuller space including a natural coordinate system. While in the second half we tackle the problem of determining the length of the shortest curve in one of these metrics on the surface. This talk will be largely picture based, with no assumed knowledge of geometry or topology.

- Wednesday 25 January 2012 (Open day)

**Sergio Morales**

*Revenue management on self organized manufacturing plants.*

In the context of manufacturing plants, revenue management focuses on creating policies (pricing, due date quoting) to determine whether an incoming job is profitable or not. The policies should not only take into account the current state of the system but also possible future jobs which might be more profitable had the current job been rejected. The talk should start by setting the problem and explaining basic --but fundamental-- concepts such as local dispatching rules (as opposed to global scheduling), opportunity cost, and disturbance. These concepts will serve to explain and justify the necessity of ADP approximate dynamic programing) as a possible tool for determining the best set of policies which account for the uncertainty inherent to the process. Furthermore optimal price sampling might be addressed as a derived sub-problem to be treated.

**Robert Tang**

*Orbifolds and musical chords*

One goal of topology is to classify all objects of a certain kind. Quite remarkably, the set of such objects can often have a natural topology associated to it. In my talk, I will describe the spaces of musical chords (with at most 3 notes) and demonstrate its nice orbifold structure. Expect lots of pictures and a chance to take home your very own hand-made orbifold!

- Wednesday 1 February 2012

**Maths Relay**

- Wednesday 8 February 2012

**Richard Webb**

*Train tracks and the curve graph*

The curve graph is a space associated to a surface that is used in the theory of hyperbolic geometry, mapping class groups, and more. The graph is locally infinite, and typically there are infinitely many geodesics between any pair of vertices. We define what a tight geodesic is and prove there are finitely many between any pair of vertices. Along the way we shall define what a train track is. Train tracks have other applications, including a proof of the Nielsen-Thurston classification of homeomorphisms up to isotopy.

- Wednesday 15 February 2012

**Taro Sano**

*Deformation-obstruction theory for algebraic varieties*

Deformation of an algebraic variety is often reduced to the study of infinitesimal deformations over Artinian rings like k[t]/(t^2), k[t]/(t^3),... There are obstructions to lift deformations further at each steps. I will explain how to study these obstructions and treat some examples of algebraic varieties with unobstructed deformations.

- Wednesday 29 February 2012

**Michael Scott**

*Stochastic Partial Differential Equations on Evolving Riemannian Manifolds*

The theory of SPDEs on evolving manifolds is devoid in the mathematical literature, at odds with the deterministic counter-part. The aim of this talk is to introduce the theory of SPDEs on evolving Riemannian manifolds, in the variational setting to the analysis of SPDEs. I will motivate the topic, look at the general theory of SPDEs in the variational setting before producing the theory in the simple case. Don't worry, there will be examples!

If I have time, I will discuss the issue of coupling the solution to the metric

- Wednesday 7 March 2012

**Maria Veretennikova**

*Control fractional dynamics*

Firstly, you will be introduced to fractional calculus which has recently gained popularity in a wide range of fields, in particular establishing itself useful in modeling anomalous diffusion by suitable continuous time random walks (CTRWs). Secondly, you will see how to write a dynamic programming equation for the payoff function for a process in our consideration which is derived from a controlled CTRW, and how scaling affects it. You will see the new equations derived in my research for the different versions on the process. We will then discuss resulting fractional Hamilton Jacobi Bellman type equations for the payoff functions.

- Wednesday 14 March 2012

**Michael Selig**

*Constructing Fano 3-folds with Tom and Jerry*

Fano varieties occur naturally in Algebraic Geometry during attempts at classification (they are a possible outcome of running the Minimal model program). Fano 3-folds are almost completely understood in codimensions upto 3, but less so in higher codimensions, due to the lack of a structure theorem for the corresponding algebras.

I explain how to construct Fanos in codimensions 1-4, and the role of the matrices Tom and Jerry in doing so. The talk will be basic: hopefully no prior knowledge of any algebraic geometry is required to follow the fairly simple calculations.

### Term 1 2011/12

- Wednesday 12 October 2011

**Thomas Collyer**

*Braid Groups are biautomatic*

The notion of an automatic group was introduced by Thurston in the 80’s and the subsequent theory was largely developed at Warwick. In this talk, we will briefly review the abstract theory before illustrating it following Thurston’s account as applied to braid groups. This work in turn was largely influenced by a remarkable paper of Garside. Only very basic knowledge of group theory will be assumed (Cayley graphs, symmetric groups etc).

- Wednesday 19 October 2011

**David Kelly**

*A gentle introduction to rough paths*

The theory of rough paths is a fairly recent one, developed by Terry Lyons and others over the last 15 years. The main aim of the theory is to solve differential equations driven by irregular signals, without doing any averaging over the signal. For example, one might want to solve a general SDE for a fixed draw of Brownian motion, which, to a young probabilist, should sound a bit odd. The task amounts to providing a pathwise definition for integrals that are driven by signals too irregular for ordinary techniques to be useful. We will go through some of the basic ideas behind rough paths with a few nice examples. We will also show how one can use rough paths to define a 'chain-rule' for this class of differential equations, which, to a probabilist, is some kind of Ito formula, where Brownian motion is replaced by an arbitrary continuous path. No knowledge of probability is required, almost surely.

- Wednesday 26 October 2011

**David Holmes**

*Links between geometry and arithmetic*

We show how to give a geometric structure to questions from number theory, and how to translate some constructions from geometry (intersection theory, homotopy theory) to this context with interesting results. This talk will be basic, with no assumed knowledge of number theory or geometry.

- Wednesday 2 November 2011

**Stephen Tate**

*Combinatorial Species of Structures and Statistical Mechanics*

The notion of Combinatorial Species of Structures was introduced late in the last century to unite various notions of structures used in Combinatorics. I will introduce the basic objects of this theory and the basic ideas. I will then proceed to introduce the important operations relevant to Statistical mechanics. From this I will present the standard relationships and how they are used to relate Virial Expansions and Mayer Expansions. I will present my own (shorter) proof of the relationship and give a simple application.

- Wednesday 9 November 2011

**David Howden**

*How to Construct a Baby Monster*

Back in the 70s when computing was still very much in its infancy, mathematicians started to develop algorithms to put this new technology to good use (and save them doing laborious calculations by hand!). At the same time a lot of work was being done by group theorists to complete the now infamous Classification of Finite Simple Groups. The talk will focus primarily on permutation groups and how they can be efficiently manipulated on computers. We'll then give a brief outline of the classification, and explain how permutation group algorithms were used to produce the first constructions of the Lyons group (1973) and Baby Monster group (1977).

- Wednesday 16 November 2011 (Open day)

**Nicholas Korpelainen**

*A Tale of Graceful Trees and Generalised Friendship: Two Conjectures by Anton Kotzig*

Once upon a time, there lived two graph theoretic conjectures, apprentices to the late Slovak-Canadian mathemagician Anton Kotzig: One of them, kind and good-natured, grew up to be crowned theorem, years after its master's death. The other one, the evil spawnchild of Kotzig and the late German co-mathemagician Gerhard Ringel, remains chaotic and unresolved. To this day, it possesses a level of tragic allure that Kotzig himself described as a 'disease'. Dear folks, now it is our turn to get sick! Prerequisites: a puzzled mind and a morbid sense of curiosity.

**Charles Brett**

*A phase field approach to image reconstruction*

We will talk about the inverse problem of reconstructing a piecewise constant function from blurred and noisy data. This has many applications, such as reading a barcode or finding the boundary between different types of tissue in a medical image. We will formulate the problem in a PDE setting using a phase field approach. We then discuss and demonstrate numerical methods for solving it

- Wednesday 23 November 2011

**Dintle Kagiso**

*Dimension of Fat and Golden Gaskets*

We look at the dimension of gaskets generated by special ratios, in particular when the ratios are some algebraic numbers. No previous knowledge of fractal geometry or measure theory is required. The talk will be basic.

- Wednesday 30 November 2011

**Sebastian Vollmer**

*Sampling, MCMC and Spectral gaps in infinite dimensions*

The problem of generating data from a probability measure is fundamental for among others integration and statistics. In many interesting cases integration can be approximated by the sample average with respect to a probability measure. In Bayesian statistics a prior believe is combined with a model and noisy data to a posterior distribution, sampling from this distribution allows to make inference from the data. In the first half of this talk we will give a basic introduction to this problem and the second half we present some new results for the same problem on an infinite dimensional Hilbert space.

- Wednesday 7 December 2011

**Dayal Strub**

*Dividing surfaces and transition states*

There are many physical problems, ranging from finding reaction rates in molecular dynamics to capture and escape processes in celestial mechanics, all of which can be studied by considering the rate of transport of phase space volume between different regions for some Hamiltonian system. I will first introduce the Hamiltonian formalism and the problem, then explain the dividing surface method used to obtain an upper bound on the rate of transport, and finally give some examples of how bifurcations can cause changes in the topology of the various submanifolds we shall consider. The talk will be full of pictures and easy to follow.