# Postgraduate Seminar 2012-13

**Organiser**: Mark Bell

### Term 3 - The seminars are held on Wednesdays 12:00-13:00 in MS.04 - Mathematics Institute

- Wednesday 19th June 2013

** Chimere Anabanti** - *Free groups as a Mathematics language*

We start with a brief description of free groups. Then visit three proofs that every subgroup of a free group is free: two from Algebra (by Otto Schreier and Jakob Nielsen), and one from Algebraic Topology using basic concepts of Graph Theory. We conclude by establishing its connection with other areas of Mathematics such as Geometric Group Theory, Hyperbolic Geometry and Presentations of Groups as we explore its connection with notions such as Commensurable groups, Hopfian groups, Hyperbolic groups, Quasi-isometry and Schottky groups.

- Wednesday 12th June 2013

**Katherine Rock** - *Vector-Borne Disease Modelling*

The transmission dynamics of vector-borne (VB) diseases such as malaria, west-nile virus and sleeping sickness are of great importance to understanding, controlling and eradicating them. I will outline existing epidemiological models for vector-borne disease and highlight the key results of these before discussing the role of vector senescence and bite rate which is largely ignored in traditional VB disease modelling. A new model incorporating age and bite structure will be developed and the effects upon disease transmission examined.

- Wednesday 5th June 2013

**Heline Deconinck** - *Modular Approach to Diophantine Equations*

Diophantine equations are polynomial equations of finite degree which only allow integer solutions. The focus of the talk will be ternary Diophantine equations, i.e. an equation of the form: .

Fermats Last Theorem (FLT), which is a ternary Diophantine equation, was the first set of equations that was solved using the modular approach method. The proof of FLT will be the motivation and focus of the overview of the modular approach method and some applications will be discussed.

- Wednesday 29th May 2013

**Tomasz Tkocz** - *Gaussian measures and inequalities*

Not only are Gaussian measures ubiquitous in probability theory and beyond, they are also the most exciting objects in mathematics. The modern theory of Gaussian measures is widely applicable in areas such as functional analysis, geometry, statistical mechanics, quantum field theory, financial mathematics, statistics, and many others. Obviously, various estimates play a crucial role in the theory. We shall review some fundamental inequalities concerning Gaussian measures mainly in the geometric context (isoperimetric, concetration, and correlation inequalities, dilations of convex sets, comparison of moments, etc.). We shall discuss some open problems as well.

Only the standard knowledge of linear algebra and analysis will be assumed.

- Wednesday 22nd May 2013

**Simon Bignold** - *Deriving Phase Field Crystals*

In this talk we will consider the relatively new area of Phase Field Crystals (PFC) and its applications. We begin by motivating the PFC model by deriving it from the older and more extensively studied area of density functional theory (DFT). This can be considered as an exercise in reducing the number of independent variables from a very large number to two and thus PFC is much simpler and hopefully more numerically tractable than DFT. This follows the method of [Elder et al, Physical Review B, 75, (2007).]

Having derived PFC we present a new method for minimising the functional of PFC in a two dimensional rectangular domain with periodic boundary conditions. This technique relies on translating our equation into Fourier space and using the Fast Fourier Transform.

We conclude by presenting some of the results of our numerical simulations. We concentrate on produce a unit hexagonal cell and demonstrate how this can be used to simulate a lattice and various defects. This talk is designed to be accessible to anyone with a reasonable degree of mathematical literacy.

- Wednesday 15th May 2013

**Christopher Williams** - *P-adic analysis and the Tate Curve*

In a previous seminar, I discussed complex uniformisation of elliptic curves (i.e. that every elliptic curve is isomorphic to a complex torus). This seminar, which is self-contained (though there will not be time to give all but the briefest exposition of the background theory for ), will follow on from that from a more number-theoretic view point, and will cover material there was not space for.

The p-adic numbers are the completion of the rationals in a different metric to the usual Euclidean one, and they have extensive applications in number theory. In this talk, I will discuss an analogue of the uniformisation theorem, due to Tate, over . Tate proved that every elliptic curve with non-integral j-invariant is isomorphic (over at worst a suitable quadratic extension of ) to a curve of form , where . To do so, I’ll develop some of the theory of p-adic analysis, showing that the field of q-periodic meromorphic functions on K* is a function field of genus one in one variable (where K is a p-adic field).

- Wednesday 8th May 2013

**Benjamin Pooley** -* Upper Bounds on the Cross-Sectional Volume of a Hypercube*

The cube is one of the simplest objects in mathematics, however the question “What is the maximum volume of a cross-section of a cube?” is not as simple as it sounds. In this talk we shall discuss the proof that in n-dimensions the largest 1-codimensional cross-sections have volume √2 together with two generalisations of this result for different codimensions. We introduce versions of the Brunn-Minkowski and Brascamp-Liebu inequalities which are used in these proofs. We may also briefly discuss the Busemann-Petty problem and the Mahler conjecture. This talk is based on an undergraduate project which was supervised by James Robinson and inspired by several results due to Keith Ball.

- Wednesday 24th April 2013

**Gareth Speight** - *Differentiability in a Null Set*

It is well known that a continuous function between Euclidean spaces may be nowhere differentiable. A function is called Lipschitz if it does not stretch distances too much. Differentiability properties of Lipschitz functions are much better than those of continuous functions; Rademacher’s theorem states that the set where a Lipschitz function is not differentiable has (Lebesgue) measure zero. We consider the converse to this question: given a set of measure zero, does there exist a Lipschitz function which is differentiable at no point of the given set? We also discuss porous sets and their connection to differentiability problems. Based on joint work with David Preiss.

### Term 2 - The seminars are held on Wednesdays 12:00-13:00 in MS.03 - Mathematics Institute

- Wednesday 13th March 2013

**Leonor Garcia-Gutierrez** - *Mesoscopic approach to capillary blood flow simulation*

Most blood flow models are macroscopic and do not reach the capillary scale or accurately represent the deformation of red blood cells, while true molecular models make the study of the long timescale behaviour of the capillary flow computationally unattainable. In between these two extreme approaches is where mesoscopic models fit in.

In this talk we will introduce a relatively new family of particle-based mesoscopic models known as Multi-Particle Collision Dynamics (also referred to as Stochastic Rotation Dynamics). We will discuss the great potential these mesoscopic methods have to simulate blood microcirculation, and also the challenges they face.

The talk will be aimed at a general audience: no specific knowledge required.

- Wednesday 6th March 2013

**Thomas Ducat** - *Mori Extremal Extractions that Blow up a Singular Curve*

A cornerstone piece of work in modern birational geometry is Mori's Minimal Model Program - an algorithm for simplifying algebraic varieties up to birational equivalence. Extremal divisoral contractions (or extractions) are one of three possible steps appearing in the MMP. In this talk I will give a basic introduction to birational geometry starting with blow ups, before describing the MMP in dimension 3 and some contractions to singular curves.

- Wednesday 27th February 2013

**Christopher Williams** - *Uniformisation of Elliptic Curves*

Elliptic curves are incredibly interesting objects in number theory. Over the complex numbers, every elliptic curve is isomorphic to a complex torus, i.e. the quotient of the plane by a lattice. In this talk, I intend to discuss the uniformisation theorem over C, before moving on to an analogue over the p-adic numbers due to Tate.

I will try not to assume anything too number theoretic, though there won’t be time to introduce the p-adic numbers in all but the briefest of detail.

- Wednesday 20th February 2013

**Simon Markett** - *Swan’s Theorem: A fundamental relation between topological and algebraic K-theory*

The subject of K-theory goes back to the work of Grothendieck, who defined the group K_{0} to formulate his Grothendieck-Riemann-Roch Theorem (’57). Shortly after Atiyah and Hirzebruch translated the concept to define topological K-theory (’59). Other applications of K-theory lie in number theory (class group), analysis (C*-algebras) and even in physics (string theory). The topic stays up to date with its close relation to motivic cohomology (Voevodsky, mid ’90s).

For this talk we are going to take step back and prove Swan’s Theorem (’62) which establishes a fruitful relation between algebraic and topological K-theory. In particular it says that projective modules are essentially the same as vector bundles over compact Hausdorff spaces, an idea which is ubiquiteous in modern algebraic topology and geometry. If time permits, we will have a glimpse at the Quillen-Lichtenbaum conjecture, which may be seen as a generalisation of Swan’s theorem for higher K-theory.

- Wednesday 13th February 2013

**Christopher Purcell** - *Boundary Properties of the Satisfiability Problem*

Satisfiability is perhaps the most well known problem in computational complexity theory, and is of course NP-complete in general, and under a variety of restrictions. Finding the strongest possible restrictions under which a problem remains NP-complete is important for establishing the NP-completeness of new problems, and for understanding the boundary between tractable and intractable instances of the problem. We use the language of graph theory to address the second issue and reveal the first boundary property of graphs representing instances of Satisfiability. Joint work with Vadim Lozin.

- Wednesday 6th February 2013

**Florian Bouyer** - *Bhargava's Cubes; or four ways to play around with cubes of integers*

In 1801 Gauss gave Binary Quadratic Forms, which can be taken from 2 by 2 integral matrices, a group structure. 40 years later Dirichlet showed a connection between this and Quadratic Rings, a bijection that is still used today to explore in details Quadratic Rings. 10 years ago, Manjul Bhargava came up with the idea of 2 by 2 by 2 cubes of integers and different ways to manipulate them to find information about Quadratic Rings.

In this talk, I will explain what Quadratic Rings and cubes of integers are, so no knowledge of algebra is required. Then we shall see how to construct at least four different objects from our cubes, and how these can be related back to Quadratic Rings.

- Wednesday 23rd January 2013

**David McCormick** - *The Navier-Stokes Equations: what’s the problem (and why should we care)?*

The Navier-Stokes equations 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 explain how far you can get with the current theory, taking a whirlwind tour through much of modern analysis, from Fourier series and functional analysis to Sobolev spaces and complicated compactness results. (No prior knowledge of PDE theory will be assumed.)

**Mark Bell** - *Minimal Train Track Atlases*

Let S be a fixed surface. A train track on S is a trivalent graph with a well defined notion of tangency at each of its vertices. It can be used to describe some, but not all, of the loops on S. By using multiple train tracks we can describe even more loops. In this talk I will discuss the problem of finding the smallest collection of train tracks which between them describe all loops on S.

- Wednesday 16th January 2013

**Polina Vytnova** - *A toy model for the fast dynamo*

The fast dynamo theory addresses the following question. Given a specific PDE, which is important in physics, with two parameters, a vector field and a small real constant ( Reynolds' number) we need to find a smooth vector field with bounded support such that one of solutions grows exponentially fast as the Reynolds number tends to zero. In real 3-dimensional space the question is open. On our way to its solution, we construct a discrete model on the real line, which turns out to be an open dynamical system with random holes, and study its properties.

### Term 1 - The seminars are held on Wednesdays 12:00-13:00 in B3.03 - Mathematics Institute

- Wednesday 5th December 2012

**Tomasz Tkocz** - *My favourite inequality*

The Loomis-Whitney inequality allows one to estimate the measure of a n-dimensional set by the measures of its (n-1)-dimensional projections. In particular, for a subset A in R^3 it reads

|A|^2 <= |A_x||A_y||A_z|,

where |*| denotes the Lebesgue measure, A_z is the projection of A on the xy-plane, etc.

We shall prove another inequality of this type. The talk will not require any more advanced tools than the Lebesgue measure.

The talk will be based on a joint work with Piotr Nayar from University of Warsaw.

- Wednesday 28th November 2012

**Barnaby Garrod** - *Systems of interacting 1–dimensional Brownian motions*

We first review some facts about Brownian motion, including the random walk construction and basic properties. We then turn to interacting particle systems and offer some motivation through modelling physical phenomena. Finally, we meet some of the techniques useful for studying such systems and touch on the known results.

**Jenny Cooley** - *Arithmetic on Cubic Surfaces*

If we take a set of rational points on a cubic surface, by considering secant lines through pairs of points and tangent lines at single points we can generate more points on the surface by looking at where these lines meet the surface again. Given this process, it is natural to ask what the size of a minimal

generating set for a given surface is, i.e. how many points do you need to start with in order to be able to generate all of the rational points on this surface. I will point to some recent results about sizes of minimal generating sets on certain cubic surfaces and a current conjecture. I will define what cubic surfaces and rational points are in my talk.

- Wednesday 21st November 2012

**Thomas Collyer** - *Mapping class group invariant sub complexes of the curve complex*

The mapping class group of a surface S, MCG(S), is the group formed by its self-homeomorphisms, up to homotopy. The curve complex of a surface, C(S), is a combinatorial object encoding information about how curves intersect on the surface. We will briefly review how these two objects interact, before turning our attention to the least complicated non-trivial example: the five-holed sphere. There is already a good large scale understanding of MCG(S) and C(S), but we shall be taking a "closer" look!

The talk will be expository and will feature lots of pictures.

- Wednesday 14th November 2012

**Michael Scott** - *PDEs on curves that form singularities in finite time*

We will look at my current research where one poses the heat equation on a evolving curve where the curve undergoes a finite time singularity and look at what happens to the solution of the equation. We see that there is a power law in the arc length parameter that is satisfied near the singular point. I will discuss the techniques one uses to prove this (from asymptotic analysis of eigenfunctions of operators to the ergodicity of diffusion processes) seeing how analysis and probability play an important role in developing analytical results.

- Wednesday 7th November 2012

**Andrew Chan** - *Gröbner bases over fields with valuations*

Gröbner bases have several nice properties that mean that certain problems in algebraic geometry can be reduced to the construction of a Gröbner basis. For example Gröbner bases allows us to easily determine whether a polynomial lives in some ideal, find the solutions to systems of polynomial equations, as well as having applications in robotics.

In this talk I shall introduce Gröbner bases and how they can be computed using Buchberger's algorithm. We will see problems that arise when trying to adapt this theory to polynomial rings over fields with valuations and how they can be overcome. Finally, we shall discuss how these Gröbner bases have important applications to tropical geometry, algebraic geometry and beyond...

- Wednesday 31st October 2012

**Benjamin Coles** - *Smooth and Discrete Morse Theories*

Classical Morse theory is a well-established method for studying the topology of smooth manifolds by considering the critical points of smooth non-degenerate real functions on such manifolds. In general the fewer critical points such a function has, the more it can tell us.

In the 90’s Robin Forman developed a discrete version of Morse theory for regular cell complexes, and reproduced many of the main results of the smooth theory in this new setting.

In this talk, we will give an introduction to discrete Morse theory, and consider situations where both the smooth and discrete theories are applicable. In particular we will prove that given a smooth Morse function on a manifold, we can construct a discrete Morse function on a regular cell decomposition of that manifold, which has as few, or fewer critical points.

- Wednesday 24th October 2012

**Rupert Swarbrick** - *Spectra and cohomology theories*

Ask me or an algebraic topologist virtually anything and we'll start mumbling about spectra at some point. Worse still, we'll then refuse to elaborate further. This talk is an attempt to make amends!

In this talk, we'll approach spectra following a round-about route, via the subject of generalized (co)homology theories. On the way, we'll take in the Spanier-Whitehead category and see why it's not a great place in which to work. The main high-point will be Brown's full-power ''representability theorem'', which shows that our definition of spectra does the right thing.

If there's time, I'll also talk about more modern models of spectra and why people are still talking about what is the right definition to use even though the first solutions appeared in the sixties.

- Wednesday 17th October 2012

**Owen Daniel** - *Random Permutations on Infinite Sets*

The study of random permutations is almost as old as probability itself, with numerous applications to gambling providing ample motivation. Classical questions include: what is the expected cycle length of a permutation? and how many fixed points are there? More recently people have considered permutation valued processes, perhaps most notably in the work of Diaconis (and others) in studying the number of shuffles a pack of cards needs before it looks mixed.

As is generally true, studying structural questions about permutations on infinite sets is hard. We will introduce one model for random permutations on an infinite set, and ask what exactly it means for such a permutation to have an infinite cycle. Upon deriving a suitable notion, we ask what the probability of such cycles occurring is.

The talk will require only a willingness to hear about probability, and no prior knowledge of random permutations. If time permits, we will treat any algebraists in the audience to a description of how probabilists have learnt to enjoy representation theory.

- Wednesday 10th October 2012

**Damiano Lupi **- *A quick proof of McShane’s identity*

McShane’s identity is an important result about the lengths of simple closed geodesics on a once-punctured torus equipped with a complete, finite-area hyperbolic structure. In this talk I will show a quick proof of the identity which is unexpectedly based on Markoff triples. Although this is just the simplest of a series of related results and generalisations, it gives a quite clear idea of the techniques that are generally used in order to prove these kinds of identities. Some very basic knowledge of hyperbolic geometry would be helpful.

- Wednesday 3rd October 2012

**Robert Fryer **-** ***Dynamics of quasiregular mappings with constant dilatation*

The dynamics of quadratic polynomials have been extensively studied. In this talk, we will investigate how features of this theory generalize to the quasiregular setting, in particular focusing on the situation of the composition of a quadratic polynomial and an affine mapping. Analogues of Bottcher coordinates are constructed for these mappings. We will show that three different types of dynamics occur and how results from complex dynamics, and in particular Blaschke products, give rise to these different situations. Relevant background material will be introduced, so this talk should be accessible to all.