# Postgraduate Seminar 2013-14

**Organiser**: Florian Bouyer

### Term 3 2013 -14 - The seminars are held on Wednesday 12:00 - 13:00 in MS.04 - Mathematics Institute

Week 9: Wednesday 18th June

**Giannis Moutsinas **- *Asymptotic series and Borel-Laplace summation*

When searching for solutions to equations in the form of series, more often than not they happen to be divergent. These divergent series can be given meaning as asymptotic to the proper solution. The Borel-Laplace summation is a method to obtain a proper solution by the asymptotic series and can be applied to a surprisingly wide range of problems. This talk aims to be an elementary introduction to the subject and no prior knowledge will be assumed.

Week 8: Wednesday 11th June

**Gareth Tracey **- *How many elements does it take to generate a finite permutation group?*

For a group $G$, let $d(G)$ denote the minimal number of elements required to generate $G$. For example, consider the group of symmetries of a regular n-gon. Every symmetry can be written as repeated compositions of a rotation $\sigma$, and a reflection $\tau$. That is, the dihedral group $D_{2n}$ can be generated by two elements, so $d(D_{2n})\le 2$ (if $n>1$, then $D_{2n}$ is not cyclic, so we clearly have $d(D_{2n})=2$).

The dihedral groups of order $2n$ are examples of permutation groups of degree $n$, so can we come up with a ''nice" upper bound on $d(G)$ (in terms of $n$) for a general finite permutation group $G$? What if we restricted our attention to special classes of permutation groups, such as primitive or transitive groups? These questions were first looked at in the late 1980s, after the classification of finite simple groups was announced, and the recent trend has been to come up with results of the form: ''there is a constant $C$ such that, for each (primitive/transitive/soluble/nilpotent/etc) permutation group $G$ of degree $n$, we have $d(G)\le Cf(n)$", where $f$ is some function of $n$.

In this talk, we will look at ways to estimate the constant $C$; we will also look at some of the ways in which $d(G)$ has been bounded for $G$ a general finite group; finally, there will be a brief digression to discuss the classification of finite simple groups.

Only basic group theory is assumed.

For slides from the talk click here

Week 7: Wednesday 4th June

**David McCormick **-* Fractional derivatives, functional analysis and fluid dynamics*

How do you take ''half'' a derivative? Taking a non-integer number of derivatives --- defined by means of Fourier analysis --- turns out to be an incredibly useful concept in PDE theory, and function spaces based on so-called "fractional derivatives" play a fundamental role, particularly in the theory of the nonlinear PDEs involved in fluid dynamics. In this talk, I will define fractional derivatives and Sobolev spaces, and explain their use in the theory of the Navier--Stokes equations, as well as in some recent work on magnetohydrodynamics (joint work with Charles Fefferman, James Robinson and Jose Rodrigo).

Week 6: Wednesday 28th May

**Jenny Cooley** - *How many points on a cubic surface over a finite field?*

Let S be a cubic surface defined over a finite field K and containing a line, l, defined over K. We can look at planes through l and consider their intersections with S. These intersections will be cubic curves containing l, so the union of l and a conic (quadratic). This is called the *conic line bundle structure* of S with respect to l.

In my talk I will give an explicit formula for the number of points on S in terms of this conic line bundle structure, and prove a slightly surprising consequence regarding the nature of the lines on a cubic surface over a finite field. As always, I will aim to make this talk very accessible and no background knowledge about number theory or algebraic geometry will be assumed.

Week 4: Wednesday 14th May

**Tomasz Tkocz **- *Colouring sets and balancing vectors*

I shall present a very simple argument of Gluskin concerning balancing vectors by adding and subtracting them. I shall show how it relates to Erdős' combinatorial question regarding colouring sets.

Week 3: Wednesday 7th May

**Daniel Rogers - ***The Sims 'Verify’ algorithm*

A presentation of a group is a method of defining a group in terms of generators and relations between those generators. In general there is no obvious way to produce a useful presentation of a given group (for instance a matrix or permutation group), but a number of algorithms exist to perform this task. We will describe one originally by Charles Sims called 'Verify’ which produces a presentation of a finite group based on its action on a suitable set. We will also introduce the notion of a base and strong generating set for a group, a useful computational tool in its own right which is also produced by 'Verify’.

This talk will only require a basic knowledge of groups and group actions – in particular, no prior knowledge of group presentations will be assumed. There will also be plenty of examples.

**Mathew Dunlop **- *Bayesian geometric inverse problems in groundwater flow*

Inverse problems arising from physical applications are often ill-posed due to insufficient data being available. The Bayesian approach to inversion acknowledges this uncertainty by treating the unknown and the data as random variables: instead of taking a single point as a solution, we seek the conditional distribution of the unknown given the data (the posterior distribution). Nonetheless, the modes of this distribution correspond to single points, which may serve as Dirac approximations to the posterior.

In our case, we are concerned with determining the permeability of a medium (modelled as a piecewise continuous scalar field) given a small number of noisy pressure measurements (solutions to a related PDE). It turns out that the modes of the posterior are given by solutions to a deterministic minimisation problem. I will outline the proof, and if time permits, discuss methods to approximate the posterior numerically.

Week 1: Wednesday 23rd April

**David Morris **- *Polynomial Straightening*

The self-similarity of the Mandelbrot set is widely know, but the underlying mathematics are generally not. I will discuss the process ofpolynomial straightening, extending a polynomial-like map on some small domain to a true polynomial, which can be used to prove this self-similarity.

### Term 2 2013-14 - The seminars are held on Wednesdays 12:00-13:00 in MS.03 - Mathematics Institute

Week 10: Wednesday 12th March

**Tomasz Tkocz** - *Can you invert a random matrix?*

I shall present a very simple argument of Tao and Vu which gave a second solution of what had been a famous problem about the invertibility of random matrices. It will be accessible to anyone with a maths undergraduate degree

Week 9: Wednesday 5th March

**Andrew Collins** - *The speed of hereditary properties and graph coding*

A graph property is an infinite class of graphs closed under taking isomorphisms. Given a property , we write for the number of graphs in with vertex set . We call this the speed of the property . In this talk I will discuss the structure of graphs within hereditary properties of 'low' speed and give a general characterisation for graphs within these classes. I will then show how the characterisation can be used to provide an optimal coding for these graphs. Finally, I will discuss some hereditary properties with a higher speed.

Week 8: Wednesday 26th February

**Francesca Iezzi **- *Complexes on surfaces and 3-manifolds (Part II)*

In my first talk I gave the definition of the curve complex and the arc complex of a surface and I introduced some of their basic properties. The construction can be generalised for higher dimension and similar complexes, such as the disc complex of handlebodies and the sphere complex of 3-manifolds, can be defined. These complexes turned out to be very useful tools in the study of the automorphisms of free groups. In this talk I will introduce the disk complex and the sphere complex and explain how all these complexes are related. Only basic knowledge about topology is required. I will also try to recap the first part.

Week 7: Wednesday 19th February

**Maria Veretennikova** - *On well-posedness and regularity of the Cauchy problem for linear and nonlinear fractional in time and space equations*

I will present an introduction for fractional calculus, explain the link to continuous time random walks, discuss the established theory for fractional differential equations and present our new results concerning fractional Hamilton-Jacobi-Bellman-type equations

Week 6: Wednesday 12th February

**James Thompson** - *Stochastic Differential Geometry: An Introduction*

In the nineteenth century, the calculus of real analysis was extended to a geometric setting by the theory of differentiable manifolds. In the twentieth, a probabilistic extension was developed using tools from stochastic analysis. Stochastic differential geometry brings these two extensions together. I will introduce the subject in the setting of a Riemannian manifold, where I will show how one can use geometric comparison theorems to deduce probabilistic inequalities for Brownian motion. So as to maximize the accessibility of this talk, I will draw lots of pictures and I won't assume specialist knowledge of either Riemannian geometry or stochastic analysis.

Week 5: Wednesday 5th February

**Pravin Madhavan **- *Numerical methods for partial differential equations on surfaces*

Partial differential equations (PDEs) on manifolds have become an active area of research in recent years due to the fact that, in many applications, models have to be formulated not on a flat Euclidean domain but on a curved surface. For example, they arise naturally in fluid dynamics and material science but have also emerged in areas as diverse as image processing and cell biology.

In this talk I will attempt to give both an introduction to partial differential equations on surfaces as well as the general ideas behind its numerical analysis.

**Celine Maistret **- *Exploring elliptic curves through the Birch and Swinnerton-Dyer Conjecture*

Below its harmless appearance, this Millennium Prize Problem could be described as highly challenging, quite surprising and rather mysterious. As it happens, stating the conjecture for an elliptic curve is already part of the challenge, owing to the fact that it involves breaking the curve down into its basic characteristics.

Abstract We propose to investigate these characteristics, pointing out on the way why elliptic curves are used in Cryptography, to finally discover that they are conjecturally linked in an intrinsic way.

Week 4: Wednesday 29th January

**Italo Cipriano **- *Some spectral properties of the Perron Frobenius operator in one dimensional dynamics*

Abstract The Perron Frobenius operator constitutes an essential tool in dynamical systems. Ergodics properties can be understand through the knowledge of the spectrum of this operator. We will discuss some recent results toward a finer spectral description in the case of one dimensional maps on the interval.

Note: Only functional analysis will be used.

Week 3: Wednesday 22nd January

**Mark Bell - ***Coordinates for curves*

One way of describing loops on a surface is to use words in the fundamental group. However, for more complicated curves these words become longer and harder to manipulate. I will discuss an alternate coordinate system which can be used to describe embedded loops on a surface. For many problems, such as determining the image of a loop under a homeomorphism, this turns out to be significantly more efficient than using the fundamental group. In fact, the calculations become so easy for a computer to do that you can even use this technique to play games based around drawing loops on a surface.

Week 2: Wednesday 15th January

**Chris Williams** - *Overconvergent modular symbols over imaginary quadratic fields*

The theory of overconvergent modular symbols, developed by Rob Pollack and Glenn Stevens, gives a beautiful and effective construction of the p-adic L-function of a modular form. They define a 'specialisation map' from the space of overconvergent modular symbols to the space of classical symbols, and the crux of their theory is a 'control theorem' that says that this map is an isomorphism on the small slope subspace. This gives an analogue of Coleman's small slope theorem in the modular symbol setting. In this talk, I will describe their results, and then discuss an analogue of the theory for the case of modular forms over imaginary quadratic fields, for which similar results exist.

### Term 1 2013-14 - The seminars are held on Wednesdays 12:00-13:00 in B3.02 - Mathematics Institute

Week 10: Wednesday 4th December

**Francesca Iezzi** - *Complexes on surfaces and 3-manifold*

Given an orientable surface S the curve graph of S is the graph whose vertices are the

homotopy classes of simple close curves on S and where two curves are adjacent if they can be realised disjointly.

Using the same idea we can construct similar graphs, like the arc graph for a surface, the disk graph for a handlebody and the sphere graph for a 3-manifold.

These complexes are useful tools for the study of the mapping class group of a surface and the automorphism group of a free group.

I will define these complexes and explain how they are related to each other.

Just basic knowledge about topology is required.

Week 9: Wednesday 27th November

**Jan Volec** - *Applications of entropy compression method to graph colorings*

In 1975, Erdős and Lovász discovered one of the most powerful tools in probablistic combinatorics -- Lovász Local Lemma (LLL). Roughly speaking, the lemma allows us to show, in a non-constructive way, existence of objects with various desired properties. It had been open for more than 30 years whether

there exists also an algorithmic version of LLL, which was finally confirmed in 2009 by Moser and Tardos.

On the way towards the algorithm for the general version, Moser found also a very simple argument that can be used for designing a different algorithm, which deals only with a special, but still fairly general, case of LLL. Although this technique differs from the one later used in the Moser-Tardos algorithm, it turned out to be useful by itself and has been already applied to various problems in combinatorics. The main idea of Moser became known as so called ''Entropy compression'' argument. In this talk, I will present an application of the argument to some graph coloring problems, mostly based on a recent work of Esperet and Parreau.

Slides of the talk

**Heline Deconinck** - *Diophantine equations in the form: *

Diophantine equations are polynomial equations of finite degree which have integer coefficients, and usually one looks for solutions in . The most famous one is probably Fermat's Last Theorem where and is a prime.

The modularity theorem states that elliptic curves over are related to modular forms. This is the key ingredient in Andrew Wiles' proof of Fermat's Last Theorem . In this talk I will look at a slightly different equation but still use the same framework as Wiles' proof, but with a twist.

Week 8: Wednesday 20th November

**Ben Pooley** - *On an Eulerian-Lagrangian formulation of the Euler equations*

We shall discuss a formulation of the Euler equations which makes use of both Eulerian and Lagrangian variables, namely the velocity from the classical equations and the back-to-labels map which is the inverse of the trajectory map. This formulation was presented by Constantin (2001 J. Amer. Math. Soc. 263--278) and he proved an existence and uniqueness result for this formulation directly in Hölder spaces. Following his method we have obtained an analogous result in Sobolev spaces. The aim of this talk will be to describe this result and explain the proof. This is joint work with James Robinson.

Week 7: Wednesday 13th November

**Florian Bouyer **-* An Introduction to K3 Surfaces and an Overview of a Certain Family*

K3 surfaces are important objects in Algebraic Geometry which are also studied by Complex Geometers, Dynamicists, Number Theorists. From a Number Theoretic point of view, they can be considered as a 2D analogy of elliptic curves. Furthermore , many results for K3 surfaces defined over fields of characteristic 0 also hold for fields of characteristic p.

In the first half of this talk, I will define K3 surfaces and give a few easily deduced important properties. To do so, I will explain some of the basic complex algebra needed (and state as facts some of the harder results). This part is based on master class I attended recently.

In the second half I will focus on a family of K3 surfaces I have been studying for a while. I will explain and prove some interesting facts I have noticed on this family.

Week 6: Wednesday 6th November

**Alejandro Argaez - ***A brief introduction to Galois Representations attached to Elliptic Curves.*

In this talk the notions of the Galois representations and Elliptic Curves over will be presented. Then we will see how, from an Elliptic curve over , a Galois representation can be constructed; giving the respective properties that arise from this construction. Also, the motivation behind the study, current conjectures and open problems of these Galois Representations will be discussed.

Week 4: Wednesday 23rd October

**Tomasz Tkocz** - *On some Sobolev-type inequality*

The Sobolev, Poincaré, Hardy, Morrey, ... inequalities say that a size of a function is controlled by its gradient. We will show a new inequality of this kind. We will discuss what it has to do with the geometry of spaces and, if time allows, we will relate it to Markov chains. Only the standard knowledge of analysis/measure theory/probability will be assumed.

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

Week 3: Wednesday 16th October

**Rosemberg Toala Enriquez **- *Relativity: Causality and Black Holes*

We will see how the invariance of the speed of light forces the structure of spacetime and introduces the Lorentz group as the natural set of symmetries for the theory of Special Relativity. Then I will present the basic tools to include gravity via the so-called Einstein's equations and study the properties of the Schwarzschild solution, which correspond to the gravitational field outside a massive object (e.g. the Sun or a Black Hole). If time permits, I will discuss the ideas behind the famous Hawking and Penrose's singularity theorems.

Week 2: Wednesday 9th October

**Chimere Anabanti** -* Nature of representatives of equivalence classes of minimal words of lengths 2,3,4 and 5 in a finitely generated free group*

In 1936, J. H. C Whitehead used topological means to introduce a theorem which can be used to decide whether two elements of a finitely generated free group are equivalent under an automorphism of the group. Twenty-two years later, Rapaport gave an algebraic proof of Whitehead's result.

In this talk, the corresponding algorithm and GAP program for classifying all minimal words of any given length in $F_{n}$ up to equivalence will be introduced. Then we present my Conjectures from my M.Sc Dissertation on the nature of representatives of resulting equivalence classes of minimal words of lengths 4 and 5 in $F_{n}$, and conclude by introducing new open problems.

Week 1 : Wednesday 2nd October

**Ben Lees** - *The Classical Heisenberg and Related Models*

It is a well known result that for finite range interactions the isotropic Heisenberg model does not exhibit symmetry breaking in dimensions one and two (Mermin-Wagner Theorem). However the remarkable result of Frölich, Simon and Spencer shows that for dimension three or greater the model does undergo a phase transition at low temperatures. This model will be introduced and a proof of the result presented.I will also comment on the similarities and differences of the Quantum system, which also undergoes a phase transition at low temperature as was proved by Dyson, Lieb and Simon. Once the theorem is proved I will present a classical nematic model, which can be viewed as a simple model of liquid crystals, and show how through a clever matrix representation of the interaction we can prove existence of a phase transition using the same method as for the Heisenberg model. If time permits we can then consider a mixture of these two models, what (limited) results are available and what open questions remain.