# Postgraduate Seminar 2014-15

**Organiser**: Daniel Rogers

### Term 3 2014 -15 - The seminars are held on Wednesday 12:00 - 13:00 in MS.04 - Mathematics Institute

Week 9: Wednesday 17th June

**Tom Ducat - ***Cluster algebras in algebraic geometry*

Cluster algebras are a special class of commutative rings discovered by Fomin and Zelevinsky some time around 2001 and defined by an elementary combinatorial rule called mutation. It turns out that cluster algebras are ubiquitous, appearing in representation theory, algebraic geometry, Teichmuller theory, integrable systems and so on and so on. I will give the basic definitions and some motivating examples (from the biased viewpoint of an algebraic geometer). In particular, I will explain who the Grassmannian Gr(2,5) is and how you can see him as a cluster algebra. The talk will be completely elementary (and hopefully even fun!).

Week 8: Wednesday 10th June

**Florian Bouyer - ***Identifying Lattices*

Do you know what a lattice is? Neither did I, so this talk will start by defining them and their basic properties (illustrated with examples). Then once we are comfortable with them, we will use several examples to play the game "here is a lattice, have I seen it before?". Along the way of playing this game, several problems will be highlighted and tricks to get around them will be shown.

Week 7: Wednesday 3rd June

**Vandita Patel - ***The Hunt for Totally Real Number Fields*

Number fields and modular forms are fundamental objects that belong to different areas of number theory, and much effort has been invested into developing algorithms for computing them. However, there is a connection between certain number fields (the so called "totally real" number fields) and modular forms via theta series. This little known connection should give useful insights into totally real number fields. We explore this connection and explain how it should help us compute totally real fields. No prior knowledge of number theory will be assumed, with most concepts illustrated with plenty of examples.

Week 6: Wednesday 27th May

**Karina Kirkina - ***Introduction to Chevalley groups*

Chevalley groups are "analogues" over arbitrary fields of complex simple Lie groups. Their construction is one of the approaches to uniformly describing the classical groups, and was an important step in the classification of finite simple groups. In this talk I will describe the basics of this construction, and illustrate the main features of these groups using the example of $B_2 (\mathbb{F}_p)$, a symplectic group of degree 4. I will then briefly talk about generalisations of this construction, in the form of Chevalley-Demazure group schemes and Kac-Moody groups. The exposition will be kept at an elementary non-specialist level.

Week 5: Wednesday 20th May

**Ben Pooley - ***Global well-posedness for the diffusive 3D Burgers equations*

We will examine the 3D vector-valued analogue of the diffusive Burgers equations. These are very closely related to the Navier-Stokes equations, however we will (hopefully) see that global well-posedness in $H^{1/2}$ is not so difficult to obtain in a periodic domain, using arguments applicable to the Navier-Stokes equations together with a maximum principle. This is based on recent joint work with James Robinson.

Week 4: Wednesday 13th May

**Matthew Thorpe - ***Asymptotic Behaviour for Phase Transitions on Random Graphs*

Given a data set with $n$ data points we construct a graph by weighting edges according to an appropriately scaled anisotropic interaction potential. We use the Ginzburg-Landau functional as a phase transition model on the graph. In particular minimizers of the Ginzburg-Landau functional define a partition. The natural question to ask is whether as $n\to \infty$ the minimizers converge, whether any limit has meaning and what is the natural scaling in the interaction potentials. Using an appropriate notion of convergence to compare functions on different graphs I will present results on the natural scaling regime and show that any sequence of minimizers are compact with any limit the minimizer of a surface energy taking binary values. The talk should be accessible to all and will focus more on the concepts than the technical details.

Week 3: Wednesday 6th May

**AlJalila Al Abri - *** Pairs of closed geodesics in metric graphs*

We discuss the counting problem of pairs of closed geodesics in metric graphs, where the difference between their geometric lengths fall in an interval which is allowed to shrink at a specific rate and can be positioned arbitrarily on the real line.

Techniques used to solve this problem start with coding the metric graphs with subshifts of finite type. Then, Fourier transforms are involved in analysing a counting function of pairs of closed geodesics.

Week 2: Wednesday 29th April

**Yuchen Pei - *** A $q$-weighted Robinson-Schensted algorithm*

This talk is about Robinson-Schensted (RS) algorithms. First I'll introduce the classical RS algorithm and show how it is related to the totally asymmetric simple exclusion point process (TASEP), then I'll talk about a $q$-weighted RS algorithm which is related to the $q$-TASEP and the $q$-Whittaker functions, which are eigenfunctions of the $q$-deformed Toda chain Hamiltonian and special cases of the Macdonald polynomials. Finally I'll show with pictures that the $q$-weighted algorithm has a symmetry property similar to that of the classical algorithm. Part of the talk is based on joint work with Neil O'Connell.

Week 1: Wednesday 22nd April

**Chris Birkbeck - *** Modular forms and the Jacquet-Langlands correspondence*

I will assume you are not a number theorist and define modular forms and give a few reasons why they are interesting. Then I’ll define modular forms on quaternion algebras and explain how the Jacquet- Langlands correspondence relates usual modular forms to those on quaternion algebras and why this is useful.

### Term 2 2014 -15 - The seminars are held on Wednesday 12:00 - 13:00 in MS.03 - Mathematics Institute

Week 10: Wednesday 11th March

**Daniel Rogers - *** Finding the maximal subgroups of $\mathrm{SL}(n,q)$*

Understanding the maximal subgroups of a group gives us a lot of insight into its structure. The maximal subgroups of $\mathrm{SL}(n,q)$, the group of invertible $n$ by $n$ matrices over the finite field with $q$ elements, fall into one of nine classes. Eight of these classes have some sort of geometric interpretation and are completely understood; the members of the ninth class, $\mathcal{C}_9$, are currently only known explicitly for $n \leq 12$. Investigating the groups in $\mathcal{C}_9$ involves looking for subgroups of $\mathrm{SL}(n,q)$ which are simple (or close to being simple). In this talk I will present the process by which we find the $\mathcal{C}_9$ candidates. Nothing beyond second-year knowledge of group theory will be assumed, and everything will be illustrated with examples.

Week 9: Wednesday 4th March

**Alex Torzewski - *** An introduction to galois actions and their Local-Global properties*

We discuss the join between two halves of number theory. After introducing the classical global objects which you may encounter in a first course in number fields, we observe that these come with a natural action of certain galois groups. We then describe some of the simpler local information in this setting and go on to look at the oldest example of a local to global theorem. Finally we examine some conjectural generalizations to account for the action of galois.

Week 8: Wednesday 25th February

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

For a group $G$, let $d(G)$ denote the minimal number of elements required to generate $G$. When $G$ is a subgroup of the symmetric group $S_{n}$, one can bound $d(G)$ in terms of $n$, but how sharp can this bound be? For arbitrary subgroups of $S_{n}$, a linear bound in $n$ is the best we can do. But is it possible to sharpen the bound by restricting our attention to special classes of permutation groups, like primitive or transitive groups? Could we come up with a bound which is logarithmic in $n$? Or even better than logarithmic? Or can we even improve the linear bound in the first place?!

In this talk, we will have a look at what is already known, and we will give some new results, paying particular attention to the primitive case. We will also look at the corresponding results for irreducible linear groups over finite fields. No permutation group theory is assumed, so everyone is welcome.

Week 7: Wednesday 18th February

**Rhiannon Dougall - *** Amenability, closed geodesics and symbolic dynamics*

We will discuss a nice relationship between the number of closed geodesics in a negatively curved covering manifold and a group-theoretic property of the cover.

Week 6: Wednesday 11th February

**Matthew Bisatt - *** Finding a normal basis for the ring of integers*

It is well known that the ring of integers of a number field has a $\mathbb{Z}$-basis, but we wish to know whether there is a particular element of it such that its Galois conjugates form a basis of this ring, ie whether the ring is a free module over the group ring $\mathbb{Z}[G]$. We will examine cyclotomic fields and use the Hilbert-Speiser Theorem to deduce things about lattices and the ramification of the extension in the global case and also in the local case. This talk will include several examples and the occasional picture and pun.

Week 5: Wednesday 4th February

**John Sylvester - ***The cover cost of a random walk on a random graph*

We introduce random walks on random graphs and their relation to electrical resistances. In this setting we can gain some results around the question: on average how long should each vertex of a random graph expect to wait before being visited by the random walk?

**Chris Birkbeck - ***Fun facts about $p$-adic numbers.*

Are you tired of not knowing what $\mathbb{Z}_p$ and $\mathbb{Q}_p$ mean? Have you been looking everywhere for a fun introduction to the $p$-adic numbers and just can’t find the right one? Then look no further! In this brief talk I will introduce you to the $p$-adic numbers and give you some facts about their algebraic and topological structures. I will assume no prior knowledge about $p$-adic numbers.

Week 4: Wednesday 28th January

**Adam Bowditch - ***Random walks on Galton-Watson trees conditioned to survive*

We begin with a description of Galton-Watson trees and an explanation of how they can be conditioned to survive. This yields an interesting structure that can be decomposed into a collection of finite i.i.d. trees attached to a backbone. Using this as a graph for a biased random walk we observe trapping phenomena in the finite trees which causes the bias to have an unusual influence on the rate at which the walk moves away from the root of the tree.

Week 3: Wednesday 21st January

**Céline Maistret - ***An approach toward the 2-parity conjecture for Jacobians of hyperelliptic curves of genus 2 admitting a Richelot isogeny.*

In this talk, we recall the $p$-parity conjecture for abelian varieties and its link to the Birch and Swinnerton-Dyer conjecture. For $p=2$, concentrating on Jacobians of hyperelliptic curves of genus 2 admitting a Richelot isogeny, we detail our approach toward this conjecture. It will imply presenting invariants of hyperelliptic curves such as their number of connected components over the reals, their deficiency and the special fibre of their minimal regular model for which we will provide examples.

Basic definitions will be recalled and illustrated by examples.

Week 2: Wednesday 14th January

**Rosemberg Toala - ***Black Holes: What do we observe?*

We will start with lots of images of astrophysical objects which are believed to harbour a Black Hole in its centre. Then, we will review some of the physical parameters behind these observations and learn how they are measured. Finally, we will outline a simplified mathematical/physical model to describe these phenomena.

Week 1: Wednesday 7th January

**Michael Selig - ***Calculating algebraic varieties using Riemann-Roch methods*

Riemann-Roch theorems give formulae for the dimension of spaces of functions of algebraic varieties, and are widely accepted (by me, at least) to be the best theorems in Mathematics. We will attempt to convey the power of these type of theorems, focusing on using them to construct abstract varieties (given just by some numerical invariants) explicitly (that is, write out equations for the varieties). We will start with some basic examples using the Riemann-Roch theorem for curves, and then move through the gears, hopefully arriving at the current state of research which I am involved in. The talk will be example-based and not focus too much on the abstract theory.

### Term 1 2014 -15 - The seminars are held on Wednesday 12:00 - 13:00 in B3.02 - Mathematics Institute

Week 10: Wednesday 3rd December

**Lorenzo Toniazzi - ***Atomless measure spaces and exchange economies*

Consider modeling the economic phenomenon of agents endowed with goods and being given the chance to exchange them at certain prices. Two important (and very different) equilibrium concepts can be formulated, namely the Core and the Walrasian equilibrium. Suppose now that the there is a continuum of agents (modelled via an atomless measure space). Then these two equilibria are identical. This is an important result in economic theory due to Robert Aumann. The intuition behind the result and the key steps for its proof will be presented.

Week 9: Wednesday 26th November

**Oliver Dunbar - ***Knot energies in the numerical detection of self-intersecting moving membranes*

Knot energies are typically used in algebra to detect the topology of certain curves, and we exploit this sensitivity to an analytical and numerical problem. By careful choice of discretisation we may approximate the energy with known accuracy and acceptable efficiency for unknown moving closed curves. We apply this as a tool to detect self intersections of cell membranes during motility modeled by surface partial differential equations.

**Vandita Patel - ***On the equation $F_n + 2 = y^p$*

Finding integer solutions to an equation may seem quite trivial at first glance, but behind this `simple' equation lies some of the deepest mathematics known to number theorists. We outline some of the techniques used to find integral solutions $(n,y,p)$ of the equations $F_n = y^p$ and $F_n +2 = y^p$ where $F_n$ is the $n$-th Fibonacci number. (Joint work with Michael Bennett - University of British Columbia and Samir Siksek - University of Warwick).

Week 8: Wednesday 19th November

**Jake Dunn - ***An Introduction to Mathematical Relativity and Stability Problems*

The stability of the black hole solutions to Einstein’s field equations is still an open problem in mathematical relativity. This talk aims to introduce some of the key notions from Einstein's theory while motivating the study of wave equations on black hole backgrounds. Once we have gone over the core framework of the theory I will introduce the Cauchy problem for the field equations and in turn the wave operator on the exterior of a Schwarzschild black hole. I will then discuss how vector field methods can be used to prove stability for this setting.

Week 7: Wednesday 12th November

**George Kenison - ***Orbit counting in conjugacy classes*

Let \(G\) be a finite connected metric graph such that the degree of each vertex is at least three. We assume that the lengths of closed geodesic paths do not lie in a discrete subgroup of \(\mathbb{R}\). The universal cover of \(G\) is an infinite tree \(\mathscr{T}\). We use \(d_{\mathscr{T}}(-, -)\) to indicate the metric that \(\mathscr{T}\) inherits from \(G\). The fundamental group of \(G\) is a free group \(F\) and this group acts freely and isometrically on \(\mathscr{T}\). For a non-identity element \(g\in F\) we denote by \(C(g)\) the conjugacy class of \(g\). Consider the counting function \(N_{C(g)}(T)\) defined by \begin{equation*} N_{C(g)}(T) =\# \{ x\in C(g) \colon d_{\mathscr{T}}(o,ox)\le T\}, \end{equation*} where \(o \in \mathscr{T}\) is a given base vertex. In this talk we shall establish the asymptotic behaviour of \(N_{C(g)}(T)\) as \(T\to\infty\).

Week 6: Wednesday 5th November

**James Thompson - ***A Probabilistic Formula for the Heat Kernel*

The heat kernel is an object which connects geometry to probability. In this talk I will discuss the heat kernel on a Riemannian manifold, where there is a formula for it in terms of stochastic processes. I will give examples and pictures and mention some of the applications of the formula.

Week 5: Wednesday 29th October

**Mark Bell - ***Planes, Trains and Automorphisms*

We will look at a technique for building 3--manifolds by taking lower dimensional pieces (planes) and gluing them together (with automorphisms). One nice feature of this construction is that we will be able to get a handle on the geometry of these manifolds (using train tracks). In particular, we will show that the complement of the L6a2 link is hyperbolic. This talk will be mainly picture based.

Week 4: Wednesday 22nd October

**Matthew Spencer - ***Brauer Relations over fields of characteristic p*

Given some finite group G it is natural to study its action on finite sets, from such a pair we can form a vector space over some field K with an action of G and thus a representation of G over K. Two distinct sets may give rise to the same representation and we call elements of this kernel Brauer relations. Brauer relations over K=Q have been entirely classified and in my talk I will describe my work towards such a classification in characteristic p.

Week 3: Wednesday 15th October

**Ben Lees - ***Probabilistic representations of Quantum spin systems*

Quantum spin systems are an interesting and challenging area of research, there are some impressive results but much is still incomplete. Probabilistic representations have been around since the 90's and have recently seen renewed interest and optimism for their usefulness. I will introduce one such representation and explore some of the things it can teach us about its connected spin system.

Week 2: Wednesday 8th October

**Tomasz Tkocz - ***Expander graphs and strange operators*

I shall present how expander graphs are useful to solve some problems in abstract functional analysis. It will be accessible to anyone with a maths undergraduate degree.

Week 1: Wednesday 1st October

**Florian Bouyer - ***Hyperelliptic curve reduction*

Hyperelliptic curves can be used in cryptography, but to be able to do so, one needs to find them. Unfortunately the known method to find hyperelliptic curves produces curves that takes a lot of memory to store.

In this talk I will go through an algorithm that Marco Streng and I developed to reduce the coefficients of the equation defining the hyperelliptic curve, hence reducing the memory to store it.

No prior knowledge will be needed for this, and I will be using examples to illustrate each steps of the talk.