# Postgraduate Seminar 2015-16

**Organiser**: Matthew Bisatt

### Term 3 2015-16 - The seminars are held on Wednesday 12:00 - 13:00 in MS.04 - Mathematics Institute

Week 10: Wednesday 29th June

**Faizan Nazar - ***Lattice Relaxation Problems
*

I will talk about my work, which aims to understand how a perfect crystal is distorted due to the presence of local defects. This is treated using the calculus of variations as an energy minimisation problem. I will use the Thomas--Fermi--von Weizsäcker (TFW) energy as a main example, which introduces a non-linear PDE to describe the energy. I will give an overview of my work on this problem, which involves techniques from analysis and also more geometric arguments, so there will be pictures too.

Week 9: Wednesday 22nd June

**George Kenison - ***STP: A Sector Tree Problem
*

Gauss' circle problem counts the number of integer lattice points of distance at most \(T\) from the origin,

\begin{equation*}

\#\{ x \in \mathbb{Z}^2 \colon d_{\mathbb{R}^2}(o, ox) \le T\},\ \text{as}\ T\to\infty.

\end{equation*}

That is, the number of lattice points inside the boundary of the circle of radius \(T\) centred at the origin as \(T\to \infty\). One may also consider the number of lattice points restricted to a given sector of the circle.

We shall discuss an analogue of the circle problem for certain metric graphs. Let \(G\) be such a metric graph. Here the fundamental group of \(G\) is a free group \(F\), and \(F\) acts freely and isometrically on \(\mathcal{T}\) the universal covering tree of \(G\). In particular, we consider the spatial distribution of the lattice points of \(F\) in \(\mathcal{T}\) by restricting the count in

\begin{equation*}

\#\{ g\in F \colon d_{\mathcal{T}}(o, og) \le T\}\ \text{as}\ T\to\infty

\end{equation*}

to a given sector. Hence the contrived title.

Week 8: Wednesday 15th June

**Natalia Jurga - ***Dimension of Bernoulli measures for some piecewise expanding maps
*

(Projected) Bernoulli measures are a natural class of invariant measures for countable branch piecewise expanding maps such as the Gauss map.

In this talk I will discuss the dimensional properties of these measures.

Week 7: Wednesday 8th June

**Lorenzo Toniazzi - ***Jump-type Markov processes and fractional differential operators
*

It is well-known that certain differential operators are intimately connected to certain stochastic processes (arguably the most famous example is the connection between the Laplacian and Brownian motion). The theories/applications of differential equations and stochastic processes strongly help each other by exploiting this connection.

In this (non-technical) talk (i) I will give two brief introductions: the first one about fractional differential operators of Riemann-Liouville and Caputo type and the second one about the related jump-type Markov processes and (ii) I will explain some of the connections between the two topics, trying to emphasise the interaction between their analytic and probabilistic aspects.

Week 6: Wednesday 1st June

**Francesco Broggi - ***Jacobians and Torelli's Theorem for Compact Riemann Surfaces
*

As Mumford wrote in his book *Curves and Their Jacobians*, “The Jacobian has always been a corner-stone in the analysis of algebraic curves and compact Riemann surfaces.” In this talk I will start from the basic definitions to give an easy introduction of the Jacobian variety associated to a compact Riemann surface.

The last part of the talk will be dedicated to the Torelli theorem for compact Riemann surfaces. There are different versions of this theorem, in this case it says that we can recover a Riemann surface starting from its Jacobian and another piece of information, the principal polarization.

Week 5: Wednesday 25th May

**Alex Wendland - ***Planar Groups
*

When studying groups, it is natural to look at their action on a topological space and ask questions about links between the space and the group. For this purpose we construct a group's Cayley graph and ask what the planarity of this tells us about the group? More specifically, my interest is if these groups have a natural structure-preserving way of breaking down into a 'nice' subclass of groups, in a way which will tell us more about the original group itself. The talk will consist of many pretty pictures (with great clashing colours) and a liberal use of hand waving.

Week 4: Wednesday 18th May

**David O'Connor - ***Focal Cell Adhesion: A Phase Field Approach
*

In adhesion of biological cells, binder molecules form surface-to-surface bonds and are confined to the cell wall, however they are mobile within the wall. In this seminar I will introduce a recently derived free boundary model for this process of focal cell adhesion starting from a series of basic assumptions on the adhesion process. In contrast with previous models we include the effects of curvature as well as allowing for free movement of binders across the entirety of the cell wall, thus generating a two sided problem. We will then introduce a phase field model that we use to approximate this free boundary problem by as well as discussing how formal asymptotics can be used to prove that the free boundary problem can be recovered in the sharp interface limit.

Week 3: Wednesday 11th May

**Daniel Rogers - ***Matrices, monsters and moonshine - an introduction to the classification of finite simple groups
*

Arguably one of the most impressive results in all of mathematics is the classification of finite simple groups, a collaborative work by hundreds of mathematicians over thousands of journal pages which was finally completed in 2004. In this talk we will follow the development of group theory from Galois and the study of integral solutions to polynomials, through to some vital results which made such a classification possible. The talk will be an overview of the classification rather than focusing on the technical details, and thus should be accessible to all.

Week 2: Wednesday 4th May

**Chris Williams - ***L-functions: number theory's “DNA”
*

For almost two centuries, L-functions have been an incredibly powerful tool in number theory. We can attach L-functions to a huge range of mathematical objects, and their special values often contain important arithmetic information. In this introductory talk, I will outline the basic theory and discuss some examples of this phenomenon, including Dirichlet's theorem on the infinitude of primes in arithmetic progressions and the Birch and Swinnerton-Dyer conjecture.

I will conclude by briefly discussing aspects of the Langlands program, a series of conjectures that can be loosely described as saying that “every suitably nice L-function comes from an automorphic form” - at a stroke creating a beautiful overarching theory that combines the worlds of number theory, analysis and geometry.

Week 1: Wednesday 27th April

**Gareth Tracey - ***Measuring the size of a finite group: Generators vs. Order
*

There are many different ways to measure the ''size" of a finite group $G$. Of course, the most obvious measure is $|G|$, the number of elements in $G$. However, the most common way to store a group in a computer is via a generating set. So in this case, perhaps $d(G)$, the minimal size of a generating set for $G$, is the most appropriate ''measure". On the other hand, there exist groups $G$ such that the number of generating sets of size $d(G)$ (compared to $|G|$) is small. Thus, instead of $d(G)$, one can study $E(G)$, the expected number of random elements to generate $G$.

In this talk, we will compare these invariants, and others, paying particular attention to the examples of permutations and matrix groups. Time permitting, we will also see some applications to graph theory and Galois theory.

### Term 2 2015-16 - The seminars are held on Wednesday 12:00 - 13:00 in MS.03 - Mathematics Institute

Week 10: Wednesday 16th March

**Ronja Kuhne - ***The geometry of 3-manifolds
*

It has been known since the nineteenth century that every closed surface admits a metric of constant curvature and that the three different metrics (positive, zero and negative curvature) can be described very simply. In the case of 3-dimensional manifolds it is in general not possible to assign one single geometric structure to the whole topological space.

However, William Thurston proposed in 1982 that every closed 3-dimensional manifold can be cut into pieces such that each of the resulting parts has precisely one of a list of eight possible geometric structures. Only recently proven by Perelman, Thurston's "geometrization conjecture" provides us with a complete classification of 3-manifolds, to which we will give an introduction in this talk.

Week 9: Wednesday 9th March

**Pedro Lemos - ***Mazur's Torsion Theorem
*

Mazur's Torsion Theorem is a remarkable result in the arithmetic theory of elliptic curves: it lists all possible torsion subgroups of the Mordell-Weil group $E(\mathbb{Q})$ of an elliptic curve E defined over $\mathbb{Q}$. In this talk, I will try to outline the main steps and ideas used in the proof of this impressive theorem -- all of this without going into details and without assuming any deep knowledge of the theory of elliptic and modular curves.

Week 8: Wednesday 2nd March

**Edward ffitch - ***Zero elements in Coxeter-idempotent monoids
*

Recall that a monoid is an algebraic structure satisfying the group axioms except possibly inverses. Given a set of subgroups of a group $G$, the set $M$ of finite-length products of these subgroups (via multiplication in $G$) has a natural monoid structure. If the set of subgroups is finite, generates $G$ and consists of subgroups of finite order, then $G$ is finite if and only if $M$ admits a $\textit{zero element}$ - an element invariant under two-sided multiplication. We show how this idea motivates the definition of a certain class of monoids through Coxeter groups, and we characterize the monoids admitting zero elements in a slightly more general class using combinatorial and algebraic techniques, building on work of S.V. Tsaranov and D. Krammer.

Week 7: Wednesday 24th February

**Rhiannon Dougall - ***Counting closed geodesics and properties of covering groups
*

For a compact negatively curved manifold $M$, there are countably many closed geodesics and these grow exponentially in their length. In this talk we look at what happens to this growth rate when we pass to bigger manifolds which cover $M$, and provide an answer based on abstract properties of the covering group.

Week 6: Wednesday 17th February

**Florian Bouyer - ***Rational Points on Quartic Surfaces (via Conics)
*

In secondary school when introduced to Pythagoras' Theorem, $a^2+b^2=c^2$, you may have wondered "How many right angled triangles have all three sides with integer length?". We will start the talk by introducing rational points on conics to answer this question. This introduction to conics will allow us to explore rational points on certain quartic (degree four) surfaces, by looking at conics on these surfaces. We will try to find (infinitely many) examples of quartic surfaces which contain conics and rational points of them.

Week 5: Wednesday 10th February

**Vandita Patel - ***Bunnies, Stars and SuperForms
*

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 via the equation $F_n + 2 = y^p$. (Joint with Michael Bennett - University of British Colombia and Samir Siksek - University of Warwick). We shall also take a look at some interesting graphs arising from studying congruences between newforms. (Joint with Samuele Anni - University of Warwick).

**Adam Griffin - ***The Evolution of Influenza
*

Much work has been done into trying to model flu, both epidemics and seasonal. This can give us an idea of how long we expect an epidemic to last, or how many people we expect to be infected. One particular area of interest is how the virus evolves over time, and how that affects immunity in individuals. This could then be used to advise better methods of vaccination, which have to be created 6 months before the 'flu season begins. To this end, we use Markov processes to model the random behaviour of epidemics in a closed population. To extend the well-studied stochastic models which assume there is only one strain of influenza, we include a mechanism to model the evolution of the influenza virus. We will investigate how this model behaves through analytic arguments and simulation through Sequential Monte Carlo methods.

Week 4: Wednesday 3rd February

**Chris Birkbeck - ***Introduction to the Langlands program
*

The Langlands program is a set of conjectures which link number theory to analysis, geometry, representation theory among others. It is currently one of the largest areas of research in number theory and in my talk I will try to give the main ideas behind these conjectures and some of their p-adic analogues. In particular, I will look at the role modular forms play in the Langlands program.

Week 3: Wednesday 27th January

**Ian Vincent - ***Bounds on the the Number of Lines on elliptic K3 surfaces*

Perhaps you know that on any smooth cubic surface in $\mathbb{P}^3$, it contains exactly 27 lines. A weaker result holds for smooth K3 surfaces: the total number of lines it contains is finite. In fact, for a randomly constructed example the surface will contain no lines, meaning that examples with many lines are special. During this talk we will see that if $X$ is a smooth K3 surface obtained as a complete intersection of three quadrics in $\mathbb{P}^5$ (over an algebraically closed field of characteristic zero), we can use an explicit elliptic fibration to combinatorially bound the number of lines in terms of elements of the Neron Severi Lattice.

Week 2: Wednesday 20th January

**Ben Pooley - ***A model for the Navier--Stokes equations in magnetization variables*

The Navier--Stokes equations can be reformulated in terms of so-called magnetization variables $w$ that satisfy $$ w_t + (\mathbb{P} w \cdot\nabla )w + (\nabla \mathbb{P} w)^{\top} w - \mathrm{\Delta} w =0,$$ and relate to the classical velocity $u$ via a Leray projection $u=\mathbb{P} w$. We will discuss the sense in which this is equivalent to the classical formulation. Modifying the second nonlinear term yields the model system $$ w_t + (\mathbb{P} w \cdot\nabla)w + \frac{1}{2}\nabla|w|^2- \mathrm{\Delta}w =0.$$ We will see that this system is globally well-posed for initial data in $H^{1/2}$ with periodic boundary conditions. This follows from a maximum principle analogous to one satisfied by the diffusive Burgers equations.

### Term 1 2015-16 - The seminars are held on Wednesday 12:00 - 13:00 in B3.02 - Mathematics Institute

Week 9: Wednesday 2nd December

**Sara Lamboglia - ***What are tropical curves and why should we care about them?
*

Even though we are not in the most tropical of the settings, we will try to get in touch with these curves. There will be an introduction to tropical geometry and the focus will be on tropical curves that have been the first objects which brought the tropical methods to the attention of the algebraic geometers. Tropical curves have been used by Mikhalkin to obtain a formula for the Gromov-Witten invariants of the plane.

Come to the talk to know more about them or only to have a tropical break in this cold winter!

Week 8: Wednesday 25th November

**Ben Lees - ***Chessboard states in dilute quantum spin systems
*

We will introduce an annealed quantum spin system on a lattice. Possible phases of this system will be considered such as the full or empty configuration and the so-called chessboard states which is characterised by half occupation on the odd/even lattice sites. It can be shown that under the right conditions the chessboard states occur. This result relies on the famous chessboard estimate and Peierls' argument.

**Marco Caselli - ***What can be drawn by ruler and compass?
*

The ancient Greek mathematicians were the first to study the plane geometry with ruler and compass. They developed many constructions, but they struggled with some cases: *"How can we double a cube? Or divide in three equal parts an angle? Can we construct a Heptagon?"*.

These (and many other) questions remained unsolved for more than two millennia, come and discover the answers!

Week 7: Wednesday 18th November

**Nick Bell - ***An Introduction to Wave Turbulence with application to reduced Magnetohydrodynamics
*

Turbulence is a field which has eluded a complete mathematical description ever since its inception. The difficulty arises from the lack of closure in the equations describing the statistics of the system. The problem of closure can be overcome by utilising Wave Turbulence. This will be introduced in this talk through a detailed application to reduced Magnetohydrodynamics which is a useful system for describing many astrophysical problems.

Week 6: Wednesday 11th November

**Nick Gale - ***An Introduction to Vassiliev Invariants
*

Recall that a knot is an embedding of the circle into 3-dimensional Euclidean space, where we consider two knots to be equivalent if there is an ambient isotopy between them. A map from the set of all equivalence classes of knots to some other set is called a knot invariant. I will define a particular type of knot invariant called a Vassiliev Invariant and then motivate their study by explaining a remarkable theorem of Kontsevich. No prior study of knot theory will be needed to understand the talk.

Week 5: Wednesday 4th November

**Matthew Spencer - ***Brauer Relations for finite groups and their applications*

We will define what it means for a finite group to admit a Brauer relation over a field $k$. Specifically this relation will encode how distinct $G$-sets may give rise to representations which are isomorphic. We will investigate relations over $\mathbb{Q}$, and $\mathbb{F}_p$ and their applications to number theory.

Week 4: Wednesday 28th October

**Katie Vokes - ***An introduction to the curve graph*

For any surface $S$, we can consider the set of all closed curves on this surface without self-intersections. Studying this set of curves is useful for understanding properties of the surface, such as the group of homeomorphisms from this surface to itself. We therefore define a graph $\mathcal{C}(S)$ called the curve graph of $S$, which has the set of curves as its vertex set and an edge between two vertices when the curves they represent are disjoint. The aim of this talk is to give an overview of some of the properties of this graph and why it is interesting. The talk will be illustrated with examples.

Week 3: Wednesday 21st October

**Heline Deconinck - ***How bilinear maps can be used for good and evil (in cryptography)*

You might remember seeing a bilinear map in your undergraduate algebra course (or you vaguely recall having seen a linear map before, but would not like to be called to the board to produce an example). However, you may not have been told that this algebraic map has exciting applications in the cryptography world. The classical scenario presents Alice and Bob who wish to communicate and an evil Eve who, as her name suggests, is a keen eavesdropper. At first we will help Eve by providing her with a bilinear map to outsmart the public key encryption scheme that Alice and Bob cooked up. Later we try to make amends by presenting a new public key encryption scheme for Alice and Bob. This scheme also uses a bilinear map, but one that improves their original recipe.

Week 2: Wednesday 14th October

**Wojciech Ożański - ***The Monge–Ampère equation with application to the 2D Navier-Stokes equation*

The Monge-Ampère equation is a fully nonlinear 2nd order PDE which arises in several areas, including the problem of prescribed Gauss curvature in riemannian geometry and the problem of optimal transport. In the talk I will discuss the application of Monge-Ampère equation in the theory of the 2D incompressible Navier-Stokes equations. We will address the problem of determining the velocity $u$ of the fluid from the pressure $p$. For this reason we will sketch the basic theory of the Monge-Ampère equation and we will present a rather neat proof of the nonexistence of velocity fields $u$ corresponding to a certain family of functions $p$. Moreover, we will observe a number of similarities between the properties of the Monge-Ampère equation and the properties of the Laplace equation.

Week 1: Wednesday 7th October

**Daniel Rogers - ***Necklaces and permutation matrices*

In combinatorics, a necklace is a $d$-tuple of beads of $n$ colours, where we consider two necklaces to be the same if one is a rotation of the other. It is possible to ask a number of nice combinatorial questions about these necklaces, and it turns out that some things we may want to know about permutation matrices can be found out purely by answering these questions. In this talk I will explain the connection, show how some of these computations can be done and (time-permitting) give some algebraic reasons why these results are useful.