# Mathematics Institute

### Term 1 2010/11 - The seminars are held on Wednesdays 12:00 in B3.02 - Mathematics Institute

#### Organiser: Sara Maloni

• Wednesday 6 October 2010
Three points on a circle - the story of a theorem.

This will be a lighthearted chat about a beautiful old theorem of Borsuk and Bott which describes the configuration space from the title. Mobius band, fundamental group, Poincare Conjecture, knots and many more - all in one talk! Not to be missed :).

• Wednesday 13 October 2010
Rigidity involving the scalar curvature.

We will explain in geometric terms the concepts of Riemannian curvature, Ricci curvature, and scalar curvature. Furthermore, we give a survey of various rigidity results involving minimal surfaces and scalar curvature. No previous knowledge of Differential Geometry will be assumed except the concept of differentiable manifold.

• Wednesday 20 October 2010
Patrick O'callaghan
Choice under uncertainty.

Many of the models in mathematical economics are based upon decision theory in the context of uncertainty. In particular, the model of von Neumman and Morgenstern (vNM) is one that is invariably used in game theory and finance. This postulates that if the preferences of the agent satisfy certain conditions, then there exists a representation which takes the form of an expected utility function. That is an order-embedding of preferences over the choice set into the real line that enjoys most of the nice properties of the standard mathematical expectation on a given state-space. The vNM approach leads to the existence of a representation which is unique up to a state independent, affine transformation. An alternative approach in the literature leads to a representation which is unique up to a state dependent, affine transformation whereby comparisons across states of neither units, nor levels of welfare have any meaning. Assuming no background in the subject, I will give a brief introduction to the above models and outline my model which lies somewhere between these two extremes, allowing units but not levels to be comparable.

• Wednesday 27 October 2010
Andrew Ferguson
On the dimension of the orthogonal projections of planar sets.

The study of the dimension of the orthogonal projection of planar sets goes back to the celebrated Marstrand projection theorem, and has been a long term object study in the field of geometric measure theory. In this talk I will discuss how recent developments within the field of dynamical systems has provided a new avenue in which to study this problem.

• Wednesday 3 November 2010
Thomas Ranner
How to solve a pde on a surface?

Familiar techniques of separation of variables and Fourier series can be used to solve a variety of pde based on domains in the plane, however these techniques do not extend naturally to surface problems. Instead we look to take a computational approach. The talk will cover the basics of finite difference and finite element approximations of the one dimensional heat equation and show how to extend these ideas on to surfaces. If time allows, we will show numerical results of an optimal partition problem based on a sphere. No background knowledge of pde or computation is required.

• Wednesday 10 November 2010
Carlos Barros
On the Lebesgue-Naguell equation.

Aim of this talk is to make a brief introduction to of some the known methods to solve a Diophantine equation, in particular the method used to solve once and for all the Fermat Last Theorem. For that I will use the Lebesgue-Nagell equation to illustrate how to apply those methods.

• Wednesday 17 November 2010
Nicholas Korpelainen
How to tile your bathroom with T-shaped Tetris pieces.

We will discuss tilings by T-shaped Tetris pieces, with an overview of some classic results, followed by some more recent and surprising connections to graph theory.

Thomas Collyer
An Excursion through the Farey Graph.

The Farey graph is a strikingly beautiful and classically studied object with many interesting properties, which makes an appearance in many different contexts. During this talk, we will (briefly!) introduce some of these properties and contexts, especially its relation to PSL(2,Z) and PGL(2,Z), before taking a more combinatorial approach to its graph structure. In particular we will show that the number of a geodesic paths between a pair of its vertices is bounded by a Fibonnaci number.

• Wednesday 24 November 2010
Colin Mayhill
Permutation Graphs and Well-Quasi-Ordering.

After a brief introduction to simple graphs, hereditary graph classes, and well-quasi-orders, we use permutations define a class of graphs. Investigating some sub-classes we find the bipartite permutation graphs are not well-quasi-ordered, and use this to show the same for split permutation graphs. We also see what these results can tell us about the permutations we used to define our graphs.

• Wednesday 1 December 2010
Umar Hayat
Gorenstein Quasi-homogeneous Affine Varieties.

We study quasi-homogeneous affine algebraic varieties, in particular their tangent bundle and canonical class, with the aim of characterising the case in which the variety is Gorenstein.

• Wednesday 8 December 2010
Andrew Duncan
A Gentle Introduction to Homogenisation.

In various fields of science, there is often cause to study physical phenomena occurring in media which contains microscopic structure. Homogenisation allows one to determine the macroscopic or effective behaviour of the system without having to solve the problem directly, which is often infeasible or impossible. The aim of this talk is to provide a very gentle introduction to the classical theory of homogenisation of PDEs with periodic microstructure and later touch upon stochastic homogenisation.

### Term 2 2010/11 - The seminars are held on Wednesdays 12:00 in MS.03 - Mathematics Institute

#### Organiser: Sara Maloni

• Wednesday 12 January 2011
Robert Tang
A talk on the curve complex not exceeding 60 minutes.

I will give an introduction to a geometric object called the curve complex. This is a simplicial complex which captures intersection information about simple closed curves on a particular surface. After going through some basic properties, I will then talk about how maps between curve complexes can be induced by some "natural" maps between their respective surfaces. Anticipate lots of pictures and hand-waving.

• Wednesday 19 January 2011
Homero Renato Gallegos Ruiz
Numerical computation of periods of genus two curves.

Let f be a degree six polynomial over the reals. The integral of 1/\sqrt{f} is hard to compute numerically to high precision when the limits of integration include the roots of f. Bost and Mestre gave a simple recursive algorithm to compute those integrals, requiring at worst computations of square roots at every step. The correctness of the algorithm can be shown by entirely elementary methods. I will describe Bost and Mestre's method and I will explain the geometry behind it.

• Wednesday 26 January 2011
Soma Purkait
Congruent Number Problem and Shimura Correspondence.

Congruent Number Problem is one of the oldest unsolved problem in Mathematics, traces of which can be found even in some anonymous 10th Century Arab manuscripts. The first famous breakthrough to this problem was given by Tunnell in 1973 using the "Shimura Correspondence" and still the final step remains -- the Birch and Swinnerton-Dyer conjecture. I will talk about the work of Tunnell and how it can be generalized to arbitrary rational elliptic curves.

• Wednesday 2 February 2011
Sebastian Helmensdorfer
Flow by Mean Curvature of Curves in the Plane.

Let F be a smooth, embedded curve in the plane. The mean curvature flow smoothly deforms F in time. The length of the curve decreases as fast as possible along the flow. I will present a theorem regarding the flow of closed curves, which is still considered to be one of the most beautiful results in the area.

• Wednesday 9 February 2011
Sara Maloni
Scissor congruence and Hilbert's Third Problem.

In this talk we'll define the notion of "scissor congruence", which is related, as the name suggests, to the process of cutting a polygon (or polyhedral) into smallest pieces and re-gluing them in a different position. In dimension 2, this is equivalent to the notion of "area congruence". In the famous lecture given in 1900, Hilbert asked, as Third Problem, if the same is true in dimension 3. In the talk we will answer his question and we'll describe generalisations, connections with algebraic K-theory and rational values of dilogarithm function and conjectures related to this topic.

• Wednesday 16 February 2011
Thotsaphon "Nook" Thongjunthug
Period Lattices and Complex Elliptic Logarithms.

In the study of elliptic curves, period lattices and elliptic logarithms are among many essential quantities which are frequently required by a number of computations. Despite their importance, computing both quantities has never been so satisfactory and only restricted to a certain type of elliptic curves. In this seminar, I will explain how to develop a complete method for computing both quantities for any elliptic curves over complex numbers in general, including a brief overview of their applications at the end. This work, which forms part of my PhD thesis, is done jointly with Professor John E. Cremona.

• Wednesday 23 February 2011
Siu Kwan Yip
Annihilating random walk, coalescing random walk and Glauber model.

ARW and CRW are two interesting instances of stochastic interacting particle systems. In this talk I will present the connection between these systems with a spin model called Glauber model. By employing the connection we can obtain the reduced probability density and show that it has a Pfaffian structure.

• Wednesday 2 March 2011
Janosch Ortmann
Random Matrices, interacting particle systems and surface growth.

We take an introductory stroll through these topics. Focussing on asymptotics we will be guided by the repeated appearance of the Tracy--Widom (TW) distributions. First we introduce the Gaussian orthogonal and unitary ensembles from random matrix theory and analyse the behaviour of the largest eigenvalue. The same asymptotic behaviour, given by the TW distributions, can be observed in certain scaling limits of an interacting particle system called totally asymmetric exclusion process (TASEP). We can relate this to a model of surface growth descriped by a stochastic partial differential equation first introduced by Khardar, Parisi and Zhang in 1986.

• Wednesday 9 March 2011
Luke Hartley
The structure of surfaces and three-manifolds.

There is a long standing classification of closed orientable surfaces by topology. In this talk I shall explain the geometric structure of these surfaces and discuss to what extent we can try to find analogous statements for higher dimensional manifolds. In dimension three this will include Thurston geometrization and the implications of the recently proven Surface Subgroup Theorem.

• Wednesday 16 March 2011
David Moxey
The onset of turbulence in pipe flow.

One of the most fundamentally important topics in fluid dynamics is understanding the transition to turbulence in shear flows. Pipe flow provides an ideal geometry in which to study transition, since it is easily modelled by computer and can be well-controlled in the laboratory, yet the underlying dynamics remain incredibly complex. The seminal works of Reynolds revealed that the transition depends on the now ubiqutous Reynolds number (Re); however, after being investigated for over 125 years, the question of a finding a critical Re below which turbulence cannot be sustained is still an open problem. Thus far, much effort has been exerted on investigating puffs -- localised pockets of turbulence found in the intermittent transitional regime. Recently, a detailed statistical survey revealed that puffs have finite lifetime regardless of Re. Whilst this may seem to suggest that all turbulence is transient, the decay process is entirely reliant on the temporal aspects of the flow and does not consider any spatial coupling. In this talk, I will introduce some basic numerical methods required to simulate the Navier-Stokes equations. Using these methods, we apply the same statistical techniques to obtain a distribution for the phenomenon of puff splitting, in which more complex spatial dynamics are naturally incorporated. By comparing this distribution to that of decay, we are able to finally obtain a value for the critical Reynolds number.

### Term 3 2010/11 - The seminars are held on Wednesdays 12:00 in MS.04 - Mathematics Institute

• Wednesday 27 April 2011
Robert Fryer
Quasiregular dynamics and the $(K,\theta)$-Mandelbrot set.

We will introduce quasiregular dynamics, which is the iteration of quasiregular mappings, and study an example. We will then introduce quasiregular versions of the Mandlebrot set and deduce some similar properties. No specific knowledge is needed and there will hopefully be lots of pictures!

• Wednesday 4 May 2011
Yuxin Yang
Commutation Formula and Weitzenbock Identity.

The commutation formula illustrates a Heisenberg-commutativity-type relationship between the gradient operator and its dual, the divergence operator. We give a simple derivation of the Weitzenbock identity from this commutative relationship, revealing an inter-play of symmetry and skew-symmetry.

• Wednesday 11 May 2011
Taro Sano
Smoothing of singular algebraic varieties by flat deformation.

We consider flat deformation of algebraic varieties to parametrize similar algebraic varieties e.g. compact Riemann surfaces of a fixed genus g. We sometimes encounter singular varieties and it is useful to know whether given singular varieties deforms to smooth ones i.e. they are smoothable or not. I will try to explain about smoothing problems in some specific cases.

• Wednesday 18 May 2011
Georg Ostrovski
Learning Dynamics in Games: Fictitious Play.

I will introduce some basic notions from Game Theory and explain how players can “learn” to play a game by repeatedly playing it. I will show how such a learning process can be modelled mathematically by a differential inclusion. One such learning dynamics is known as “Fictitious Play”. I will demonstrate some of the remarkable features of this dynamical system and some of the mathematical questions it gives rise to.

• Wednesday 25 May 2011
Shengtian Zhou
Hilbert polynomials and Hilbert schemes.

Algebraic Geometry studies objects defined by polynomials in certain spaces (affine or projective). We study different properties of such objects and classify them according to certain invariants. Among others, the Hilbert polynomial is such an invariant for closed subschemes in a projective space. I will introduce notions, like, Hilbert functions, Hilbert polynomials and Hilbert series, and discuss the relations between them. When time permits, I will talk about Hilbert schemes, that is, the scheme parametrizes the subschemes with the same Hilbert polynomial, and give some very simple examples of such schemes. It will be a very basic talk.

• Wednesday 15 June 2011
Damon McDougall
Monte Carlo Sampling.

This talk will be basic. Monte Carlo sampling, in a nutshell, is an algorithm that utilises random samples from a probability distribution to compute some desired quantity. I will cover some aspects of convergence from these samples and prove them. There will be videos and visual aids to go along with the ideas I will present and the knowledge I assume will be minimal. I will also recap some first year probability and statistics to refresh your minds.

• Wednesday 22 June 2011
Rupert Swarbrick
Stable Homotopy Theory for Spheres.

This talk should be a gentle tour of the subject of stable homotopy theory, focusing on the computation of the homotopy groups of what seem to be the simplest of all spaces: the spheres, Sⁿ. I shall talk briefly about homotopy groups in general and those of spheres in particular. Then I shall talk about stabilisation in this context and try to give some idea of how one might go about computing these groups. And also why it's a really hard problem...

• Wednesday 29 June 2011