# Mathematics Postgraduate Seminar Series - 2016

## Seminars in 2016, Semester 1

Monday February 29

A nonlinear interpolation result with application to nonlinear semigroups

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In this talk, I want to present a new nonlinear interpolation theorem, which improves Peetre's (Theorem 3.1 in [Mathematica1970]) and Tartar's (Theoreme 4 in JFA1972) nonlinear interpolation results. In order to highlight the strength of this result I will provide some applications to nonlinear semigroups.

Monday March 7

Computing the Spectrum of an Infinite Matrix

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The go-to method for computing the eigenvalues of a (finite) matrix is the QR algorithm, but for a matrix with infinitely many entries (indexed by the natural numbers), you wouldn't think this approach would be possible. In this talk (after briefly introducing the QR algorithm) I will discuss some nontrivial situations in which QR-type algorithms can indeed be used for finding the eigenvalues of an infinite matrix.

Monday March 14

An Introduction to Toric Varieties

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What is a Toric Variety and why is interesting? In this talk we will study a family of varieties with the property that they all contain an n-dimensional torus in a nice way. Then we will explore their strict relationship with cones and toric ideals.

Monday March 21

Chern's proof of Gauss-Bonnet

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We state the generalized version of Chern's proof of the generalized Gauss-Bonnet formula, and provide the context into which this proof fits.

Monday April 4

Diophantine Approximation and Liouville's Number

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A transcendental number is a number in the complex numbers that is not a root of any polynomial with rational coefficients. In other words it is not algebraic of any degree. Showing that a number is transcendental is not a trivial task. However Liouville realised that there was a connection with algebraic numbers and approximating by rationals. This connection might seem odd at first however it is one of the main tools used in showing the existence of transcendental numbers.

Monday April 11

Handlebodies and the high-dimensional Poincare conjecture

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The notion of a handlebody is a combinatorial way to describe a manifold, analogous to the idea of a cell complex. One of the reasons they are useful is that natural topological operations performed on handlebodies translate very directly to algebraic operations performed on chain complexes, reducing problems in topology to simple linear algebra. In this talk, weÕll explore some basic manipulations of handlebodies and their links to algebraic invariants. Time permitting, we will illustrate the power of this technique by outlining (a version of) SmaleÕs proof of the high-dimensional Poincare conjecture. Some exposure to algebraic topology is helpful (in particular, familiarity with cell complexes and cellular homology) but not assumed.

Monday April 18

Young tableaux and their application in the representation theory of the symmetric group

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The aim of this talk is to introduce Young tableaux and study their combinatorial properties. Ultimately aiming to use them to describe the representation theory of the symmetric group over the complex numbers. If there is enough time I'll outline other applications and properties of Young tableaux as well.

Monday May 2

An Introduction to Discrete Morse Theory

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In the 1990s Robin Forman introduced a discrete version of Morse theory, for application to CW complexes. The aim of this talk is to introduce some of the fundamental tools used by Forman, showing that as in the classical case one may use real-valued functions to efficiently compute topological information. I will demonstrate the combinatorial analogue of the Morse complex and its use in computing the homology groups of a CW complex.

Monday May 9

Mean Curvature Flow Surgery - How to deal with necks

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I wish to develop the notion of a surgery procedure for Mean Curvature Flow. This idea was first developed by Hamilton for Ricci Flow and used by Perelman to solve the PoincarŽ conjecture, it was adapted by Huisken and Sinestrari for Mean Curvature flow to deal with 2-convex surfaces. Surgery allows us to continue the flow past a singular time T and keeps track of the changes in topology that occur for the surface, allowing us to classify the possible geometries of the initial surface. In this talk I'd like to give a quick overview of the surgery process and how to deal with "necks". I will assume a background knowledge of Riemannian Geometry, but no prior knowledge of geometric flows.

Monday May 16

Lax pair equations and harmonic maps

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Lax pairs are a simple Lie algebra differential and a common object in integrable systems. We wish to introduce Lax pair equations as a natural way to construct harmonic maps, and we wish to look at their solutions. Furthermore we'll look at methods at constructing additional solutions via a method known as dressing.

Monday May 23

Stratified Morse Theory

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Morse Theory is a powerful technique for computing the topology of manifolds. However, not all spaces are manifolds. Stratified spaces are a those that, roughly speaking, are manifolds of different dimensions glued together. This picture heavy introduction to stratified Morse theory will introduce the basic notions and theorems, and compare them to their more familiar counterparts.

Monday May 30

Rational trigonometry of a tetrahedron

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This talk outlines a trigonometry for a general tetrahedron in three-dimensional space, using the framework of rational trigonometry (as developed by NJ Wildberger). Concepts from one-dimensional and two-dimensional affine trigonometry are built on to obtain some results pertaining to the trigonometric quantities of the tetrahedron. Some examples are given.

## Seminars in 2016, Semester 2

Monday July 25

Introduction To The Painleve' Equations

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We will show that when studying nonlinear ODEs, spontaneous singularities are inevitable. Nonetheless, Painleve' et al have classified all second order equations where these singularities are minimal in some sense (being poles). If time allows, we will study the first Painleve' equation, and it's tritronquee solution in detail.

Monday August 1

Wall spaces and CAT(0)-cube complexes

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A wall space is a set X together with a collection W of "walls" -- partitions of X into two non-empty subsets. A simple example is a tree: X is the vertex set, and each edge naturally determines a wall by separating the tree into two components. I will describe a construction due to Sageev which associates to a wall space its dual cube complex C(X,W). This dual cube complex admits some nice metrics, and captures many combinatorial properties of the original wall space geometrically. I will go through several examples and, time permitting, some applications to topology and group theory.

Monday August 8

Computational modelling of the lymphatic vascular system

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The lymphatic vascular system consists of networks of numerous vessels which play a key role in immune surveillance by transporting lymph (a colourless liquid that contains white blood cells that helps to purge undesirable materials and toxins from the body) and protein from the tissue space back to the circulatory system. In contrast to the cardiovascular system which has a central pump, the lymphatic vascular system has no central pump hence the transport of fluid against gravity is driven by extrinsic and intrinsic pumping mechanisms. Disorder of the lymphatic vascular system results in a fluid build-up in the tissues which leads to lymphoedema (swelling of the limbs due to build-up of fluid) and unfortunately there is no known cure for lymphoedema which is partly due to inadequate knowledge of the lymphatic vascular system and its transport mechanisms. In my talk I will look at lumped parameter models that are used to describe the transport mechanism of lymphatic vessels. I will also explore some of the parameters of these models to see how they affect the flow-rate.

Monday August 15

What is "mathematics education" and why should we care?

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This talk will provide a brief introduction to the research field of mathematics education, focussing somewhat on the tertiary level as it is most applicable to members of our School. A selection of educational theories and frameworks will be introduced in general and then applied to the context of mathematics. Finally, we will discuss what motivates and emanates from academic research in tertiary mathematics education, and why it may be beneficial (or even essential) to be on our individual agendas and the School as a whole (with some interesting case studies presented).

Monday August 22

An introduction to scale theory for totally disconnected locally compact groups.

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In the early 90's Willis jumpstarted the study of totally disconnected locally compact (tdlc) groups with the introduction of the scale function and tidy subgroups. With the help of the automorphism group of a regular tree, we will see the definitions of these concepts and see how they can be used to gain insight into geometry of tdlc groups.

Monday August 29

Stiefel-Whitney classes and real division algebras

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We will first describe what a vector bundle is and then describe what a characteristic class is. Roughly, the latter is a natural assignment to each vector bundle of a cohomology class of the base. Next, we will provide Stiefel-Whitney classes as examples of characteristic classes and describe an application of these classes to the existence of real division algebras.

Monday September 5

Equidistribution and Unique Ergodicity

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We all know that if $$\alpha \in \mathbb{R}/\mathbb{Z}$$ is irrational, then the sequence $$n \alpha$$ is dense. The goal of this talk is to prove that this is also true for the sequence $$n^2 \alpha$$. In fact, we will show that this sequence is equidistributed (a result due to Weyl). To do so, we will introduce some basic notions in Dynamics and Ergodic Theory, culminating in a proof of Furstenberg's result on the unique ergodicity of skew products, which implies Weyl's result.

Monday September 12

The McKay Correspondence: An introduction to ADE Classification

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Dynkin diagrams (sometimes referred to as Coxeter-Dynkin diagrams) appear in many independent classification theorems, particularly those diagrams of ADE type. In this talk we will characterise these simply laced diagrams, which can be thought of as graphs, and then look at an example of an ADE classification: the McKay correspondence that connects the classification of finite subgroups of SL(2) to the classification of affine Dynkin diagrams.

Monday September 19

Geometric Structures and FG Moduli Space

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The aim of this talk is to introduce the audience to the concept of Geometry as Geometric Structure. We will mainly refer to $$2$$--dimensional examples, with particular focus on Projective Structures. We will conclude with a famous theorem of Fock and Goncharov (FG), which parametrises framed convex projective structures with geodesic boundary on surfaces.

Monday October 10

Music: A Mathematical Offering

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Since the time of the Ancient Greeks, much has been written about the relation between mathematics and music: from harmony and number theory, to musical patterns and group theory. Mark will prove an introduction to the real interplay between these two ancient disciplines, using examples drawn from Dave Benson's book, Music: A Mathematical Offering. Time permitting, he will get thoroughly side-tracked on the topic of mathematical modelling of tube amplifiers, and why they sound so good.

Monday October 17

Numerically solving Fractional Fokker-Planck equations

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Fractional Fokker-Planck equations are partial integro-differential equations that model the anomalous diffusion of certain populations of particles. They find application in a myriad of scientific contexts such as biology, ecology and physics. I will outline a numerical method for solving such equations based on unbiased density estimation of an associated stochastic process, and outline some of the insights into the nature of anomalous diffusion processes gained by this estimation method.

Monday October 24

On Property (T) and amenability

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Both Property (T) and amenability, introduced by Kazhdan and von Neumann in 1967 and 1929 respectively, are properties of locally compact group. They are at the heart of rigidity theory of locally compact groups and their lattices, and have become vital in many other respects. This talk provides an introduction to these properties and their interplay along with basic examples, featuring Kazhdan's original interest in Property (T): it implies finite generation of the fundamental groups of certain locally symmetric spaces.