## Seminars in 2017, Semester 1

The seminars will be held at 5:00 pm on Mondays in Carslaw Room 535A.

Monday March 6

Introduction to (G, X)-Structures

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In 1872 Felix Klein outlined the Erlangen Program with the aim of studying the intrinsic geometry of surfaces using group theory and projective geometry. This idea is formalised in the notion of (G, X)-structures wherein the geometric features of a given surface are locally modelled on that of a smooth manifold X and preserved by the action of a group G on X. I plan to give an introduction to the language and fundamental features of (G, X)-structures, including the Developing Map and Holonomy Representation. This will provide the framework for a discussion of the Teichmuller space and its parameterisation through Fenchel-Nielsen coordinates.

Monday March 13

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Dead water refers to a mysterious increase in resistance experienced by boats in density-stratified waters. The problem has been documented since ancient times, and studied scientifically for over a century. However, past investigations have been limited in several important ways. For the first time, we study the phenomenon using state-of-the-art numerical simulations. We reproduce the effect and demonstrate that it is most pronounced in strongly nonlinear regimes poorly modelled by current theory. The most exciting development is a new trailing vortex found behind the boat experiencing the effect. This robust structure is consistent with sailors accounts, but has been missed in previous scientific studies. We expect these results to lead to actionable ways to mitigate dead water in the real world.

Monday March 20

Triangulated spheres and Pachner moves

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In the world of discrete geometry-somewhere between combinatorics and geometry- Pachner moves (or bistellar flips) are an important tool to "build up" complex triangulations from more simple ones while preserving topological properties. I will give an introduction of basic objects in discrete geometry such as polytopes and simplicial complexes and show with examples how Pachner moves can help both to devise new questions and to answer others, spanning topics from combinatorics to topology.

Monday March 27

Gaussian Product Conjecture

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The aim is to present Gaussian Product Conjecture which proposes lower bound for moments of multivariate normal distribution. The conjecture is related to problems in functional analysis and combinatorics. I will present partial results and possible lines of attack.

Monday April 03

Mean curvature flow with surgeries and the level set flow

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An intuitive talk on mean curvature flow with surgeries, developed by Huisken and Sinestrari. And the techniques used to show that this flow converges to the well known weak solution of level-set flow studied by Evans and Spruck. This talk is based on the work of John Head.

Monday April 10

Inflammation and PDEs Structured in Cellular Quantities

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Macrophages are cells which rapidly populate inflamed tissue and clear pro-inflammatory material which enables the inflammatory response to resolve. Funnily, these cells eat their dead and recycle accumulated substances. When this cannibalistic cell population is expressed as a distribution (using PDEs) across accumulated substances, we observe that a substantial proportion of cells contain an extraordinary amount of material. This is due to biomagnification where accumulated substances increase in concentration along a chain of cells which have consumed at least one dead cell. Certain substances are cytotoxic in excess and induce necrotic (bad) cell death which sustains cell recruitment to the site of inflammation. This creates a positive feedback loop which sustains both cell recruitment and necrosis and hence amplifies inflammation in time. I will support and explore this theory using a non-local partial integro-differential equation whose steady state can be determined analytically. I will conclude with the analysis of a non-local partial differential equation which suggests that cell division/proliferation can halt biomagnification and promote inflammation resolution.

Monday April 24

Solving polynomial equations with radicals or why there are no general solutions for polynomial equations of degree 5 and higher

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The history of solving polynomial equations dates back to about 2000 BC for which we have written evidence that the old Babylonians already solved quadradic equations. Throughout the centuries people tried to formulize and solve these equations in general. Finally, in Italy during the 16th century scholars discovered the general solutions for cubic and quartic equations but the general quintic could not be solved. Nowadays we know that there is no solution using radicals for the general quintic and higher degree polynomial equations but historically it took until the early 19th century to give a proof for this fact. In 1799 Ruffini and Gauß were the first to formulate that there is no general solution for degree 5 and higher. Following them Cauchy, Wantzel, and especially Abel worked to help to finish Ruffini's first draft of a proof and led to the famous Abel-Ruffini Theorem in 1824. Independently and without knowing about Abel's proof a young Frenchman named Évariste Galois laid the groundwork of what is known today as Galois theory. Galois gave us a beautiful general approach to deal with solvability of polynomial equations of any kind and, moreover, his work led to solve two of the three classical problems of ancient mathematics. Unfortunately, he died way to young at the age of 20 after being severly injured in a duel. My talk will aim to lead throughout the centuries of the quest to solve polynomial equations and explain why there can't be a solution for the general quintic.

Monday May 01

TBA

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TBA

Monday May 08

Modular Forms and Number Theory - an insight into the Ramanujan conjectures

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TBA

Monday May 15

Tetrahedron centres over a general metrical framework

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In this talk, the three main centres of tetrahedra (centroid, circumcentre and Monge point) will be obtained over an arbitrary symmetric bilinear form. I will also talk about the Euler line (unifying the three main centres of a general tetrahedron) and the twelve-point sphere, as well as discuss the existence of orthocentres in a general tetrahedron.

Monday May 22

TBA

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TBA

Monday May 29

A different approach to representations of the symmetric group

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The representation theory of the symmetric group is closely connected to the combinatorics of Young tableaux, however the usual way in which this is realised can seem unnatural. The aim of this talk is to present a more recent approach to the subject by Okounkov and Vershik, which pays close attention to the structure of the nested group algebras $$\mathbb{C}S_{n} \subseteq \mathbb{C}S_{n + 1}$$ and the restriction of $$S_{n+1}$$-representations to $$S_n$$. By constructing an extremely nice basis of any irreducible representation of $$S_n$$, along with a corresponding diagonal subalgebra of $$\mathbb{C}S_n$$, we will be able to "do weight theory" to a representation, from which the connection with Young tableaux will emerge naturally.

Monday June 05

TBA

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TBA

## 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.

## 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 Poincar 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 2015, Semester 2

Wednesday August 5

Morse Theory and Morse Homology: Critical Points

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Morse theory begins with the observation that studying the right function on a (compact, oriented) manifold can give a surprisingly rich amount of information about its topology, including its homotopy type and singular homology. In this talk, we will take a tour through some of the foundational results which form the backbone of Morse theory, and use these as tools to tackle some first applications.

Wednesday August 12

Binary quadratic forms after Conway and Eindhoven

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We derive an algorithm for finding the minimum nonzero value of a positive-definite quadratic form on a free rank-two $$\mathbb{Z}$$-module (i.e. a lattice), in the Eindhoven style---deriving the algorithm and its proof of correctness almost hand-in-hand. Time permitting, we may also give but not prove an algorithm for finding the collection of primitive elements in said lattice with bounded q-value. This work is inspired by Conway's approach to binary quadratic forms.

Wednesday August 18

An Introduction to Representation Theory

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If groups are thought of as a formalisation of symmetries, then representation theory (of groups) is the study of how symmetries can arise in various spaces. We'll discuss some basic properties of representations of finite groups, focusing on representations over $$\mathbb{C}$$, where character theory proves to be an extremely powerful tool. We'll also discuss what happens over a field of positive characteristic (the modular case). Time permitting, we'll talk about some further topics (such as the case of symmetric groups, ways of constructing representations, or reasons to consider representations of things other than groups).

Wednesday August 26

The ideal class group: An introduction to algebraic number theory

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The ideal class group of a number field describes the obstruction for the ring of integers being a UFD, and carries information on the multiplicative structure of the ring. In this talk, I will give an introduction to algebraic number theory with the focus on ideal class groups. Time permitting I will discuss its role in class field theory.

Wednesday September 2 - Social Event

Wednesday September 9 - Fourth Year Joint Talk

Proof of the Fundamental Gap Conjecture & Witten's Deformation and the Morse Inequalities

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We consider Schr\âdinger Operators of the form $$-\Delta + V$$ on a bounded and convex domain \$$Omega \subset \mathbb{R}^n$$. Further we assume that $$V$$ is a convex potential. Such operators have an increasing sequence of eigenvalues. The difference of the first two is called the fundamental gap and has many physical and mathematical implications. The question arises as to whether or not a natural and optimal lower bound exists for the fundamental gap of all such operators. The answer is yes. In this talk, we closely follow the work of Ben Andrews and Julie Clutterbuck in their 2011 proof of such a result, outlining the major theorems and their connections.

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In 1982, E. Witten introduced a modification of the de Rham-Hodge operator and used it to give a new proof of the Morse inequalities, as well as a number of other results. His ideas have since been studied extensively, and "Witten's deformation" is now a relatively well-known tool in differential geometry, providing connections between de Rham-Hodge theory, Morse theory and index theory. In this talk, we define Witten's deformation and describe some of its key properties, before sketching the proof of the Morse inequalities.

Wednesday September 16 - Fourth Year Joint Talk

Categorification of Quantum sl_2 & Teichmuller space on surfaces

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To begin, we introduce the notion of categorification and surrounding research areas. The main focus of the talk is then to explore some of the more innovative methods which were used by Lauda in 2008 to construct a categorification of the quantum enveloping algebra /( sl_2 /). If there is time, some remarks will be made on a second characterisation of the categorification via inverse limits in an appropriate 2-category.

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In so far as mathematics is concerned with the art of classification and enumeration, Teichmuller space is a natural object of study. Named for work of Oswald Teichmuller in the early 1940s, this is the space of complete, marked complex structures with geodesic boundary on a given surface. I will relate a simple parameterisation of the Teichmuller space of the once punctured torus given by Bernard Maskit in 1989 and an introduction to Fenchel Nielsen coordinates for compact surfaces. Having constructed such a space one may consider questions relating to topology and metrics on the space and in the case of hyperbolic structures, the way in which the space of hyperbolic structures embeds into the space of projective structures.

Wednesday September 23

Geometric phase and periodic orbits in the equal-mass, planar three-body problem with vanishing angular momentum

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Geometric phase is the rotation of a dynamical system separated from angular momentum. The canonical example of such is a cat (a non-rigid body with an inbuilt control system), falling from an inverted position, being able to re-orient itself with negligible total angular momentum so as to land on its feet. The system of three bodies moving under gravity is similarly non-rigid, and capable of changing size and shape. Using coordinates that reduce by translations and rotations and simultaneously regularise all binary collisions, which separate shape dynamics from rotational dynamics, we show how certain discrete symmetries of the new Hamiltonian (including both reversing and non-reversing symmetries of the resulting equations of motion) can force the geometric phase of motion periodic in these coordinates to vanish. This result is illustrated with periodic orbits discovered numerically, many of which we believe are heretofore unknown.

Wednesday September 30 - Mid-Semester Break

Wednesday October 7 - Mid-Semester Break

Wednesday October 14

A baby step towards (linear) Lie Groups and Lie Algebras for non--experts

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In this talk we hope to give a very soft introduction to Lie Groups and Lie Algebras from a slightly geometric point of view. In particular we will touch the following tool with some examples along the way:

a) Basic definitions;

b) The exponential map and its properties;

c) Von Neumann and Cartan Theorem;

The talk is based on the book: "Notes on differential geometry and Lie Groups - Jean Gallier"

Wednesday October 21

A (mega)-brief introduction to Ergodic Theory

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We will discuss some basic tools of Ergodic Theory and their use in number theory. We hope to tell you what is

a) Ergodic theorems;

b) Normal numbers in [0,1], and their connection to Birkhoff ergodic theorem;

c) Poincare recurrence;

d) Multiple recurrence and non-standard ergodic averages;

e) Szemeredi theorem on arbitrary long arithmetic progressions in sets of positive density.

Wednesday October 28

Introduction to cellular algebras and their representation theory

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Roughly speaking, a cellular Algebra /(A/) is just an algebra with a distinguished cellular basis. However, this basis gives rise to a rich 'cell'-structure which is a powerful tool to study the representation theory of /(A/). So, the aim of this talk is to introduce the concept of cellularity on algebras, unfold the strength of this structure and then use this to fully develop the representation theory of these algebras.

## Seminars in 2015, Semester 1

The seminars will be held at 5:00 pm on Thursdays in Carslaw Room 535A.

Thursday March 5

Mathematical model of cancer treatments using oncolytic viruses and immunotherapy.

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Oncolytic virotherapy -- the use of genetically engineered viruses to treat tumours -- is a promising approach to finding a cure for cancer. However, progress is often hindered by the cost of clinical trials; mathematical modelling provides the solution to this problem. We present a computational model for a genetically engineered virus, and compare this to real world data. We also investigate potential improvements to the virus through a stochastic simulation utilising the Gilespie algorithm.

Thursday March 12

Some steady states solutions of the Euler equations on a toroidal domain and their stability.

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The Euler equations describing inviscid flow are a cornerstone of fluid mechanics. We will look at some steady states solutions of the Euler equations on a toroidal domain, and analyse their stability. The talk will begin with an introduction to non-canonical Hamiltonian systems, which are often used in Fluid Mechanics. The key concepts we use are a special truncation introduced by Zeitlin and a decoupling introduced by Li. Interestingly, the question of stability reduces down to one that can be described from a simple geometric viewpoint. This is based on work completed with Holger Dullin and Robby Marangell.

Thursday March 19

Introduction to Random Matrix Theory.

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This talk aims to provide an introduction to the exciting area of Random Matrix Theory. The focus of the talk will be on the concept of universality encountered in random matrix theory. I will discuss the recent progress made in understanding universality in Wigner random matrices; I will also highlight the curious connection between random matrix models, the Riemann zeta function, neutron resonances in heavy nuclei, and orthogonal polynomials.

Friday March 27

3-manifolds and hyperbolic geometry.

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3-manifolds and hyperbolic geometry are well known to provide strong connections between Topology and Geometry. In this talk, we aim to give a simple and clean overview on the connection between classification of three manifolds and hyperbolic manifolds for non experts. We will informally look at the decomposition scheme and focus on the ideal triangulation technique. In particular, we will show how to use such a method to construct a complete hyperbolic structure on a triangulated hyperbolic three manifold.

Thursday April 3

Free Lie Algebras and the Symmetric Group.

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A Free Lie Algebra is the "most general" Lie Algebra generated by a given set. The talk will cover some key ideas about Free Lie Algebras including the PBW Theorem and Witt's Formula for the dimension of the homogeneous components, including a charming proof. Time permitting we will continue this line of reasoning and view the Free Lie Algebra as a representation of the Symmetric Group and generalise Witt's Formula to describe its Character. Only very basic algebra will be assumed.

Friday April 17

A connection between formal power series for solutions to ODEs and the combinatorics of rooted trees.

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We will discuss a connection between formal power series for solutions to ODEs and the combinatorics of rooted trees. Given an ODE of the form $$\frac{dx}{dt} = f(x)$$ for some $$f(x)$$, we may expand $$x(t)$$ as a power series in $$t$$, where the coefficients will be functions of the coefficients of the power series of $$f$$. We will find a general formula for the power series of $$x(t)$$ involving a sum parametrised by rooted trees. From this, we will solve some ODEs to prove some combinatorial facts.

Friday April 24

Computing the universal cover and fundamental group of triangulated 3-manifolds.

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I will describe a method to compute a presentation for the fundamental group of a triangulated 3-manifold. (Note: every 3-manifold admits a triangulation) This will be done via the dual 2-skeleton, as opposed to the more standard method (described, e.g., in Hatcher) via the usual 2-skeleton. I will also describe how to construct a triangulation of the universal cover given one of the original 3-manifold. This triangulation will have the important property that the deck group acts on the collection of 3-simplices in such a way that the covering map is equivariant.

Thursday April 30

Forms and Semigroups: Modern Methods for Evolution Equations

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This talk will give a brief introduction to the theory of strongly continuous semigroups. These semigroups are a powerful tool to study parabolic equations (such as the heat equation) in a more abstract setting. Using some basic form methods (such as the Lax-Milgram lemma) as well as the Lumer-Phillips theorem, we can show that divergence form elliptic operators generate strongly continuous semigroups and these can be used to solve our parabolic equations. If time permits, we may briefly consider the non-autonomous case and discuss how forms continue to play a part.

Thursday May 7

An Introduction to Geometric Integration via Runge-Kutta Methods

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Geometric Integration is concerned with the following. Suppose we have a differential equation which we have proved has certain geometric properties e.g. a conserved energy or that solutions are constrained to lie on a certain surface. Can we design numerical approximations to the solution that also possess these properties, or is it a fact of life that they will only be approximately respected? In this talk I will explain the properties of certain Runge-Kutta methods (which will be defined), and how they can be used to solve Hamiltonian systems effectively ("effectively" will also be defined). Towards the end of the talk I will discuss some recent work on volume preservation i.e. systems in which measurable sets of initial conditions are mapped to sets of equal volume for all times.

Thursday May 14

Anomalous diffusion processes

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The stochastic representation of diffusion processes is well known and is aptly characterised by the Brownian motion. Perhaps less well known is the equivalent for so-called anomalous diffusion processes, in which the linear time-scaling of variance seen in regular diffusion no longer holds. We will present a construction of a stochastic process which characterises random walkers in anomalous diffusions. Further, we will use this stochastic process to construct a numerical solution to the Fractional Fokker-Planck equation- the differential equation which characterises anomalous diffusion processes, framed as an initial value problem. Some knowledge of statistics is probably required, however we will endeavour to make the talk as elementary as possible.

Thursday May 21

A gentle introduction to topological quantum computation

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This talk presents a physics-light introduction to quantum error correction and topological codes. After a brief review of the relevant quantum mechanics, we discuss basic classical linear coding theory and how it can be "lifted" to the quantum case in order to deal with the wider range of errors that can affect a qubit as opposed to a classical bit. Drawing inspiration from statistical mechanicals, we then introduce a class of quantum error correcting codes known as topological codes, illustrating many of their most important properties through Kitaev's toric code. Time permitting, we will also discuss the concept of universal quantum computation and how certain topological codes are capable of achieving this.

Thursday May 28

Ergodic Plunnecke inequalities

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Furstenberg's ergodic theoretic proof of Szemeredi's theorem gave rise to a fruitful connection between Ergodic Theory and Combinatorial Number Theory. After a very brief historical introduction to Ergodic Ramsey Theory, I will present some ergodic-theoretic extensions of the classical PlÃŒnnecke inequalities from Additive Combinatorics (joint work with A. Fish).