# Mathematics Postgraduate Seminar Series - 2015

## Seminars in 2015, Semester 1

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).

## 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.