# Postgraduate Seminar

Organiser: Aidan Browne

#### Term 1 2018-19 - The seminars are held on Wednesday 12:00 - 13:00 in B3.02 - Mathematics Institute

Week 1: Wednesday 3rd October

**Stephen Cantrell **- Ergodic Theory, Symbolic Dynamics and their Applications to Geometry

This talk will be a gentle introduction to a branch of ergodic theory known as symbolic dynamics. We will introduce basic ideas and notions from ergodic theory and look at how they can be used to understand the long term behaviour of dynamical systems. We will then explore how techniques from symbolic dynamics can be used to tackle problems from geometry. In particular, we will discuss 'comparison theorems' and a 'dynamical version' of the prime number theorem.

This talk will be accessible to mathematicians from all areas - no prior knowledge of ergodic theory required!

Week 2: Wednesday 10th October

**Sami Al-Izzi** - Active Membrane Tubes: Geometry, Fluids & Elasticity in Cell Biology

Lipid membranes are an interesting biological material as they display 2D viscous behaviour laterally, but behave as an elastic surface in the normal direction. The coupling of these processes (along with hydrodynamics of the solvent) can lead to novel phenomena which can be key to shaping cell and organelle morphology.

We will focus on the morphology of membrane tubes, which are found throughout cells, and are involved in many processes vital to cell survival. I will start by reviewing some of the basic properties of membrane tubes and their formation, in particular their similarities and differences from columns of fluid with surface tension. We will then outline two ways in which "active" processes can act on membrane tubes to change their morphology. The first of these is related to the problem of volume conservation in single celled organisms and the second is related to the scissoring of membrane tubes in cells more generally.

This talk will be accessible to a general mathematics audience; no knowledge of biology is assumed.

Week 3: Wednesday 17th October

**Rob Chamberlain** - Computing With Permutation Representations of Finite Groups

How we represent a group significantly affects the speed of computations involving the group. In this talk we concentrate on permutation representations of finite groups. The aim is to present the main mathematical approaches to speeding up such computations. The focus will be on the 'best' representation of a finite group and how one might use the structure of the group to find a 'good' representation, though other mathematical approaches are introduced.

Week 4: Wednesday 24th October

**Sophie Meakin** - Stochastic modelling of infectious diseases in interacting populations

Heterogeneity in the interaction between individuals plays an important role in the dynamics, persistence, evolution and control of infectious diseases. This can be incorporated into epidemiological models by dividing the population into multiple interacting subpopulations. In this talk I will discuss two results based around this framework: firstly, a method for inferring the level of interaction between populations using data on disease incidence; and secondly, a model for a recent outbreak of Ebola in the Democratic Republic of the Congo.

Week 5: Wednesday 31st October

**Philip Herbert** - An introduction to PDEs on surfaces with surface finite elements.

We start by talking about what it means in general to solve a PDE (weak solutions) in a flat domain. We then move on to discuss some of the basic notions from Finite Element Methods (FEM), and see how this gets translated to the surface finite element case. If time permits I will discuss some of my related research.

Week 6: Wednesday 7th November

**Aidan Browne **- A disk-type solution to the Plateau problem.

The general Plateau problem of finding a surface of least area with a given boundary has a long history. It is named after the 19th century physicist Joseph Plateau, who showed empirically that a solution exists by creating it with a soap film and a wire boundary. In this talk I will present a proof of the restricted disk type version of the problem, originally due to Richard Courant in the 1950s.

Week 7: Wednesday 14th Novermber

**Quirin Vogel **- Hot and cold loop Soups.

A seminar on soups, filled with random loops. The question not to hate, is whether they do percolate. A few results did yield, coupling with the Free Field. Some things will be shown, whilst others still rest unknown. Afterwards, you'll have free food, and chat about what you've viewed. No pre-knowledge is required, so please do come and be inspired.

Week 8: Wednesday 21st November

**Nikos Alexandrakis **- Embedded Solitons and Complex Analysis

In the late nineties, a new type of Soliton was discovered, the Embedded Soliton, which is a wave packet that propagates with the same velocity as linear (sinusoidal) waves in the medium. Moreover, they can exist in families as solutions to certain parametric models. Their existence can be detected using tools from Complex Analysis and Asymptotics. In this talk, we will explain the basic ideas behind the corresponding techniques; no pre-knowledge is required.

**Quirin Vogel **- Rare Events and Swiss Chesse

Did you know that the holes in Swiss cheese are created by sausages, Wiener sausages to be precise? If not, why don't you come to the postgraduate seminar to see how, using the theory of large deviations and random walks, we can give rigorous meaning to the above statement. No foreknowledge required and Swiss cheese may even be served afterwards.

Week 9: NO TALK

Week 10: NO TALK

#### Term 2 2018-19 - The seminars are held on Wednesday 12:00 - 13:00 in B3.02 - Mathematics Institute

Week 3: Wednesday 23rd January

**Daniele Mastrostefano **- Heuristics for the distribution of primes in short intervals

"God may not play dice with universe, but something strange is going on with the prime

numbers." (Paul Erdos). The prime numbers are deterministic objects and indeed we are able to detect them using sieve techniques or approximate their counting function by means of tools from complex analysis. However, they somehow behaves like random objects, exhibiting properties typical of a suitable sequence of random variables. For this reason we refer to this aspect as pseudorandomness. In this talk we will show how to randomly model the prime numbers to obtain heuristics for their distribution in short intervals and we will compare these predictions with the known results on this problem.

Week 4: Wednesday 30th January

**David Woodford **- Random Randomness: An Introduction to Markov Additive Processes

There are many scenarios where the evolution of a stochastic process depends on some underlying state which changes randomly. We will disucss a particular class of examples, Markov additive processes, where the underlying state is given by the value of a continuous time discrete state space Markov chain and the stochastic process of interest is a Levy process. This is an example that naturally arises in queueing, finance and theoretical probability.

**Darion Mayes **- Random Permutations and Edge Subgraphs in Graphs of Diverging Degree

The random interchange process is a random walk on the set of permutations of the vertices of a graph. This may be studied via a connection to the theory of percolation. Inspired by their study on the complete graph, we conjecture a surprising similarity between two generalisations of these models.

Week 7: Wednesday 20th February

**Augustin Moinat **- Regularity Structures for $Phi_3^4$

I will attempt to give a gentle introduction to regularity structures as introduced by Hairer in 2014 through some motivating examples. This theory has led to great results over the past half decade and Hairer was awarded a Fields medal in 2014. I will show the need for so-called "positive" and "negative" "renormalisation" and give some details of the construction of solution to the dynamic $Phi^4$ model in $3$ dimensions.

Week 8: Wednesday 27th February

**Anna Parlak **- Pseudo-Anosov Homeomorphisms of Surfaces

Among all types of surface homeomorphisms pseudo-Anosovs draw the greatest attention of mathematicians working in geometry, topology and dynamical systems. The definition of a pseudo-Anosov homeomorphism requires the existence of a pair of transverse measured singular foliations on a surface that satisfy some special conditions with respect to the homeomorphism. During the talk we will work with an explicit example of a pseudo-Anosov map of a genus two surface. Using this example we will show how to prove that a given map is pseudo-Anosov directly from the definition, without referring to any computationally convenient characterisations of such homeomorphisms.

In order to meet this goal we will need some additional tools, in particular half-translation structures on surfaces and train tracks. The latter objects, introduced by Thurston, are closely related to singular foliations of surfaces and provide a way to encode them combinatorially.

I also hope to mention some results about pseudo-Anosov mapping tori, which are, as shown by Thurston, hyperbolic 3-manifolds as long as the surface fibre has negative Euler characteristic.

Week 10: Wednesday 13th March

**Alessandro Arlandini -**** **Euler Systems, Selmer Groups and L-functions

Euler systems are one of the most recent techniques developed by number theorists to tackle arithmetic problems from an algebraic point of view. In particular, they are used in the study of representations of Galois groups, and consist of "families" of elements attached to a representation. By constructing an Euler system one hopes to extract the arithmetic content that a representation shows at any finite field extension, in a "combined" way.

In this talk I will introduce the notion of Euler system and explain two of their most important applications: to the study of Selmer groups and (p-adic) L-functions. I will mention the link with a few open conjectures with the help of concrete examples. If time permits I will discuss some of my related research.

#### Term 3 2018-19 - The seminars are held on Wednesday 12:00 - 13:00 in MS.04 - Mathematics Institute

Week 6: Wednesday 29th May

**Jaromir Sant **- Bayesian Inference for Genetics - A Tale of Two Alleles

Mathematical population genetics aims at developing a mathematical framework through which one can better understand evolutionary phenomena and patterns of genetic variation. The Wright-Fisher model is perhaps one of the most well-known models, and converges (subject to suitable scaling) to a diffusion process limit termed the Wright-Fisher diffusion. Important information regarding the genetic evolution of a population (such as mutation and selection parameters) is encompassed within the drift coefficient, which then becomes the main object of interest when one conducts inference.

In this talk I will give a brief introduction to mathematical population genetics, introduce the Wright-Fisher model and diffusion, and focus on work that has been done on the Bayesian Estimator for selection. No prior knowledge of biology or genetics is required.

Week 8: Wednesday 12th June

**Hanson Bharth **- The Lord of the Metal Rings: The Fellowship of the Spin

Week 9: Wednesday 19th June

**Mnerh Alqahtani **- Rare Events Computation in Fluid Dynamics

Despite their small probabilities, rare events can have major impacts, such as economic crises, natural disasters and epidemics. If a large deviation principle (LDP) holds, then the probability of these tail events decays exponentially, at a certain rate, which depends on the rate function. Our main interest is the long-term behaviour of weak noise-driven dynamical systems; more precisely, this research addresses the probability of a path of random process that hits a certain set.

In this talk, I will introduce the relevant theorems from large deviations theory (LDT), and present the instanton approach that provides the maximum likelihood pathway (MLP) of unlikely events of a system comes from fluid dynamics, including the roots of this approach in quantum mechanics (QM). Finally, I will illustrate the numerical implementations of Instanton equations which are equivalent to the minimisation of the rate function. No prior knowledge of LDT or QM is assumed.

Week 10: Wednesday 26th June

**Nik Alexandrakis -** Solitons, Kinks, and Singular Perturbation Theory

The peculiar nature of Solitary waves has drawn the attention of physicists and engineers due to the variety of possible applications in industry as well as their seemingly "impossible" physical properties. From a mathematical point of view, the interest lies in certain types of nonlinear differential equations. Furthermore, things get more intriguing when singular perturbations come into play. In this talk, we will discuss whether localized traveling waves, i.e. traveling waves that asymptote to "sea states", like Solitons and Kinks (Plasma/Magnetohydrodynamics), survive under certain singular perturbations. The mathematical toolbox we are going to use combines Complex Analysis, Asymptotics and of course Perturbation Theory.