# Postgraduate Seminar

Organiser:

#### 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: Wednesday 28th November

Week 10: Wednesday 5th December