## The Era of Duhart

This is the compilation of the talks given in the 2013/14 Postgraduate Seminar Series which was organised by Horacio G. Duhart.

# Semester 1 2013/14

1 Oct 2013, Horacio G. Duhart

### The risks of becoming an actuary: an introduction to life insurance maths

In this talk we'll explain what is an actuary and what they have to do with life insurance. Then we'll see the most basic concepts needed for calculating the premiums of life insurance which involve techniques from the theory of interest and probability

8 Oct 2013, Istvan Redl

### A glimpse into mathematical finance? The realm of option pricing models

After introducing one of the most important concepts of mathematical finance, the fundamental theorem of asset pricing (FTAP) and the related no arbitrage pricing theory (NAPT), I will briefly discuss the main techniques and tools extensively used in option pricing, namely Monte Carlo, Fourier Transform and PDE methods. In order to give a fairly well-structured overview of a great chunk of currently preferred models, through a simple example the hierarchy of the mathematical models will be demonstrated by going from the basic Black-Scholes to some more advanced models, e.g. Stochastic Volatility with jumps. (Even those people, who are familiar with these concepts, might find the main focus, i.e. structured overview, of this talk beneficial).

### Base-station subset selection for full coverage

Rural internet access is still limited in the UK. One possible solution proposed by BT is to place long range telecommunication masts at some of the existing telephone exchange points. We design an algorithm which returns the smallest possible subset of the masts such that they will still cover the total possible coverable area. We implement the algorithm in parallel in Python/C++ and show promising results: for our practical context the problem is solved in a matter of hours even for a large number of masts and large choices of radius. This is achieved by reducing the problem in polynomial time, and solving the sub-problems using dynamic programming. This final stage has an empirical running time or approximately $\algrowth^n$. Furthermore we relax the implementation to accept arbitrarily-shaped coverage regions, not just Disks.

22 Oct 2013, George Frost

### Relativity for the Inertial Observer

Einstein's theory of (general) relativity is one of the cornerstones of modern physics, but it also has a rich mathematical foundation. Following a brief overview of Special Relativity, we will discuss the postulates and introduce the mathematical framework of general relativity. This will include a non-technical introduction to manifolds, curvature and Einstein's gravitational field equations.

29 Oct 2013, Katy Gaythorpe

### Systems Epidemiology

Systems Biology studies the elements of a biological system before and after perturbations. I will introduce this field and examine parallels with epidemiological system modeling with an aim to use the machinery of systems biology on a macro scale. This will hopefully bring some new insights (and issues) to the world of epidemiology.

5 Nov 2013, James Roberts

### Harmonic maps

Our focus will be the techniques and tools required to conclude the Hölder continuity of minimisers of the Dirichlet Energy functional associated to an appropriate class of maps between Euclidean space (of dimension greater than 2) and a smooth, compact Riemannian manifold. I will explain the setting, present the main results required and give an overview their foundation.

12 Nov 2013, Kuntalee Chaisee

### Bayesian point estimation as a decision problem

One of the most common questions in statistics is what is the best point estimate of the unknown parameter theta? I will introduce you how to give the best guess of theta by using the Bayesian expected loss principle. I aim for an efficient computational approach to establish the point estimate related to a complex loss function.

19 Nov 2013, Ben Boyle

### Perturbation and multiscale methods in differential equations

APerturbation techniques are something of a dark art but when applied with sufficient cunning can give us a sideways glance at behaviours of solutions of differential equations which defy many other less morally ambiguous attempts at analysis. In this talk I will illustrate their strengths and weaknesses with a series of examples culminating in a demonstration of the unreasonably effective technique of multiscale analysis, in which we pretend that things that clearly aren't independent are.

25 Nov 2013, Matt Pressland

### Introduction to Galois Theory

This talk serves as a (very) brief introduction to Galois theory. We warm up by exploring the limitations of ruler and compass constructions, and prove that they cannot be used to trisect angles. Glossing over some technical details, we will outline the fundamental theorem of Galois theory. This theorem is the first example of a Galois correspondence. Such correspondences are somewhat ubiquitous, and we illustrate the general idea with a second (less technical) example, involving homogeneous spaces.

3 Dec 2013, Matteo Fasiolo

### A Synthetic Likelihood approach for stochastic ecological models.

(Wood, 2010) proposed an approach, called Synthetic Likelihood (SL), that can be used to do statistical inference for highly non-linear or near-chaotic dynamical systems. The approach creates an approximate likelihood, based on a Multivariate Normal approximation, which can be used to do inference about the models parameters. After introducing SL I will describe two extension of this approach. The first is meant to accelerate parameter estimation using SL, by employing a stochastic optimization routine to efficiently maximize the Synthetic Likelihood. The second extension consists in using an Empirical Saddlepoint Approximation to relax the normality assumption upon which SL is based.

10 Dec 2013, Ray Fernandes

### Welcome to the Xmaths Special #625 (approx)

We look at asymptotic properties of nonlinear dispersion equations, and their (2k+1)th-order analogies. The global in time similarity solutions, which lead to "nonlinear eigenfunctions" of the rescaled ODEs, are constructed. The basic mathematical tools include a "homotopy-deformation" approach, where the limit as n goes to 0+ turns out to be fruitful. At n=0, we see that the problem is reduced to the linear dispersion one, where the corresponding Hermitian linear non-self-adjoint spectral theory, giving a complete countable family of eigenfunctions, is known.

Erm, possibly. Come along for the usual xmaths mayhem, where as ever, there will be little in the way of maths and talk (possibly quite literally this year as things are going quite badly for me...). Join me for the PSS of a lifetime*, with fabulous prizes to be won**!

*The lifetime of an alcoholic, chain-smoking, mayfly.
**Disclaimer: prizes may be non-existent or may include what I can scrounge from the department cupboard (which as you know, it means it's pretty much non-existent anyway).

# Semester 2 2013/14

4 Feb 2014, Matt Dawes

### Modular forms and other monstrosities

In 1978, John McKay made a remarkable conjecture, based on limited numerical evidence, relating two totally separate worlds: the monster group and modular forms. This audacious conjecture was named 'monstrous moonshine' and, in 1998, Richard Borcherds won a fields medal for proving it. I will give a brief introduction to modular forms, and the monster group, and will discuss the history of the conjecture and discuss the work of some of the characters involved.

11 Feb 2014, Samuel Gamlin

### Descendants in $$\mathbb{Z}^d$$ spanning forests

I will explain how to construct a spanning tree using simple random walks. This then means we can use known results for random walks to deduce results about infinite spanning trees. To demonstrate this i will describe how to bound the descendants of a set and how well they can be approximated by looking at finite subgraphs, in the case of $$\mathbb{Z}^d$$.

18 Feb 2014, Horacio G. Duhart

### Large Deviations: an introduction to its theory and applications

Although the first problems and results can be traced to the beginning of the 20th century, the theory was unified by Varadhan in 1966. Surprisingly, the study of the probabilities of very rare events arise in other subjects such as Risk Management, Information Theory, and Statistical Mechanics among other applications. In this talk we'll go through the definition, main theorems, applications and basically everything everybody wants too know about them in a very introductory way.
The only prerequisite for the talk is to know how to toss a coin!

25 Feb 2014, Maren Eckhoff

### High-speed travel through a random environment

Many real-world networks carry a flow between different parts of the network.
These networks are well-described by graphs with edge weights, representing the cost or time required to traverse the edges.
In this talk we discuss an important network model and properties of the optimal path between two vertices in that model.

4 Mar 2014, Jack Blake

### Mixing asymptotic analysis with numerical analysis

We will take a beginners look at using asymptotics within an integro-PDE, and try to answer the age-old question: what can asymptotics do for us? After partially answering that for one specific application, we might move on to look at an example of equivalence between preconditioning and acceleration of iterative methods. I will do my best to define everything in a useful way without overly complicating anything.

11 Mar 2014, Jesús Tapia Amador

### G-igsaw Transformations

Given a directed graph, we can define a G-igsaw transformation as a deformation of one of its subgraphs into another. By working through an example, I'll introduce this process in a combinatorial way and explain how it is useful in constructing certain smooth geometric objects.

18 Mar 2014, Ernesto "The Legend" Padilla

### A marketer and a statistician walk into a bar

A little known fact about market research (which is, by itself, a little known subject) is that it is almost entirely based on statistics. True, a lot of it is simple cross tabulation, but there are some more advanced techniques involving actual maths. As a long-time veteran of the Research and Development & Marketing Sciences*, I will give a very brief overview of these more advanced techniques, their conceptual framework and applications. Since there are a lot of them and only a little time, don't expect heavy maths. Or medium maths. In fact, try to not expect maths (or seriousness).

*Yes, there is such a thing as Marketing Sciences and no, there is no actual science involved.

25 Mar 2014, Steve Cook

An informal introduction to dynamic mesh adaptation through mesh refinement (h-adaptivity) or mesh-point relocation (r-adaptivity) for weather and ocean modelling.

1 Apr 2014, Steven Pagett

### Coalescence and Fragmentation of Terrorist Cells

Applying a coalescence/fragmentation model to the size and strength of terrorist cells involved in long term warfare.

8 Apr 2014, Mason Pember

### Bubble trouble

We will define the mean curvature of a surface in Euclidean 3-space and discuss a class of surfaces whose mean curvature is zero - minimal surfaces. These surfaces locally minimize surface area and physical models of these surfaces are obtained using soap films. We will then introduce the Weierstrass representation which gives us a way to construct many examples of these surfaces.

15 Apr 2014, Hannah Woodall

### Extinction thresholds in epidemic models

By looking at extinction thresholds we will find the probability of an epidemic for different stochastic epidemic models. We will start by introducing a simple stochastic model and explaining the methodology. We will then consider some well-known models and extend these to incorporate age-structure.