# Semester 1 2016/17

### 6th October 2016 - Dan Green

#### A Spoonful of Observations (helps the forecast error go down)

In this PSS talk we will investigate the theory behind weather forecasting, GPS navigation, seismology, and avoiding being hit by a ball. Data assimilation is a way of using and combining observations into a model to create a better guess for the truth.
Here we will see why weather forecasts can be terribly wrong, but why it’s impressive that they exist in the first place. We introduce the idea of data assimilation, and the maths behind it with some visual examples of where it goes right… and where it doesn’t.

Hopefully it'll be very clear to see that a spoonful of observations helps the forecast error go down … in a most delightful way.

### 13th October 2016 - Tom Burnett

My PhD is about optimising Adaptive Enrichment designs, they are a form of randomised control trial intended to deal more effectively with patient sub- populations. We will conduct some trials using the audience to demonstrate the differences between adaptive enrichment and standard designs. Don’t worry I did a very similar talk over summer and there were no harmful side effects (the “treatment” is sweets, if you have any requests let me know).

We will be using R to analyse the results of our trials so we can see the different hypothesis testing structures that become necessary. Further to this R will allow us to simulate the operating characteristics of various designs allowing us to optimise adaptive enrichment and compare the overall performance of different designs.

### 20th October 2016 - Pablo Vinuesa

#### The blue-eyed islanders puzzle and the Kakeya conjecture

The first half of this talk will be dedicated to the blue-eyed islanders puzzle. We will see that it has a rather surprising solution, and proving this requires us to apply mathematics and logic. For the second half, I will discuss the Kakeya conjecture which I have always found to be very interesting. I will not go into a lot of detail, but I will present the main definitions and results to provide the audience with an idea of what the problem is about and why we became interested in it in the first place. We will see, for example, the Hausdorff dimension and the Minkowski dimension, otherwise known as the box-counting dimension.

### 27th October 2016 - Xavier Pellet

#### Gradient Flow in the Monge Kantorovich metric and application to the Patlak Keller Segel system

The Patlak Keller Segel system is used in biology (contamination) and economy (dynamic of population). It's model aggregation or diffusion of a flow of particles. I will introduce optimal transport, gradient flow and energy of a system in order to understand this PDE. This is an amazing approach connecting analysis and probability. This presentation is based on the work of R. Jordan, D. Kinderlehrer, F. Otto, A. Blanchet, V. Calvez and J. Carillo.

### 3rd November 2016 - Francisco de Melo Viríssimo

#### On a Masters degree taken in Brazil

This talk will highlight the research I have done in Brazil, which resulted in the thesis 'Hydrodynamic interaction between rigid bodies: the potential case'. Higher education degrees (BSc, MSc, PhD, etc.) can differ a lot from country to country, so even if you are not interested in the mathematics of my monograph, come to hear some intriguing tales about the student life in my motherland.

### 10th November 2016 - Jack Betteridge

#### Spiral Waves in Excitable Media

Solutions to the reaction-diffusion equations which model excitable media include, as one of the more interesting cases, rotating spiral waves. By varying parameters of the models that determine these equations, these solutions we observe produce beautiful, complicated and intriguing behaviour.

This talk will be an introduction into a popular and very active research area, without assuming more than undergraduate mathematical knowledge. It will cover the reaction-diffusion equations which model excitable media, possible computational models and the behaviour of solutions. I endeavour to talk a little bit about some of my own (previous) research as well as current research and open ended problems in this area.

Warning: May contain traces of biology, chemistry and physics. In addition; flashing hypnotic visuals and possible audience participation.

### 17th November 2016 - Robbie Peck

#### Sequential tests and the random assortment of mathematics needed to build one

As outlined in the Bible, Noah sequentially sent forward doves from the ark to test whether they came back with olive branches to see if there was land. In a journey through some Markov Processes, numerical integration, and some of Chris’ book, this presentation will look at why Noah knew what he was doing by using sequential, rather than fixed sample testing. This presentation should be accessible and of interest to all Maths PhD students.

By testing sequentially, we have rules to stop part of the way through a test process, rather than having to wait until the end. Group Sequential Methods can produce tests with desirable properties and are extremely important in industry where unnecessary testing can be considered unethical and inefficient.

The presentation starts by saying why Group Sequential Methods are better than fixed sample methods, introduces a general canonical form to explore properties of, builds a sequential test, and leads one through the daunting task of the numerical analysis required to give it desired properties. It also has plenty of nice plots and diagrams in. At the end of the presentation I will make a dramatic exit and get on a plane to China.

### 24th November 2016 - Claudio Onorati

#### An informal, down-to-earth and soft (attempt of) introduction to algebraic geometry

I will start with commutative algebra: are you able to find all the zeros of a polynomial? This simple question is indeed very hard to answer in general and it is considered by someone the basic motivation to introduce algebraic geometry. I will give the (possibly non rigorous) definition of an algebraic variety and I will try to do it in the most down-to-earth possible way. At the end of the talk you probably will not know what an algebraic geometer really does, but for sure you know when it's time to escape in case you meet one of them!

### 1st December 2016 - Owen Pembery

#### The Helmholtz Equation: Sounds easy?

The Helmholtz equation describes the propagation of sound waves of a single frequency. Therefore, we might expect that solving it numerically would be straightforward. However, if we thought that, we'd be wrong.

In this talk I'll give an overview of why we should care about solving the Helmholtz equation (this includes imaging the rocks underneath the sea and escaping from Taylor Swift), some of the reasons why it is difficult to solve numerically, and what current research is being done to make solving it easier.

As this talk will be given on the first of December, we'll conclude by playing `Can you guess the Christmas Carol from its Fourier Transform?'

Come along for Fourier transforms, frustration, finite elements, festivities, fun and much, much more.

### 8th December 2016 - Amy Middleton

#### Chrismography

As you may have guessed from the title, this talk will have a festive theme. I am going to give a brief introduction to cryptography, which I have renamed chrismography for the rest of this month.

This introduction will include both some Number Theory and some Statistics, showing how both these areas are used together. I will relate everything to Christmas as much as possible.

There will be a small competition where the audience will use the methods discussed to encrypt some text of their own before attempting to decrypt other peoples.

### 15th December 2016 - Dan Green & Leonard Hardiman

#### PSS XMaths Special (Pointless Edition)

Alexander Armstrong & Richard Osman, also known as Leonard Hardiman & Dan Green host this year’s Christmas edition of the PSS. Some people may refer to PSS as pointless (which it isn’t), but this week it is Pointless with a capital P. We gave 100 maths undergraduates 100 seconds to name as many answers to our questions as possible. Competing in teams, it is your teams goal to try and find the most obscure answers!
Questions range from Maths, to Christmas with little in between. Prizes will include (and may be limited to), that warm feeling of destroying the competition.

# Semester 2 2016/17

### 9th February 2017 - Kieran Jarrett

#### Fatou, Julia and Mandelbrot Sets

My talk will be a short introduction to complex dynamics - an area of mathematics which exhibits beauty arising from simplicity as well as any other I know.

I will explain how by simply considering a polynomial $$p$$ and the orbits $$z$$, $$p(z)$$, $$p^2(z)$$, $$\ldots$$ of each point $$z$$ in the complex plane one can describe a large variety of beautiful and intricate fractals called Julia sets. In fact, most of my examples will be drawn from just the case $$p(z) = z^2+c$$ where $$c$$ is a complex number. I’ll state some results which explain some of the structure of these fractals and, to finish, I’ll tell you how Julia sets are linked to the famous Mandelbrot set. Hopefully I’ll have time to give a few properties of the Mandelbrot set, and show you a cool fractal drawing program I’ve been playing with the last few weeks.

Prerequisites: If you know that quadratic equations can be solved and what it means to say a sequence is bounded you'll be fine.

### 16th February 2017 - Gianluca Detommaso

#### Continuous Level Monte Carlo

Multilevel Monte Carlo (MLMC) is a recently developed and well established Monte Carlo (MC) method to estimate statistics of some quantity of interest $$Q$$. It exploits a sequence of approximations $$(Q_\ell)^L_{\ell = 0}$$ of the quantity $$Q$$ , where the indices $$\ell$$'s are levels of accuracies, from coarsest to finest. By exploiting the different computational cost of each approximation $$Q_\ell$$ and combining them in differences to get variance reduction, a Complexity Thereom ensures that, under some hypotheses, the order of convergence of MLMC drastically beats the MC's one.

Continuous Level Monte Carlo (CLMC) extends MLMC by considering a continuous sequence of approximations $$(Q_\lambda)_{\lambda \in [\Lambda_0, \Lambda]}$$ of the quantity $$Q$$ , where $$\lambda$$ represents a continuous index. By replacing sums with integrals and differences with derivatives, we are able to show that an analogous Complexity Theorem holds also in this case. However, from a practical point of view, the integral and the derivative must be again . This arises a large class of approximated CLMC method, that MLMC belongs to as a particular case. Here the concept of Asymptotic Perfect Variance Reduction (APVR) is introduced, studying the conditions that the approximated CLMC method should satisfy to optimally reduce the variance. As an example, we then compare MLMC with a particular approximated CLMC method that we address as 5-stencil CLMC. We see theoretically and numerically that, asymptotically in the level $$\lambda$$ , 5-stencil CLMC produces a double reduction in the variance with respect to MLMC.

### 23rd February 2017 - Ben Robinson

#### Random motion and PDEs: What’s the connection?

As mathematicians, we all have some idea of how heat diffuses. However, we may use different languages to describe this phenomenon. For example, an analyst will most likely write down the heat equation, derived from Fourier’s law on the transfer of heat. Whereas a probabilist may talk about individual particles moving at random, inspired by Brown’s observations of the movement of pollen grains.
In this talk, I will aim to convince you that these two formulations are in fact equivalent. More generally, I will describe how random motion can be characterised by partial differential equations and, conversely, how solutions of certain elliptic and parabolic PDEs can be written down in terms of corresponding random processes. Beyond the diffusion of heat, I will give examples from other applications, ranging from acoustics to finance.
Along the way, I will introduce the framework which probabilists use to describe random motion in continuous space, and explain how this gives rise to differential operators.
I will also briefly touch on my own research area, where we see relationships between more complicated random behaviour and another class of PDEs.

### 2nd March 2017 - Joel Cawte

#### Cawte's Captivating Colloquium/Conferral of the Complexity of Quantum Computers

Word on the Scientific Grapevine is that Quantum Computers are just around the chronological corner. Whispers of immense computing power are spreading, and are causing people to lose their minds in awe and fear. Many questions are tearing the world apart: How do they work? What can they do? What even is Quantum? Can I still run 20XXTE?

In an attempt to calm everyone down, political and economic order may be restored if I try to half answer these questions by skim-reading some old lecture notes clear up the ambiguities and concerns with unmatched authoritative knowledge, and explain how one can teleport information and defuse bombs.

### 9th March 2017 - Emma Horton

#### An Introduction to Schramm-Loewner Evolution

Schramm-Loewner evolution (SLE) is a family of random curves that start at the origin and evolve in the upper half of the complex plane.
In this talk, I will introduce SLE via a solution to Loewner’s differential equation

$$\frac{\partial z_t}{\partial t} = \frac{2}{z_t - \sqrt{\kappa}B_t}$$

where $$(B_t)_{t\ge 0}$$ is a Brownian motion and $$\kappa \in [0, \infty)$$ is a diffusivity parameter, which controls how much the curve turns as it works its way through the upper half-plane. For certain values of $$\kappa$$, the corresponding SLE is the scaling limit of a discrete probabilistic model, which I will also briefly discuss. For example, when $$\kappa = 2$$, SLE(2) is the scaling limit of a certain type of random walk, and if $$\kappa = 6$$, SLE(6) is the scaling limit of a percolation model.

As with most maths involving processes in the complex plane, there will be plenty of pretty pictures!

### 16th March 2017 - Cameron Smith

#### Hybrid Models: Innovative Methods or a Waste of Time? You Decide!

In this somewhat maths deficient PSS talk, I plan to convince you that hybrid models are very useful for simulating multi-scale systems. And if not, well at least there is cake afterwards!
Join me for a (mostly pictorial) journey through several different hybrid approaches to simulate reaction-diffusion systems; an important group of models for explaining, predicting and answering the big questions in biology such as:

• Why do some mice have belly spots?
• (Overdramatic voice) How can we stop the next big pandemic from destroying us all?
• Why can’t a leopard change it’s spots (into stripes at the very least)?

We will then move onwards to some of my own work, creating new models which add in extra biological realism or simply fill a gap in the market.
Health warning: may contain traces of maths, a pinch of biology and some weird images. Fun cannot be guaranteed.

### 23rd March 2017 - Tom Crawley

#### Introduction to Algebraic Topology

This talk will give you an introduction to one of the big ideas in 20th century mathematics. I will describe the fundamental group of a topological space and use it to distinguish an apple from a doughnut. I will talk about some applications including the fundamental theorem of algebra, a fixed point theorem and slicing a theoretical(?) ham sandwich exactly in half. Unfortunately there will not be time to discuss hairy balls.

### 30 Mar 2017 - Emiko Dupont

#### We're gammin' - we hope you like gammin' too

Nestled in the Swiss mountains lies the small municipality of Gams, population size 3,296 (anno Dec 2015). Apart from the name, this has no particular relevance to my talk. The GAMs I'll be talking about are Generalised Additive Models which are a very flexible class of statistical models that generalise linear regression models and GLMs (Generalised Linear Models), some of the most widely used models in applied statistics. All of these models describe the relationship between a variable of interest $$Y$$ and one or more explanatory variables (or predictors) - and are very cool because they enable us, for example, to identify which predictors have a significant effect on $$Y$$ or to make predictions of $$Y$$ at unobserved values of the explanatory variables. However, linear models and GLMs require prior specification of the functional form of the relationship between $$Y$$ and the predictors (e.g. linear, quadratic, exponential etc). The advantage of GAMs is that they estimate the functions as part of the output. This is useful because for many applications, the functional form is unknown or quite complex.

### 6th April 2017 - Dorka Fekete

#### Skeletal SDEs for continuous-state branching processes

Continuous state branching processes (CSBPs) are used to model the evolution of (renormalised) large populations on a large time scale. It is well understood that we can give a pathwise representation of a CSBP as immigration along an embedded discrete tree, called the skeleton. Equally well understood is the notion of a spine or immortal particle dressed in a Poissonian way with immigration, which emerges when conditioning the process to survive forever.

In this talk, after introducing the different types of decompositions, I will show how to put them in a common framework using the language of coupled stochastic differential equations. I will also explain how the skeleton thins out (in the sense of weak convergence) to become the spine when conditioned on survival. This is joint work with Joaquin Fontbona and Andreas Kyprianou.

### 27th April 2017 - Matt Lee

#### Tunnels and Vacuums

The quantum world is miasma of bizarre, counter-intuitive, and insanely improbable phenomena. However, such things are incredibly useful to every one of us - at least, some of the phenomena are. I will be briefly discussing two of the more interesting results of quantum physics - tunnelling and quantum vacuums.
If you ever wanted to consider why two uncharged conductive plates will move towards either other in a vacuum and whether Platform 9 and 3/4 might actually exist, then I will attempt to satisfy your curiosity.

### 4th May 2017 - Alge Wallis

#### Algebraic Geometry in Computer Vision.

The real world needs machines that can see. There is an infamous story that the task of solving this problem of computer vision was considered only a summer project. 50 years later, it is still a hot topic of research and sits at a cross roads of pure maths, applied maths, and some pretty lucrative jobs in industry.

In the last few decades the employment of computational algebraic geometry has made feasible and practical solutions of pose estimation and 3D reconstruction.

We will look briefly at some key computer vision problems and how Groebner bases, finite fields, and eigenvectors can be used to produce efficient and stable algorithms. We also look at a few avenues for further research.

Knowledge of Groebner bases is not assumed. This talk is based on a research project I undertook at the National Institute of Informatics, Japan.