### Postgraduate Seminar Series (PSS) 2016/17

This is the compilation of the talks given in the 2016/17 Postgraduate Seminar Series which was organised by Dan Green.

This is the compilation of the talks given in the 2016/17 Postgraduate Seminar Series which was organised by Dan Green.

6 Oct 2016, **Dan Green**

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.

13 Oct 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.

20 Oct 2016, **Pablo Vinuesa**

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.

27 Oct 2016, **Xavier Pellet**

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.

3 Nov 2016, **Francisco de Melo Virissimo**

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.

10 Nov 2016, **Jack Betteridge**

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.

17 Nov 2016, **Robbie Peck**

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.

24 Nov 2016, **Claudio Onorati**

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!

1 Dec 2016, **Owen Pembery**

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.

8 Dec 2016, **Amy Middleton**

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.

15 Dec 2016, **Dan Green & Leonard Hardiman**

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.

9 Feb 2017, **Kieran Jarrett**

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.

16 Feb 2017, **Gianluca Detommaso**

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.

23 Feb 2017, **Ben Robinson**

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.

2 Mar 2017, **Joel Cawte**

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.

✔ Vegan

✔ Dairy Free
**?** Nut Free

✔ Gluten Free

✔ Rigour Free
**X ** Maths Free
**?** Humour Free
**?** Other Allergens

9 Mar 2017, **Emma Horton**

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

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!

16 Mar 2017, **Cameron Smith**

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.

23 Mar 2017, **Tom Crawley**

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**

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.

6 Apr 2017, **Dorka Fekete**

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.

27 Apr 2017, **Matt Lee**

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.

4 May 2017, **Alge Wallis**

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.