## Online PSS

Given the current COVID-19 situation, we moved the Postgrad Seminar online. Until further notice, all seminar sessions will be held on Microsoft Teams. On top of that, we invite all maths PhD students from Bristol, Cardiff and Exeter to join the remote sessions in order to overcome this isolating phase and get the opportunity to build a network with other GW4 students!

Please send an email to get remote access to the seminar.

## General information

This seminar is organised for and by maths students that takes place every **Thursday at 10:15 in Wolfson (4W 1.7) on Microsoft Teams**.
Each week a talk is given on an area of maths of interest to the speaker (

**not**necessarily their research area). The talks are meant to be accessible for a non-specialised audience and invite maths students with any background. Moreover, the friendly and relaxed environment offers a great opportunity for students to practice their presentation skills.

Below you find the schedule of talks for the current semester. A list with titles and detailed abstracts will be updated continuously. At the end of this page you can access a list of talks from previous years.

PSS is currently jointly organised by **Jeremy Worsfold** (jw3286@bath.ac.uk) and **Piotr Morawiecki** (pwm27@bath.ac.uk).

# Schedule for Academic Year 2020/21

## Semester 1

Speaker | Title | Date |
---|---|---|

Piotr Morawiecki | From alco-rockets and neural networks in a can to international political incidents: Why running science projects in schools is fun? | 08/10/2020 |

Ed Gallagher | Curve Shortening Flow | 15/10/2020 |

Cameron Smith | Hybrid methods: innovative or a waste of time? You decide! | 22/10/2020 |

Jeremy Worsfold | Multi-armed bandits: how tech giants are force-feeding us annoyingly useful adverts | 29/10/2020 |

Will Graham | What is Measure Theory, and How does it Affect Me? | 05/11/2020 |

Zoë Dennison | Concentrated Vorticity in the Euler equations | 12/11/2020 |

Matt Turner | Blowups: An algebraic-geometric approach to resolving singularities | 19/11/2020 |

Yvonne Krumbeck | Will a Large Complex System be Stable? | 26/11/2020 |

Yi Sheng Lim | An Invitation to Probability Theory | 03/12/2020 |

Shahzeb Raja Noureen | Modelling hair follicle development and formation of periodic patterns of cells | 10/12/2020 |

Christmas PSS | 17/12/2020 |

# List of Abstracts

## Semester 1

### Modelling hair follicle development and formation of periodic patterns of cells

10 Dec 2020,**Shahzeb Raja Noureen**

The development of hair follicles starts in the early embryonic days. In this talk, I will take you through an exciting journey of what contributes to the formation of hair follicles and how pigment (colour) producing cell behave in response to these follicles to form cool periodic patterns. Don’t need an A* in GCSE biology or any complicated maths as you will see how complex biological behaviours can be modelled using some simple mathematical tools.

### An Invitation to Probability Theory

03 Dec 2020,**Yi Sheng Lim**

Probability is a big subject. My goal for the session will be to give a tour of what kind of maths happens at its core. The basic objects are probability spaces and random variables, which are built on the language of measure theory. My take, however, is that measure theory is just half of the story -- you do not want to do probability on *any* space ... it must be on a space as nice as R. In other words, I would say that the core of probability is about "Probability Measures on Metric Spaces". This is the title of a book by K R Parthasarathy.

So, the tour centres around this book. I will take some time to explain how one should go from A-Level probability and statistics + mathematical analysis (point set topology, metric spaces) to the starting point of this book. Once we have reached the start of the book, I will give a whirlwind tour on the most important ideas in it. The second part is intended to be a touch-and-go, as a cultural appreciation of the language and the results that one can expect in this field. Come by and have fun with probability!

### Will a Large Complex System be Stable?

26 Nov 2020,**Yvonne Krumbeck**

We can intuitively tell what makes a complex system with interacting components stable. For example in ecosystems, we know that the extinction of a prey species can lead to a mass extinction of predator species that feed on prey to sustain themselves, and genetic diversity helps organisms adapting to changing environments and rapidly evolving diseases. But is there a way to quantify stability with maths?

When mathematicians speak about the stability of ecosystems, they usually refer to the asymptotic stability of an equilibrium point, characterised by the eigenvalues of a species interaction matrix. In reality, however, these interaction coefficients are difficult - if not impossible - to measure. Therefore in 1972, Robert M. May introduced a community matrix model, where coefficients are sampled from a random distribution, and derived a stability criterion based on the distribution of the eigenvalues using random matrix theory. For nearly 50 years, this model has been improved and applied in theoretical ecology.

Long story short: I will talk you through the following review paper by S. Allesina and S. Tang: https://doi.org/10.1007/s10144-014-0471-0, and briefly discuss other studies on random matrix theory and ecosystem stability. (Expect pictures of cute animals!)

### Blowups: An algebraic-geometric approach to resolving singularities

19 Nov 2020,**Matt Turner**

It is rare that blowing something up solves a problem - but when it comes to objects with singularities, it turns out to be a good approach! In this talk, I will outline the method of blowing up a singularity of an algebraic variety in order to produce a new variety. This new geometrical object will have very similar properties, but with the added benefit of being less singular, or even smooth.

Along the way, I will introduce the notion of projective space and how embedded objects can be seen using charts. Unlike my own research, this geometry can be visualised easily and so there will be plenty of pictures and nice examples to see how blowups work.

### Concentrated Vorticity in the Euler equations

12 Nov 2020,**Zoë Dennison**

In this talk I will give an overview of what I spend my time looking at. I will begin with the Euler and Navier Stokes equations and bring you all the way to the specific case that I am interested in - incompressible, stationary, Euler with helical symmetry and concentrated vorticity. I will try to give an overview and highlight the problems with the research so far whilst trying not to bore you with too many mentions of epsilon.

### What is Measure Theory, and How does it Affect Me?

05 Nov 2020,**Will Graham**

Whilst having an intricate knowledge of the ins and outs of measure theory is by no means a requirement for many applications of maths in the real world; it does secretly underpin a lot of the more familiar (and friendly!) things that we use in applied maths, and probability.

- Have you ever wondered why PDFs are such a big deal when working in probability?
- Have you ever used FEM or Fourier transforms to solve a problem?
- Have you ever tried to break up a chocolate bar, and then reassemble the pieces into two chocolate bars of equal size to the first????

In this talk I will aim to provide a crash-course overview of the key concepts behind measure theory, focusing on the construction of the Lebesgue measure. This measure is the one that we all use each day without knowing it - it tells us that the area of a circle is πr2, and justifies the existence of rulers. At the end of this talk everyone will (hopefully) leave with an understanding of how measure theory permeates into other areas of mathematics, and an introductory understanding to the theory as a whole.

Disclaimer: I will be limiting the technical details to the bare minimum I need, and will include diagrams and analogies wherever possible. I’m also not going to be assuming any prior knowledge of measure theory, so anyone should be able to follow along even if they’ve never seen, never heard, or have actively avoided this topic before. It is not my intention to give a technical analysis talk at 10:15 in the morning, no-one wants that, not even me!

### Multi-armed bandits: how tech giants are force-feeding us annoyingly useful adverts

29 Oct 2020,**Jeremy Worsfold**

You may have seen or heard recently about the Netflix documentary called the Social Dilemma, exploring the potential problems of companies constantly bombarding us with suggestions and adverts. But how do they pick which adverts to show us? I’ll introduce you to recommender systems and show through a simplified example called the multi-armed bandit problem how these companies can learn what the most effective adverts are. I’ll then discuss where this could go in the future and probably sound like a maniac telling you that we’re all doomed to be controlled by the machines. It should be a lot of fun!

Disclaimer: I am no expert in statistics or reinforcement learning so I will try to keep the serious maths to a minimum and cover any holes in my knowledge with pretty pictures.

### Hybrid methods: innovative or a waste of time? You decide!

22 Oct 2020,**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 (virtual) 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? (If only we had paid attention back then!)
- Why can’t a leopard change its 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.

### Curve Shortening Flow

15 Oct 2020,**Ed Gallagher**

Differential geometry originated as the study of 'curved' or 'bent' spaces fixed in time; over the last 4 decades, however, geometers have overseen huge developments in the study of spaces which are not fixed but change, or 'flow', over time. One of the simplest and perhaps most natural examples is curve shortening flow, the topic I did my master's dissertation on, where essentially a curve "moves inwards with speed proportional to its bendiness."

In this talk we will give an introduction to curve shortening flow and look at some of the (surprising?) ways in which it behaves, starting in the plane before moving onto surfaces. We will then briefly touch on mean curvature flow, a generalisation of curve shortening flow to higher dimensions, and see some of the applications of curve shortening flow in both pure maths and the real world.

There will be some equations and overviews of proofs, but it won't be super technical, and there'll also be nice pictures and diagrams so it should be possible to follow the talk without any knowledge of differential geometry at all!

### From alco-rockets and neural networks in a can to international political incidents: Why running science projects in schools is fun?

08 Oct 2020,**Piotr Morawiecki**

During the talk I will try to infect you with my passion for running science projects in schools, which may be a great thing to do in parallel with the standard academic research. The talk will include three stories from my past school projects full of risk and danger, but also fun and satisfaction.

For example, you will learn how not to construct home-made rockets or how by a "small" miscalculation you can almost cause a serious international political incident. The talk won’t include any academic math, but surely you won’t regret coming!

# Talks from Previous Years

### Academic year (organisers)

- 2019/20 (Allen Hart & Yvonne Krumbeck)
- 2018/19 (Matthew Griffith & Leonard Hardiman)
- 2017/18 (Ben Robinson)
- 2016/17 (Dan Green)
- 2015/16 (Kieran Jarrett)
- 2014/15 (James Roberts)
- 2013/14 (Horatio Duhart)
- 2012/13 (Matt Pressland)

Here is a spreadsheet detailing every PSS talk ever.