The Postgraduate Seminar Series (PSS) is a series of talks given by postgraduate students in the Department of Mathematical Sciences, to an audience of postgraduate students. It is an opportunity for postgraduates to gain experience of presenting in a relaxed environment, to learn about other areas of mathematics by listening to accessible talks, and to get to know more about other postgraduates in the department and their research interests.

The styles and subjects vary greatly, as you can probably tell from the list of talks below. Due to the ongoing pandemic, these talks happen on Microsoft Teams at 10:15am on Thursdays . If you want to get involved, whether that be to give a PSS talk or just to watch then email either me at jw3286@bath.ac.uk or Piotr at pwm27@bath.ac.uk.

For information of the PSS talks from last year, visit the PSS website 2019/20 or download a complete list of every PSS ever here.

I will talk about the work I have done over the past 3 months in the UK Government Office for Science, the department responsible for ensuring the UK Gov has access to the best and most up to date scientific advice during the COVID-19 Pandemic. I will talk about the role of science in government, the importance of an international cooperation during a global crisis, and how to write a briefing!

The CDIMA reaction is a naturally photosensitive system in which the reaction rates depend on the intensity and presence of white light. Striking spatial and temporal heterogeneous patterns are observed as the reaction progresses, and the photosensitivity allows a great deal of control in modulating these oscillations. In this talk, I will show the work I did at the Oxford Modelling Camp. Here we started by analysing a simpler system of chemical reactions that show pattern formation. I will explain how we formulated a general process for analytical and numerical analysis of chemical reactions of this type and how we used the simple set of reactions to test the general method and study pattern growth. Finally, I will talk about how we can apply the general methods to the CDIMA reaction.

Small defects on a train track can have big impact on high-speed trains. However, as those defects are small, detecting and classifying them by visual inspection is difficult and time consuming. On the other hand, a train moving over small defects can shake significantly so we can measure train movements to detect track problems. In this talk, we will look at a short project I did in the Oxford Model Camp. We have a direct data extraction model and a forward model to understand how camera images can help us to understand the movement of trains. I will explain how images is formed how 3D objects is projected onto a 2D surface. There are a few nice videos to show how an old but useful algorithm - the Hough Transformation, can give us train tracks from videos.

Many natural phenomena are modelled by a stochastic process. Since no model can be completely correct, it is useful to be able to measure the difference between the model and reality. In this talk, we will look at an appropriate notion of distance between two discrete-time stochastic processes. This distance comes from a variation of optimal transport, a classical problem in analysis. We will see how this distance can be applied to optimal stopping problems, and if there’s time we’ll discuss extending it to cover continuous-time processes like Brownian motion. This is the topic that I’m starting to work on in my postdoc. Everything I will talk about is work done by someone else (usually some of my colleagues in Vienna) and I’m no expert in this, so the talk should be a (fairly) accessible introduction!

Have you ever wondered exactly how many integer lattice points are contained in a circle centred at the origin of a given radius? Or have you pondered if it is possible to find pi from counting primes? Are the integers getting a bit boring for you and you’re looking for a cool new number system that relates circles and the complex plane in an exciting new way? Then this is the talk for you! Spend an hour (45 minutes) with me as we discover not one, not two, but **three** ways to solve Gauss’ circle problem and learn a few of the classic themes and concepts within Number theory as we go.

Why do traffic jams appear on a stretch of road when there’s no crash or bottleneck? Why do fish decide to swim in the same direction? This talk will give an overview on the work I’ve been doing on stochastic interacting particle systems. We’ll see how we can take a relatively simple, general model for a group of particles and describe how it behaves on a large scale and why this could answer the questions above. I’ll show you some examples, some grounded in real world scenarios and one which is simply bizarre, but all will be accompanied with pictures and animations like the one below.

In most practical image segmentation tasks, the image to be segmented will need to first be reconstructed from indirect, damaged, and/or noisy observations. Traditionally, this reconstruction-segmentation task would be done in sequence: first apply the reconstruction method, and then the segmentation method. Joint reconstruction-segmentation is a method for using segmentation and reconstruction techniques simultaneously, to use information from the segmentation to guide the reconstruction, and vice versa. In this talk, we will demonstrate how this can be done using the graph-PDE-based segmentation techniques developed by (Bertozzi, Flenner, 2012) and (Merkurjev, Kostic, Bertozzi, 2013), with ideas drawn from (Budd, Van Gennip, 2020) and (Budd, Van Gennip, Latz, 2020).

In this talk, I will shamelessly repeat a presentation I gave at a conference in 2020 about Echo State Networks and Dynamical Systems. Envision a room M full of objects, that evolve according to a system of ODEs ϕ. A protagonist (who is an Echo State Neural Network) stands outside the room, partially observing the objects' dynamics through a narrow window. With these observations alone, can the protagonist learn the dynamics of the objects in the room, and predict their future trajectory?

Did you ever wonder the river network evolve through years? Probably not. But you can still come to PSS and learn why! During the talk I will show how a beautiful fractal-like river networks we can see around us emerges from the most basic principles of fluid dynamics and how can we use satellite imaginary to further understand this phenomon. I’ll start from very basics, so even those for whom fluid dynamics is still a witchcraft, can learn it and appreciate its potential.

Solving inverse problems in imaging can be very challenging due to the inherent ill-posedness and high dimensionality. To deal with this, one often employs efficient regularisation methods that enforce some additional known properties of the solution. In this talk, we shall consider Krylov subspace-based regularization approaches that combine direct matrix factorization methods on small subproblems with iterative solvers.

In this talk I am going to review the main concepts underlying moving mesh strategies in 1D and their applications in numerical analysis. In particular, I will treat the notion of *equidistribution* and *monitor function*, which are essential to prove the optimality of the resulting mesh in terms of interpolation error. Finally, I will introduce the complications that arise when we move to 2D problems and mention different moving mesh strategies that are used in research.

Random growth processes are paradigmatic in probability theory and describe phenomena in many contexts, such as cancer growth, rumour propagation and population dynamics. In this talk, we will explore some classical growth processes and provide an overview for intriguing mathematical results related to them. In particular, we will see diffusion limited aggregation (DLA), a growth process that has defied a strong mathematical treatment for decades. We will discuss known results for DLA and its variations, culminating in an astonishing recent result by Elboim, Nam, and Sly that describes critical behaviour for the model in dimension one, the only dimension that is known to observe a phase transition.

Investigating the formation of hair follicles

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!

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.

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.

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.

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.

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? 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.

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.