Welcome to the section of my website regarding PSS! This academic year, the series is being hosted by Corin Lee and myself. Here, you'll find some general information about PSS, and a list of all the speakers so far!

What is PSS?

PSS is a series of talks given by postgraduate students in the Department of Mathematical Sciences, to an audience of fellow postgraduates. It is an opportunity to gain experience of presenting, and learn about other areas of mathematics, all in a relaxed and chill environment.

Where is PSS?

Seminars occur on Thursdays at 10:15 in Wolfson (4W 1.7). A hybrid component exists too, for which details are given in the weekly emails.

Here is the list of everyone that has spoken so far in 2022/23, along with the title and abstracts of their talks. Interested in having your name added to this list of awesome people? Sign up for a PSS talk today!

Semester 2 2022/23:

Model spaces and the compressed shift operator

Just as how understanding the multiplication operator Mz on L^{2} is key to the study of self-adjoint operators, the same is true for the compressed shift operator Su on a model space Ku, in the study of completely non-unitary (CNU) contractions.

The goal of this session is to introduce model spaces and the compressed shift, covering topics of interest from the book “Introduction to Model Spaces and their Operators” by Stephan Ramon Garcia, Javad Mashreghi, and William T. Ross. In the first half, I will discuss function theoretic aspects of the Hardy space H^{2}(D), with the goal to introduce the model spaces Ku. In the second half, I will discuss the compressed shift operator Su, with the goal to define the H^{∞}-functional calculus for Su (developed by Sz.-Nagy-Foiaş in the 1950s).

How to predict the future with a combination of maths, finance and wisdom of the crowd? A layman introduction to prediction markets

Prediction markets are a simple, yet the most effective tool we currently know, for aggregating the wisdom of the crowd in order to predict future events. It is nothing else than a financial market, on which players bet money on whether a given event will occur or not, with stakes dependent on their personal beliefs. In Poland, I was on the team which developed a prediction market, that our research funding agency used to predict the outcome of graphene-based research projects. In this talk, I would like to introduce you to the idea of prediction market, their history, and show you how interesting, yet unexplored this area is, from a mathematical perspective.

Oh, and to get most out of the talk, I encourage everyone to join the seminar in person if possible - you will see very soon why 😉

Tackle Trying Problems (Or Watch Rugby Instead)

Reviews:

"- Talk is not great with concrete advice. A newbie is going to ask "I've got £200 to spend, I want great filter coffee, how should I spend it?", this talk doesn't have a direct answer. I understand why the talk doesn't recommend specific brands, but a newbie doesn't know, and that is frustrating." - Nic Küpper

"- I felt the talk laboured too extensively with examples from the past and repeatedly failed to get succinctly to whatever point was being made: as if the speaker was being paid by the word." - Bill Gates

"- Lack of pro tips. Henry gets all the way through the milk steaming section without mentioning that you can use water with a single drop of dish soap to practice steaming without wasting milk. There's plenty of room and opportunity to drop in tips which can help even the most knowledgeable viewers, but these tips don't exist." - Warren Buffett

"- Offers nothing in my opinion. Investment advice for 2 year olds." - Henry Elsom

Multilevel Monte Carlo: This one simple trick will save you hours of computation time!

Have you ever been sat in front of your laptop waiting for code to finish, staring blankly at the progress bar praying that this time it will work? Well I have and so in this talk I will introduce the concept of Multilevel Monte Carlo. I'll show that for many Monte Carlo simulations which approximate a random process, we can cleverly sample at different levels of accuracy to greatly reduce the overall computational cost or runtime. It might not make your code correct, but at least you'll know it's wrong sooner. Initially developed for stochastic differential equations applied to financial mathematics, this approach can be used in various other applications. The main focus of this talk will be biological systems modelled with Poisson processes.

Pivot! Using real algebraic geometry to move a sofa.

How many times has this happened to you? You want to move your infinitely-thin ladder of length l through an L-shaped corridor of width 1, but you’re not sure if it’ll fit and don’t want to bother trying if it won’t. I know it happens to me all the time.

This seminar looks at the idea of a cylindrical algebraic decomposition, a method and algorithm to decomposing real space into regions of constant sign, along with some examples and pretty pictures, and an application of this to The Piano Movers Problem. Come and join if you like pictures with colours and dots on them, and incredibly dated pop culture references.

A taste of Primal-Dual with Alternating Projections and Optimal Transport

Primal-dual approach in convex analysis is like a dynamic duo - Batman and Robin, Abbott and Costello, Peanut Butter and Jelly. In this approach, we form the convex dual of a primal problem and solve them simultaneously. It’s like having two brains working together to tackle a problem! In this talk, we will delve into primal-dual alternating projections, acceleration methods on them, and their applications. To spice things up, we will also talk about an exciting accelerating scheme for optimal transport problems. So if you are curious about primal-dual problems and acceleration methods, join us for our presentation and see how these duos can save the day!

A brief introduction to quantum computation and Hamiltonian simulation

Quantum computation is a rapidly growing field that has the potential to revolutionize the way we solve computational problems. In this talk, we will start with a brief introduction to quantum computation, including mathematical foundations about qubits and multi-qubit systems. We will also introduce the current state of quantum computation and its applications. Furthermore, we will delve into quantum Hamiltonian simulation with a focus on my current work on time-dependent Hamiltonian simulation. This talk is intended for mathematicians who are new to quantum computation, and no prior knowledge of quantum mechanics or computer science is assumed.

A brief introduction to Physics-Informed Neural Networks and how to solve image registration problems with WarpPINN

In this talk, I will introduce the idea of Physics-Informed Neural Networks (PINNs) and the related Deep-Ritz Method, two novel approaches where the deep learning machinery is employed for solving Partial Differential Equations, PDE-Constrained Optimisation Problems and Inverse Problems with physical constraints!
The application I will show is in cardiac imaging, where the goal is to determine the deformation field of the heart during the cardiac cycle from magnetic resonance images. Solving this inverse problem is very useful since it allows for assessing cardiac diseases from non-invasive measurements (a quite challenging task). To solve this problem, we make use of WarpPINN, a neural network designed to solve image registration problems.

If you are interested in PDEs, neural networks, and want to see funny animations of hearts beating then join us!

Ants navigate from their nest (N) towards a food source (F) via pheromones deposited by previous ants. In this talk I will present a probabilistic model for this process on a simple connected graph and show that the ants find the shortest path between N and F when they are connected by a single edge.

It has been suggested that the distance from the initial inoculum to the nearest blood vessel is important in determining whether an individual exposed to the Mycobacterium tuberculosis (Mtb) pathogen will develop the disease tuberculosis (TB). Additionally, dormant Mtb bacteria have been suggested, in some circumstances, to cause reactivation and/or relapse of the disease. In this talk, I will present a multi-cell, multiscale model of TB progression at the cellular level to investigate these hypotheses. I have developed the model using CompuCell3D, an open-source computer software used for simulating cellular biological processes; the aim is to compare the generated results
with those from a previously developed within-host infectious disease model. This work should prove useful for researchers looking to build models using
CompuCell3D and will highlight the differences between this software and other agent-based modelling approaches.

tl;dr - I made a model of TB progression using CompuCell3D. Come along to find out more!

Circumnutating Cucurbits — Weird and Wonderful Winding

Round like a circle in a spiral, like a wheel within a wheel
Never ending or beginning on an ever spinning reel
Like a grape vine on a trellis, or a hop bine on a string
Like a tendril overwinding, as it grows reversing springs
Like a twining plant seeking an anchor as it whirls about in place
or new sunflower seedlings nutating silently in space
There are helices to be found
in plants growing all around!

In this (very) non-technical talk, I will do my best to convince you that plants are really really really really cool. I will try to explain three ways in which certain types of plants grow in spirals, and outline some mathematical modelling that can help us understand them. There will be pretty timelapse videos, sweets-based demonstrations of instabilities of elastic filaments, and biscuits.

I love minimising functions. I also love imaging problems. In this talk I motivate how you can phrase denoising images as a minimisation problem. The types of functions that I want to minimise typically involve some unknown parameters that need to be specified beforehand. While there are some general methods on how you can go about choosing said parameters, here I motivate bilevel optimisation, a framework in which you can determine suitable parameter values - and the topic of my PhD! I will cover some cute heuristics and theoretical results and promise to include many pictures of my dog(s).

What is the shortest path between two points? How much land can you enclose with a given amount of fencing? What happens to an elastic rod if we try and bend it? What shape do we get if we dip a wire frame in soapy water? Why are red blood cells biconcave discs? Which donut has the lowest energy?

The answers to such questions can be found in the field of calculus of variations, which concerns the minimisation of functionals: that is, we want to find the function which minimises some given quantity, usually an integral. Calculus of Variations in L∞, on the other hand, also known as L∞ CoV, focuses on problems where the functional we want to minimise is not an integral but instead a supremum (an L^{∞} norm). Because integral functionals are sensitive to small changes-- i.e. they are differentiable-- and supremal functionals are not, the theory of L^{∞} CoV is very different to the usual theory, as in fact the main tool we use to solve problems is no longer available!

In this talk, given at a level such that it is accessible for those without a deep analysis background, I will discuss some of the history and motivation behind L^{∞} CoV problems and introduce the so-called L^{p} approximation, a common tool used to solve L^{∞} problems. I will go through some specific examples to give a feeling of how variational problems work, while sweeping all the technical analysis underneath the rug, and I will discuss how my own research fits into the wider picture.

How to do OK at GeoGuessr with simple statistics (and 3 CPU years of data processing)

GeoGuessr is a popular online game where players guess locations based on Google Street View imagery: the closer you get to the actual location, the more points you score. The problem of geolocating a photograph is challenging for humans (play the game to see for yourself), but can computers do any better? I will show how a very simple learning algorithm, k-nearest neighbours, has been applied to the problem and discuss various statistical concepts that arise along the way.

From Skeletons to Supercontinents: How Fluid Dynamics Can Model It All.

Many materials we encounter in life are really quite complicated! One such class of material exhibits viscoelastic properties, that is, they display behaviours of viscous liquids (such as oil and water) and elastic solids (such as rubber). These viscoelastic materials arise almost everywhere, from the tendons in your body, right through to the Earth’s tectonic plates!

But how do we model them? In this talk, I aim to chart the development of some simple viscoelastic models (and one not-so-simple one), and assess whether they are actually any good. On the way, we’ll encounter continuum mechanics, some funky ‘circuit’ diagrams, and even a weird looking time derivative. Naturally, there’ll be some scary looking equations, but despite the fact it’s almost Halloween, these will be kept to a minimum.

P.S. If you’re looking for 40 minutes of hardline asymptotics and/or Rick-Rolling, then you’re going to be sorely disappointed.

Kat Does Maths or: How I learned to stop worrying and love the Comm

If you've ever been curious about maths communication this is the talk for you. I will be going over a brief history of what how and why I ended up with a successful twitch channel as well as the ethos of what maths communication effective, and some resources tailored to Bath Students if you're particularly swayed.

SSLC Forum

This week, we were joined by the postgraduate student reps — Guannan Chen, Nic Küpper and Pawel Rudnicki, who provided feedback on issues concerning general aspects of postgraduate life.

Phase space reconstruction using Echo State Networks

In this talk, we will explore how to use an Echo State Neural Network to forecast the future of a time series, where the time series is a sequence of scalar observations from a dynamical system. I gave this talk right at the start of my PhD, and for my final PSS, I will give it again with some commentary on how this shaped the rest of thesis.

Defensive symbionts and me: an unexpected tale about an uneasy alliance

Most complex organisms can be infected by pathogens, a common class of which are parasites. These lovely critters require a host in order to reproduce by utilising their host's bodies and cells, causing damage and possibly death. On the other hand, defensive symbionts work to protect the host and in return, gain an increased ability to reproduce and transmit. This mutualistic arrangement can evolve to become more or less protective towards the host depending on conditions.

On this sunny/rainy/cloudy (delete as appropriate) Thursday morning, I will let you know what happens when a defensive symbiont and parasite go to battle in a population of hosts. We will find out if both can coexist or if one kills the other off, and we will see if the defensive symbiont decides to protect the host at all or whether it thinks better of it. Finally, we will establish what the effect of this is on the host population, and whether the hosts become stronger from their uneasy alliance, or if they are better off suffering from the parasite alone.

Extremism, segregation and oscillatory states emerge through collective opinion dynamics in a novel agent-based model

Humans are pretty complex creatures. Well, some of us at least. And so the way we think and form opinions is undoubtedly even more complex, we couldn't possibly begin to quantify and recreate all of the underlying mechanisms and processes that go into opinion formation. Enter mathematician. In this PSS, I'll be talking you through (and simultaneously reminding myself of) the emergent phenomena produced by an agent-based model of human opinion dynamics. Despite it's simplicity and limiting assumptions, the model is capable of mimicking many socio-psychological phenomena including consensus, segregation, extremism and oscillatory states. So, can we really use mathematics to accurately model the collective behaviour of human opinion dynamics? Maybe I'll let you form your own opinion on that after you've endured my talk. Exit mathematician, pursued by a bear.

How to derive deterministic approximations from the stochastic model of enzyme kinetics?

Enzymes are proteins that accelerate the conversion of other molecules (called substrates) into products, but they themselves are not changed by the reaction. Enzymatic reactions are ubiquitous in biologic systems, and they constitute a basic step in the mathematical modelling of biochemical networks. From a modelling perspective, one can either understand an enzymatic reaction as a stochastic interacting particle system (where a finite number of molecules of substrate, product and enzyme interact to each other) or as a deterministic dynamical system where the densities of the substrate and product are modelled as continuous functions. In this talk, we are going to verify how to derive the deterministic dynamics from the stochastic process. We also going to give conditions based on the number of molecules to compute “how different both models are from each other”. Although the word martingale will be employed, hopefully no previous knowledge of probability will be required. This talk will be aimed to mathematical biologists, probabilists and anybody who likes Super Mario 😊.

In a change from our regularly scheduled programming, our resident quizmaster Wilfred brought the maths department together with a quiz concerning all things Christmas! With topics including film reviews, Christmas music and (seemingly simple) mathematics, this was a great way to round off the semester!

Want to go back even further? Then here's a list of all the PSS speakers since 2012!