Term 3 2023/2024

Week 2: 1st May

William O'Regan On the Erdős distance problem and fractal variants

The Erdős distance problem asks us to find the number of distinct distances that can be found between points in the plane. This is more or less resolved by Guth and Katz: n points generate about n distinct distances. In this talk I will, time allowing, introduce this problem, give some simple arguments that give some reasonable upper and lower bounds, discuss ideas of the proof of Guth and Katz, and give fractal variants of the problem. An A-level in mathematics, or equivalent,will be required to follow this talk.

Week 3: 8th May

Lucas Araujo Bonomo Acoustic Liners for Turbofan Aircraft Engines: Duct Acoustics, Experimental Methods and What on earth a Mechanical Engineer is doing in WMI

Noise stands as a key factor in aircraft regulation. In the case of modern turbofan aircraft, the fan emerges as a primary source of noise. Due to its distinctive acoustic signature, characterised by a profusion of tones spanning a broadband spectrum, fan noise incurs significant penalties in aircraft noise metrics, given its pronounced annoyance to humans. Acoustic liners are passive devices installed on the interior walls of turbofan nacelles to mitigate fan noise. This presentation aims to introduce the mathematical problem inherent in the engineering challenge of modelling and optimizing such devices. The talk covers the fundamentals of Duct Acoustics and the current methodologies for modelling these devices into numerical models of turbofan engines. Additionally, I talk about the experimental techniques utilised to characterise liners under realistic conditions. Finally, the current debate on the academic liner community is introduced. By the end of this talk, I hope you will understand what an experimental mechanical engineer is doing in a maths department.

Week 5: 22nd May

Grega Saksida What do we care about in statistical mechanics?

Statistical mechanics is a field of mathematics and physics that describes matter by applying probability theory to microscopic particles; hence the name? We will illustrate this approach on an example. We will see what kind of mathematical objects and properties we generally care about in statistical mechanics, and we will finish by showing how one can predict a particular phase transition by studying random walks.

Week 6: 29th May

Marco Milanesi How many manifolds are out there?

We say that two smooth closed manifolds are bordant if there is a bigger smooth compact manifold whose boundary is the disjoint union of the starting two. This nice geometric definition gives rise to an equivalence relation on the set of all closed smooth manifolds and it turns out that we can completely understand the structure of the equivalence classes. We will explore a very fascinating idea behind this calculation that passes through the notions of generalized homology theories and homotopy groups of sequential spectra.

Week 7: 5th June

Ryan Acosta Babb The royal road to geometry: 2000 years of history of the Parallel Postulate

We all know how the story ends: Gauss, Lobachevsky and others worked out the consequences and consistency of non-Euclidean geometry, finally settling the question of whether the Fifth Postulate could be derived from the rest. But how did we get here? Why did the Greeks regard this postulate with mistrust, and why did the question obsess some of the most prominent mathematicians in history? Let's find out through this tantalising tale of deceiving diagrams, bitter betrayals and envious epistles!

Week 9: 19th June

Hamdi Dervodeli Does tropical geometry know how to factor classical polynomials?

The factoring locus of a polynomial is a list of conditions on its coefficients under which the polynomial factors. The aim of this talk is to explore potential connections tropical geometry has with this factoring locus. More generally, we want to know if the reducibility of a variety is detected by the tropicalization of its defining ideal.

Week 10: 26th June

Marc Homs Dones Circularity vs Directionality

Have you ever wondered how one can measure how circular a graph/network is? This question arises in such important contexts as when assessing the health of financial systems and recyclability. In this talk, I’ll instead tackle the epidemic of climbing chalk usage in the department. This noble pursuit certainly justifies the usage of Markov chains, linear programming and even a bit of topology.

Term 2 2023/2024

Week 1: 10th January

Andrew Nugent Approaches to modelling opinion formation

Do you prefer an ODE or a PDE? A computer simulation or a proof? An analytic approximation or a data-driven discovery? Whatever your preference it's (probably) got an application to opinion dynamics. This talk will give a whistle-stop tour through various techniques that have been used to model opinion formation, with a focus on how different models are connected and complementary. Throughout we'll be trying to answer the big question: when do people agree? And if not, why not?

Week 2: 17th January

Joris Koefler (Max-Planck Institute) Exploring positive geometries and their link to Physics

Join us for an accessible introduction to positive geometries—an emerging field blending algebraic/polytopal geometry, combinatorics, and particle Physics. We will begin by exploring key concepts of positive geometry and in the latter part we will showcase its surprising connection to Quantum Field theories.

Week 3: 24th January

Glen Salter Continued Fractions, Chaos and The Modular Surface

Week 4: 31st January

William O'Regan Introduction to Kolmogorov complexity and applications to problems in fractal geometry

Week 5: 7th February

Philip Holdridge  I see threes of green: Colouring results in number theory

Week 6: 14th February

Nicola Rosetti Random walks and electrical circuits

Did you know that electrons in a circuit and drunk people on the street have something in common? Well this is not properly true, however the so called "drunkard walk" and electric circuits, share somehow some common features; indeed in the course of the seminar, we will see how both can be modelled by a random walk on a graph and we will discover how such connection will help us proving a beautiful theorem: Polya's recurrence and transience theorem. Approaching the problem from the point of view of resistor networks will in fact allow us to prove the result in a more elegant way, moreover it will lead us to prove the same result in the more general setting where the conductances of the network are random.

Week 7: 21st February

Tarek Acila Qualitative Mathematical Modelling of Inverse Blebbing

Biological cells live in a hostile environment in which they are constantly exposed to external forces. To preserve their functionality, they employ their mechanical and chemical abilities to change their shapes accordingly, thanks to their elastic membranes. Shape change is a biophysical mechanism used by cells to counteract any applied forces. In this talk, we will focus on a specific type of shape change known as Inverse Blebbing. We will start with a biological introduction and then slowly make our way to the mathematical set-up of this problem. We derive a qualitative continuum mathematical PDE model that describes inverse blebbing using techniques from differential geometry and continuum mechanics. To conclude our presentation, we discuss why and how this model can be improved in the future.

Week 8: 28th February

Kristian Romano Real Time Telemetric Monitoring of the Circadian Rhythm via Hidden Markov Models for Advanced Pancreatic Cancer Patients undergoing Chemotherapy

Disruptions of the Circadian Timing System in cancer patients are associated with poorer treatment outcomes, and short progression-free and overall-survival. The MultiDom clinical trial (NCT04263948) telemonitors the physical activity and body temperature of patients with advanced or metastatic pancreatic cancer undergoing chemotherapy aiming to reduce toxicity-related emergency hospitalizations (Bouchahda et al. 2023BMJ Open). We develop methodology based on Hidden Markov Models (HMM) that allows for near-real time estimation of circadian parameters dynamics to inform proactive decision making by the medical team. For that aim we developed a HMM with a time changing transition matrix that we estimated via state of the art gradient based methods. As we assumed Zero Inflated Gamma emissions, we will employ a Metropolis-Hastings for the shape parameter with an educated proposal distribution. The work presented provides a first step into the reproducible and fully automatic Bayesian computation in real time of circadian parameters of interest for the purpose of telemonitoring.  The approach also facilitates a systematically updated daily quantification of uncertainty. We will demonstrate the use of our methodology in the context of tele monitoring patients undergoing chemotherapeutic treatment, thus highlighting the potential for circadian digital and precision medicine.

Week 9: 6th March

Isabella Gonçalves de Alvarenga Contact-and-Barrier

We consider a model representing a dynamic of a species reproducing on the right side of a randomly positioned barrier. We define it formally and establish some connections of this model with the multitype contact process. The main question is: given some instant of time, how close is the nearest living individual to the barrier? We prove that, under certain conditions, this distance is tight, and sketch the proof. Under the same conditions, we also show some results on convergence of distribution of this process as seen by the barrier. We finish with some open questions about this model.

Week 10: 13th March

Marc Truter What is that doughnut doing in my variety? Let's blow things up with toric geometry

Let's go through an introductory picture-by-picture tour of a powerful tool in algebraic geometry, toric geometry. Varieties are the main object of study in algebraic geometry, and many of them contain doughnuts, which make them what we call toric! For toric varieties, we can draw some neat pictures that allow us to turn some pretty complex procedures into neat combinatorial pictures. In particular, removing nasty singular points by blowing up our variety becomes a simple combinatorial exercise.

The best part is, its all pictures!

Term 1 2023/2024

Week 1 : 4th October

Katerina Santicola Reverse-Engineering

The simplest way to prove something exists is not always by constructing it. Such proofs are called ineffective proofs. Falting's Theorem is an example of an ineffective result: it tells us that every curve (of genus at least 2) "belongs" to a finite set of rational points. In this talk, I will convince you that the converse is true: that every finite set of rational points "belongs" to a curve. Our proof is, by contrast, effective. I will also convince you that this has nothing to do with reverse-engineering.

Week 2: 11th October

Luke Murray Kearney Data-driven age structured contact networks

Network structure provides a powerful mechanism through which to analyse social interactions at an individual level. Many respiratory and sexually transmitted diseases travel through these social connections, allowing epidemiologists to use contact networks to analyse disease transmission in a population. Much work in network epidemiology attempts to reproduce an exact contact network from outbreak data to aid in control or surveillance activities such as contact tracing or aims to create a representative network for use in outbreak simulation. However, there is no standardised approach to creating a suitable network from the type of individual data that are routinely collected. In this project, we propose two novel approaches to create age-specific network representations from real world data that are adaptable to times of social restriction in response to an outbreak. We test these approaches using two periods of varied lockdown strictness during the Covid-19 pandemic in the UK, using the degree distributions of the egocentric data provided by the CoMix study and the between demographic contact rates. We further analyse the scale-ability and characteristics of our approaches through their computation times and the structure of the resulting structure of the network.

Nathan Van der Riet From pancakes to cups: the physically driven structure of the Golgi Apparatus

The Golgi Apparatus is a key organelle found in eukaryotic cells responsible for modifying proteins and lipids and distributing them to their target destinations. Akin to an amped up post office, the Golgi is in a constant state of flux, receiving and distributing packages, and yet it has a remarkably stable structure. This structure is of interest to biologists and now, hopefully, to mathematicians. This is due to its two distinct compartments which can be characterised by their distinct Gaussian curvatures. We seek to explain this distinction by means of a minimal physical model, assuming that the Golgi’s lipid membrane consists of two molecules with opposing affinities for Gaussian curvature. By following through a chain of reasoning on vesiculation events and by modelling the Golgi as a dynamical system we come up with a plausible explanation for the existence of the negative-Gaussian curvature structure. Encouraged, we seek then to produce from a handful of assumptions and deductions a physically motivated dynamical system whose equilibrium is consistent with qualitative observations of the Golgi’s structure.

Week 3: 18th October

Layne Hall Between chaotic flows and combinatorics

A flow is a continuous dynamical system, like a set of ODEs. Especially in dimension three, there are deep relationships between the dynamics of flows and the topological properties of the underlying space. We will introduce, through examples, a large class of flows which exemplify this connection. We will see that these flows have robust, chaotic dynamics. Then, we will discuss how the flows can be elegantly characterised through combinatorial topology.

Week 4: 25th October

Abigail Hollingsworth Hyperbolic Knot Theory

A key question in knot theory is how to differentiate two knots. The knot invariant we will study is the knot complement. Knots can be classified into three groups, one of which is hyperbolic knots, which have hyperbolic knot complements. To understand when a knot complement is hyperbolic, we will see an overview of hyperbolic geometry in two and three dimensions and define the three key equations that decide if a knot complement is hyperbolic. These equations define how the tetrahedral decomposition of the knot complement glues together. This talk is inspired by Jessica Purcell’s book on Hyperbolic Knot Theory.

Week 5: 1st November

Hefin Lambley What does it mean to be most likely?

The concept of a most likely value, or mode, appears frequently in statistics and applied mathematics. Sometimes the definition is obvious. Often it is not, e.g. in large-deviation theory or the Bayesian approach to inverse problems.

This talk will introduce a general theory for modes of probability distributions on metric spaces. In most cases, this theory works very well, and admits an intuitive order-theoretic formulation. However, we'll also see some pathological distributions where the definition of a mode is very unclear, and where there might not even exist a good approximation to a mode.

Week 6: 8th November

Patience Ablett How many points to verify a Hilbert polynomial?

Week 7: 15th November

Yorick Fuhrmann From formal groups to elliptic cohomology

Give me a ring and a formal group law over it, then I'll give you a cohomology theory! Although... I have to admit, your ring should be Landweber flat as an -module. In this talk we will find out what all that means and how complex cobordism is able to classify cohomology theories which have Euler classes for complex vector bundles. This is a celebrated result of Daniel Quillen which opened up the world of chromatic homotopy theory. We will use it to rephrase topological -theory in terms of complex cobordism and see how it can be utilized to get new cohomology theories from elliptic curves.

Week 8: 22nd November

David Hubbard Torsion Points on Curves in the Torus

Lang posed the following question on solutions of Diophantine equations which consist of pairs of roots of unity:

“If f(x,y)=0 is a Diophantine equation, then when is the set {(a,b) | f(a,b)=0 a,b roots of unity} finite?”

Geometrically, this set is the intersection of the curve X={f=0} with torsion points of the algebraic 2-torus (C*)^2. Generically, this set is finite unless X is ‘special’.

This problem belongs to the ‘Unlikely Intersections’ of arithmetic geometry which contains the Mordell, André-Oort and Manin-Mumford Conjecture. A great deal of recent research has aimed to find uniform versions of these conjectures, and I will present the uniform version of Lang’s problem by Beukers and Smyth using methods from convex geometry.

Week 9: 29th November

Laura Bradby If I made a list of every doughnut, would it be compact?

Suppose I wanted to make a list of all compact genus 1 surfaces. How many are there? Well, given such a torus (or indeed, any genus g surface), if you consider its topological structure and work up to homeomorphism, then there’s only one. How many if you consider smooth structures, up to diffeomorphism? Surprisingly, still just one. What if you care about complex structure, and work up to biholomorphism? Uncountably many! The last case is the only interesting one from a list-making perspective, and has the major problem that uncountably many things are famously impossible to list. So instead, we extend our idea of “list”, and do what any sensible person would do with uncountably many things: shove them all together and give that a topology. This gives us the concept of a moduli space, and in this talk we will discuss the moduli space of the torus, and then the moduli spaces of surfaces of higher genus, using the genus 1 case as a guiding example. Our aim will be to see how to build these spaces in a sensible way, and to begin to understand what they look like, including how “close” they are to being compact.

Week 10: 6th December

Muhammad Manji Local-Global Principles

When studying geometric objects, sometimes we care about questions which exist only globally, for example; where is the maximum of a function, if such a thing exists? At other times we care about questions which are only local; what is the derivative at a point of interest? If a property holds everywhere locally, will it necessarily give us a global condition? By patching together local information we can sometimes tackle global problems, and at other times we can't. We will discuss some of these questions and deal with the basics of arithmetic geometry.