Mathematical Sciences Seminar Abstracts 2023-24








Steven Schiff (Yale University, USA)

Title: Rate-Induced Tipping in Brain Stimulation – from the Curse of Joule to Siberian Peat Fires

Abstract: All electric and magnetic stimulation of the brain deposits thermal energy in the brain. This occurs through Joule heating of the conductors carrying current through electrodes or magnetic coils, or through dissipation of energy in the resistive brain. Magnetic induction lets us separate Joule heating from induction effects by contrasting AC and DC driving of magnetic coils using equivalent energy dissipation. Although neurons accommodate to very slow changes in temperature, small temperature increases and decreases of 1°C on time scales of 5 seconds caused consistent transient suppression and excitation of neuronal activity. Modeling the biophysics demonstrated that the Na-K pump, and to a lesser extent the Nernst potential, are responsible for these transient effects which depend upon compartmental ion fluxes. We note a remarkable similarity of these physics with other thermal rate-dependent tipping points in planetary warming dynamics. In both cases, an exponential relationship with temperature of either the rate of ion transport across compartments, or the rate of microbial decomposition of soil carbon compounds, leads to a loss of an equilibrium state of system. From a clinical medicine perspective, these experimental and theoretical findings demonstrate that stimulation of the brain must account for small thermal effects that are always present in electrical and magnetic stimulation.



















Anne Skeldon (University of Surrey, UK)

Title: The fast-slow dynamics of sleep regulation

Abstract: The regulation of sleep and wake occurs through the interaction of specific neuronal populations in the brain with macroscale internal rhythms and external environmental cues, such as the light-dark cycle. Mathematical models of sleep-wake regulation have been very successful at providing a conceptual framework for understanding the timing and duration of sleep, with implications for public policy on design of our living environments, the timing of our clocks (e.g. daylight saving time versus standard time) and the timing of our work schedules. Not surprisingly, since we remain in the states of sleep or wake for some time, but switch between states rapidly, models typically have multiple time scales. Here I’ll give a brief overview of the fast-slow dynamics of some recent models of sleep-wake regulation and the insights that understanding that structure brings.



















Jan Sieber (University of Exeter, UK)

Title: Continuation with non-invasive control schemes

Abstract: Tracking unstable equilibria or periodic orbits in an experiment requires application of some type of stabilizing feedback control. This feedback control has to be non-invasive in the sense that its effect vanishes asymptotically. While in principle the most efficient way to achieve non-invasiveness is application of a (possibly simplified) Newton iteration, in practice many other approaches that are "naturally" non-invasive without requiring an iteration have been used. During work on controlling pedestrian flow simulations we discovered a generalisation of several commonly used techniques for non-invasively controlling equilibria that makes these techniques applicable to every generic scenario.

Reference: Panagiotopoulos, I., Starke, J., Sieber, J., & Just, W. (2023). Continuation with noninvasive control schemes: Revealing unstable states in a pedestrian evacuation scenario. SIAM Journal on Applied Dynamical Systems, 22(1), 1-36.



















Katrin Wendland (Trinity College Dublin, Ireland)

Title: Generic data can refine invariants

Abstract: In this talk, we explain how some of the classical geometric invariants can be investigated and refined by 'counting' generic features in geometry. In particular, on some complex surfaces we will 'count' certain types of cohomology classes, which in physics are known as 'BPS states'.



















Noemi Picco (Swansea University, UK)

Title: Cooperation and competition in the cancer microenvironment shapes the emergence of drug resistance

Abstract: The environment in which the tumour lives is known to modulate drug resistance, offering protective and survival cues to cancer cells. However, cancer and non-cancer cells also must compete for shared resources such as space and nutrient. The resulting dynamics of complex interactions between the cancer cells, the host tissue, and the drug can be explored using a mathematical and computational framework which integrates experimental observations. In this talk I will use a range of modelling approaches that can explicitly capture the spatio-temporal dynamics of tumour growth, invasion and resistance observed clinically and experimentally. Elucidating the contributions of the tumour microenvironment in regulating treatment response, these models can offer insight into therapeutic strategies that can successfully control the emergence of drug resistance.



















Graham Smith (Pontifical Catholic University of Rio de Janeiro, Brazil)

Title: Clifford structures constant extrinsic curvature surfaces

Abstract: Constant extrinsic curvature surfaces in space forms are best understood as Sasakian objects living in the unitary bundle. In this framework they become bilegendrian surfaces. We discuss the insights this perspective provides on the study of constant extrinsic curvature surfaces.



















Joris Labarbe (Université Côte d'Azur, France)

Title: Surface recovery of rotational overhanging waves from bottom pressure measurements

Abstract: We derive equations relating the pressure at a flat seabed and the free-surface profile for steady gravity waves with constant vorticity. The set of integro-differential equations enables the recovery of overhanging (or not) rotational (or not) water waves from pressure measurements at the bottom. Furthermore, the flow vorticity (unknown a priori) is determined solely from the bottom pressure as part of the recovery method. This approach is applicable even in the presence of stagnation points and its efficiency is illustrated via numerical examples.



















Ayrse Arik (Heriot-Watt University, UK)

Title: Modelling cancer risk: inequalities, COVID-19 impact and future projections

Abstract: Reliable modelling of cancer risk is important for insurance purposes since it can impact pricing and reserving in areas such as critical illness insurance. It also has a significant direct impact on health and social care. In this study, we focus on investigating cancer rates based on the population data of England. We consider a Bayesian setting to identify trends and differences in the most prevalent types of cancer, whilst also accounting for heterogeneity and uncertainty in the data. We have found marked socioeconomic differences in some of the cancer types, such as lung cancer, and that the deprivation gap across the most and least deprived groups has been worsening over time.
Our research has further revealed that increases in average age-at-diagnosis, that can be associated with diagnostic delays due to COVID-19 health disruptions, could lead to significant increases in cancer mortality, also exhibiting uneven outcomes at regional level. Under a Bayesian framework, our predicted rates suggest persistent inequalities in some of the most prevalent cancer types.



















Robert Osburn (University College Dublin, Ireland)

Title: From knots to quantum modularity

Abstract: Knots are objects which appear in nature, science and the arts. We see them while untying our shoelaces, looking under a microscope or admiring the Book of Kells. Knot invariants are quantities defined for each knot which are the same for equivalent knots.
Modular forms are analytic objects with intrinsic symmetric properties. Over the past two hundred years, the study of modular forms has enjoyed fruitful interdependencies with many areas such as number theory, algebraic geometry, combinatorics and mathematical physics. In particular, they were the key players in Maryna Viazovska's spectacular result on the sphere packing problem for which she was awarded the 2022 Fields Medal.
Over the past two decades, there have been intriguing connections between these two seemingly disparate areas. In this talk, we discuss historical developments and recent striking interactions between quantum knot invariants and a new spectrum of modular forms, namely mock modular and quantum modular forms.



















Ioannis Parissis (University of the Basque Country UPV/EHU, Spain)

Title: Directional singular integrals in two and higher dimensions

Abstract: In this talk I will give an overview of the theory of directional averages and singular integrals, firstly in dimension 2. We will see the main obstructions to the boundedness of these objects which will naturally lead us to the discussion of the connections with the Kakeya conjecture and the Stein and Zygmund conjectures. Finally, I will present a sharp estimate for directional singular integrals in codimension one, and general ambient dimension.



















Hildeberto Jardón Kojakhmetov (University of Groningen, the Netherlands)

Title: On singularities of network dynamics

Abstract: In the realm of low-dimensional dynamical systems, the presence of singularities often serves as a precursor to intricate dynamics unfolding in their vicinity. Remarkably, network dynamics, owing to their inherent graph structure, can exhibit singularities that were traditionally considered degenerate but arise as a generic feature in this context. This presentation focuses primarily on the exploration of nilpotent singularities within network dynamics, shedding light on recent endeavors aimed at unraveling their perturbations and implications.



















Constantin Arnscheidt (University of Cambridge, UK)

Title: Nonlinear Earth system dynamics: stability and catastrophe

Abstract: Together with all other life and the global environment, we humans exist within a single complex coupled "Earth system". Nonlinear dynamics play a key role in this system at all scales, and simple mathematics can be used to obtain profound insights. In this talk I demonstrate this through two case studies.
First, I consider the question of how Earth has maintained habitable surface conditions over its billion-year lifespan. Stabilising long-term (i.e. >100,000 year) feedbacks have long been hypothesised, but the evidence has remained surprisingly contentious. Using ideas from stochastic dynamics, I show that evidence regarding such feedbacks can be directly extracted from time series of Earth's past temperatures.
Second, motivated by past mass extinctions and some present-day concerns, I consider large-scale catastrophes for life. Specifically, I argue that "rate-induced tipping" --- an abrupt transition when external forcings change faster than a critical rate --- is a ubiquitous pathway towards collapse in evolutionary systems (or complex adaptive systems more generally). I support this using a simple dynamical-system model, a more complex agent-based model with emergent dynamics, and evidence from nature at a vast range of scales. If true, this is relevant to some of the most catastrophic risks facing humanity today, and there is much important work to be done.



















Mahir Hadzic (University College London, UK)

Title: Self-similar collapse for self-gravitating fluids

Abstract: We will review several recent results on global dynamics of radial self-gravitating compressible Euler flows, which arise in the mathematical description of stars. We will discuss classes of smooth initial data that lead to the formation of imploding finite-time singularities. Our main focus is on the decisive role of scaling invariances and their interaction with the nonlinearities.



















Trung Nghia Vu (Karolinska Institutet, Sweden)

Title: Analysing omics data using statistical and computational methods

Abstract: Omics data generated by high-throughput next-generation sequencing has been widely used in molecular data analysis and cancer research. We have developed multiple bioinformatics tools to analyze the data from both single-cell- and tissue- levels. In this seminar, I will introduce our recent studies on omics data analysis including detection of gene fusions and circular RNAs, quantification of isoform expression with extension work for single-cell data analysis, and finally, pathway activation scores with potential applications in pharmacogenomics.



















Kyle Wedgwood (University of Exeter, UK)

Title: Interface approaches for studying travelling waves in spiking neural networks

Abstract: Certain neural systems show computation through patterned activity: persistent localised activity, in the form of bumps, has been linked to working memory, whilst the propagation of activity in the form of waves has been associated with binocular rivalry tasks. Individual neurons typically exhibit an all-or-nothing response, dependent on the summation of signals they receive from the rest of the network. This fact, coupled with the desire to understand coherent patterns of activity across the network has resulted in the widespread use of non-smooth neural models that greatly simplify the complex dynamics of individual cells. Whilst these descriptions often provide tractable models of neural tissue, their non-smooth nature presents its own mathematical challenges.
We will show how localised bumps of activity and travelling waves are generated in a synaptically coupled neural network and how they lose stability through bifurcations of both smooth and non-smooth type. This will be done exemplified using two different model approaches. The first, taking the form of a discrete time Markov chain model, the second forming a continuum approximation of a discrete neural network. In both cases, analysis will be facilitated via the construction of interface equations that take advantage of the non-smooth nature of the model. Armed with this framework, we compute existence and linear stability properties of waves and bumps in such networks and further show how numerical coarse-graining procedures can be used to assess the impact of spatial and temporal noise. Finally, we will examine how this can be used to explain the experimentally observed activity in grid cells, which are a significant component of the brain's representation of spatial location within an environment.



















Dietmar Bisch (Vanderbilt University, USA)

Title: Subfactors and noncommutativity

Abstract: Vaughan Jones' theory of subfactors has profound applications in operator algebras, knot theory, noncommutative probability theory and quantum physics. Subfactors are algebras of operators on Hilbert spaces with a very rich representation theory that, at its most basic level, is given by the Temperley-Lieb algebras. I will explain a natural notion of noncommutativity for a subfactor and illustrate it with a theorem that provides the first examples of subfactors with distinct degrees of noncommutativity. This idea might also be of interest in quantum information theory.



















Marvin Anas Hahn (Trinity College Dublin, Ireland)

Title: A tropical spin on Hurwitz numbers

Abstract: Hurwitz numbers count branched morphism between Riemann surface with fixed numerical data. While a classical invariant, having been introduced in the 19th century, Hurwitz numbers are an active topic of study, among others due to their interplay with Gromov-Witten theory and their role in mirror symmetry. In recent years, many variants of Hurwitz numbers were introduced and studied. In 2005, Eskin, Okounkov and Pandharipande introduced what is now called spin Hurwitz numbers, which enumerate maps between spin curves, i.e. Riemann surfaces with a spin structure. In this talk, we employ tropical geometry to obtain a combinatorial description of spin Hurwitz numbers and use it to study their properties.
This is a joint work in progress with Loujean Cobigo.



















David O’Sullivan (University of Limerick, Ireland)

Title: A mathematically tractable model for information diffusion between communities

Abstract: Online social networks such as Twitter, Facebook, Instagram, and TikTok serve as media for the spread of information among their users. There has been significant progress in the study of community structure within these social networks, where much of the research focused on methods to identify community structure [5, 3], the quantification of the role of communities [1], and the modelling of community structures’ impact on spreading [2]. However, there’s room for further study on how information spreads within networks marked by distinct community structures. In this context, we introduce a novel approach using multi-type branching processes that allow us to capture community structure and model information diffusion accurately on the network.

We are interested in developing models for information diffusion in the presence of community structure to gain a deeper understanding of its drivers via multi-type branching processes. We build on previous work [4], where a simple branching process was used to capture the dynamics of simple contagion. We extend this model by tracking the evolution of a cascade inside and between different communities where the types, in the multitype branching process, correspond to the nodes in each community. Employing a multi-type branching processes allows us to calculate many key quantities of interest readily. Importantly, we show how our multi-type branching process description allows us, using multivariate probability-generating functions, to account for the community structure changes features like the cascade size distribution size in comparison to network with no community structure. Additionally, the use of a multi-type branching processes allows us to calculate the probability of extinction each community individually. We are also able to derive other statistical properties of interest, such as the probability of reintroduction into a community where information was once circulating but has died out.

[1]   M. Beguerisse-Diaz, A. K. McLennan, G. Garduno-Hernandez, M. Barahona, and S. J. Ulijaszek. The ‘who’ and ‘what’ of #diabetes on Twitter. DIGITAL HEALTH, 3:205520761668884, Jan. 2017.
[2]   G. Curato and F. Lillo. Optimal information diffusion in stochastic block models. Physical Review E, 94(3):032310, Sept. 2016.
[3]   T. Funke and T. Becker. Stochastic block models: A comparison of variants and inference methods. PLOS ONE, 14(4):e0215296, Apr. 2019.
[4]   J. P. Gleeson, T. Onaga, P. Fennell, J. Cotter, R. Burke, and D. J. O’Sullivan. Branching process descriptions of information cascades on twitter. Journal of Complex Networks, 8(6):cnab002, 2020.
[5]   X. Que, F. Checconi, F. Petrini, and J. A. Gunnels. Scalable Community Detection with the Louvain Algorithm. In 2015 IEEE International Parallel and Distributed Processing Symposium, pages 28–37, Hyderabad, May 2015. IEEE.



















Niall Madden (University of Galway, Ireland)

Title: An enriched finite element space for boundary layer problems

Abstract: The numerical solution of singularly perturbed PDEs is both interesting and challenging. In this talk, I'll first give a general introduction to these problems, and some of the numerous attempts to construct specialised numerical schemes that take into account the presence of any layers in the solutions. In the second part of the talk, I'll present a fitted operator-type approach, where we incorporate information on boundary layers into an otherwise standard finite element method.
This is joint work with Kirk Soodhalter, TCD.












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