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# Mathematical Sciences Seminar Abstracts 2022-23

**Louis M. Pecora (US Naval Research Laboratory)**

Title: Statistics of Attractor Embeddings in Reservoir Computing

Abstract: A recent branch of AI or Neural Networks that can handle time-varying signals often in real time has emerged as a new direction for signal analysis. These dynamical systems are usually referred to as reservoir computers. A central question in the operation of these systems is why a reservoir computer (RC), driven by only one time series from a driving or source system of many time-dependent components, can be trained to recreate all dynamical time series signals from the drive leads to the idea that the RC must be internally recreating all the drive dynamics or attractor. In addition, there have been some speculations that RCs may be a fundamental type of system that describes how neuronal networks in biology process sensory input. This has led to the possibility that the RC is creating an embedding of the drive attractor in the RC dynamics. There have been some mathematical advances that move that argument closer to a general theorem. However, for RCs constructed from actual physical systems like interacting lasers or analog circuits or possibly actual neuronal networks, the RC dynamics may not be known well or known at all. And many of the existing embedding theorems have restrictive assumptions on the dynamics. We first show that the best way to analyze RC behavior is to first treat it properly like a dynamical system, which it is. This will lead to some conflict with existing ideas about RCs, but also a clarification of those ideas. Secondly, in the absence of complete theories on RCs and attractor embeddings, we show several ways to analyze the RC behavior to help understand what underlying processes are in place, especially regarding if there are good embeddings of the drive system in the RC. We show that a statistic we developed for other uses can help test for homeomorphisms between a drive system and the RC by using the time series from both systems. This statistic is called the continuity statistic and it is modeled on the mathematical definition of a continuous function. We show the interplay of dynamical quantities (e.g. Lyapunov exponents, Kaplan-Yorke dimensions, generalized synchronization, etc.) and embeddings as exposed by the continuity statistic and other statistics based on ideas from nonlinear dynamical systems theory. These viewpoints and results lead to a clarification of various currently vague concepts about RCs, such as fading memory, stability, and types of dynamics that are useful.

This is joint work with Thomas L. Carroll.

**Benjamin Taylor (Lancaster University, UK)**

Title: Inference for aggregated spatiotemporal log-Gaussian Cox processes under changing and uncertain support

Abstract: Aggregated point processes data are common in epidemiological applications. They arise when the true disease process is continuous in space-time, but only data from aggregation units, e.g. health facilities, or administrative regions, are available. The challenges posed by such data are often ignored, or substantially simplified in practice. In this talk, I will introduce solutions to the pragmatic challenges typically encountered through an example concerning the modelling of case counts of malaria at the health facility level in Zambia. Health facilities in Zambia have fuzzy catchment areas, they report irregularly and change in number and size over time. We treat the underlying data-generation process as a spatio-temporally continuous point process, capturing aggregation through an additional model hierarchy and using a GPU-accelerated data-augmentation scheme for inference. Along the way, I will share my thoughts on the ecological fallacy.

**Jean-Claude Cuenin (University of Loughborough, UK)**

Title: Effective bounds on scattering resonances

Abstract: The celebrated Weyl law describes the asymptotic distribution of eigenvalues of the Laplacian on a compact manifold. Scattering resonances are analogues of eigenvalues when the underlying manifold is non-compact. The simplest case concerns Schrödinger operators *-Δ+V* on Euclidean space *R ^{d}* with compactly supported potential

*V*. The object of interest is the resonance counting function

*n*, that is, the number of resonances in a disk of radius

_{V}(r)*r*. In dimensions greater than one, asymptotics are known only in a few special cases. The topic of this talk are polynomial upper bounds on the resonance counting function. These have a long history, starting in the 80’s with work of Melrose. The sharp upper bound

*n*was proved by Zworski. In this talk I will present effective versions of this upper bound for non compactly supported potentials. Effective means that

_{V}(r)≤C_{V}r^{d}*C*does not depend on

_{V}*V*itself but only on some weighted norms. The proof of this result features a combination of harmonic, functional and complex analysis.

**Kaustubh Agarwal (Indiana University-Purdue University Indianapolis, USA/ University College Cork, Ireland)**

Title: Signatures of Parity and Time reversal symmetry breaking on a single LC Oscillator

Abstract: What is the fate of an oscillator when its inductance and capacitance are varied while its frequency is kept constant? Inspired by this question, we propose a protocol to implement parity-time (PT) symmetry in a lone oscillator. Different forms of constrained variations lead to static, periodic, or arbitrary balanced gain and loss profiles, that can be interpreted as purely imaginary gauge fields. With a state-of-the-art, dynamically tunable LC oscillator comprising synthetic circuit elements, we demonstrate static and Floquet PT breaking transitions, including those at vanishingly small gain and loss, by tracking the circuit energy. Concurrently, we derive and observe conserved quantities in this open, balanced gain-loss system, both in the static and Floquet cases. Lastly, by measuring the circuit energy, we unveil a giant dynamical asymmetry along exceptional point (EP) contours that emerge symmetrically from the Hermitian degeneracies at Floquet resonances. Distinct from material or parametric gain and loss mechanisms, our protocol enables on-demand parity-time symmetry in a minimal classical system - a single oscillator - and may be ported to other realizations including metamaterials and optomechanical systems. [1]

[1] M. A. Quiroz-Juárez, K. S. Agarwal, Z. A. Cochran, J. L. Aragón, Y. N. Joglekar, and R. d. J. León-Montiel, "On-demand parity-time symmetry in a lone oscillator through complex, synthetic gauge fields," arXiv preprint arXiv:2109.03846, 2021.

**Piotr Suffczynski (University of Warsaw, Poland)**

Title: Computational modeling of epileptic seizures with ion concentration dynamics

Abstract: Human and animal EEG data demonstrate that epileptic seizures are not stationary events but evolve with dynamics in the range of tens of seconds. We investigate the processes associated with seizure dynamics by complementing the Hodgkin-Huxley mathematical model with the physical laws that dictate ion movement. A computer model showed that seizure initiation, maintenance and autonomous termination can be explained by feedback mechanisms between ion concentration changes and neuronal activity. The model predicted a specific scaling law of inter-bursting intervals observed at the end of seizures, which was validated experimentally.

**Mine Caglar (Koç University, Istanbul, Turkey) **

Title: Hedging Portfolio for a Market Model of Degenerate Diffusions

Abstract: We consider a semimartingale market model when the underlying diffusion has a singular volatility matrix and compute the hedging portfolio for a given payoff function. Recently, the representation problem for such degenerate diffusions as a stochastic integral with respect to a martingale has been completely settled. This representation and Malliavin calculus established further for the functionals of a degenerate diffusion process constitute the basis of the present work. Using the Clark-Hausmann-Bismut-Ocone type representation formula derived for these functionals, we prove a version of this formula under an equivalent martingale measure. This allows us to derive the hedging portfolio as a solution of a system of linear equations.

This is joint work with I. Demirel and S.A. Ustunel.

**Daniel Meyer (University of Liverpool, UK)**

Title: Fractal spheres, visual metrics, and rational maps

Abstract: Quasisymmetric maps map ratios of distances in a controlled way. They generalize conformal maps. The quasisymmetric uniformization theorem asks if a certain metric space is quasisymmetrically equivalent to some model space. Of particular interest in this context is the question to characterize quasispheres, i.e., metric spaces that are quasisymmetrically equivalent to the standard 2-sphere. A simple class of fractal spheres are "snowballs", which are topologically 2-dimensional analogues of the van Koch snowflake curve.

A Thurston map is a topological analogue of a rational map (i.e., a holomorphic self-map of the Riemann sphere). Thurston gave a criterion when such a map "is" rational. Given such a map *f* that is expanding, we can equip the sphere with a "visual metric". With respect to this metric, the sphere is a quasisphere if and only if *f* "is" rational.

This is joint work with Mario Bonk (UCLA).

**David Andrade (University of Plymouth, UK)**

Title: The nonlinear Benjamin-Feir instability

Abstract: In this talk we revisit the classical framework of the Benjamin-Feir instability for deep water waves. The hydrodynamical system is recast in terms of a Hamiltonian which has only two variables: a modal amplitude and a dynamic phase, and only two free parameters: the total wave action and mode separation. The mode separation serves as a bifurcation parameter which allows us to classify the dynamics. We found steady-state nearly-resonant quartets of waves, corresponding to center points of the Hamiltonian system and new types of breather-like solutions corresponding to heteroclinic orbits. One of such heteroclinic solutions is an analog of the famed Akhmediev breather of the nonlinear Schrödinger equation. Other heteroclinic orbits correspond to new types of solutions of the water wave problem which, to the best of our knowledge, have not been found before.

**Áine Byrne (University College Dublin, Ireland)**

Title: Next generation mesoscopic models for neural activity

Abstract: The use of mathematics has many historical successes, especially in the fields of physics and engineering, where mathematical concepts have been put to good use to address challenges far beyond the context in which they were originally developed. More recently, mathematics has been employed to further our understanding of biological systems, such as the human brain. Despite the immense complexity of the brain, mathematical modelling has allowed for major advances to be made towards understanding behaviour, consciousness and disease. Assuming no specific neuroscience knowledge, this talk introduces the general ideas behind mathematically modelling the human brain. I will briefly review seminal work in the field, such as the Hodgkin-Huxley and Wilson-Cowan models, before discussing the development of the next generation of mesoscopic models for neural activity. This new class of model provides an explicit relationship between the neural activity and the level of within population synchrony at the mesoscopic level. To highlight the usefulness of such models, I will show how they can be deployed in a number of neurobiological contexts, such as providing understanding of the changes in power-spectra observed in EEG/MEG neuroimaging studies of motor-cortex during movement, insights into patterns of functional-connectivity observed during rest and their disruption by transcranial magnetic stimulation, and to describe wave propagation across cortex.

**Mario Chavez (Paris Brain Institute, France)**

Title: The intrinsic geometry of brain networks as a biomarker in epilepsy

Abstract: Epilepsy is a condition of recurrent unprovoked seizures resulting from different causes. This neurological disorder is nowadays conceptualized as a network disease with functionally and/or structurally aberrant connections on virtually all spatial scales. In epilepsy, brain networks generate and sustain normal, physiological brain dynamics during the seizure-free interval and are involved in the generation, maintenance, spread, and termination of pathophysiological activities such as seizures. Connectivity (network) analysis in epilepsy has provided valuable information on seizure onset, propagation and termination, as well on the functional organization of the brain after a resection surgery. Nevertheless, traditional (Euclidean) network embeddings are unable to fully capture the rich structural organization of brain connectivity, which motivates the quest for a latent geometry of the brain connectivity. In this talk I will show how non-Euclidean (hyperbolic) geometries can be used to represent brain networks of epileptic patients, and how these embeddings can provide an appropriate representation to unveil properties that could potentially result in robust biomarkers for surgery outcome. Namely, representation of brain networks in hyperbolic space can also identify regions of interest responsible or implicated in the surgery failure that could help understanding the origin of the unfavorable surgery outcomes for some patients.

**Frank Kwasniok (University of Exeter, UK)**

Title: Data-driven dimension reduction, prediction and modelling of complex dynamical systems

Abstract: The talk discusses methodologies for extracting from high-dimensional data sets leading spatial modes together with their dynamics. Dynamic mode decomposition (DMD) or linear inverse modelling (LIM) is a widely used tool; a linear model is fitted to the leading principal components and characteristic damping timescales and oscillation frequencies are obtained. Here, we investigate extensions of DMD. In optimal mode decomposition (OMD), the modes are optimised simultaneously with the linear model. We also look at non-Markovian and nonlinear dynamics as well as non-autonomous dynamics which allows for analysis and modelling of critical transitions. The techniques are exemplified on data sets from the Lorenz 1996 system, the Swift-Hohenberg equation, intermediate-complexity atmospheric models and a comprehensive climate model.

**Soizic Terrien (Laboratoire d'Acoustique de l'Universite du Mans, France)**

Title: Emergence of complex pulsing dynamics in excitable systems with delayed feedback

Abstract: Excitability is a very general phenomenon encountered in contexts as diverse as biology, chemical sciences or optics. While at rest state, an excitable system can react to a sufficiently large external perturbation by releasing a pulse response. In the presence of a feedback loop, excitable pulses can regenerate themselves when reinjected after a delay time. As demonstrated recently in optical and biological excitable systems, this can lead to the emergence of multistable periodic pulsing regimes corresponding to different number of equidistant pulses in the feedback loop. Here, we consider an excitable microlaser subject to delayed optical feedback. Experimentally and numerically, we investigate the emergence of multifrequency dynamics. An in-depth bifurcation analysis of a suitable mathematical model, written as a system of three delay-differential equations, unveils that resonance tongues play a key role in the emergence of complex dynamics, including periodic regimes with non-equidistant pulsing patterns and chaotic pulsing. A merging process of resonance tongues is shown to result in unexpectedly large regions of locked dynamics in the parameter plane of feedback delay and feedback strength. These locking regions may disconnect from the relevant torus bifurcation curve in a general mechanism of interaction between resonance tongues at an extremum of the rotation number on the torus bifurcation curve. The existence of such unconnected large regions of locked dynamics is in excellent agreement with experimental observations.

**Jane Hutton (University of Warwick, UK)**

Title: Exploring Missing Data Using Chain Event Graphs: The Treacherous Subtlety of Missingness

Abstract: Applied researchers are familiar with working with data. The most popular characterisation of missingness is through its generating mechanisms. Several methods have been developed for handling of missing values. However, they rely on the correct identification of the missingness mechanism. This is severely hindered by two main factors: the missingness mechanisms are ill-defined when two or more variables have missing values, and two different but related versions of the definitions for the missingness mechanisms have been used interchangeably in the literature. We review problems encountered in identifying the missingness mechanism. Further, we also demonstrate that by using the tools of probabilistic graphical models, in particular Chain Event Graphs, it is possible to represent the various missingness mechanisms. The graphs assist in evaluating models for missingness, and provide tests which distinguish generating mechanisms.

K Hemming and JL Hutton (2012) Bayesian sensitivity models for missing covariates in the analysis of survival data. J Eval. Clin. Pract.

LM Barclay, JL Hutton and JQ Smith (2014) Chain Event Graphs for Informed Missingness. Bayesian Analysis 9:53-76

**Serhiy Yanchuk (Potsdam Institute for Climate Impact Research, Germany)**

Title: Adaptive dynamical networks: from multiclusters to recurrent synchronization

Abstract: It is a fundamental challenge to understand how the function of a network is related to its structural organization. Adaptive dynamical networks represent a broad class of systems that can change their connectivity over time depending on their dynamical state. The most important feature of such systems is that their function depends on their structure and vice versa. While the properties of static networks have been extensively investigated in the past, the study of adaptive networks is much more challenging. Moreover, adaptive dynamical networks are of tremendous importance for various application fields. For example, models for neuronal synaptic plasticity, adaptive networks in chemical, epidemic, biological, transport, and social systems, to name a few. In this talk I highlight the dynamical phenomena arising in adaptive networks.

**Shane Whelan (University College Dublin, Ireland)**

Title: Mortality and Longevity in Ireland

Abstract: Shane will take us through some of the highlights of the research in his recent book, *Mortality and Longevity in Ireland*, recently published by Dublin University Press with the support of the Society of Actuaries in Ireland. The talk will be an informal overview, focussing on some important issues with those interested in the detail directed to the different chapters.

Shane will attempt to answer two questions:

- How long can those now alive in Ireland expect to live?
- How long can their unborn children expect to live?

He will overview official mortality forecasts (on which he advises the CSO). A simple two-parameter model that well described late-life mortality will be outlined. He will mention stochastic approaches and show how to build scenarios under a deterministic model, with associated probability levels. He will discuss the wisdom, or not, of relying on longevity forecasts to change the State retirement age. Putting a value on human life, he will review how actuaries estimate lump sum compensation for future loss. Finishing on a much-needed action, he highlights that compensation for negligence in the delivery of maternity services in Ireland is now higher than its day-to-day running costs and persistent recommendations for reform of the services remain unimplemented.

**Gabriella Clemente (Masaryk University, Brno, Czechia)**

Title: Almost-complex geometry: extrinsic vs. intrinsic perspectives

Abstract: I will discuss extrinsic and intrinsic approaches to study the question of (non-)existence of high dimensional almost-complex but not complex manifolds. I will explain how universal embedding results for compact almost-complex manifolds could potentially be used to prove the non-existence of complex structures up to homotopy in real dimension at least 6: Then, I will present some curvature obstruction equations to the integrability of almost-complex structures that find applications to the main existence problem in the almost-hermitian setting.

**Gerry Kileen (University College Cork, Ireland)**

Title: The key role of statisticians, data scientists and mathematical scientists in enabling globally inclusive science to address gross global inequalities

Abstract: Malaria is an exceptionally important example of an underlying cause of poverty and underdevelopment that specifically affects Africa in a disproportionate manner for purely environmental and biological reasons we can only blame mother nature for. Regrettably, much of the purported investment in scientific solutions to the malaria problem has actually exacerbated pre-existing global inequalities of scientific capacity and opportunity. At present, most of the advanced analyses of malaria-related data are conducted by specialist investigators based at globally renowned institutions in very comfortable, malaria-free locations that are dangerously remote from the processes they are trying to capture and the front-line professionals they most need to engage with.

Furthermore, many vector biologists and other non-specialists in malaria-endemic countries who lack sufficient appreciation of how statistics, mathematics and data science work, or what they can and cannot realistically tell us, often make poor use of the evidence generated by these advanced analytical disciplines. In particular, policy makers lacking adequate expert specialist advisory support from local institutions often disregard unsettling insights from analyses they don’t understand well enough to trust, while also accepting the conclusions of others based on their palatability rather than technical validity. Improving advanced analytical skills on the continent isn’t just about enabling African experts to independently apply them to their own data, it is also about empowering them to play their natural role in local, regional and global discourse on the vector-borne diseases that affect their own countries most directly.

I will discuss the challenges and opportunities at hand for specialists in advanced analytical disciplines all over the world to collectively address this grossly unfair global imbalance through inclusive scientific collaboration that puts real meaning behind the term *global solidarity*.

**Jerome V Moloney (University of Arizona, USA)**

Title: Extreme Nonlinear Optics and Nonequilibrium Dynamics in 2D and 3D Solids

Abstract: Semiconductor crystal structures, when exposed to intense ultrashort optical or THz pulses, are driven into highly nonequilibrium states that are dominated by higher order many-body carrier correlations. In this talk, I will combine a description of the theoretical foundation of these interactions with experimental realizations of continuous wave and pulsed optically pumped semiconductor disk lasers (SDLs) as illustrations. Historically, we used the theory to design SDLs and experimentally observed a record > 100W high brightness continuous wave emission from a single chip, a record > 16W single frequency operation and a room temperature tunable THz source. Under mode-locked pulsed operation we observed a record less than 100fs pulse duration and demonstrated a novel offset-free tunable mid-IR frequency comb. The microscopic description of both resonant (gain/absorption) interactions and highly detuned THz pulse driven nonperturbative HHG in detuned semiconductors is governed by the microscopic Semiconductor Bloch equations. We will show that higher order correlations beyond the Hartree-Fock limit play a role in both CW and pulsed SDL operation. The second part of my talk will address the theory of quasi-2D Transition Metal Dichalcogenides (TMDCs). Because of the very large in-plane 2D Coulomb potential, these materials display prominent room temperature exciton features below the bandgap – in stark contrast to 3D confinement where the latter are only observed at cryogenic temperatures. Out of plane interactions with other materials are governed by much weaker Van der Waal’s interactions, making them ideal candidates for sensing and electronic communications. The physics of these materials is captured by the Semiconductor Dirac Bloch equations – in contrast to 3D, higher order interactions such as Auger process and pair correlations all appear at the linear level.

**Jim Hanley (McGill University, Canada)**

Title: Excursions in Statistical History: Highlights

Abstract: Over the last 20 years, the speaker has delved into the origins of 'regression'; the development of the 't' and 'Poisson' distributions; forerunners of the 'hazard' function; and the statistical design and conduct of US Selective Service lotteries from 1917 onwards. This talk will recount the stories, data and simulations behind some of these, and provide some modern-day re-enactments.