# 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.