Seminars

15th November 2023

Erik Koelink, Radboud Universiteit (Netherlands).


Abstract:

25th January 2023 (ELA102b)

Sergey Dyachenko, Department of Mathematics, University at Buffalo.
Instabilities of almost limiting Stokes waves

Abstract:
The Stokes wave is a water wave that travels over a free surface of water without changing shape. When a time-varying fluid domain is mapped to a fixed geometry, such as a periodic strip in the lower half-plane, the equation for the Stokes wave is a nonlinear integro-differential ODE whose solutions are found numerically to arbitrary precision. The spectral stability of Stokes waves is studied by linearization of the equations of motion for the free surface around a Stokes wave, and studying the spectrum of the associated Fourier-Floquet-Hill (FFH) eigenvalue problem. We developed a novel approach to studying the eigenvalue spectrum by combining the conformal Hamiltonian canonical variables, the FFH technique built into a matrix-free Krylov-Schur eigenvalue solver. We report new results for the Benjamin-Feir instability as well as the high-frequency, and localized (superharmonic) instabilities of the waves close to the limiting Stokes wave.

15th February 2023 (ELA002)

Timm Kruger, University of Edinburgh.
Formation and stability of pairs of particles in inertial microfluidics

Abstract:
Inertial particle microfluidics is an emerging technology for microfluidic particle separation and manipulation. Some applications, such as flow cytometry and cell encapsulation, require the formation of stable particle trains with consistent inter-particle spacing along the flow axis. In order to understand the formation and properties of particle trains, we have investigated the formation and stability of the simplest case of particle trains - particle pairs - in straight channels under mild inertia through immersed-boundary-lattice-Boltzmann-finite-element simulations. We found that initial conditions, particle softness and size heterogeneity play important roles in the formation and stability of particle pairs. Our results demonstrate that all these factors must be considered in the design of inertial microfluidic devices for manipulating inter-particle spacing.

22nd February 2023 (ELA002)

Maia Angelova, Data Analytics Research Lab, School of Information Technology, Deakin University, Melbourne Burwood Campus.
Data-driven models for precision healthcare

Abstract:
In this talk, I will share our latest research and describe how our data-driven models, based on physical and physiological data, can assist in the early detection of insomnia and diabetes and provide a robust basis for pre-screening of individuals with wearable devices. I will consider the following case studies based on our most recent projects: (a) Glucose-insulin regulation models: glucose-insulin dynamics is central for understanding the regulation mechanisms between different organs in the human body and is key to maintain healthy life and prevent diabetes. We combine dynamical systems approach with machine learning algorithms to model the regulation between glucose and insulin and predict glucose dynamics and insulin utilisation in healthy and pre-diabetic regimes [1,2]; (b)  Classification of chronic back pain from multi-modal data: we use image, questionnaires and physical activities data to  classify chronic low back pain [3]. (c) Classification of insomnia and sleep from physical activities (actigraphy) data. Our models can distinguish acute from chronic insomnia and healthy sleep [4-6]. They provide a robust basis for pre-screening of insomnia with wearable devices in a home environment.
References:
1.Angelova, M., Beliakov, G., Ivanov, A. Shelyag, S. 2021. Global stability and periodicity in a glucose-insulin regulation model with a single delay. CNSNS 95, 105659, https://doi.org/10.1016/j.cnsns.2020.105659.
2. Huard, B., Bridgewater, A. and Angelova, M. 2017. Mathematical investigation of diabetically-impaired ultradian oscillations in the glucose-insulin regulation. J Theor Biol 418: 66-76, doi:10.1016/j.jtbi.2017.01.039.
3.Tagliaferri, S.D., Wilkin, T., Angelova, M. et al. Chronic back pain sub-grouped via psychosocial, brain and physical factors using machine learning. Sci Rep 12, 15194 (2022). https://doi.org/10.1038/s41598-022-19542-5.
4.Rani, S., Shelyag, S., Karmakar, C., Zhu, Ye, Fossion, R., Ellis, Jason, Drummond, S. P. A. and Angelova, M. (2022) Differentiating acute from chronic insomnia with machine learning from actigraphy time series data. Frontiers in Network Physiology, 2. p. 1036832. ISSN 2674-0109.
5.M. Angelova, C. Karmakar, Y. Zhu, S. P. A. Drummond and J. Ellis, ""Automated Method for Detecting Acute Insomnia Using Multi-Night Actigraphy Data, in IEEE Access, vol. 8, pp. 74413-74422, 2020, doi: 10.1109/ACCESS.2020.2988722.
6.Kusmakar, Shitanshu, et al. "A machine learning model for multi-night actigraphic detection of chronic insomnia: development and validation of a pre-screening tool." Royal Society open science 8.6 (2021): 202264, https://doi.org/10.1109/TBME.2018.2845865.

8th March 2023 (SAN 313)

Thibault Bonnemain, Kings College London.
Quadratic mean field games: Schrödinger and electrostatic representations

Abstract:
Mean Field Games provide a powerful theoretical framework to deal with stochastic optimization problems involving a large number of coupled subsystems. They can find applications in several fields, be it finance, economics, sociology, engineering ... Though these models are much simpler than the underlying differential games they describe in some limit, their precise behaviour is not yet fully understood. After introducing the basic tenets of the formalism, I will focus on a toy model from a particular class of games, for which there is a deep connection between the associated system of PDEs and the nonlinear Schrödinger equation. This allows me to identify limiting regimes that can be dealt with explicitly. In particular one such regime yields insight on the intrinsic forward-backward structure of Mean Field Games through a mapping to an electrostatic problem.

15th March 2023 (ELA002)

Casper Oelen, Department of Mathematical Sciences, Loughborough University.
Automorphic Lie algebras on complex tori

Abstract:
An automorphic Lie algebra is a Lie algebra of certain invariants, initially arising in the theory of integrable systems, or more specifically, in the context of algebraic reduction of Lax pairs. They are defined as follows. Let a finite group $\Gamma$ act on a compact Riemann surface and on a complex finite dimensional Lie algebra $\mathfrak{g}$, both by automorphisms. Consider the space of meromorphic maps from the Riemann surface to the Lie algebra with poles restricted to an orbit of $\Gamma$. The subspace of $\Gamma$-equivariant maps is an automorphic Lie algebra. It is an infinite dimensional Lie algebra over the complex numbers and it can be seen as a generalisation of the famous Onsager algebra. We present a classification for $\mathfrak{g} = \mathfrak{sl}_2(\mathbb{C})$, in which case it turns out that $\Gamma$ can be naturally restricted to be cyclic, dihedral or the alternating group on four symbols. We shall give a construction of normal forms for the case of $\mathfrak{g}=\mathfrak{sl}_2\mathbb{C})$, which is one of the key elements in the proof of the classification result. Finally, we will mention some applications to integrable systems. This talk is based on my PhD thesis, supervised by Sara Lombardo and Vincent Knibbeler.

29th March 2023 (ELA002)

Thomas Bothner, School of Mathematics, University of Bristol.
Bulk spacings in non-Hermitian matrix models

Abstract:
Random matrix eigenvalue spacings tend to show up in problems not directly related to random matrices: for instance, bumper to bumper distances of parked cars in a numberof roads in central London are well represented by the so-called eigenvalue bulk spacing distribution of a suitable Hermitian matrix model. In this talk we will first survey several occurrences of these Hermitian spacing distributions and afterwards try to generalise them to non-Hermitian models. As it turns out, the theory of integrable systems, especially Painlev´especial function theory, plays a crucial role in this field. Based on arXiv:2212.00525, jointwork with Alex Little (Bristol).

19th April 2023 (ELA002)

Tom Bridges, Department of Mathematics University of Surrey.
Superharmonic instability and water wave breaking

Abstract:

26th April 2023 (ELA002)

Peter Clarkson, University of Kent.
Orthogonal Polynomials, Integrable Systems and Random Matrices

Abstract:
In this talk I will discuss the relationship between orthogonal polynomials and integrable systems. The main example will involve Freud weights which have the form exp(U(x)), where U(x) is an even polynomial, which arise in the context of Hermitian matrix models and random symmetric matrix ensembles. I will describe properties of the recurrence coefficients in the three-term recurrence relation associated with the orthogonal polynomials. For Freud weights recurrence coefficients satisfy discrete equation which are members of the first discrete Painleve hierarchy, also known as the string equation, and also satisfy integrable nonlinear differential equations. This is joint work with Kerstin Jordaan (University of South Africa) and Ana Loureiro (University of Kent).

3rd May 2023 (ELA002)

Andrew Hone, Kent.
Elliptic analogues of Chebyshev polynomials

Abstract:
After reviewing some classical facts about continued fractions and orthogonal polynomials, using the Chebyshev polynomials of the 2nd kind as the main example, we present a construction of the continued fraction expansion of a function on a hyperelliptic curve of genus g. We show how this is connected to discrete integrable systems, Toda lattices and integer sequences, using the elliptic case (g=1) to illustrate the main ideas.

10th May 2023 (ELA002)

Daniel Ratliff, Northumbria.
What Can Water Waves Teach Us About Chorus Waves?

Abstract:
For over a century, the linear and nonlinear analysis of oceanic waves has led to great strides in our understanding of their evolution and phenomenology, and in turn had admitted a host of mathematics in the form of nonlinear PDEs and methods by which wave phenomena can be described elsewhere. In contrast, there is comparatively less understood and formulated for plasmas in our near-Earth environment. Here electromagnetic effects yield a whole new host of physics that can manifest throughout wave motion. One such wave family that is of interest are Whistler-Mode Chorus (WMC) waves, which are responsible for particle energisation (potentially to dangerous levels!) all the while 'singing' due to coherent frequency sweeping. What about the plasma could be causing this, how can we understand it, and could the whole process be an extreme event in disguise? To answer these questions, we dust off our tools used in ocean waves to make progress. In doing so, we will demonstrate that wave-particle interactions are analogous to wave-mean flow ones and are at the heart of the dynamics which emerge. A combination of classical modulation theory and multiple scales point to higher order effects which are responsible for this singing behaviour, which for plasmas become comparable to leading order effects. The wavepackets which emerge alter the wave's frequency via their interaction, and thus we determine leading order estimates of the frequency chirp rate for WMC waves. We then observe via numerical observation that the singing events to kinetic energy amplification, making these more siren songs than heavenly chorus.

17th May 2023 (ELA002)

Ted Johnson, UCL.


Abstract:

24th May 2023 (ELA002)

Francesco Giglio, Glaglow.


Abstract: