MaPLe
The Mathematical Physics at Leeds Seminar Series is aimed at bringing together researchers at any level from across the University of Leeds — from mathematics and physics departments alike — to give talks on themes in mathematical physics, broadly construed. On occasion, seminars also include those by researchers from outside the University of Leeds.
The series was co-founded, and organised between February 2024 and May 2025, by Linden Disney-Hogg and me. A list of all MaPLe seminars held in this period can be found below. You can check out this webpage for up-to-date information on the series.
February 20, 2024 | 11:00-12:00
Path integral formulation of stochastic processes
Steve Fitzgerald
School of Mathematics
University of Leeds
Abstract
Traditionally, stochastic processes are modelled one of two ways: a continuum Fokker-Planck approach, where a PDE is solved to determine the time evolution of the probability density, or a Langevin approach, where the SDE describing the system is sampled, and multiple simulations are used to collect statistics. There is also a third way: the functional or path integral. Originally developed by Wiener in the 1920s to model Brownian motion, path integrals were famously applied to quantum mechanics by Feynman in the 1950s. However, they also have much to offer to classical stochastic processes (and statistical physics).In this talk, I will introduce the formalism at a physicist’s level of rigour, and focus on determining the dominant contribution to the path integral when the noise is weak. There exists a remarkable correspondence between the most-probable stochastic paths and Hamiltonian dynamics in an effective potential [1, 2]. I will then discuss some applications as time permits, including reaction pathways conditioned on finite time [2]. We demonstrate that the most probable pathway at a finite time may be very different from the usual minimum energy path used to calculate the average reaction rate.
[1] Ge, Hao, and Qian, Hong. Int. J. Mod. Phys. B 26.24 1230012 (2012)
[2] Fitzgerald, Steve, et al. J. Chem. Phys. 158.12 (2023)
March 05, 2024 | 11:00-12:00
The geodesic approximation and the \(L^2\)-geometry of vortex moduli spaces
Cas Chaudhuri
School of Mathematics
University of Leeds
Abstract
The geodesic approximation is a method by which the low-energy/non-relativistic dynamics of solitons in a classical field theory are modelled by geodesics on a related moduli space. In practical terms, this reduces the problem of understanding soliton dynamics to studying the Riemannian geometry of the associated moduli space, often a more tractable problem. The moduli space constructed is also an object worthy of study in its own right, possessing canonical geometric structures beyond the Riemannian metric which can affect the soliton dynamics.In this talk, I will introduce the geodesic approximation in the particular context of the dynamics of vortices in Abelian Yang-Mills-Higgs theory. We will begin with a brief overview of Abelian YMH theory and the existence of vortex solitons, moving onto the existence and structure of static vortex moduli spaces, and the validity of the geodesic approximation in the low-energy regime. The second half of the talk will focus on finer details about the vortex moduli space including the construction of the L²-metric and some key geometric properties. Time permitting, we will mention some new results on how the vortex metric can itself be approximated in certain parametric limits.
March 19, 2024 | 11:00-12:00
Beyond the eigenstate thermalisation hypothesis: deep thermalisation in constrained quantum systems
Tanmay Bhore
School of Physics and Astronomy
University of Leeds
Abstract
The Eigenstate Thermalisation Hypothesis (ETH) is a powerful conjecture that explains the emergence of thermodynamics in isolated quantum systems. By postulating a connection between random matrix ensembles and deterministic unitary dynamics, ETH postulates that the reduced density matrix of a generic quantum system evolves to the universal form of a Gibbs ensemble. Then, "thermalisation" occurs as entanglement builds up between a subsystem and its complement.Performing measurements on a complementary subsystem, however, can reveal finer nuances in the system's ability to thermalise. This concept, dubbed as "deep thermalisation", promises to generalize ETH and has been recently realised in experiments on Rydberg atom arrays [1, 2]. In this talk, I will give a brief introduction to ETH and introduce this new formalism. I will also present the idea that systems which look "thermal" in the ETH sense can be highly "non-thermal" when probed through the lens of deep thermalisation [3]. This finding will be illustrated on several constrained models that describe slow relaxation in quantum glasses and quantum many-body scars in Rydberg atom arrays.
[1] https://journals.aps.org/prxquantum/abstract/10.1103/PRXQuantum.4.010311
[2] https://www.nature.com/articles/s41586-022-05442-1
[3] https://journals.aps.org/prb/abstract/10.1103/PhysRevB.108.104317
April 30, 2024 | 11:00-12:00
On the action principle for integrable systems
Vincent Caudrelier
School of Mathematics
University of Leeds
Abstract
The principle of least action associated to Lagrangians is a fundamental notion in many areas of science. Its alter ego, the Hamiltonian formalism, is just as fundamental. In many instances, one can pass from one to the other (Legendre transform) and choose what is best suited to the task at hand. A famous development of the 20th century is quantum mechanics, where one saw the Lagrangian formulation come back in full force with Feynman's breakthrough after canonical quantisation based on the Hamiltonian formalism had been the method of reference since the birth of the theory. When it comes to integrable systems, which possess a large amount of symmetries, the picture has been skewed towards the Hamiltonian formulation where the Liouville-Arnold theorem plays a crucial role. It was only in 2009, here in Leeds, that a Lagrangian framework emerged which encodes integrability via a generalised variational principle. I will present this framework and illustrate it in the simplest context of finite-dimensional systems (classical mechanics). I will sketch how the main ideas go over to field theory. Finally, I will briefly touch upon an important motivation for this programme: the quantisation of integrable systems via Feynman's path integral.May 14, 2024 | 11:00-12:00
Yang–Mills instantons in dimensions 7 and 8
Tathagata Ghosh
School of Mathematics
University of Leeds
Abstract
In this talk I will gently introduce the notion of Yang–Mills instantons in higher dimensions, in particular, in dimensions 7 and 8. I will also briefly discuss the current research in this area, including my own, and how it fits into the bigger picture.After reviewing 4-dimensional instantons, I will discuss the main physical motivations behind higher-dimensional instantons, by following the historical development of the subject. Then, I will introduce Güraydin–Nicolai instantons and Fairlie–Nuyts–Fubini–Nicolai (FNFN) instantons on ℝ⁷ and ℝ⁸ respectively. These are the earliest examples of instantons in dimensions 7 and 8 respectively, analogous to the BPST instantons on ℝ⁴.
Finally, I will briefly explain how my own research on the deformation theory of instantons on asymptotically conical manifolds can provide many important properties of these instantons.
June 04, 2024 | 11:00-12:00
Kähler manifolds, the Calabi construction, and harmonic spinors
Guido Franchetti
Department of Mathematical Sciences
University of Bath
Abstract
Harmonic spinors, that is, solutions of the massless Dirac equation, have been the object of considerable interest for both the mathematical and physical communities. In the talk I will show how the rich structure of Kähler manifolds allows to recast the Dirac equation in a way which makes obtaining explicit solutions easier. The method will be applied to the Eguchi-Hanson manifold, for which we show how to reproduce known solutions, and to more general Ricci-flat Kähler manifolds obtained via the Calabi construction, for which we present new solutions.October 22, 2024 | 10:00-11:00
Anyons
Jiannis K. Pachos
School of Physics and Astronomy
University of Leeds
Abstract
Anyons are quasiparticles in two-dimensional systems that show statistical properties very distinct from those of bosons or fermions. While their isolated observation has not yet been achieved, it is possible to perform quantum simulations with physical systems that reveal central properties of anyons. In this talk I will present encoding and manipulation of anyons with quantum technology platforms that reveal their exotic statistical properties with the goal of eventually employing them for topological quantum computation.November 05, 2024 | 10:00-11:00
How to cheat at billiards: a new open classical and quantum billiard model
Katherine Holmes
Department of Mathematics
Imperial College London
Abstract
The classical billiard model has been used to study dynamical systems and chaos theory. Its quantum counterpart is the quantum billiard model, a toy model of quantum optical systems in QED cavities and quantum dots. The billiard model in both the classical and quantum regimes has been well-documented in the literature, with a multitude of variations having been constructed. In recent years, we have seen the introduction of leaky billiards, billiards with loss mechanics such as internal holes and permeable boundaries.In this talk, I will introduce the billiard model and debut a new classical leaky billiard model with a permeable internal region. This model allows for the study of intricate structures on the Poincaré-Birkhoff phase space via intensity landscapes. The talk will conclude with a discussion of what may be the quantum and semiclassical counterpart to this classical leaky billiard.
This talk is based on a paper soon to be released on arXiv: Intensity landscapes in elliptical and oval billiards with a circular absorbing region. The final section will implement semiclassical methods inspired by a recent PRL: Husimi dynamics generated by non-Hermitian Hamiltonians, 2023.
November 19, 2024 | 10:00-11:00
The Yang-Baxter equation and quantum group symmetry
Benjamin Morris
School of Mathematics
University of Leeds
Abstract
We begin with an introduction to the Yang-Baxter equation, as a master equation for integrability in 2D lattice models in statistical mechanics. We will see that through the classification of (classes of) solutions to this equation it is natural to consider solutions related to physical symmetries known as quantum groups. We will then discuss a scheme for obtaining factorised solutions to the Yang-Baxter equation in a class of infinite-dimensional representations of the quantum group U𝑞sl(n).December 03, 2024 | 10:00-11:00
Random geometric graphs – from discs to scale-free models
Peter Gracar
School of Mathematics
University of Leeds
Abstract
We take a look at several random geometric graphs (RGG) with increasing levels of complexity, starting from the classical Gilbert disc model with fixed radius and up to the weight-dependent random connection model. At each step, we discuss the heuristics of what the newly added complexity changes in the behaviour of the models and how it affects the criticality of the largest connected component and the typical distance between two points of this component.January 28, 2025 | 10:00-11:00
Quantum many-body scars: a new paradigm of order amidst quantum chaos
Zlatko Papic
School of Physics and Astronomy
University of Leeds
Abstract
The quest to understand out-of-equilibrium behaviour of complex quantum systems represents one of the frontiers of contemporary quantum science. For a long time, the prevailing belief has been that complex quantum systems, comprising many interacting degrees of freedom, all suffer the same inevitable fate: that of thermalisation, whereby the system relaxes towards a featureless thermal state, completely "forgetting" its initial condition. However, a flurry of recent works has unearthed a new paradigm of behaviour in many well-known physical systems, including Rydberg atoms, lattice gauge theories, and certain kinds of frustrated magnets. Such systems have been understood to possess a subtle breakdown of ergodicity, now commonly known as "quantum many-body scars". Quantum many-body scars exhibit fascinating properties, such as extreme sensitivity to initial conditions: while a system initialised randomly undergoes chaotic dynamics and thermalisation, specific initial conditions can result in persistent dynamical revivals, surpassing native thermalisation timescales. The discovery of quantum many-body scars has not only deepened our understanding of many-body quantum mechanics, but it also has direct practical relevance for improving the control over the delicate physical phenomena underpinning quantum technologies. In this talk, I will present a pedagogical overview of this fascinating new field of physics, highlighting a few of the remaining mysteries for theory and future experiments.February 11, 2025 | 10:00-11:00
Discrete approximations for the induced connection
Derek Harland
School of Mathematics
University of Leeds
Abstract
The induced connection is a natural connection on a subbundle of a vector bundle. In physics, it is known as the Berry connection, and its parallel transport operators give rise to the Berry phase. In this talk I will explain exactly what the Berry/induced connection is and present some work I have done on finding numerical approximations to its parallel transport. This will lead to some interesting(?) questions for the algebraists in the audience!February 25, 2025 | 10:00-11:00
Integrable vortices in the Abelian Higgs model
Nora Gavrea
School of Mathematics
University of Leeds
Abstract
Vortices are 2-dimensional topological solitons defined on a Riemann surface in the context of the Abelian Higgs model. Physically, they model magnetic flux tubes in superconductors. At critical coupling, they satisfy a 1st order system of PDEs called the Bogomolny equations. I will first review the derivation of these equations using a Bogomolny argument, and then derive the Taubes equation. Next, I will introduce a generalised Abelian Higgs energy functional, which gives rise to 5 different vortex equations (this has been carefully investigated by Nicholas Manton). For a constant curvature base surface, these equations turn out to be integrable, reducing to a Liouville equation. One can obtain further integrable vortex equations by choosing suitable conformal factors, and in this case, the Taubes equation becomes the sinh-Gordon or Tzitzeica equation. If we assume radial symmetry, these are equivalent to a Painlevé III ODE. I will discuss the construction of these vortices for the rest of the talk, which is a joint work with Maciej Dunajski.March 11, 2025 | 10:00-11:00
Magnetic buoyancy in the solar tachocline
Lucas Gosling
School of Mathematics
University of Leeds
Abstract
Magnetic buoyancy is the phenomenon for strong magnetic fields to reduce the pressure of electrically-conducting plasma, which can lead to gravitational instabilities. Starting from a toy model primarily of academic concern, magnetic buoyancy was seen as an interesting phenomenon with no known applications. Just one year after its conception, Eugene N. Parker hypothesised that magnetic buoyancy could be a component of the solar dynamo, explaining how the Sun uses this mechanism to redistribute its magnetic field generated deep beneath its surface. The first part of this talk will be a brief literature review summarising how our knowledge of magnetic buoyancy evolved over time, and how its applications were discovered.The second part of the talk will focus on overstability, i.e., states which exhibit oscillations which grow in time. There are two well-known physical mechanisms within the magnetic buoyancy instability (MBI) which drive overstable modes. I will describe these physical mechanisms and present the findings of my second paper, including the discovery of a third mechanism for overstability. Furthermore, generalising MBI to include variable diffusion restricts overstability, and our newly discovered mechanism is the only one possible in solar interiors. However, our fluid model does not provide sufficient physical insight to describe it, and we instead create a secondary flux tube model to capture the physics of the problem.
March 18, 2025 | 10:00-11:00
From the quantum sine-Gordon model to number theory via partitions
Clare Dunning
School of Engineering, Mathematics and Physics
University of Kent
Abstract
Partitions of integers play a role in a variety of fields including number theory, representation theory and random matrix theory as well as being of independent interest in enumerative combinatorics. I will present several key concepts and discuss various places where partitions have arisen in my research.May 06, 2025 | 10:00-11:00
Regularizing 3D conformal field theories via the fuzzy sphere
Cristian Voinea
School of Physics and Astronomy
University of Leeds