# MaPLe

#### Spring-Summer 2024

##### 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**

**Gautam 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

*L*²-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