Thermalization is the process by which a physical system evolves toward a state of maximal entropy, as permitted by conservation laws. I will begin by outlining the framework used to understand this phenomenon in quantum systems with unitary evolution (Eigenstate Thermalization Hypothesis). Next, I will discuss factors that can hinder or slow down thermalization. One example is long-lived prethermalization, where certain effective (or pseudo-conserved) quantities significantly delay thermalization depending on specific model parameters. This theory is particularly relevant for periodically driven systems, which can exhibit remarkable resistance to heating over extended timescales. I will then explore the possibility of systems that robustly fail to thermalize. Here, robustness refers to the fact that no fine-tuning is required, in contrast with integrable models. Many-body localization (MBL) is the most well-known, and possibly the only example of systems that fail to thermalize on their own. I will examine MBL from both theoretical and numerical perspectives, covering its description in terms of local integrals of motion, the destabilizing effect of quantum avalanches, and recent mathematical advancements. These later developments are welcome given the challenges in properly interpreting numerical results in this field.

Quantum Chromodynamics (QCD) has been a profound source of inspiration for theoretical physics, driving the development of key concepts such as string theory, effective field theories, instantons, anomalies, and lattice gauge theories. In these lectures, I will explore two distinct regimes of QCD - its infrared (IR) and ultraviolet (UV) limits - and the theoretical tools used to study them.

In the IR regime, where perturbative techniques break down, Effective Field Theories (EFTs) provide a powerful framework. I will introduce the pion EFT as a tool to study non-linearly realized symmetries and soft theorems. In the UV regime, where QCD becomes amenable to perturbative analysis, I will discuss the Operator Product Expansion and renormalization group equations, focusing on their application to deep inelastic scattering, a cornerstone in the discovery of quarks and gluons.

These two regimes illustrate the richness of QCD and its pivotal role in shaping our understanding of fundamental physics.

The equation

y^3 = x^6 + 23x^5 + 37x^4 + 691x^3 ��� 631204x^2 + 5169373941

obviously has the solution (1, 1729). Are there any other solutions in rational numbers? The study of integral or rational solutions to polynomial equations, sometimes known as the theory of Diophantine equations, is among the oldest pursuits in mathematics. This lecture will give an idiosyncratic survey of the remarkable advances made in the 20th and 21st century for the special case of equations of two variables. The emphasis will be on the techniques of arithmetic geometry, the study of spaces built up of finitely-generated systems of numbers.

It is common to transform data to stationarity, such as by differencing and demeaning, before estimating factor models in macroeconomics. Imposing these transformations, however, limit opportunities to learn about trending behaviour. Trends and deterministic processes can play a central role in the behaviour of macroeconomic processes and so it is important to be able to characterise these features of the data. In this paper, we develop a model of common and idiosyncratic deterministic and stochastic processes in a factor model. A judicious choice of parameter expansion and post-processing ensures the model avoids a non-invariant specification imposed before estimation. This renders inference data-driven and makes computation efficient.

We extend the powerful property of Yangian invariance to a new large class of conformally invariant multi-loop Feynman integrals. This leads to new highly constraining differential equations for them, making integrability visible at the level of individual Feynman graphs. Our results apply to planar Feynman diagrams in any spacetime dimension dual to an arbitrary network of intersecting straight lines on a plane (Baxter lattice), with propagator powers determined by the geometry. The graphs we consider determine correlators in the recently proposed "loom" fishnet CFTs. The construction unifies and greatly extends the known special cases of Yangian invariance to likely the most general family of integrable scalar planar graphs. We also relate these equations in certain cases to famous GKZ (Gelfand-Kapranov-Zelevinsky) hypergeometric operators, opening the way to using new powerful solution methods.

The traditional mathematics route to produce lecture notes by compiling LaTeX to PDF gives outputs which suffer from accessibility problems and are often not optimized for screen viewing, especially on the mobile devices favoured by many of our students. Various markdown-based solutions have recently been developed to address this issue and I will report on my own attempts to get to grips with Quarto (https://quarto.org/) and use it to simultaneously produce lecture notes in HTML and PDF formats with properly typeset equations, cross-linked chapters and citations. I will demonstrate the features (and possible pitfalls) of this approach in the context of a set of notes produced for a short course given at the African Institute for Mathematical Sciences (AIMS) in Ghana\DSEMIC along the way, I hope to convey something of the work of AIMS and what I learnt from teaching in that environment.

Given a Riemannian surface, the study of sharp upper bounds for Laplacian eigenvalues under the area constraint is a classical problem of spectral geometry going back to J. Hersch, P. Li, S.-T. Yau and N. Nadirashvili. The particular interest in this problem stems from the remarkable fact that the optimal metrics for such bounds arise as metrics on minimal surfaces in spheres. In the talk I will survey recent results on the subject with an emphasis on the fruitful interaction between the geometry and spectral bounds. In particular, I will describe a surprisingly effective method of constructing new minimal surfaces based on the eigenvalue optimisation with a prescribed symmetry group.

Rough path theory provides a framework for the study of nonlinear systems driven by highly oscillatory (deterministic) signals. The corresponding analysis is inherently distinct from that of classical stochastic calculus, and neither theory alone is able to satisfactorily handle hybrid systems driven by both rough and stochastic noise. The introduction of the stochastic sewing lemma (Khoa L��, 2020) has paved the way for a theory which can efficiently handle such hybrid systems. In this talk, we will discuss how this can be done in a general setting which allows for jump discontinuities in both sources of noise.