Week 10.03.2025 – 16.03.2025

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.

The characteristic polynomial of a random unitary matrix is a much studied and mathematically rich object in random matrix theory. In this talk I will discuss the secular coefficients, those obtained after expanding the characteristic polynomial in terms of its monomial powers. These coefficients turn out to have interesting structure, related to combinatorial objects known as magic squares and to a holomorphic counterpart of Gaussian multiplicative chaos. I will discuss recent work where we obtain their limiting distributions in the Circular \beta Ensemble, for any \beta > 2. This is joint work with Joseph Najnudel, Elliot Paquette and Truong Vu.

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 Lê, 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.

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.

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.

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.

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.