Week 17.03.2025 – 23.03.2025

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.

Kernel-based random graphs (KBRGs) are a class of random graph models that account for inhomogeneity among  vertices. We consider KBRGs on a discrete d-dimensional torus. Conditionally on an i.i.d. sequence of Pareto weights, we connect any two points independently with a probability that increases in the points' weights and decreases in the distance between the points. We focus on the adjacency matrix of this graph and study its empirical spectral distribution. In the dense regime we show that a limiting distribution with non-trivial second moment exists as the size of the torus goes to infinity, and that the corresponding measure is absolutely continuous with respect to the Lebesgue measure. We also derive a fixed-point equation for its Stieltjes transform in an appropriate Banach space. In the case corresponding to so-called scale-free percolation we can explicitly describe the limiting measure and study its tail. Based on a joint work with R. S. Hazra, N. Malhotra and M. Salvi.

The Thurston norm of a 3-manifold M measures the minimal topological complexity of a surface dual to a character of M . In this talk, we will introduce a real-valued function on the first cohomology of an arbitrary group which generalises the Thurston norm. We will propose a strategy for proving that such a function defines a seminorm using the theory of L2-invariants. Finally, we will implement this strategy for some classes of right-angled Artin groups.

We introduce and study the planted directed polymer, in which the path of a random walker is inferred from noisy "images" accumulated at each time step. Formulated as a nonlinear problem of Bayesian inference for a hidden Markov model, this problem is a generalization of the directed polymer problem of statistical physics, coinciding with it in the limit of zero signal to noise. For a one-dimensional walker we present numerical investigations and analytical arguments that no phase transition is present. When formulated on a Cayley tree, methods developed for the directed polymer are used to show that there is a transition with decreasing signal to noise where effective inference becomes impossible, meaning that the average fractional overlap between the inferred and true paths falls from one to zero.

This will be an expository lecture intended for a general mathematical audience to illustrate, through examples, the theme of p-adic variation in the classical theory of modular forms. Classically, modular forms are complex analytic objects, but because their fourier coefficients are typically integral, it is possible to also do elementary arithmetic with them. Early examples arose already in the work of Ramanujan. Today one knows that modular forms encode deep arithmetic information about elliptic curves and Galois representations. Our main goal will be to illustrate these ideas through simple concrete examples.

Title: Mock modular forms from meromorphic Jacobi forms

Abstract: Mock modular forms (mock theta functions) were first discussed by Ramanujan over a century ago, but only in this millennium, due to work of S. Zwegers and others, has a theory been developed. I will present a theorem which shows how mock modular forms appear from meromorphic Jacobi forms (after settiing up the various elements). This is based on joint work with A. Dabholkar and D. Zagier, and meant as a taster of one problem relating physics and number theory. The discussion will be at an elementary level. Over tea, I would be happy to discuss in more detail the relevance of this theorem in physics and also other problems relating physics and number theory.