Week 16.03.2025 – 22.03.2025

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.

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.

I will provide a rather lengthy introduction in oder to highlight interest in exploring QFts on AdS spaces (without dynamical gravity). The aspects involve the dyanmics of boundaries and interfaces in normal QFTs in flat space, the actual dynamics of confining gauge theories on AdS, the question of prximity in the pace of QFRTs, a more general notion of holography and its connection to S-matrices and finally Euclidean wormholes.

All these issues will connect in the effort to describe holographic QFTs on AdS.

We shall investigate in a concrete example how the related classical solutions explore the space of QFTs and we construct the general solutions that interpolate between the same or different CFTs with arbitrary couplings. The solution space contains many exotic RG flow solutions that realize unusual asymptotics, as boundaries of different regions in the space of solutions. We find phenomena like "walking" flows and the generation of extra boundaries via "flow fragmentation".



We will then move on and describe an example of a holographic theory that confines on flat space, when we put it on AdS.

We will find three types of regular solutions are found. Theories with two AdS boundaries provide interfaces between two confining theories. Theories with a single AdS boundary correspond to ground states of a single confining theory on AdS. We find solutions without a boundary, whose interpretation is

probably as interfaces between topological theories. We analyze in detail the holographic dictionary for the one-boundary solutions and compute the free energy. No (quantum) phase transitions are found when we change the curvature. We find an infinite number of pure vev solutions, but no CFT solution without a vev. We also compute the free energy of the interface solutions. We find that the product saddle points have always lower free energy than the connected solutions. Finally we will comment on the spectrum of propa gating states of holographic theories on AdS and dS.

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.