L6

One curious little fact about the Riemann zeta function is that if you evaluate its derivatives at the zeros of zeta, then on average this is real and positive (even though the function is complex). This has been proven for some time now, but the aim of this talk is to generalise the question further (higher derivatives, complex moments) and gain insight using random matrix theory. The takeaway message will be that there are a multitude of different proof techniques in RMT, each with their own advantages

This brief pedagogical talk introduces key concepts of Carroll geometries, which arise as the limit of relativistic spacetimes in the vanishing speed of light regime. In this limit, light cones collapse along a timelike direction, resulting in a manifold equipped with a degenerate metric. Consequently, physics in such spacetimes exhibits peculiar properties. Despite this, the Carroll contraction is relevant to a wide range of applications, from flat-space holography to condensed matter physics. To complement this introduction, and depending on the audience’s interests, I can discuss Carroll affine connections, symmetry groups, conservation laws, and Carroll-invariant field theories.

In this talk I will review recent results on the development of a form factor program for integrable quantum field theories (IQFTs) perturbed by irrelevant operators. It has been known for a long time that under such perturbations integrability is preserved and that the two-body scattering phase gets deformed in a simple manner. The consequences of such a deformation are stark, leading to theories that exhibit a so-called Hagedorn transition and no UV completion. These phenomena manifest physically in several distinct ways. In our work we have mainly asked the question of how the deformation of the S-matrix translates into the correlation functions of the deformed theory. Does the scaling of correlators at long and short distances capture any of the "pathologies" mentioned above? Can our understanding of irrelevant perturbations tell us something about the space of IQFTs and about their form factors? In this talk I will answer these questions in the afirmative, summarising work in collaboration with Stefano Negro, Fabio Sailis and István M. Szécsényi.

The time evolution of quantum matter systems toward their thermal equilibria, characterized by their entanglement entropy (EE), is a question that permeates many areas of modern physics. The dynamic of EE in generic chaotic many-body systems has an effective description in terms of a minimal membrane described by its membrane tension function. For strongly coupled systems with a gravity dual, the membrane tension can be obtained by projecting the bulk Hubeny-Rangamani-Ryu-Takayanagi (HRT) surfaces to the boundary along constant infalling time. In this talk, I will introduce the membrane theory of entanglement dynamics, its generalization to 2d CFT, as well as several applications. Based on arXiv: 1803.10244 and arXiv: 2411.16542.

It was recently argued that topological operators (at least those associated with continuous symmetries) need regularization. However, such regularization seems to be ill-defined when the underlying QFT is coupled to gravity. If both of these claims are correct, it means that charges cannot be meaningfully measured in the presence of gravity. I will review the evidence supporting these claims as discussed in [arXiv:2411.08858]. Given the audience's high level of expertise, I hope this will spark discussion about whether this is a promising approach to understanding the fate of global symmetries in quantum gravity.

Given an FP-infinity subgroup G of SL(2,C), we are interested in the asymptotic behavior of the cohomology of G with coefficients in an irreducible complex representation V of SL(2,C). We prove that, as the dimension of V grows, the dimensions of these cohomology groups approximate the L2-Betti numbers of G. We make no further assumptions on G, extending a previous result of W. Fu. This yields a new method to compute those Betti numbers for finitely generated hyperbolic 3-manifold groups. We will give a brief idea of the proof, whose main tool is a completion of the universal enveloping algebra of the p-adic Lie algebra sl(2, Zp).

Black holes play a central role in our understanding of quantum gravity, but identifying their precise counterparts in a dual QFT remains a tricky business. These states are heavy, chaotic, and encode various universal aspects — but are also notoriously hard to characterise. In this talk, we’ll explore how supersymmetric field theories provide a controlled setting to study black hole states. In particular, we’ll introduce the idea of fortuitous states as a useful criterion for identifying BPS black hole states. We’ll then illustrate this concept with concrete examples, including the (supersymmetric) SYK model and the D1-D5 CFT.

The discussion will be based on the following recent papers:
arXiv:2402.10129, arXiv:2412.06902, and arXiv:2501.05448.

For some values of degrees d=(d_1,...,d_c), we construct a compactification of a Hilbert scheme of complete intersections of type d. We present both a quotient and a direct construction. Then we work towards the construction of a quasiprojective coarse moduli space of smooth complete intersections via Geometric Invariant Theory.

Quotients of hyperbolic groups (groups that act geometrically on a hyperbolic space) and their generalizations have long been a powerful tool for proving strong algebraic results. In this talk, I will describe the geometry of random quotients of certain of groups, that is, a quotient by a subgroup normally generated by k independent random walks.  I will focus on the class of hierarchically hyperbolic groups (HHGs), a generalization of hyperbolic groups that includes hyperbolic groups, mapping class groups, most CAT(0) cubical groups including right-angled Artin and Coxeter groups, many 3–manifold groups, and various combinations of such groups.  In this context, I will explain why a random quotient of an HHG that does not split as a direct product is again an HHG, definitively showing that the class of HHGs is quite broad.  I will also describe how the result can also be applied to understand the geometry of random quotients of hyperbolic and relatively hyperbolic groups. This is joint work with Giorgio Mangioni, Thomas Ng, and Alexander Rasmussen.

Let X be a smooth rigid-analytic variety. Ardakov and Wadsley introduced the sheaf D-cap of infinite order differential operators on X, along with the category of coadmissible D-cap-modules. In this talk, we present a Riemann–Hilbert correspondence for these coadmissible D-cap-modules. Specifically, we interpret a coadmissible D-cap-module as a p-adic differential equation, explain what it means to solve such an equation, and describe how to reconstruct the module from its solutions.