Memories of Jean-Bernard

• Philippe di Francesco (IPhT Saclay and UIUC)
• Andrea Cappelli (INFN Florence)

Room: Amphithéatre Hermite

Nimreps, old and new

• Robert Coquereaux (I2M Marseille)

Room: Amphithéatre Hermite After recalling basic properties of fusion algebras and nimreps in a modern setting (module-categories), we describe exotic nimreps associated with an old exceptional invariant of F4 at level 6 that cannot be obtained from a conformal embedding of affine Lie algebras.

Concentration of measure

• Michel Talagrand (IMJ-PRG, Sorbonne Université, CNRS)

Room: Amphithéatre Hermite

Asymptotics and HCIZ integrals

• Alice Guionnet (UMPA, ENS de Lyon)

Room: Amphithéatre Hermite

Entanglement at Quantum Criticality

• Benoit Estienne (LPTHE, Sorbonne Université)

Room: Amphithéatre Hermite Over the past twenty years, quantum entanglement has established itself as a central theme of research across a wide spectrum of physics, from high-energy theory to condensed matter and quantum information. In the context of this talk, the key point is that entanglement measures have proven to be a remarkably powerful tool for investigating the physics of one-dimensional quantum critical systems. I will begin by introducing the relevant concepts and reviewing the replica approach to computing Rényi entropies. For one-dimensional quantum systems in particular, the entanglement entropy can be evaluated within two complementary frameworks. In the geometric perspective, the scaling limit reduces the problem to computing a partition function on a branched covering of certain Riemann surfaces—effectively a replication of space-time. In contrast, the algebraic perspective replicates the degrees of freedom themselves, leading to the formulation of a cyclic orbifold. Following this review, I will highlight the main obstacles to evaluating entanglement entropies in one-dimensional quantum critical systems, and outline some recent developments.

Counting trees and maps: Cyclic sieving phenomena

• Mireille Bousquet-Mélou (LaBRI, U. Bordeaux)

Room: Amphithéatre Hermite Some families of combinatorial objects, on which a cyclic group G acts, exhibit a "cyclic sieving phenomenon": this means that for any size n, there exists a nice polynomial P(q), with integer coefficients, such that the value of P at any primitive d-th root of unity gives... the number of objects of size n that are fixed by any element of G of order d. Strange! In particular, P(1) is the total number of objects of size n. We will show such phenomena for various actions of a cyclic group on families plane trees (this is joint work with Christian Krattenthaler, from Vienna) and on plane maps (this is joint work with Marie Albenque, from Paris).

Greedy Tamari intervals and constellations

• Philippe Biane (Universite Gustave Eiffel, Paris)

Room: Amphithéatre Hermite We prove a conjecture of Bousquet-Mélou and Chapoton about the enumeration of intervals of a greedy version of the Tamari order, by providing a recursive bijection with rooted bipartite planar maps. This bijection extends to constellations. Joint work with W. Fang.

Knot enumeration and crossing loop models coupled to gravity

• Paul Zinn-Justin (University of Melbourne)

Room: Amphithéatre Hermite 24 years ago, Jean-Bernard suggested we work together on the enumeration of knots, tangles and links using matrix models and related combinatorial tools. Though we managed to derive some exact results, many questions remain unanswered, including the important issue of the asymptotic enumeration of knot diagrams as the number of crossings goes to infinity. In this talk I will give an update on these questions, both on the theoretical side (precise predictions from QFT coupled to 2D gravity) and on the numerical side.

Cryptozoology of Special Functions

• Colin McSwiggen (Academia Sinica, Taiwan)

Room: Amphithéatre Hermite Cyptozoology is the study of cryptids, animals whose existence has been conjectured but never proven. Well-known cryptids include the Loch Ness monster, the jackalope, the chupacabra, and — perhaps most famously — the sasquatch, AKA bigfoot, believed by many enthusiasts to be a "missing link" between apes and humans. This talk will introduce the cryptozoology of integral representations for special functions, beginning with the HCIZ integral formula proved by our dear friend Jean-Bernard, and ending with new results from joint work with Siddhartha Sahi. Reference: https://arxiv.org/abs/2606.15185

Taking a trip to cumuland with Jean-Bernard

• Sylvain Lacroix (LPTHE, Sorbonne Université)

Room: Amphithéatre Hermite In this talk, I will discuss our recent work with Jean-Bernard Zuber on finite N precursors of the free cumulants. These are polynomials on the space of NxN matrices, invariant under conjugations and extracted from the so-called HCIZ integral. Their main feature is their additivity with respect to the average over sums of U(N)-conjugacy orbits. Moreover, they admit natural expansions in terms of Newton and Schur polynomials, related to the representation theory of the unitary and symmetric groups and recovering a previous construction by Capitaine and Casalis. I will further discuss their large N limit, which yields the free cumulants, with 1/N² corrections expressed in terms of Hurwitz numbers. Finally, expanding on the main additivity property of these objects, I will describe the behaviour of general U(N)-invariant polynomials under sums of conjugacy orbits.

From divergent series to moduli spaces

• Marcos Marino (Université de Genève)

Room: Amphithéatre Hermite Jean-Bernard Zuber has made contributions to an extraordinarily broad range of topics. Among these are two apparently distant subjects: the large-order behavior of perturbative series in quantum field theory, and integration over moduli spaces of Riemann surfaces. In this talk, I will describe recent developments that bring these two subjects together. In particular, I will show how physical ideas about large-order behavior lead to predictions for enumerative invariants of moduli spaces.

Jean-Bernard and his defects

• Costas Bachas (ENS Paris)

Room: Amphithéatre Hermite Defects, to whose study JB has made pioneering contributions, are ubiquitous in modern Quantum Filed Theory. I will present a curious identity obeyed by all superconformal defects, and discuss its physics implications.

JBZ, ADE, Defects & All That: A Lattice Perspective

• Paul Pearce (University of Melbourne)

Room: Amphithéatre Hermite Jean-Bernard has always been passionate about all things ADE. In this talk I will consider the construction of defects in the context of Yang-Baxter integrable classical, affine and twisted affine ADE RSOS lattice models. I will also discuss their associated fusion algebras with an emphasis on the $E_6$ and $D_4^{(1)}$ examples. For the classical ADE models, the defects satisfy the Ocneanu algebra for arbitrary system sizes and thus reproduce, in the continuum scaling limit, the well known CFT results of Petkova and Zuber 2001 for topological defects.

Large N is solvable — and even integrable : correlation functions in N=4 SYM at strong coupling

• Didina Serban (IPHT Saclay)

Room: Amphithéatre Hermite