Mathematical Bootstrap: From Conformal Fields to L-Functions

• Dalimil Mazáč (IPhT Saclay & IHES)

Room: Centre de conférences Marilyn et James Simons In this course, I will revisit the conformal bootstrap from a mathematical perspective and describe its recent applications in number theory and spectral geometry. The conformal bootstrap is an algebraic formulation of conformal field theory, valid in general dimension. While its axioms define a precise mathematical structure, realizing the corresponding objects has posed a significant challenge. The main idea presented in this course is that an almost identical structure has a natural realization in number theory, specifically in spectral theory of automorphic forms. The Hilbert space of conformal field theory becomes a direct sum of automorphic representations, and the structure constants of the operator algebra are related to L-functions. This link allows us to import the bootstrap mindset and methods into mathematics, which has led to new progress on important open problems. I will describe two recent applications: bounds on spectral gaps of hyperbolic manifolds, and subconvex bounds on L-functions. (Course given in English.) References: https://arxiv.org/abs/2111.12716 https://arxiv.org/abs/2308.11174 https://arxiv.org/abs/2508.20576

Problèmes ouverts et progrès récents en géométrie spectrale hyperbolique

• Frédéric Naud (Sorbonne Université, IMJ-PRG)

Room: Centre de conférences Marilyn et James Simons Dans cet exposé, on fera un survey des progrès récents sur la compréhension du trou spectral des surfaces hyperboliques, dans la limite de grand genre. On fera le lien avec la théorie des champs via le déterminant du laplacien dont le comportement moyen sur l'espace des modules est un des problèmes clés en théorie des cordes. On conclura sur des conjectures sur le spectre en genre "infini", inspirées par la physique du solide et les quasi-cristaux. (Exposé en français)

Mathematical Approaches to Conformal Bootstrap

• Balt van Rees (CPHT, École polytechnique)

Room: Centre de conférences Marilyn et James Simons We will review the framework of conformal field theory and explain how its operator algebra leads to a natural arena for well-posed mathematical questions with physical importance. We will present some results concerning the so-called lightcone bootstrap and discuss some important open problems. (Seminar given in English).