Séminaire Amplitudes et Gravitation sur l’Yvette (IHES/IPhT)
Reconstructing the metric of a perturbed black hole due to the presence of a companion, including both the stationary and radiative contributions, is a highly non-trivial problem. In this talk, we address this to linear order in perturbation theory through the characteristic initial value formulation, prescribing data on two intersecting null hypersurfaces, one of which is a perturbed isolated horizon.   By including the possibility of small amounts of infalling radiation at the horizon, we show that the ringdown modes arise naturally in this formalism when there is no incoming radiation from null infinity. This therefore establishes analytically strong correlations between the usual quasi-normal modes observed in the outgoing radiation with the horizon geometry and the infalling radiation at the horizon. This can be viewed as a demonstration of black hole tomography in a perturbative setting, where we are able to determine the detailed dynamics of the horizon geometry based on observations of gravitational waves from the late stage of a binary black hole merger.
 
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The goal of this conference is to advance the dialogue and interactions between the LLM community and the larger world of mathematics in order to further the mathematical understanding of LLMs and contribute to solving some of the outstanding problems in the new field of LLMs.
 
In particular we intend to investigate mathematical structures that can be used to understand LLMs in terms of what they implicitly learn and how. 
 
At the same time, in the opposite direction the use of LLMs in order to do mathematics will be investigated.
 
Registration is free and open until May 15, 2025.
Invited speakers:Katie Collins (Cambridge University)Yann Fleureau (Numina)Fabian Glöckle (Ecole Nationale des Ponts et Chaussées & FAIR at Meta, Paris)Javier Gómez-Serrano (Brown University)Timothy Gowers (Collège de France)Yann Ollivier (FAIR at Meta, Paris)Simon Frieder (Oxford University)
Organizers: François Charton (Meta AI Research), Michael Douglas (Harvard University & IHES), Amaury Hayat (CERMICS) & Yiannis Vlassopoulos (Athena Research Center & IHES)
 

Running Seminar
 
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Probability and analysis informal seminar
The $phi^4$ model is a real-valued spin system with quartic potential. This model has deep connections with the classical Ising model, and both are expected to belong to the same universality class. We construct a random cluster representation for $phi^4$, analogous to that of the Ising model. For this percolation model, we prove that local uniqueness of macroscopic cluster holds throughout the supercritical phase. The corresponding result for the Ising model was proved by Bodineau (2005) and serves as the crucial ingredient in renormalization arguments used to study fine properties of the supercritical behaviour, such as surface order large deviations, the Wulff construction and exponential decay of truncated correlations. The unboundedness of spins in the $phi^4$ model imposes considerable difficulties when compared with the Ising model. This is particularly the case when handling boundary conditions, which we do by relying on the recently constructed random current representation of the model.Joint work with Trishen Gunaratnam, Christoforos Panagiotis and Romain Panis.
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 Although there are several ways to  »choose a compact hyperbolic surface at random », putting the Weil-Petersson probability measure on the moduli space of hyperbolic surfaces of a given topology is certainly the most natural.
The work of M. Mirzakhani has made possible the study of this probabilistic model: it is one of the only models of  »random Riemannian manifolds » where some explicit calculations are actually possible. One may thus ask questions about of the geometry and the spectral statistics of the Laplacian of a randomly chosen surface — in analogy with what is usually asked for models of random graphs.
I will be interested in the spectral gap of the Laplacian for a random compact hyperbolic surface, in the limit of large genus (joint work with Laura Monk).

Branched manifolds were introduced by Bob Williams in his study of dynamical systems, and generalize the notion of train tracks. Let G be one of the eight Thurston geometries. There is a branched 3-manifold W(G) with a finite triangulation such that a closed 3-manifold M immerses into W(G) if and only if M admits a G-structure. Equivalently, there is a combinatorial characterization of those manifolds that admit a G-structure. This is joint with Priyam Patel and Leslie Mavrakis.

Running Seminar
The analogues of mapping tori in arithmetic are curves over finite fields; flat connections on 3-manifolds are similar to unramified Galois representations. In order to find flat connections on a mapping torus defined by a diffeomorphism F we used the automorphism of the character surface induced by F. I will state a counterpart of this problem (finding Galois representations explicitly) in the world of p-adic differential equations, and will try to explain how to turn it into an effective computation following some ideas of F. Beukers.
 
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Amplitudes on the Coulomb branch of N=4 Super-Yang Mills (SYM) provide a rich playground to understand scattering amplitudes involving massive particles. We will discuss a symplectic Grassmannian integral representation of these amplitudes with a particular focus on three and four point amplitudes. We will discuss on-shell functions and the BCFW bridge on the Coulomb branch and possible future directions to obtain an amplituhedron-like geometry for this theory. Based on arxiv:2311.17763, and ongoing work with Veronica Calvo Cortes, Yassine El Maazouz and Amit Suthar.
 
 
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Séminaire de géométrie arithmétique
I will describe an arithmetic analogue of the theory of differential equations in which derivation operators acting on  functions are replaced  by Fermat quotient operators acting on numbers. I will then review a series of arithmetic applications of the theory including: the effective Manin-Mumford conjecture, finiteness  results for Heegner points (joint work with B. Poonen), construction of quotients of moduli spaces of abelian varieties by Hecke actions (joint work with A.Vasiu), and the curvature of the spectrum of the ring of integers (joint work with L.Miller).
 
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Séminaire Amplitudes et Gravitation sur l’Yvette (IHES/IPhT)
In this talk I present a new worldline action in which the worldline spin degrees of freedom are carried by bosonic oscillators. First, we construct a Hamiltonian for the worldline accounting for all spin-multipole moments at linear order in curvature, investigating the constraints on the Hamiltonian from a specific choice for the spin supplementary condition (SSC). Converting to the Lagrangian, we find a simple kinetic term for the spin DoFs in terms of bosonic oscillators, and also break the quadratic-in-spin ceiling of previous WQFT approaches involving Grassmann spinors. Injecting insight from the previously constructed Hamiltonian reveals a new class of operator redundancies in the EFT not related to equations of motion or integration by parts. Finally, a simple on-shell condition neatly ties up considerations related to the SSC, and paves the way for future higher-PM calculations involving spin.
 
 
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I will explain work joint with Aaron Landesman where we prove that for a finite group G and conjugacy invariant subset c, Hurwitz spaces parameterizing connected G-covers of the complement of a configuration of points on a disk with monodromy in c satisfy homological stability. We moreover compute the dominant part of the stable homology after inverting finitely many primes. This has applications to Malle’s conjecture over function fields, the Cohen—Lenstra—Martinet heuristics over function fields, as well as to the Picard rank conjecture.
 

Running Seminar
General Discussion 
Traces, Torsions, and the Neumann-Zagier Data for Specific Knots as Examples
 
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