Séminaire de géométrie arithmétique
In the setting of algebraic geometry in characteristic zero (or of complex geometry), the Bernstein-Sato polynomial is a polynomial defined for a function on a smooth variety and has deep connections with several invariants attached to the singularities of the zero locus of the function, among which the focus in the talk is on the connection with the monodromy eigenvalues on the nearby cycle sheaf, known as the theorem of Kashiwara and Malgrange. 
There have been attempts to develop a Bernstein-Sato-type theory also in the setting of positive characteristic, and these have led to a definition of Bernstein-Sato roots (but not their multiplicities), which again has deep connections to the theory of singularities. However, it has not been well studied how this theory is related to the monodromy eigenvalues on a nearby cycle sheaf. In this talk, I will explain an observation that the Bernstein-Sato roots seem to recover some of the monodromy eigenvalues on a suitable nearby cycle sheaf but only those on the unit root part, which we think suggests a better definition of Bernstein-Sato roots that captures all the monodromy eigenvalues and produces finer information about the singularities of the zero locus. This is a joint work in progress with Eamon Quinlan-Gallego and Daichi Takeuchi. 
 
 
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Séminaire Amplitudes et Gravitation sur l’Yvette (IHES/IPhT)
Recent advances in modelling unbound binary black hole interactions have been driven by the application of scattering amplitude methods to generate results within the post-Minkowskian (PM) expansion. However, this expansion breaks down when approaching the strong-field where the large curvature effects become non-negligible. In this talk, I will show how numerical information from self-force (SF), an expansion in the small mass ratio of the system, can inform higher-order coefficients in the PM expansion. I will also show how a single point of SF scattering data can be used to resum the PM series, providing accurate predictions for scattering angles across all separations. Additionally, I will present results from unbound numerical relativity simulations, where the full Einstein field equations are solved for comparable mass systems, and compare these with predictions from PM-informed effective one body models.
 
 
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Probability and analysis informal seminar
We start by reviewing the classical Sherrington-Kirkpatrick (SK) model. In this model, +1/-1-valued spins interact with each other subject to random coupling constants. The covariance of the random interaction is quadratic in terms of spin overlaps. Parisi proposed the celebrated variational formula for the limit of free energy of the SK model in the 80s, which was later rigorously verified in the works by Guerra and Talagrand. This formula has been generalized in various settings, for instance, to vector-valued spins, by Panchenko. However, in these cases, the convexity of the interaction is crucial. In general, the limit of free energy in non-convex models is not known and we do not have variational formulas as valid candidates. Here, we report recent progress through the lens of the Hamilton-Jacobi equation. Under the assumption that the limit of free energy exists, we show that the value of the limit is prescribed by a characteristic line; and the limit (as a function) satisfies an infinite-dimensional Hamilton-Jacobi equation « almost everywhere ». This talk is based on a joint work with Jean-Christophe Mourrat.​
 
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In this talk, we bootstrap various objects in planar maximally supersymmetric Yang- Mills theory. Focusing on the four-point correlation function of stress-tensor, we first demonstrate why the conventional bootstrap approach fails and new techniques are required. Next, we introduce a set of sum rules that are tailored for this problem as they are sensitive only to single-traces in the OPE expansion. Integrability enters at this stage and provides information on the spectrum of these operators. Their OPE coefficients, however, remain elusive. We then discuss how these sum rules can be employed in numerical bootstrap to nonperturbatively bound the OPE coefficients, the four-point correlation function and the energy-energy correlator. We show, for the first time, rigorous non-perturbative results for the planar OPE coefficient of single-trace operators as well as the correlation function at various points in cross-ratio space. Additionally, focusing on the energy-energy correlator (EEC), we present rigorous bounds for its spin 2 and spin 3 Legendre coefficients as well as the full EEC function at various angles. These results were obtained for a wide range of  t’Hooft couplings, highlighting the power of the bootstrap in probing non-perturbative aspects of planar N=4 SYM theory.
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Séminaire Amplitudes et Gravitation sur l’Yvette (IHES/IPhT)
In conventional world-line formalism for spinning binaries in general relativity, one assumes that the dynamical degrees of freedom for spin are the completely captured by the rest frame canonical spin. A spin supplementary condition (SSC) is then necessary to remove redundancies. We study this problem from an amplitude-based field theory perspective. In higher spin field theories, it is notoriously difficult to impose transverse and traceless conditions when interactions are included. We take an alternative approach and keep the additional degrees of freedom. We see that for generic Wilson coefficients, we obtain a system with a dynamical mass dipole that has physical effect starting at the quadrupole level. It will decouple when we choose special values for Wilson coefficients, and we land back on the dynamics of conventional spinning binaries. The situation is very similar to a symmetry breaking in the classical limit. We also construct a world-line Lagrangian and a classical effective Hamiltonian that completely match the physics mentioned above, which incorporates a dynamical mass dipole as the additional dynamical degree of freedom. The mass dipole has physical effects, and its significance is a question for phenomenology. On the other hand, the dipole can be removed by an emergent world-line shift symmetry when Wilson coefficients take special values. From this perspective, our formalism can simplify the calculation for conventional spinning binaries, as the SSC constraint can be effectively relaxed.
 
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Given a compact hyperbolic surface together with a suitable choice of orthonormal basis of Laplace eigenforms, one can consider two natural spectral invariants: 1) the Laplace spectrum $Lambda$, and 2) the 3-tensor Cijk representing pointwise multiplication (as a densely defined map L2 x L2  $to$ L2) in the given basis. Which pairs ($Lambda$,C) arise this way? Both $Lambda$ and C are highly transcendental objects. Nevertheless, we will give a concrete and almost completely algebraic answer to this question, by writing down necessary and sufficient conditions in the form of equations satisfied by the Laplace eigenvalues and the Cijk. This answer was suggested by physicists Kravchuk, Mazac, and Pal, who introduced these equations (in an equivalent form) as a rigorous model for the crossing equations in conformal field theory.
 
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Séminaire Amplitudes et Gravitation sur l’Yvette (IHES/IPhT)
In this talk, I will show how one can study gravitational perturbations from the near-horizon region of extremal and near-extremal rotating black holes in a general higher-derivative extension of Einstein gravity. I will explain how the near-horizon Teukolsky equation is modified via a correction to the angular separation constant. The near-horizon region also provides constraints on the form of the full modified Teukolsky radial equation, which serve as a stepping stone towards the study of quasinormal modes of near-extremal black holes. In the second part of the talk, I will show how this EFT can be constrained, motivated by preserving two fundamental properties of GR: gravitational waves are non-birefringent, and black hole quasinormal modes are isospectral. This leads to a novel class of EFT extensions, which remarkably, coincides with predictions from string theory and implies a previously unknown feature of string theory effective actions.
 
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We use Teichmüller theory to construct a new geometric model for the classifying space of the mapping class group of a three-dimensional handlebody. Two consequences are obtained: (i) Chan-Galatius-Payne have recently shown that the homology of Kontsevich’s commutative graph complex injects into the homology of the mapping class groups of surfaces, producing an enormous amount of highly unstable homology classes. We show that this homomorphism factors through the homology of the corresponding handlebody mapping class groups. (ii) The handlebody mapping class group is a virtual duality group in the sense of Bieri-Eckmann, with dualizing module given by a certain complex of nonsimple disk systems; the analogous result for mapping class groups of surfaces is a theorem of Harer. (Joint with Louis Hainaut and with Ric Wade.) 
 
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Séminaire de géométrie arithmétique
For an absolutely unramified extension $K/mathbb{Q}_p$ with perfect residue field, by the works of Fontaine, Colmez, Wach and Berger, it is well known that the category of Wach modules over a certain integral period ring is equivalent to the category of lattices inside crystalline representations of $G_K$ (the absolute Galois group of $K$). Moreover, by the recent works of Bhatt and Scholze, we also know that lattices inside crystalline representations of $G_K$ are equivalent to the category of prismatic $F$-crystals on the absolute prismatic site of $O_K$, the ring of integers of $K$. The goal of this talk is to present a direct construction of the categorical equivalence between Wach modules and prismatic $F$-crystals over the absolute prismatic site of $O_K$. If time permits, we will also mention a natural generalisation of these results to the case of a « small » base ring and intended application (work in progress).
 
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Probability and analysis informal seminar
The Laplacian is a central operator in the analysis of surfaces (and life in general). In this talk, we investigate how small its small eigenvalues can be, giving a sharp, quadratic bound on the k-th eigenvalue of a surface in terms of k, the surface’s genus g, and its global geometry via the injectivity radius. The techniques involve extremal length, spectral embedding, and volume arguments.Joint work with Guy Lachman and Asaf Nachmias, based on the paper: https://arxiv.org/abs/2407.21780For an exposition and overview of the paper, see here:

New paper on arXiv: A sharp lower bound on the eigenvalues of surfaces

 
 
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An introduction to examples of SL2(R) – invariant subvarieties V of the moduli space of holomorphic 1-forms, Omega Mg , and their connections to topology, Hodge theory and algebraic geometry.

Introduction of translation surfaces (definitions and examples) and their moduli spaces, period coordinates, SL(2,R)-action (examples: pseudo-Anosov), Kontsevich-Zorich cocycle (examples of concrete matrices via Thurston-Veech and/or Rauzy-Veech algorithm), some applications (of Zorich phenomenon in Delecroix-Hubert-Lelievre and of Lyapunov exponents in Kappes-Moeller).