Séminaire de géométrie arithmétique
From the 1980s to the 1990s, Jean-Marc Fontaine introduced the theory of (phi, Gamma)-modules to study p-adic Galois representations. They are simpler than p-adic Galois representations, but he showed an equivalence between them. Among p-adic Galois representations, some classes are particularly important in number theory. Main examples are crystalline representations, semi-stable representations and de Rham representations. In this talk, I will explain how we can determine the (phi, Gamma)-modules corresponding to these representations. These results can be seen, in a sense, as generalizations of Wach modules.
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In this talk, we first review the local monodromy at infinity of the Bessel F-isocrystals following Dwork, Sperber. Then we explain a generalization of this story for theta connections. Theta connections are certain rigid connections over P1 minus two points, related to epipelagic representations under the geometric Langlands correspondence. As an application, we verify a conjecture of Reeder-Yu on the epipelagic Langlands parameters under some technical conditions. The talk is based on my joint work with Xinwen Zhu and a work in progress with Lingfei Yi. 
 
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Motivated by the study of spontaneously broken non-invertible symmetries, I will examine the connection between N=1* gauge theories and elliptic Calogero-Moser (CM) integrable systems, with a focus on the well-studied type A case. N=1* gauge theories are mass-deformations of N=4 super Yang-Mills theories, while elliptic Calogero-Moser systems describe integrable systems of particles on a torus associated with compact or complex simple Lie algebras. A puzzling feature of this correspondence is the fact that there seems to be a one-to-one mapping between the isolated extrema of CM systems and the massive vacua of N=1* theories on R4. These extrema, however, should rather be in one-to-one correspondence with the massive vacua of N=1∗ theories compactified on a circle, which are typically more numerous. I will explain how this apparent discrepancy is resolved by distinguishing global variants of N=1* theories, leading to an association of CM systems with global variants of Lie groups rather than Lie algebras.
 

Séminaire de géométrie arithmétique
The Brauer group ${rm Br}(X)$ of an algebraic variety $X$ is defined as the group of Azumaya algebras on $X$ up to Morita equivalence. There is an injective map (the Brauer map) ${rm Br}(X) hookrightarrow {rm H}^2_{mbox{ét}} (X,{mathbb G}_m)$. Understanding the image of this map constitutes the so-called Brauer problem.Toën introduced the notion of derived Azumaya algebra, later also developed by Lurie. Derived Azumaya algebras modulo Morita equivalence form the derived Brauer group dBr(X), which contains Br(X) and admits a map $phi : {rm Br}(X) hookrightarrow {rm H}^2_{mbox{ét}} (X,{mathbb G}_m)$ extending the classical Brauer map. Unlike that, however, $phi$ is an isomorphism, and thus offers a natural way to describe those cohomology classes not contained in the image of the Brauer map.With Michele Pernice (KTH Stockholm) we gave a more concrete description of $phi$ and its inverse, by using the interpretation of ${rm H}^2_{mbox{ét}} (X,{mathbb G}_m)$ via ${mathbb G}_m$-gerbes and by implementing the notion of twisted sheaves in the derived setting.I will explain this result and give some perspectives on ongoing work regarding the interaction of the derived Brauer group with Beilinson’s theory of adèles, in the case of a curve.
 
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Séminaire Amplitudes et Gravitation sur l’Yvette (IHES/IPhT)
The hierarchical limit of the three-body problem is relevant to many systems in astrophysics and provides a regime where solutions to the notorious three-body problem can be obtained using techniques from scattering amplitudes and effective field theory. In this talk, I will describe how to model the dynamics of hierarchical triples, which are three-body systems composed of two bodies separated by a distance $r$ and a third body a distance $rho$ away, with $r ll rho$. I will show the application of the method of regions for evaluating Fourier transform integrals by systematically expanding in the small ratio $r/rho$. With this technique, I will present our results for the conservative three-body potential at $mathcal{O}(G^2)$ at leading and next-to-leading order in $r/rho$. Our results are exact in velocity, and can be used in analyses involving both bound and unbound hierarchical triples in astrophysical systems.
 
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I present the results of a new numerical conformal bootstrap study of the 3d Ising CFT involving mixed correlator constraints of the operators σ, ε, and the stress tensor T. These constraints produce new precise determinations of the scaling dimensions (∆σ, ∆ϵ) = (0.518148806(24), 1.41262528(29)) as well as the OPE coefficients involving σ, ε, T. In addition, with the subset of the system only involving the stress tensor, we find universal bounds (i.e. which apply to all local unitary CFTs in 3d) on similar data, as well as hints of universal structures in the space of CFT data. I will also present the variety of methodological and technical challenges that this work entailed which resulted in several improvements and upgrades to the algorithms and software of the numerical bootstrap. This talk is based on 2411.15300 of the same title as well as ongoing work, done in collaboration with Cyuan-Han Chang, Vasiliy Dommes, Alexandre Homrich, Petr Kravchuk, Aike Liu, Matthew S. Mitchell, David Poland, and David Simmons-Duffin.
 
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The space of finitely additive measures on compact convex bodies (also called convex valuations) is an infinite-dimensional vector space that shares many properties with the cohomology algebra of a compact Kaehler manifold. In particular it is a graded algebra that satisfies a version of Poincaré duality. In a recent work with Jan Kotrbatý (Prague) and Thomas Wannerer (Jena) we prove a version of the mixed hard Lefschetz theorem and mixed Hodge-Riemann relations. The latter can be translated into new higher-order versions of the famous Alexandrov-Fenchel inequality for mixed volumes. The proof combines techniques from differential geometry and functional analysis.
 
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Séminaire Laurent Schwartz — EDP et applications
 

Séminaire Laurent Schwartz — EDP et applications
 

Séminaire Laurent Schwartz — EDP et applications
 

The perturbative expansion of quantum field theory associates numbers (or functions) to combinatorial graphs. These Feynman integrals are often transcendental and hard to evaluate. I will review various combinatorial invariants of graphs that behave similar to these integrals. In particular, I will explain a relation between spanning tree partitions and circuit partitions. It allows for efficient counting of these partitions, producing for every graph an integer sequence that determines the Feynman integral, conjecturally through an Apery-like limit. This is joint work with Francis Brown and Karen Yeats.
 
(based on https://arxiv.org/abs/2304.05299 and ongoing work)
 
 
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Séminaire Amplitudes et Gravitation sur l’Yvette (IHES/IPhT)
I will present work in progress on asymptotic detector operators and their measurements in perturbative quantum gravity in asymptotically-flat spacetimes. The simplest example is the energy detector, which in gravity detects outgoing graviton radiation and records its energy. I will review more general detector operators in QFT, defined as light-ray operators, and construct their analogs in gravitational theories. Then I will review how to compute event shapes — correlation functions of detector operators in suitable states — in perturbation theory, repackaged in a modern language. Utilizing this, I will present the first computation of an energy-energy correlator (EEC) in perturbative quantum gravity, to leading nontrivial order in the gravitational coupling. One can think of such observables as capturing a universal part of quantum correlations between idealized LIGO detectors — as unlikely as it may be to be measured in the real world. As a bonus, I will present some results on the two-point correlators of detectors measuring powers of energy and mention work towards the three-point energy correlator (EEEC) in gravity. Along the way, I will comment on the « space of asymptotic detector operators » in a gravitational theory.
 
 
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