Publications

2024

• Stratification d'Ekedahl-Oort pour les modèles de Pappas-Rapoport des variétés de Shimura
  
  • Berger Diego
  , 2024. Dans cette thèse nous étudions la géométrie de la réduction de certainesvariétés de Shimura modulo un nombre premier p. Plus précisément on considèrela réduction modulo p des modèles entiers des variétés de Shimura de type PELconstruits par Pappas et Rapoport. Dans le cas d’une donnée PEL de type Hilbert,on montre que la stratification induite par le polygone de Hodge est une bonnestratification (l’adhérence d’une strate est une union disjointe de strates). Ensuitenous calculons les G-orbites de la fibre spéciale du modèle local de Pappas-Raporport dans le cas Hilbert, où G est le groupe associé à la donnée PEL.Ces orbites induisent une bonne stratification de la fibre spéciale de la variétéde Shimura, que l’on appelle stratification de Kottwitz-Rapoport (analogue à lastratification de Kottwitz-Rapoport des modèles entiers de Kottwitz). Dans untravail récent, Xu Shen et Yuqiang Zheng ont défini une stratification d’Ekedahl-Oort des modèles entiers de Pappas-Rapoport. Dans le cas Hilbert nous montronsque « l’intersection » de leur stratification avec la straitification de Kottwitz-Rapoport est une bonne stratification.Dans la seconde partie de cette thèse nous nous intéressons aux modèleslocaux dans le contexte de la théorie de Hodge p-adique. Nous définissons unplongement en niveau entier des modèles locaux de Pappas-Rapoport dans unecertaine Grassmannienne affine de type Beilinson-Drinfeld, analogue au plongementdéfinit Scholze et Weinstein pour les modèles entiers de Kottwitz.
• The Kinetic Fokker-Planck equation in a domain: Ultracontractivity, hypocoercivity and long-time asymptotic behavior
  
  • Carrapatoso Kleber
  • Mischler Stéphane
  , 2024. We consider the Kinetic Fokker-Planck (FKP) equation in a domain with Maxwell reflection condition on the boundary. We establish the ultracontractivity of the associated semigroup and the hypocoercivity of the associated operator. We deduce the convergence with constructive rate of the solution to the KFP equation towards the stationary state with same mass as the initial datum.
• TRILINEAR COMPENSATED COMPACTNESS AND BURNETT'S CONJECTURE IN GENERAL RELATIVITY
  
  • Huneau Cécile
  • Luk Jonathan
  Annales Scientifiques de l'École Normale Supérieure, Gauthier-Villars ; Société mathématique de France, 2024. Consider a sequence of C 4 Lorentzian metrics {hn} +∞ n=1 on a manifold M satisfying the Einstein vacuum equation Ric(hn) = 0. Suppose there exists a smooth Lorentzian metric h 0 on M such that hn → h 0 uniformly on compact sets. Assume also that on any compact set K ⊂ M, there is a decreasing sequence of positive numbers λn → 0 such that ∂ α (hn − h 0) L ∞ (K) λ 1−|α| n , |α| ≥ 4. It is well-known that h 0 , which represents a "high-frequency limit", is not necessarily a solution to the Einstein vacuum equation. Nevertheless, Burnett conjectured that h 0 must be isometric to a solution to the Einstein-massless Vlasov system. In this paper, we prove Burnett's conjecture assuming that {hn} +∞ n=1 and h 0 in addition admit a U(1) symmetry and obey an elliptic gauge condition. The proof uses microlocal defect measures-we identify an appropriately defined microlocal defect measure to be the Vlasov measure of the limit spacetime. In order to show that this measure indeed obeys the Vlasov equation, we need some special cancellations which rely on the precise structure of the Einstein equations. These cancellations are related to a new "trilinear compensated compactness" phenomenon for solutions to (semilinear) elliptic and (quasilinear) hyperbolic equations. (10.24033/asens.2577)
  DOI : 10.24033/asens.2577
• Sidon Sets in Algebraic Geometry
  
  • Forey Arthur
  • Fresán Javier
  • Kowalski Emmanuel
  International Mathematics Research Notices, Oxford University Press (OUP), 2024, Volume 2024 (Issue 8), pp.6400–6421. We report new examples of Sidon sets in abelian groups arising from generalized Jacobians of curves, and discuss some of their properties with respect to size and structure. (10.1093/imrn/rnad169)
  DOI : 10.1093/imrn/rnad169
• Lower bounds on fibered Yang-Mills functionals: generic nefness and semistability of direct images
  
  • Finski Siarhei
  , 2024. The main goal of this paper is to generalize a part of the relationship between mean curvature and Harder-Narasimhan filtrations of holomorphic vector bundles to arbitrary polarized fibrations. More precisely, for a polarized family of complex projective manifolds, we establish lower bounds on a fibered version of Yang-Mills functionals in terms of the Harder-Narasimhan slopes of direct image sheaves associated with high tensor powers of the polarization. We discuss the optimality of these lower bounds and, as an application, provide an analytic characterisation of a fibered version of generic nefness. As another application, we refine the existent obstructions for finding metrics with constant horizontal mean curvature. The study of the semiclassical limit of Hermitian Yang-Mills functionals lies at the heart of our approach.
• QUANTUM K-THEORY OF IG(2, 2n)
  
  • Benedetti Vladimiro
  • Perrin Nicolas
  • Xu Weihong
  , 2024. We prove that the Schubert structure constants of the quantum K-theory rings of symplectic Grassmannians of lines have signs that alternate with codimension and vanish for degrees at least 3. We also give closed formulas that characterize the multiplicative structure of these rings, including the Seidel representation and a Chevalley formula.
• Kodaira-Saito vanishing for the irregular Hodge filtration
  
  • Sabbah Claude
  , 2024. After making correct, and then improving, our definition of the category of irregular mixed Hodge modules thanks to Mochizuki's recent results arXiv:2108.03843, we show how these results allow us to obtain Kodaira-Saito-type vanishing theorems for the irregular Hodge filtration of irregular mixed Hodge modules.
• Singularities of functions: a global point of view
  
  • Sabbah Claude
  , 2024. This text surveys cohomological properties of pairs $(U,f)$ consisting of a smooth complex quasi-projective variety $U$ together with a regular function on~it. On the one hand, one tries to mimic the case of a germ of holomorphic function in its Milnor ball and, on the other hand, one takes advantage of the algebraicity of~$U$ and $f$ to apply technique of algebraic geometry, in particular Hodge theory. The monodromy properties are expressed by means of tools provided by the theory of linear differential equations, by mimicking the Stokes phenomenon. In the case of tame functions on smooth affine varieties, which is an algebraic analogue of that of a holomorphic function with an isolated critical point, the theory simplifies much and the formulation of the results are nicer. Examples of such tame functions are exhibited.
• Pd-catalyzed S -glycosylation of cysteine-containing peptides at room temperature
  
  • Shen Linhua
  • Le Bideau Franck
  • Chen Gong
  • Messaoudi Samir
  Organic Chemistry Frontiers, Royal Society of Chemistry, 2024, 11 (22), pp.6347-6352. This study reports the synthesis of thioglycopeptides via a Pd-catalyzed coupling of cysteine-containing peptides with iodoglycals. A variety of cysteine-containing peptides and other thiol nucleophiles including thiosugars and thiophenols were used. (10.1039/d4qo01389a)
  DOI : 10.1039/d4qo01389a
• Decay and absorption for the Vlasov-Navier-Stokes system with gravity in a half-space
  
  • Ertzbischoff Lucas
  Indiana University Mathematics Journal, Indiana University Mathematics Journal, 2024, 73 (1). This paper is devoted to the large time behavior of weak solutions to the three-dimensional Vlasov-Navier-Stokes system set on the half-space, with an external gravity force. This fluid-kinetic coupling arises in the modeling of sedimentation phenomena. We prove that the local density of the particles and the fluid velocity enjoy a convergence to 0 in large time and at a polynomial rate. In order to overcome the effect of the gravity, we rely on a fine analysis of the absorption phenomenon at the boundary. We obtain a family of decay estimates for the moments of the kinetic distribution, provided that the initial distribution function has a sufficient decay in the phase space. (10.1512/iumj.2024.73.9538)
  DOI : 10.1512/iumj.2024.73.9538