Publications

2023

• Validité de la théorie cinétique des gaz: au-delà de l'équation de Boltzmann
  
  • Golse François
  , 2023. L'obtention d'une justification rigoureuse de la théorie cinétique des gaz à partir du principe fondamental de la dynamique, dû à Newton, pour un grand nombre de sphères identiques interagissant par collisions binaires élastiques, est un problème formulé par Hilbert en 1900 (6ème problème). En 1975, Lanford a démontré la validité de l'équation de Boltzmann sur un intervalle de temps très court, de l'ordre d'une fraction du laps de temps moyen entre deux collisions successives subies par une même particule. Ce résultat de Lanford peut être interprété comme une sorte de "loi des grands nombres" lorsque le nombre de particules tend vers l'infini. Ce point de vue pose plusieurs questions. D'abord, le c\oe ur de l'argument utilisé par Boltzmann pour aboutir à l'équation portant son nom est l'hypothèse que deux particules sur le point d'entrer en collision sont presque indépendantes statistiquement. Ceci suggère d'examiner la validité de cette hypothèse en étudiant la dynamique des corrélations entre particules. D'autre part, l'interprétation de l'équation de Boltzmann comme loi des grands nombres conduit à étudier précisément les fluctuations de la mesure empirique dans l'espace des phases autour de sa moyenne (dont l'évolution est décrite par l'équation de Boltzmann). Une série d'articles récents de T. Bodineau, I. Gallagher, L. Saint-Raymond et S. Simonella répond à ces diverses questions et permet d'aller au-delà de l'équation de Boltzmann dans la compréhension de la théorie cinétique des gaz.
• Pickl's Proof of the Quantum Mean-Field Limit and Quantum Klimontovich Solutions
  
  • Ben Porat Immanuel
  • Golse François
  , 2023. This paper discusses the mean-field limit for the quantum dynamics of N identical bosons in the three-dimensional Euclidean space interacting via a binary potential with Coulomb type singularity. Our approach is based on the theory of quantum Klimontovich solutions defined in [F. Golse, T. Paul, Commun. Math. Phys. 369 (2019), 1021–1053]. Our first main result is a definition of the interaction nonlinearity in the equation governing the dynamics of quantum Klimontovich solutions for a class of interaction potentials slightly less general than those considered in [T. Kato, Trans. Amer. Math. Soc. 70 (1951), 195–211]. Our second main result is a new operator inequality satisfied by the quantum Klimontovich solution in the case of an interaction potential with Coulomb type singularity. When evaluated on an initial bosonic pure state, this operator inequality reduces to a Gronwall inequality for a functional introduced in [P. Pickl, Lett. Math. Phys. 97 (2011), 151–164], resulting in a convergence rate estimate for the quantum mean-field limit leading to the time-dependent Hartree equation.
• A construction of the polylogarithm motive
  
  • Dupont Clément
  • Fresán Javier
  , 2023. Classical polylogarithms give rise to a variation of mixed Hodge-Tate structures on the punctured projective line $S=\mathbb{P}^1\setminus \{0, 1, \infty\}$, which is an extension of the symmetric power of the Kummer variation by a trivial variation. By results of Beilinson-Deligne, Huber-Wildeshaus, and Ayoub, this polylogarithm variation has a lift to the category of mixed Tate motives over $S$, whose existence is proved by computing the corresponding space of extensions in both the motivic and the Hodge settings. In this paper, we construct the polylogarithm motive as an explicit relative cohomology motive, namely that of the complement of the hypersurface $\{1-zt_1\cdots t_n=0\}$ in affine space $\mathbb{A}^n_S$ relative to the union of the hyperplanes $\{t_i=0\}$ and $\{t_i=1\}$.
• Marked Length Spectral determination of analytic chaotic billiards with axial symmetries
  
  • de Simoi Jacopo
  • Kaloshin Vadim
  • Leguil Martin
  Inventiones Mathematicae, Springer Verlag, 2023, 233 (2), pp.829-901. We consider billiards obtained by removing from the plane finitely many strictly convex analytic obstacles satisfying the non-eclipse condition. The restriction of the dynamics to the set of non-escaping orbits is conjugated to a subshift, which provides a natural labeling of periodic orbits. We show that under suitable symmetry and genericity assumptions, the Marked Length Spectrum determines the geometry of the billiard table. (10.1007/s00222-023-01191-8)
  DOI : 10.1007/s00222-023-01191-8
• Quadratic relations between Bessel moments
  
  • Fresán Javier
  • Sabbah Claude
  • Yu Jeng-Daw
  Algebra & Number Theory, Mathematical Sciences Publishers, 2023, 17 (3), pp.541-602. Motivated by the computation of some Feynman amplitudes, Broadhurst and Roberts recently conjectured and checked numerically to high precision a set of remarkable quadratic relations between the Bessel moments ∫∞0I0(t)iK0(t)k−it2j−1dt(i,j=1,…,⌊(k−1)/2⌋), where k≥1 is a fixed integer and I0 and K0 denote the modified Bessel functions. We interpret these integrals and variants thereof as coefficients of the period pairing between middle de Rham cohomology and twisted homology of symmetric powers of the Kloosterman connection. Building on the general framework developed by Fresan, Sabbah and Yu (2020), this enables us to prove quadratic relations of the form suggested by Broadhurst and Roberts, which conjecturally comprise all algebraic relations between these numbers. We also make Deligne’s conjecture explicit, thus explaining many evaluations of critical values of L-functions of symmetric power moments of Kloosterman sums in terms of determinants of Bessel moments. (10.2140/ant.2023.17.541)
  DOI : 10.2140/ant.2023.17.541
• Symplectic Homogenization
  
  • Viterbo Claude
  , 2023. Let $H(q,p)$ be a Hamiltonian on $T^*T^n$. We show that the sequence $H_{k}(q,p)=H(kq,p)$ converges for the $\gamma$ topology defined by the author, to $\overline{H}(p)$. This is extended to the case where only some of the variables are homogenized, that is the sequence $H(kx,y,q,p)$ where the limit is of the type ${\overline H}(y,q,p)$ and thus yields an ``effective Hamiltonian''. We give here the proof of the convergence, and the first properties of the homogenization operator, and give some immediate consequences for solutions of Hamilton-Jacobi equations, construction of quasi-states, etc. We also prove that the function $\overline H$ coincides with Mather's $\alpha$ function which gives a new proof of its symplectic invariance proved by P. Bernard.
• Toric geometry and integral affine structures in non-archimedean mirror symmetry
  
  • Pille-Schneider Léonard
  • Mazzon Enrica
  , 2023.
• Non-cutoff Boltzmann equation with soft potentials in the whole space
  
  • Carrapatoso Kleber
  • Gervais Pierre
  , 2023. We prove the existence and uniqueness of global solutions to the Boltzmann equation with non-cutoff soft potentials in the whole space when the initial data is a small perturbation of a Maxwellian with polynomial decay in velocity. Our method is based in the decomposition of the desired solution into two parts: one with polynomial decay in velocity satisfying the Boltzmann equation with only a dissipative part of the linearized operator ; the other with Gaussian decay in velocity verifying the Boltzmann equation with a coupling term.
• On toric Hermitian ALF gravitational instantons
  
  • Biquard Olivier
  • Gauduchon Paul
  Communications in Mathematical Physics, Springer Verlag, 2023, 399, pp.389-422. We give a classification of toric, Hermitian, Ricci flat, ALF Riemannian metrics in dimension 4, including metrics with conical singularities. The only smooth examples are on one hand the hyperKaehler ALF metrics, on the other hand, the Kerr, Taub-NUT and Chen-Teo metrics. There are examples with conical singularities with infinitely many distinct topologies. We provide explicit formulas. (10.1007/s00220-022-04562-z)
  DOI : 10.1007/s00220-022-04562-z
• The Dirac-Klein-Gordon system in the strong coupling limit
  
  • Lampart Jonas
  • Le Treust Loïc
  • Rota Nodari Simona
  • Sabin Julien
  Annales Henri Lebesgue, UFR de Mathématiques - IRMAR, 2023, 6, pp.541-573. We study the Dirac equation coupled to scalar and vector Klein-Gordon fields in the limit of strong coupling and large masses of the fields. We prove convergence of the solutions to those of a cubic non-linear Dirac equation, given that the initial spinors coincide. This shows that in this parameter regime, which is relevant to the relativistic mean-field theory of nuclei, the retarded interaction is well approximated by an instantaneous, local self-interaction. We generalize this result to a many-body Dirac-Fock equation on the space of Hilbert-Schmidt operators. (10.5802/ahl.171)
  DOI : 10.5802/ahl.171
• On Lipschitz Normally Embedded Singularities
  
  • Fantini Lorenzo
  • Pichon Anne
  , 2023. Any subanalytic germ ($X,0$) ⊂ ($\mathbb{R}^n,0$) is equipped with two natural metrics: its outer metric, induced by the standard Euclidean metric of the ambient space, and its inner metric, which is defined by measuring the shortest length of paths on the germ ($X,0$). The germs for which these two metrics are equivalent up to a bilipschitz homeomorphism, which are called Lipschitz Normally Embedded, have attracted a lot of interest in the last decade. In this survey we discuss many general facts about Lipschitz Normally Embedded singularities, before moving our focus to some recent developments on criteria, examples, and properties of Lipschitz Normally Embedded complex surfaces. We conclude the manuscript with a list of open questions which we believe to be worth of interest. (10.1007/978-3-031-31925-9_10)
  DOI : 10.1007/978-3-031-31925-9_10
• Hypocoercivity for kinetic linear equations in bounded domains with general Maxwell boundary condition
  
  • Bernou Armand
  • Carrapatoso Kleber
  • Mischler Stéphane
  • Tristani Isabelle
  Annales de l'Institut Henri Poincaré (C), Analyse non linéaire (Nonlinear Analysis), EMS, 2023, 40 (2), pp.287–338. We establish the convergence to the equilibrium for various linear collisional kinetic equations (including linearized Boltzmann and Landau equations) with physical local conservation laws in bounded domains with general Maxwell boundary condition. Our proof consists in establishing an hypocoercivity result for the associated operator, in other words, we exhibit a convenient Hilbert norm for which the associated operator is coercive in the orthogonal of the global conservation laws. Our approach allows us to treat general domains with all type of boundary conditions in a unified framework. In particular, our result includes the case of vanishing accommodation coefficient and thus the specific case of the specular reflection boundary condition. (10.4171/AIHPC/44)
  DOI : 10.4171/AIHPC/44
• Geometry at infinity of the space of Kähler potentials and asymptotic properties of filtrations
  
  • Finski Siarhei
  Journal für die reine und angewandte Mathematik, Walter de Gruyter, 2023. The main goal of this article is to describe a relation between the asymptotic properties of filtrations on section rings and the geometry at infinity of the space of Kähler potentials. More precisely, for a polarized projective manifold and an ample test configuration, Phong and Sturm associated a geodesic ray of plurisubharmonic metrics on the polarizing line bundle. On the other hand, for the same data, Witt Nyström associated a filtration on the section ring of the polarized manifold. In this article, we establish a folklore conjecture that the pluripotential chordal distance between the geodesic rays associated with two ample test configurations coincides with the spectral distance between the associated filtrations on the section ring. This gives an algebraic description of the boundary at infinity of the space of positive metrics, viewed – as it is usually done for spaces of negative curvature – through geodesic rays. (10.1515/crelle-2024-0076)
  DOI : 10.1515/crelle-2024-0076
• $\Lambda$-buildings associated to quasi-split groups over $\Lambda$-valued fields
  
  • Hébert Auguste
  • Izquierdo Diego
  • Loisel Benoit
  Münster Journal of Mathematics, Münster Mathematical Institutes, Universität Münster, 2023, 16 2. Let $\mathbf{G}$ be a quasi-split reductive group and $\mathbb{K}$ be a Henselian field equipped with a valuation $\omega:\mathbb{K}^{\times}\rightarrow \Lambda$, where $\Lambda$ is a totally ordered abelian group. In 1972, Bruhat and Tits constructed a building on which the group $\mathbf{G}(\mathbb{K})$ acts provided that $\Lambda$ is a subgroup of $\mathbb{R}$. In this paper, we deal with the general case where there are no assumptions on $\Lambda$ and we construct a set on which $\mathbf{G}(\mathbb{K})$ acts. We then prove that it is a $\Lambda$-building, in the sense of Bennett. (10.48550/arXiv.2001.01542)
  DOI : 10.48550/arXiv.2001.01542