Publications

2023

• On the dynamical Manin-Mumford conjecture for plane polynomial maps
  
  • Dujardin Romain
  • Favre Charles
  • Ruggiero Matteo
  , 2023. We prove the dynamical Manin-Mumford conjecture for regular polynomial maps of A^2 and irreducible curves avoiding super-attracting orbits at infinity, over any field of characteristic 0.
• Partial regularity in time for the space-homogeneous Boltzmann equation with very soft potentials
  
  • Golse François
  • Imbert Cyril
  • Silvestre Luis
  , 2023. We prove that the set of singular times for weak solutions of the homogeneous Boltzmann equation with very soft potentials constructed as in Villani (1998) has Hausdorff dimension at most $\frac{|\gamma+2s|}{2s}$ with $\gamma \in [-4s,-2s)$ and $s \in (0,1)$.
• Remarks on rigid irreducible meromorphic connections on the projective line
  
  • Sabbah Claude
  Documenta Mathematica, Universität Bielefeld, 2023, 28 (6), pp.1473-1492. (10.4171/dm/937)
  DOI : 10.4171/dm/937
• On composition of torsors
  
  • Florence Mathieu
  • Arteche Giancarlo Lucchini
  • Izquierdo Diego
  International Mathematics Research Notices, Oxford University Press (OUP), 2023. Let K be a field, let X be a connected smooth K-scheme and let G, H be two smooth connected K-group schemes. Given Y → X a G-torsor and Z → Y an H-torsor, we study whether one can find an extension E of G by H so that the composite Z → X is an E-torsor. We give both positive and negative results, depending on the nature of the groups G and H. (10.1093/imrn/rnad068)
  DOI : 10.1093/imrn/rnad068
• Non-linear bi-algebraic curves and surfaces in moduli spaces of Abelian differentials
  
  • Deroin Bertrand
  • Matheus Carlos
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2023, 361 (G10), pp.1691-1698. (10.5802/crmath.529)
  DOI : 10.5802/crmath.529
• On the self-similar stability of the parabolic-parabolic Keller-Segel equation
  
  • Borges Frank Alvarez
  • Carrapatoso Kleber
  • Mischler Stéphane
  , 2023. We consider the parabolic-parabolic Keller-Segel equation in the plane and prove the nonlinear exponential stability of the self-similar profile in a quasi parabolic-elliptic regime. We first perform a perturbation argument in order to obtain exponential stability for the semigroup associated to part of the first component of the linearized operator, by exploiting the exponential stability of the linearized operator for the parabolic-elliptic Keller-Segel equation. We finally employ a purely semigroup analysis to prove linear, and then nonlinear, exponential stability of the system in appropriated functional spaces with polynomial weights.
• The γ -support as a micro-support
  
  • Asano Tomohiro
  • Guillermou Stéphane
  • Humilière Vincent
  • Ike Yuichi
  • Viterbo Claude
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2023, 361 (G8), pp.1333-1340. We prove that for any element L in the completion of the space of smooth compact exact Lagrangian submanifolds of a cotangent bundle equipped with the spectral distance, the γ -support of L coincides with the reduced micro-support of its sheaf quantization. As an application, we give a characterization of the Vichery subdifferential in terms of γ -support. (10.5802/crmath.499)
  DOI : 10.5802/crmath.499
• Topological entropy of a rational map over a complete metrized field
  
  • Favre Charles
  • Truong Tuyen Trung
  • Xie Junyi
  , 2022. We prove that the topological entropy of any dominant rational self-map of a projective variety defined over a complete non-Archimedean field is bounded from above by the maximum of its dynamical degrees, thereby extending a theorem of Gromov and Dinh-Sibony from the complex to the non-Archimedean setting. We proceed by proving that any regular self-map which admits a regular extension to a projective model defined over the valuation ring has necessarily zero entropy. To this end we introduce the e-reduction of a Berkovich analytic space, a notion of independent interest. (10.48550/arXiv.2208.00668)
  DOI : 10.48550/arXiv.2208.00668
• Families of automorphisms of abelian varieties
  
  • Favre Charles
  • Kuznetsova Alexandra
  , 2023. We consider some algebraic aspects of the dynamics of an automorphism on a family of polarized abelian varieties parameterized by the complex unit disk. When the action on the cohomology of the generic fiber has no cyclotomic factor, we prove that such a map can be made regular only if the family of abelian varieties does not degenerate. As a contrast, we show that families of translations are always regularizable. We further describe the closure of the orbits of such maps, inspired by results of Cantat and Amerik-Verbitsky. (10.48550/arXiv.2309.13730)
  DOI : 10.48550/arXiv.2309.13730
• About a Family of ALF Instantons with Conical Singularities
  
  • Biquard Olivier
  • Gauduchon Paul
  Symmetry, Integrability and Geometry : Methods and Applications, National Academy of Science of Ukraine, 2023, 19, pp.079. We apply the techniques developed in our previous article to describe some interesting families of ALF gravitational instantons with conical singularities. In particular, we completely understand the 5-dimensional family of Chen-Teo metrics and prove that only 4-dimensional subfamilies can be smoothly compactified so that the metric has conical singularities. (10.3842/SIGMA.2023.079)
  DOI : 10.3842/SIGMA.2023.079
• Complex embeddings, Toeplitz operators and transitivity of optimal holomorphic extensions
  
  • Finski Siarhei
  , 2022. In a setting of a complex manifold with a fixed positive line bundle and a submanifold, we consider the optimal Ohsawa-Takegoshi extension operator, sending a holomorphic section of the line bundle on the submanifold to the holomorphic extension of it on the ambient manifold with the minimal $L^2$-norm. We show that for a tower of submanifolds in the semiclassical setting, i.e. when we consider a large tensor power of the line bundle, the extension operators satisfy transitivity property modulo some small defect, which can be expressed through Toeplitz type operators. We calculate the first significant term in the asymptotic expansion of this "transitivity defect". As a byproduct, we deduce the composition rules for Toeplitz type operators, the extension and restriction operators, and calculate the second term in the asymptotic expansion of the optimal constant in the semi-classical version of Ohsawa-Takegoshi extension theorem. (10.48550/arXiv.2201.04102)
  DOI : 10.48550/arXiv.2201.04102
• On the metric structure of section ring
  
  • Finski Siarhei
  , 2022. The main goal of this article is to study for a projective manifold and an ample line bundle over it the relation between metric and algebraic structures on the associated section ring. More precisely, we prove that once the kernel is factored out, the multiplication operator of the section ring becomes an approximate isometry (up to normalization) with respect to the $L^2$-norms and the induced Hermitian tensor product norm. We also show that the analogous result holds for the $L^1$ and $L^{\infty}$-norms if instead of the Hermitian tensor product norm, we consider the projective and injective tensor norms induced by $L^1$ and $L^{\infty}$-norms respectively. Then we show that $L^2$-norms associated with continuous plurisubharmonic metrics are actually characterized by the multiplicativity properties of this type. Using this, we refine the theorem of Phong-Sturm about quantization of Mabuchi geodesics from the weaker level of Fubini-Study convergence to the stronger level of norm equivalences.
• Submultiplicative norms and filtrations on section rings
  
  • Finski Siarhei
  , 2022. We show that submultiplicative norms on section rings of polarised projective manifolds are asymptotically equivalent to sup-norms associated with metrics on the polarisation. As an application, we establish that over canonically polarised manifolds, convex hull of Narasimhan-Simha pseudonorm over pluricanonical sections is asymptotically equivalent to the sup-norm associated with the supercanonical metric of Tsuji, refining a result of Berman-Demailly. As another application, we deduce that the jumping measures associated with bounded submultiplicative filtrations on section rings converge to the spectral measures of their Bergman geodesic rays, generalizing previous results of Witt Nyström and Hisamoto. We show also that the latter measures can be described using pluripotential theory. More precisely, we establish that Bergman geodesic rays are maximal, i.e. they can be constructed through plurisubharmonic envelopes. As a final application, for non-continuous metrics on ample line bundles over projective manifolds, we prove that a weak version of semiclassical holomorphic extension theorem holds for generic submanifolds. This means that up to a negligible portion, the totality of holomorphic sections over generic submanifolds extends in an effective way to the ambient manifold. As an unexpected byproduct, we show that injective and projective tensor norms on symmetric algebras of finitely dimensional complex normed vector spaces are asymptotically equivalent. (10.48550/arXiv.2210.03039)
  DOI : 10.48550/arXiv.2210.03039
• Hydrodynamic limit for the non-cutoff Boltzmann equation
  
  • Cao Chuqi
  • Carrapatoso Kleber
  , 2023. This work deals with the non-cutoff Boltzmann equation for all type of potentials, in both the torus $\mathbf{T}^3$ and in the whole space $\mathbf{R}^3$, under the incompressible Navier-Stokes scaling. We first establish the well-posedness and decay of global mild solutions to this rescaled Boltzmann equation in a perturbative framework, that is for solutions close to the Maxwellian, obtaining in particular integrated-in-time regularization estimates. We then combine these estimates with spectral-type estimates in order to obtain the strong convergence of solutions to the non-cutoff Boltzmannn equation towards the incompressible Navier-Stokes-Fourier system.
• The Physisorbate-Layer Problem Arising in Kinetic Theory of Gas-Surface Interaction
  
  • Aoki Kazuo
  • Giovangigli Vincent
  • Golse François
  • Kosuge Shingo
  , 2023. A half-space problem of a linear kinetic equation for gas molecules physisorbed close to a solid surface, relevant to a kinetic model of gas-surface interactions and derived by Aoki et al. (K. Aoki et al., in: Phys. Rev. E 106:035306, 2022), is considered. The equation contains a confinement potential in the vicinity of the solid surface and an interaction term between gas molecules and phonons. It is proved that a unique solution exists when the incoming molecular flux is specified at infinity. This validates the natural observation that the half-space problem serves as the boundary condition for the Boltzmann equation. It is also proved that the sequence of approximate solutions used for the existence proof converges exponentially fast. In addition, numerical results showing the details of the solution to the half-space problem are presented.
• Fixed-point statistics from spectral measures on tensor envelope categories
  
  • Forey Arthur
  • Fresán Javier
  • Kowalski Emmanuel
  , 2023. We prove some old and new convergence statements for fixed-points statistics using tensor envelope categories, such as the Deligne--Knop category of representations of the "symmetric group" $S_t$ for an indeterminate~$t$. We also discuss some arithmetic speculations related to Chebotarev's density theorem.
• Quantum Optimal Transport: Quantum Couplings and Many-Body Problems
  
  • Golse François
  , 2023. This text is a set of lecture notes for a 4.5-hour course given at the Erd\H os Center (Rényi Institute, Budapest) during the Summer School ``Optimal Transport on Quantum Structures'' (September 19th-23rd, 2023). Lecture I introduces the quantum analogue of the Wasserstein distance of exponent 2 defined in [F. Golse, C. Mouhot, T. Paul: Comm. Math. Phys. 343 (2016), 165-205], and in [F. Golse, T. Paul: Arch. Ration. Mech. Anal. 223 (2017) 57-94]. Lecture II presents various applications of this quantum analogue of the Wasserstein distance of exponent 2, while Lecture III discusses several of its most important properties, such as the triangle inequality, and the Kantorovich duality in the quantum setting, together with some of their implications.
• On ergodic properties of some flows on moduli spaces of flat and hyperbolic surfaces and their uses on counting problems
  
  • Bonnafoux Etienne
  , 2023. This thesis presents two results combining tools from ergodic theory and geometry, hyperbolic for the first result and flat for the second. Motivated by Mirzakhani’s theorem giving a polynomial asymptotics for the number of simple and closed geodesics on a hyperbolic surface, we wanted to better understand the behavior of the earthquake flow. Indeed, it is its ergodicity that Mirzakhani uses for her theorem. The idea is that a knowledge of a mixing rate for the earthquake flow, for a good class of observables, would allow to refine Mirzakhani’s counting, with an error term for example. Our first result is a step in this direction. It bounds the possible mixing rate of the earthquake to a polynomial one whose degree depends on the surface topology. To know precisely its mixing speed, an idea would be to look at the action of the special linear group of order 2, on flat surfaces. Indeed, there is a link between the earthquake flow and the flow of the unipotent subgroup called horocyclic flow. In this perspective we interested ourselves in refining a result of Athreya, Fairchild and Masur, giving the quadratic count of holonomy vectors of pair of saddle connection whose virtual area is bounded by a given constant for almost any flat surface with respect to a natural measure called Masur-Veech measure. One of the key points of their result is an ergodic theorem due to Nevo. We have deepened this result by giving an error term. Finally we will discuss a lemma which allow to extend the result to a larger family of measures.
• Mathematical analysis of some fluid-kinetic systems of equations
  
  • Ertzbischoff Lucas
  , 2023. This thesis delves into the mathematical analysis of fluid-kinetic systems, which describe the evolution of a suspension of particles in an ambient fluid. In this framework, a kinetic equation is coupled with the standard equations of fluid mechanics.The focus of Chapters 2 and 3 is the long-time behaviour of the Vlasov-Navier-Stokes equations in a domain, with absorption boundary condition for the particles. In Chapter 2, we analyse the competition between concentration in velocity and absorption in a bounded domain. We show that the particle distribution function has a monokinetic behaviour in velocity, and exhibit a wide variety of scenarios for the spatial asymptotic profile.In Chapter 3, we consider the half-space case, taking into account the action of the gravity force on the particles. The stability of the trivial solution for the system is explored and provenby combining both the absorption at the boundary and the gravity effects. Our approach is based on time-decay estimates for all moments in velocity of the distribution function, obtained by introducing an appropriate geometric control condition.Chapter 4 builds on the previous ideas to study a hydrodynamic limit of the Vlasov-Navier-Stokes equations with gravity, in a high-friction regime. We obtain the global in time derivation of a Boussinesq-Navier-Stokes type system.Chapter 5 is dedicated to the mathematical study of a thick spray system, which is a singular coupling between a kinetic equation and the compressible fluid equations. In the case of a viscous fluid, we prove the local in time strong well-posedness of the equations with Sobolev regularity, for initial data satisfying a Penrose stability condition. This is the first rigorous construction of solutions to this type of system, in the spirit of some recent works on singular Vlasov equations
• Arithmetic subgroups of Chevalley group schemes over function fields II: Conjugacy classes of maximal unipotent subgroups
  
  • Bravo Claudio
  • Loisel Benoit
  , 2023. Let $\mathcal{C}$ be a smooth, projective, geometrically integral curve defined over a perfect field $\mathbb{F}$. Let $k=\mathbb{F}(\mathcal{C})$ be the function field of $\mathcal{C}$. Let $\mathbf{G}$ be a split simply connected semisimple $\mathbb{Z}$-group scheme. Let $\mathcal{S}$ be a finite set of places of $\mathcal{C}$. In this paper, we investigate on the conjugacy classes of maximal unipotents subgroups of $\mathcal{S}$-arithmetic subgroups. These are parameterized thanks to the Picard group of $\mathcal{O}_{\mathcal{S}}$ and the rank of $\mathbf{G}$. Furthermore, these maximal unipotent subgroups can be realized as the unipotent part of natural stabilizer, that are the stabilizers of sectors of the associated Bruhat-Tits building. We decompose these natural stabilizers in terms of their diagonalisable part and unipotent part, and we precise the group structure of the diagonalisable part.
• A Duality-Based Proof of the Triangle Inequality for the Wasserstein Distances
  
  • Golse François
  , 2023. This short note gives a proof of the triangle inequality based on the Kantorovich duality formula for the Wasserstein distances of finite exponent p≥1 in the case of a general Polish space. In particular it avoids the ``glueing of couplings'' procedure used in most textbooks on optimal transport.
• On the large scale geometry of semisimple groups : vanishing and non-vanishing of L^p-cohomology of Archimedean and non-Archimedean groups
  
  • López Neumann Antonio
  , 2023. This thesis deals with large scale geometric invariants in Lie theory. More precisely, we study an invariant of cohomological nature called L^p-cohomology (where p>1) that gives quasi-isometry invariants in different settings and generalizes L^2-cohomology. We are interested in computing L^p-cohomology for semisimple groups over local fields (both Archimedean and non-Archimedean) and different types of buildings. The manuscript is divided into four chapters. The first introduces the concepts on Lie theory, group cohomology and L^p-cohomology that will be needed in subsequent chapters. The second reproduces the preprint "Vanishing of the second L^p-cohomology group for most higher rank semisimple Lie groups". Its main result is vanishing of L^p-cohomology in degree 2 for most semisimple groups over local fields of split rank at least 3. The third reproduces the preprint "Top degree l^p-homology and conformal dimension of buildings". In it, we study top degree ell^p-homology of buildings, its relation with other invariants such as virtual cohomological dimension of Coxeter groups and conformal dimension of Gromov-hyperbolic buildings. The fourth reproduces the article "Finitely presented simple groups and measure equivalence", where, using L^2-Betti numbers of groups acting on products of buildings, we exhibit a first family of finitely presented simple groups that lie in infinitely many measure equivalence classes.
• Relative regular Riemann–Hilbert correspondence II
  
  • Fiorot Luisa
  • Fernandes Teresa Monteiro
  • Sabbah Claude
  Compositio Mathematica, Foundation Compositio Mathematica / Cambridge University Press, Cambridge, 2023, 159 (7), pp.1413-1465. We develop the theory of relative regular holonomic $\mathcal {D}$ -modules with a smooth complex manifold $S$ of arbitrary dimension as parameter space, together with their main functorial properties. In particular, we establish in this general setting the relative Riemann–Hilbert correspondence proved in a previous work in the one-dimensional case. (10.1112/S0010437X23007224)
  DOI : 10.1112/S0010437X23007224
• Hodge properties of Airy moments
  
  • Sabbah Claude
  • Yu Jeng-Daw
  Tunisian Journal of Mathematics, Mathematical Science Publishers, 2023, 5 (2), pp.215-271. (10.2140/tunis.2023.5.215)
  DOI : 10.2140/tunis.2023.5.215
• 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.