Publications

2024

• Local Conservation Laws and Entropy Inequality for Kinetic Models with Delocalized Collision Integrals
  
  • Charles Frédérique
  • Chen Zhe
  • Golse François
  , 2024. This article presents a common setting for the collision integrals $\mathrm{St}$ appearing in the kinetic theory of dense gases. It includes the collision integrals of the Enskog equation, of (a variant of) the Povzner equation, and of a model for soft sphere collisions proposed by Cercignani [Comm. Pure Appl. Math. \textbf{36} (1983), 479--494]. All these collision integrals are ``delocalized'', in the sense that they involve products of the distribution functions of gas molecules evaluated at positions whose distance is of the order of the molecular radius. Our first main result is to express these collision integrals as the divergence in $v$ of some mass current, where $v$ is the velocity variable, while $v_i\mathrm{St}$ and $|v|^2\mathrm{St}$ are expressed as the phase space divergence (i.e divergence in both position and velocity) of appropriate momentum and energy currents. This extends to the case of dense gases an earlier result by Villani [Math. Modelling Numer. Anal. M2AN \textbf{33} (1999), 209--227] in the case of the classical Boltzmann equation (where the collision integral is involves products of the distribution function of gas molecules evaluated at different velocities, but at the same position. Applications of this conservative formulation of delocalized collision integrals include the possibility of obtaining the local conservation laws of momentum and energy starting from this kinetic theory of denses gases. Similarly a local variant of the Boltzmann H Theorem, involving some kind of free energy instead of Boltzmann's H function, can be obtained in the form of an expression for the entropy production in terms of the phase space divergence of some phase space current, and of a nonpositive term.
• Nonlinear interpolation and the flow map for quasilinear equations
  
  • Alazard Thomas
  • Burq Nicolas
  • Ifrim Mihaela
  • Tataru Daniel
  • Zuily Claude
  , 2024. We prove an interpolation theorem for nonlinear functionals defined on scales of Banach spaces that generalize Besov spaces. It applies to functionals defined only locally, requiring only some weak Lipschitz conditions, extending those introduced by Lions and Peetre. Our analysis is self-contained and independent of any previous results about interpolation theory. It depends solely on the concepts of Friedrichs' mollifiers, seen through the formalism introduced by Hamilton, combined with the frequency envelopes introduced by Tao and used recently by two of the authors and others to study the Cauchy problem for various quasilinear evolutions in partial differential equations. Inspired by this latter work, our main application states that, for an abstract flow map of a quasilinear problem, both the continuity of the flow as a function of time and the continuity of the data to solution map follow automatically from the estimates that are usually proven when establishing the existence of solutions: propagation of regularity via tame a priori estimates for higher regularities and contraction for weaker norms.
• Paracomposition Operators and Paradifferential Reducibility
  
  • Alazard Thomas
  • Shao Chengyang
  , 2024. Reducibility methods, aiming to simplify systems by conjugating them to those with constant coefficients, are crucial for studying the existence of quasiperiodic solutions. In KAM theory for PDEs, these methods help address the invertibility of linearized operators that arise in a Nash-Moser/KAM type scheme. The goal of this paper is to prove paradifferential reducibility results, enabling the reduction of nonlinear equations themselves, rather than just their linearizations, to constant coefficient form, modulo smoothing terms. As an initial application, we demonstrate the existence of quasiperiodic solutions for certain hyperbolic systems. Despite the small denominator problem, our proof does not rely on traditional Nash-Moser/KAM-type schemes. To achieve this, we develop two key toolsets. The first focuses on the calculus of paracomposition operators introduced by Alinhac, interpreted as the flow map of a paraproduct vector field. We refine this approach to establish new estimates that precisely capture the dependence on the diffeomorphism in question. The second toolset addresses two classical reducibility problems: one for matrix differential operators and another for nearly parallel vector fields on the torus. We resolve these problems by paralinearizing the conjugacy equation and exploiting, at the paradifferential level, the specific algebraic structure of conjugacy problems, akin to Zehnder's approximate Nash-Moser approach.
• Tessellations of an affine apartment by affine weight polytopes
  
  • Hebert Auguste
  • Bravo Claudio
  • Izquierdo Diego
  • Loisel Benoit
  , 2024. Let A be a finite dimensional vector space and Φ be a finite root system in A. To this data is associated an affine poly-simplicial complex. Motivated by a forthcoming construction of connectified higher buildings, we study "affine weight polytopes" associated to these data. We prove that these polytopes tesselate A. We also prove a kind of "mixed" tessellation, involving the affine weight polytopes and the poly-simplical structure on A. (10.48550/arXiv.2411.10282)
  DOI : 10.48550/arXiv.2411.10282
• About Wess-Zumino-Witten equation and Harder-Narasimhan potentials
  
  • Finski Siarhei
  , 2024. For a polarized family of complex projective manifolds, we identify the algebraic obstructions that govern the existence of approximate solutions to the Wess-Zumino-Witten equation. When this is specialized to the fibration associated with a projectivization of a vector bundle, we recover a version of Kobayashi-Hitchin correspondence. More broadly, we demonstrate that a certain auxiliary Monge-Ampère type equation, generalizing the Wess-Zumino-Witten equation by taking into account the weighted Bergman kernel associated with the Harder-Narasimhan filtrations of direct image sheaves, admits approximate solutions over any polarized family. These approximate solutions are shown to be the closest counterparts to true solutions of the Wess-Zumino-Witten equation whenever the latter do not exist, as they minimize the associated Yang-Mills functional. As an application, in a fibered setting, we prove an asymptotic converse to the Andreotti-Grauert theorem conjectured by Demailly.
• On Harder-Narasimhan slopes of direct images
  
  • Finski Siarhei
  , 2024. For a polarized family of complex projective manifolds, we study the asymptotic distribution of Harder-Narasimhan slopes of direct image sheaves associated with high tensor powers of the polarization. We establish a theorem of Mehta-Ramanathan type, showing that this asymptotic distribution can be recovered from the analogous asymptotic distributions associated with base changes of the family over generic curves.
• 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.
• Toeplitz operators, submultiplicative filtrations and weighted Bergman kernels
  
  • Finski Siarhei
  , 2024. We demonstrate that the weight operator associated with a submultiplicative filtration on the section ring of a polarized complex projective manifold is a Toeplitz operator. We further analyze the asymptotics of the associated weighted Bergman kernel, presenting the local refinement of earlier results on the convergence of jumping measures for submultiplicative filtrations towards the pushforward measure defined by the corresponding geodesic ray.
• Generic vanishing for holonomic D-modules : a study via Cartier duality
  
  • Ribeiro Gabriel
  , 2024. Motivated by questions in analytic number theory and complex geometry, this thesis studies a generic vanishing theorem for the de Rham cohomology of holonomic D-modules on a commutative connected algebraic group G.We start by constructing an algebraic space G^flat that parametrizes multiplicative line bundles with flat connection on G, referred to as character sheaves. For abelian varieties, this provides a new construction of Simpson's moduli space of line bundles with integrable connection. Importantly, our construction also applies to non-proper groups G. The novelty of our approach lies in the systematic use of a stacky incarnation of Cartier duality, leading to the study of various extensions of abelian sheaves, which may be of independent interest.Examining the geometry of the moduli space G^flat, we identify a class of subspaces known as linear subspaces. Then, the generic vanishing theorem states that, for each holonomic D-module, there exists a finite union of translates of these linear subspaces such that the de Rham cohomology of twists by character sheaves in its complement is concentrated in degree zero.Using the generic vanishing theorem, we deduce that a certain quotient category of holonomic D-modules is tannakian. In other words, each such D-module amounts to a representation of some algebraic group. Notably, for unipotent groups or tori, we prove a comparison result identifying these algebraic groups respectively to a differential or difference Galois groups.
• Tropical linear series, combinatorial flag arrangements and applications to the study of Weierstrass points
  
  • Gierczak-Galle Lucas
  , 2024. We first introduce new combinatorial objects called “matricubes”, a natural generalization of matroids. In the same way that matroids provide a combinatorial axiomatization of hyperplane arrangements in a vector space, matricubes abstract arrangements of flags. As for matroids, we provide cryptomorphic definitions of matricubes in terms of rank function, flats, circuits, and independent sets. We provide precise connections between matricubes and permutation arrays, and propose a description of matricubes in terms of local matroids.We then use matricubes to develop a purely combinatorial theory of limit linear series on metric graphs. This is based as well on the formalism of slope structures, which constrains the slopes of tropical meromorphic functions. We show that combinatorial linear series arise naturally by tropicalizing linear series on algebraic curves. We explore their topological properties and develop tools to study them. We provide a full classification of combinatorial linear series of rank one, showing that they are in one-to-one correspondence with harmonic morphisms from the graph to metric trees. This entails a smoothing theorem.Finally, we study tropical Weierstrass points, which are analogues, on tropical curves, of ramification points of line bundles on algebraic curves. The tropical Weierstrass locus of a divisor on a metric graph can be infinite. Nevertheless, we associate intrinsic weights to each of its connected components. We prove that the total weight of the tropical Weierstrass locus depends only on the degree of the divisor, its rank, and the genus of the tropical curve. Furthermore, in the case the metric graph is the tropicalization of an algebraic curve, we show, using combinatorial linear series, that our weights count the number of algebraic Weierstrass points which are tropicalized to each connected component of the tropical locus.In each of these contributions, we discuss multiple examples and ask open questions leading to other perspectives of research.
• NON-ARCHIMEDEAN TECHNIQUES AND DYNAMICAL DEGENERATIONS
  
  • Favre Charles
  • Gong Chen
  , 2024. We develop non-Archimedean techniques to analyze the degeneration of a sequence of rational maps of the complex projective line. We provide an alternative to Luo's method which was based on ultralimits of the hyperbolic 3-space. We build hybrid spaces using Berkovich theory which enable us to prove the convergence of equilibrium measures, and to determine the asymptotics of Lyapunov exponents.
• Semiclassical Ohsawa-Takegoshi extension theorem and asymptotics of the orthogonal Bergman kernel
  
  • Finski Siarhei
  Journal of Differential Geometry, International Press, 2024, 128 (2), pp.639-721. In this paper, we study the asymptotics of Ohsawa-Takegoshi extension operator and orthogonal Bergman projector associated with high tensor powers of a positive line bundle. More precisely, for a fixed submanifold in a complex manifold, we consider the operator which associates to a given holomorphic section of a positive line bundle over the submanifold the holomorphic extension of it to the ambient manifold with the minimal $L^2$-norm. When the tensor power of the line bundle tends to infinity, we prove an exponential estimate for the Schwartz kernel of this extension operator, and show that it admits a full asymptotic expansion in powers of the line bundle. Similarly, we study the asymptotics of the orthogonal Bergman kernel associated to the projection onto the holomorphic sections orthogonal to those which vanish along the submanifold. All our results are stated in the setting of manifolds and embeddings of bounded geometry. (10.4310/jdg/1727712891)
  DOI : 10.4310/jdg/1727712891
• Birkhoff attractors of dissipative billiards
  
  • Bernardi Olga
  • Florio Anna
  • Leguil Martin
  Ergodic Theory and Dynamical Systems, Cambridge University Press (CUP), 2024, 45 (4), pp.989-1047. We study the dynamics of dissipative billiard maps within planar convex domains. Such maps have a global attractor. We are interested in the topological and dynamical complexity of the attractor, in terms both of the geometry of the billiard table and of the strength of the dissipation. We focus on the study of an invariant subset of the attractor, the so-called Birkhoff attractor. On the one hand, we show that for a generic convex table with "pinched" curvature, the Birkhoff attractor is a normally contracted manifold when the dissipation is strong. On the other hand, for a mild dissipation, we prove that generically the Birkhoff attractor is complicated, both from the topological and the dynamical point of view. (10.1017/etds.2024.68)
  DOI : 10.1017/etds.2024.68
• Gravitational Instantons, Weyl Curvature, and Conformally Kähler Geometry
  
  • Biquard Olivier
  • Gauduchon Paul
  • Lebrun Claude
  International Mathematics Research Notices, Oxford University Press (OUP), 2024, 2024 (20), pp.13295-13311. In a previous paper [7], the first two authors classified complete Ricci-flat ALF Riemannian 4-manifolds that are toric and Hermitian, but non-Kähler. In this article, we consider general Ricci-flat metrics on these spaces that are close to a given such gravitational instanton with respect to a norm that imposes reasonable fall-off conditions at infinity. We prove that any such Ricci-flat perturbation is necessarily Hermitian and carries a bounded, non-trivial Killing vector field. With mild additional hypotheses, we are then able to show that the new Ricci-flat metric must actually be one of the known gravitational instantons classified in [7]. (10.1093/imrn/rnae200)
  DOI : 10.1093/imrn/rnae200
• On the integral part of A -motivic cohomology
  
  • Gazda Quentin
  Compositio Mathematica, Foundation Compositio Mathematica / Cambridge University Press, Cambridge, 2024, 160 (8), pp.1715-1783. The deepest arithmetic invariants attached to an algebraic variety defined over a number field $F$ are conjecturally captured by the integral part of its motivic cohomology . There are essentially two ways of defining it when $X$ is a smooth projective variety: one is via the $K$ -theory of a regular integral model, the other is through its $\ell$ -adic realization. Both approaches are conjectured to coincide. This paper initiates the study of motivic cohomology for global fields of positive characteristic, hereafter named $A$ -motivic cohomology , where classical mixed motives are replaced by mixed Anderson $A$ -motives. Our main objective is to set the definitions of the integral part and the good $\ell$ -adic part of the $A$ -motivic cohomology using Gardeyn's notion of maximal models as the analogue of regular integral models of varieties. Our main result states that the integral part is contained in the good $\ell$ -adic part . As opposed to what is expected in the number field setting, we show that the two approaches do not match in general. We conclude this work by introducing the submodule of regulated extensions of mixed Anderson $A$ -motives, for which we expect the two approaches to match, and solve some particular cases of this expectation. (10.1112/S0010437X24007218)
  DOI : 10.1112/S0010437X24007218
• Non-ergodicity on SU(2) and SU(3) character varieties of the once-punctured torus
  
  • Forni Giovanni
  • Goldman William
  • Lawton Sean
  • Matheus Carlos
  Annales Henri Lebesgue, UFR de Mathématiques - IRMAR, 2024, 7, pp.1099-1130. Utilizing KAM theory, we show that there are certain levels in relative SU (2) and SU (3 ) character varieties of the once-punctured torus where the action of a single hyperbolic element is not ergodic. (10.5802/ahl.216)
  DOI : 10.5802/ahl.216
• Constructive Krein-Rutman result for Kinetic Fokker-Planck equations in a domain
  
  • Carrapatoso Kleber
  • Gabriel Pierre
  • Medina Richard
  • Mischler Stéphane
  , 2024. We consider a general Kinetic Fokker-Planck (KFP) equation in a domain with Maxwell reflection condition on the boundary, not necessarily with conservation of mass. We establish the wellposedness in many spaces including Radon measures spaces, and in particular the existence and uniqueness of fundamental solutions. We also establish a Krein-Rutman theorem with constructive rate of convergence in an abstract setting that we use for proving that the solutions to the KFP equation converge toward the conveniently normalized first eigenfunction. Both results use the ultracontractivity of the associated semigroup in a fundamental way.
• Families of automorphisms on abelian varieties
  
  • Favre Charles
  • Kuznetsova Alexandra
  Mathematische Annalen, Springer Verlag, 2024. 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. (10.1007/s00208-024-02943-4)
  DOI : 10.1007/s00208-024-02943-4
• Global well-posedness for a system of quasilinear wave equations on a product space
  
  • Huneau Cécile
  • Stingo Annalaura
  Analysis & PDE, Mathematical Sciences Publishers, 2024, 17 (6), pp.2033-2075. We consider a system of quasilinear wave equations on the product space $\mathbb{R}^{1+3}\times \mathbb{S}^1$, which we want to see as a toy model for Einstein equations with additional compact dimensions. We show global existence for small and regular initial data with polynomial decay at infinity. The method combines energy estimates on hyperboloids inside the light cone and weighted energy estimates outside the light cone. (10.2140/apde.2024.17.2033)
  DOI : 10.2140/apde.2024.17.2033
• The Landau equation in a domain
  
  • Carrapatoso Kleber
  • Mischler Stéphane
  , 2024. This work deals with the Landau equation in a bounded domain with the Maxwell reflection condition on the boundary for any (possibly smoothly position dependent) accommodation coefficient and for the full range of interaction potentials, including the Coulomb case. We establish the global existence and a constructive asymptotic decay of solutions in a close-to-equilibrium regime. This is the first existence result for a Maxwell reflection condition on the boundary and that generalizes the similar results established for the Landau equation for other geometries in \cite{GuoLandau1,GS1,GS2,MR3625186,MR4076068}. We also answer to Villani's program \cite{MR2116276,MR2407976} about constructive accurate rate of convergence to the equilibrium {(quantitative H-Theorem)} for solutions to collisional kinetic equations satisfying a priori uniform bounds. The proofs rely on the study of a suitably linear problem for which we prove that the associated operator is hypocoercive, the associated semigroup is ultracontractive, and finally that it is asymptotically stable in many weighted $L^\infty$ spaces.
• Multi-phase high frequency solutions to Klein-Gordon-Maxwell equations in Lorenz gauge in (3+1) Minkowski spacetime
  
  • Salvi Tony
  , 2024. We study a 1-parameter family (A{\lambda}, {\Phi}{\lambda}){\lambda} of multi-phase high frequency solutions to Klein-Gordon-Maxwell equations in Lorenz gauge in the (3+1)-dimensional Minkowski spacetime. This family is based on an initial ansatz. We prove that for {\lambda} small enough the family of solutions exists on an interval uniform in {\lambda} only function of the initial ansatz. These solutions are close to an approximate solution constructed by geometric optics. The initial ansatz needs to be regular enough, to satisfy a polarization condition and to satisfy the constraints for Maxwell null-transport in Lorenz gauge, but there is no need for smallness of any kind. The phases need to interact in a coherent way. We also observe that the limit (A0, {\Phi}0) is not solution to Klein-Gordon-Maxwell equations but to a Klein-Gordon-Maxwell null-transport type system.
• Residue of special functions of Anderson A-modules at the characteristic graph
  
  • Gazda Quentin
  • Maurischat Andreas
  Journal of Number Theory, Elsevier, 2024, 260, pp.1-28. (10.1016/j.jnt.2024.01.013)
  DOI : 10.1016/j.jnt.2024.01.013
• Special macroscopic modes and hypocoercivity
  
  • Carrapatoso Kleber
  • Dolbeault Jean
  • Hérau Frédéric
  • Mischler Stéphane
  • Mouhot Clément
  • Schmeiser Christian
  Journal of the European Mathematical Society, European Mathematical Society, 2024. We study linear inhomogeneous kinetic equations with an external confining potential and a collision operator admitting several local conservation laws (local density, momentum and energy). We classify all special macroscopic modes (stationary solutions and time-periodic solutions). We also prove the convergence of all solutions of the evolution equation to such non-trivial modes, with a quantitative exponential rate. This is the first hypocoercivity result with multiple special macroscopic modes with constructive estimates depending on the geometry of the potential. (10.4171/JEMS/1502)
  DOI : 10.4171/JEMS/1502
• Semi-classical limit for Klein-Gordon equation toward relativistic Euler equations via an adapted modulated energy method
  
  • Salvi Tony
  , 2024. We show the convergence of the solutions to the massive nonlinear Klein-Gordon equation toward solutions to a relativistic Euler with potential type system in the semi-classical limit. In particular, the momentum and the density of Klein-Gordon converge to the the momentum and the density of the relativistic Euler system in Lebesgue norms. The relativistic Euler with potential is equivalent to the usual relativistic Euler with pressure up to a rescaling. The proof relies on the modulated energy method adapted to the wave equation and the relativistic setting.
• Braids in Low-Dimensional Hamiltonian Dynamics
  
  • Morabito Francesco
  , 2024. In this thesis we study Hamiltonian systems using the topology of their closed orbits. The results we present deal, on the one hand, with properties of the generating functions associated with a particular Hamiltonian diffeomorphism, and on the other hand, with the Hofer distance between two diffeomorphisms that realise braids of different types. In the first context, we will rely on results by Patrice Le Calvez to show that any (up to stabilisation) generating function of a Hamiltonian diffeomorphism with compact support in the plane admits a filtration into the second tensor power of the Morse complex. By “linking filtration” we mean a filtration which associates an integer with any pair of critical points, and such when the two points are distinct the associatedvalue is exactly the linking number of the two orbits corresponding to the critical points. It is possible to define such filtration in the context of Hamiltonian Floer theory as well, and to study its behaviour with respect to the product in homology. The author’s results in this direction are still unpublished. On the other hand, we consider the set of Hamiltonian diffeomorphisms with compact support of a surface with boundary which preserves a predetermined configuration of circles. We give estimates of the Hofer energy of such a diffeomorphism based on the complexity of a type of braid that we assign to each diffeomorphism in this class. The tool we use here is quantitative Heegaard-Floer theory, recently developed by Cristofaro-Gardiner, Humilière, Mak, Seyfaddini andSmith. The results in this direction are already contained in a work by the author, and in one in collaboration with Ibrahim Trifa.