Publications

2019

• Barcodes and area-preserving homeomorphisms
  
  • Roux Frederic Le
  • Seyfaddini Sobhan
  • Viterbo Claude
  , 2019. In this paper we use the theory of barcodes as a new tool for studying dynamics of area-preserving homeomorphisms. We will show that the barcode of a Hamiltonian diffeomorphism of a surface depends continuously on the diffeomorphism, and furthermore define barcodes for Hamiltonian homeomorphisms. Our main dynamical application concerns the notion of {\it weak conjugacy}, an equivalence relation which arises naturally in connection to $C^0$ continuous conjugacy invariants of Hamiltonian homeomorphisms. We show that for a large class of Hamiltonian homeomorphisms with a finite number of fixed points, the number of fixed points, counted with multiplicity, is a weak conjugacy invariant. The proof relies, in addition to the theory of barcodes, on techniques from surface dynamics such as Le Calvez's theory of transverse foliations. In our exposition of barcodes and persistence modules, we present a proof of the Isometry Theorem which incorporates Barannikov's theory of simple Morse complexes.
• B. Rémy - Génération de groupes topologiques simples
  
  • Remy Bertrand
  • Bastien Fanny
  , 2019. Les groupes finis simples sont connus pour être engendrés par des paires d’éléments bien choisies. On peut se poser la même question avec des groupes topologiques : que peut-on espérer comme partie engendrant un sous-groupe dense ? Évidemment, la réponse dépend des groupes considérés ; on y répondra partiellement pour des groupes de matrices, et on évoquera les nombreuses questions ouvertes dans le domaine.
• On profinite subgroups of an algebraic group over a local field
  
  • Loisel Benoit
  Transformation Groups, Springer Verlag, 2019, 24 (4), pp.1173–1211. The purpose of this paper is to link anisotropy properties of an algebraic group together with compactness issues in the topological group of its rational points. We nd equivalent conditions on a smooth ane algebraic group scheme over a non-Archimedean local eld for the associated rational points to admit maximal compact subgroups. We use the structure theory of pseudo-reductive groups provided, whatever the characteristic, by Conrad, Gabber and Prasad. We also investigate thoroughly maximal prop subgroups in the semisimple case, using Bruhat-Tits theory.
• A non-Archimedean approach to K-stability
  
  • Boucksom Sébastien
  • Jonsson Mattias
  , 2019. We study K-stability properties of a smooth Fano variety X using non-Archi-medean geometry, specifically the Berkovich analytification of X with respect to the trivial absolute value on the ground field. More precisely, we view K-semistability and uniform K-stability as conditions on the space of plurisubharmonic (psh) metrics on the anticanonical bundle of X. Using the non-Archimedean Calabi-Yau theorem and the Legendre transform, this allows us to give a new proof that K-stability is equivalent to Ding stability. By choosing suitable psh metrics, we also recover the valuative criterion of K-stability by Fujita and Li. Finally, we study the asymptotic Fubini-Study operator, which associates a psh metric to any graded filtration (or norm) on the anticanonical ring. Our results hold for arbitrary smooth polarized varieties, and suitable adjoint/twisted notions of K-stability and Ding stability. They do not rely on the Minimal Model Program.
• A NOTE ON LANG'S CONJECTURE FOR QUOTIENTS OF BOUNDED DOMAINS
  
  • Boucksom Sébastien
  • Diverio Simone
  , 2019. It was conjectured by Lang that a complex projective man-ifold is Kobayashi hyperbolic if and only if it is of general type together with all of its subvarieties. We verify this conjecture for projective mani-folds whose universal cover carries a bounded, strictly plurisubharmonic function. This includes in particular compact free quotients of bounded domains.
• On congruence half-factorial Krull monoids with cyclic class group
  
  • Plagne Alain
  • Schmid Wolfgang
  Journal of Combinatorial Algebra, European Mathematical Society Press, 2019, 3 (4), pp.331-400. We carry out a detailed investigation of congruence half-factorial Krull monoids of various orders with finite cyclic class group and related problems. Specifically, we determine precisely all relatively large values that can occur as a minimal distance of a Krull monoid with finite cyclic class group, as well as the exact distribution of prime divisors over the ideal classes in these cases. Our results apply to various classical objects, including maximal orders and certain semi-groups of modules. In addition, we present applications to quantitative problems in factorization theory. More specifically, we determine exponents in the asymptotic formulas for the number of algebraic integers whose sets of lengths have a large difference. (10.4171/jca/34)
  DOI : 10.4171/jca/34
• Inhibition of receptor-interacting protein kinase 1 improves experimental non-alcoholic fatty liver disease
  
  • Majdi Amine
  • Aoudjehane Lynda
  • Ratziu Vlad
  • Islam Tawhidul
  • Afonso Marta
  • Conti Filoména
  • Mestiri Taïeb
  • Lagouge Marie
  • Foufelle Fabienne
  • Ballenghien Florine
  • Ledent Tatiana
  • Moldes Marthe
  • Cadoret Axelle
  • Fouassier Laura
  • Delaunay Jean-Louis
  • Aït-Slimane Tounsia
  • Courtois Gilles
  • Fève Bruno
  • Scatton Olivier
  • Prip-Buus Carina
  • Rodrigues Cecília
  • Housset Chantal
  • Gautheron Jérémie
  • Rodrigues Cecília M.P.
  Journal of Hepatology, Elsevier, 2019, 72 (4), pp.627-635. Background & aims: In non-alcoholic fatty liver disease (NAFLD), hepatocytes can undergo necroptosis: a regulated form of necrotic cell death mediated by the receptor-interacting protein kinase (RIPK) 1. Herein, we assessed the potential for RIPK1 and its downstream effector mixed lineage kinase domain-like protein (MLKL) to act as therapeutic targets and markers of activity in NAFLD. Methods: C57/BL6J-mice were fed a normal chow diet or a high-fat diet (HFD). The effect of RIPA-56, a highly specific inhibitor of RIPK1, was evaluated in HFD-fed mice and in primary human steatotic hepatocytes. RIPK1 and MLKL concentrations were measured in the serum of patients with NAFLD. Results: When used as either a prophylactic or curative treatment for HFD-fed mice, RIPA-56 caused a downregulation of MLKL and a reduction of liver injury, inflammation and fibrosis, characteristic of non-alcoholic steatohepatitis (NASH), as well as of steatosis. This latter effect was reproduced by treating primary human steatotic hepatocytes with RIPA-56 or necrosulfonamide, a specific inhibitor of human MLKL, and by knockout (KO) of Mlkl in fat-loaded AML-12 mouse hepatocytes. Mlkl-KO led to activation of mitochondrial respiration and an increase in β-oxidation in steatotic hepatocytes. Along with decreased MLKL activation, Ripk3-KO mice exhibited increased activities of the liver mitochondrial respiratory chain complexes in experimental NASH. In patients with NAFLD, serum concentrations of RIPK1 and MLKL increased in correlation with activity. Conclusion: The inhibition of RIPK1 improves NASH features in HFD-fed mice and reverses steatosis via an MLKL-dependent mechanism that, at least partly, involves an increase in mitochondrial respiration. RIPK1 and MLKL are potential serum markers of activity and promising therapeutic targets in NAFLD. Lay summary: There are currently no pharmacological treatment options for non-alcoholic fatty liver disease (NAFLD), which is now the most frequent liver disease. Necroptosis is a regulated process of cell death that can occur in hepatocytes during NAFLD. Herein, we show that RIPK1, a gatekeeper of the necroptosis pathway that is activated in NAFLD, can be inhibited by RIPA-56 to reduce not only liver injury, inflammation and fibrosis, but also steatosis in experimental models. These results highlight the potential of RIPK1 as a therapeutic target in NAFLD. (10.1016/j.jhep.2019.11.008)
  DOI : 10.1016/j.jhep.2019.11.008
• A BOUNDARY-PARTITION-BASED DIAGRAM OF D-DIMENSIONAL BALLS: DEFINITION, PROPERTIES AND APPLICATIONS
  
  • Duan Xianglong
  • Quan Chaoyu
  • Stamm Benjamin
  , 2019. In computational geometry, different ways of space partitioning have been developed, including the Voronoi diagram of points and the power diagram of balls. In this article, a generalized Voronoi partition of overlapping d-dimensional balls, called the boundary-partition-based diagram, is proposed. The definition, properties and applications of this diagram are presented. Compared to the power diagram, this boundary-partition-based diagram is straightforward in the computation of the volume of overlapping balls, which avoids the possibly complicated construction of power cells. Furthermore, it can be applied to characterize singularities on molecular surfaces and to compute the medial axis that can potentially be used to classify molecular structures.
• Lyapunov-type characterisation of exponential dichotomies with applications to the heat and Klein–Gordon equations
  
  • Chen Gong
  • Jendrej Jacek
  Transactions of the American Mathematical Society, American Mathematical Society, 2019, 372 (10), pp.7461-7496. (10.1090/tran/7923)
  DOI : 10.1090/tran/7923
• Trilinear compensated compactness and Burnett's conjecture in general relativity
  
  • Huneau Cécile
  • Luk Jonathan
  , 2019. Consider a sequence of $C^4$ Lorentzian metrics $\{h_n\}_{n=1}^{+\infty}$ on a manifold $\mathcal M$ satisfying the Einstein vacuum equation $\mathrm{Ric}(h_n)=0$. Suppose there exists a smooth Lorentzian metric $h_0$ on $\mathcal M$ such that $h_n\to h_0$ uniformly on compact sets. Assume also that on any compact set $K\subset \mathcal M$, there is a decreasing sequence of positive numbers $\lambda_n \to 0$ such that $$\|\partial^{\alpha} (h_n - h_0)\|_{L^{\infty}(K)} \lesssim \lambda_n^{1-|\alpha|},\quad |\alpha|\geq 4.$$ 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 $\{h_n\}_{n=1}^{+\infty}$ and $h_0$ in addition admit a $\mathbb U(1)$ 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.
• Collet, Eckmann and the bifurcation measure
  
  • Astorg Matthieu
  • Gauthier Thomas
  • Mihalache Nicolae
  • Vigny Gabriel
  Inventiones Mathematicae, Springer Verlag, 2019, 217 (3), pp.749-797. The moduli space M_d of degree d ≥ 2 rational maps can naturally be endowed with a measure µ_bif detecting maximal bifurcations, called the bifurcation measure. We prove that the support of the bifurcation measure µ_bif has positive Lebesgue measure. To do so, we establish a general sufficient condition for the conjugacy class of a rational map to belong to the support of µ_bif and we exhibit a large set of Collet-Eckmann rational maps which satisfy this condition. As a consequence, we get a set of Collet-Eckmann rational maps of positive Lebesgue measure which are approximated by hyperbolic rational maps. (10.1007/s00222-019-00874-5)
  DOI : 10.1007/s00222-019-00874-5
• ON THE BOUNDARY LAYER EQUATIONS WITH PHASE TRANSITION IN THE KINETIC THEORY OF GASES
  
  • Bernhoff Niclas
  • Golse François
  , 2019. Consider the steady Boltzmann equation with slab symmetry for a monatomic, hard sphere gas in a half space. At the boundary of the half space, it is assumed that the gas is in contact with its condensed phase. The present paper discusses the existence and uniqueness of a uniformly decaying boundary layer type solution of the Boltzmann equation in this situation, in the vicinity of the Maxwellian equilibrium with zero bulk velocity, with the same temperature as that of the condensed phase, and whose pressure is the saturating vapor pressure at the temperature of the interface. This problem has been extensively studied first by Y. Sone, K. Aoki and their collaborators, by means of careful numerical simulations. See section 2 of [C. Bardos, F. Golse, Y. Sone: J. Stat. Phys. 124 (2006), 275-300] for a very detailed presentation of these works. More recently T.-P. Liu and S.-H. Yu [Arch. Rational Mech. Anal. 209 (2013), 869-997] have proposed an extensive mathematical strategy to handle the problems studied numerically by Y. Sone, K. Aoki and their group. The present paper offers an alternative, possibly simpler proof of one of the results discussed in [T.P. Liu, S.-H. Yu, loc. cit.]
• A local model for the trianguline variety and applications
  
  • Breuil Christophe
  • Hellmann Eugen
  • Schraen Benjamin
  Publications mathematiques de l' IHES, IHES, 2019, 130, pp.299--412. We describe the completed local rings of the trianguline variety at certain points of integral weights in terms of completed local rings of algebraic varieties related to Grothendieck's simultaneous resolution of singularities. We derive several local consequences at these points for the trianguline variety: local irreducibility, description of all local companion points in the crystalline case, combinatorial description of the completed local rings of the fiber over the weight map, etc. Combined with the patched Hecke eigenvariety (under the usual Taylor-Wiles assumptions), these results in turn have several global consequences: classicality of crystalline strictly dominant points on global Hecke eigenvarieties, existence of all expected companion constituents in the completed cohomology, existence of singularities on global Hecke eigenvarieties. (10.1007/s10240-019-00111-y)
  DOI : 10.1007/s10240-019-00111-y
• Propagation of Moments and Semiclassical Limit from Hartree to Vlasov Equation
  
  • Lafleche Laurent
  Journal of Statistical Physics, Springer Verlag, 2019, 177 (1), pp.20-60. In this paper, we prove a quantitative version of the semiclassical limit from the Hartree to the Vlasov equation with singular interaction, including the Coulomb potential. To reach this objective, we also prove the propagation of velocity moments and weighted Schatten norms which implies the boundedness of the space density of particles uniformly in the Planck constant . (10.1007/s10955-019-02356-7)
  DOI : 10.1007/s10955-019-02356-7
• EMPIRICAL MEASURES AND QUANTUM MECHANICS: APPLICATION TO THE MEAN-FIELD LIMIT
  
  • Golse François
  • Paul Thierry
  Communications in Mathematical Physics, Springer Verlag, 2019, 369 (3), pp.1021-1053. In this paper, we define a quantum analogue of the notion of empirical measure in the classical mechanics of $N$-particle systems. We establish an equation governing the evolution of our quantum analogue of the $N$-particle empirical measure, and we prove that this equation contains the Hartree equation as a special case. Our main application of this new notion to the mean-field limit of the $N$-particle Schrödinger equation is an $O(1/\sqrt{N})$ convergence rate in some appropriate dual Sobolev norm for the Wigner transform of the single-particle marginal of the $N$-particle density operator, uniform in $\hbar\in(0,1]$ (where $\hbar$ is the Planck constant) provided that $V$ and $(−∆)^{3+d/2}V$ have integrable Fourier transforms. (10.1007/s00220-019-03357-z)
  DOI : 10.1007/s00220-019-03357-z
• INSTABILITY OF INFINITELY-MANY STATIONARY SOLUTIONS OF THE SU (2) YANG-MILLS FIELDS ON THE EXTERIOR OF THE SCHWARZSCHILD BLACK HOLE
  
  • Huneau Cécile
  • Häfner Dietrich
  Advances in Differential Equations, Khayyam Publishing, 2019, 24 (7/8), pp.435-464. We consider the spherically symmetric SU (2) Yang-Mills fields on the Schwarzschild metric. Within the so called purely magnetic Ansatz we show that there exists a countable number of stationary solutions which are all nonlinearly unstable. (10.57262/ade/1556762455)
  DOI : 10.57262/ade/1556762455
• ALEX GROSSMANN
  
  • Paul Thierry
  Gazette des Mathématiciens, Société Mathématique de France, 2019, 161, pp.78-81. un homme multidisciplinaire
• Real algebraic curves in real del Pezzo surfaces
  
  • Manzaroli Matilde
  , 2019. The study of the topology of real algebraic varieties dates back to the work of Harnack, Klein and Hilbert in the 19th century; in particular, the isotopy type classification of real algebraic curves with a fixed degree in RP2 is a classical subject that has undergone considerable evolution. On the other hand, apart from studies concerning Hirzebruch surfaces and at most degree 3 surfaces in RP3, not much is known for more general ambient surfaces. In particular, this is because varieties constructed using the patchworking method are hypersurfaces of toric varieties. However, there are many other real algebraic surfaces. Among these are the real rational surfaces, and more particularly the R-minimal surfaces. In this thesis, we extend the study of the topological types realized by real algebraic curves to the real minimal del Pezzo surfaces of degree 1 and 2. Furthermore, we end the classification of separating and non-separating real algebraic curves of bidegree (5,5) in the quadric ellipsoid.
• Construction of dynamics with strongly interacting for non-linear dispersive PDE (Partial differential equation).
  
  • Nguyen Tien Vinh
  , 2019. This thesis deals with long time dynamics of soliton solutions for nonlinear dispersive partial differential equation (PDE). Through typical examples of such equations, the nonlinear Schrödinger equation (NLS), the generalized Korteweg-de Vries equation (gKdV) and the coupled system of Schrödinger, we study the behavior of solutions, when time goes to infinity, towards sums of solitons (multi-solitons). First, we show that in the symmetric setting, with strong interactions, the behavior of logarithmic separation in time between solitons is universal in both subcritical and supercritical case. Next, adapting previous techniques to (gKdV) equation, we prove a similar result of existence of multi-solitons with logarithmic relative distance; for (gKdV), the solitons are repulsive in the subcritical case and attractive in the supercritical case. Finally, we identify a new logarithmic regime where the solitons are non-symmetric for the non-integrable coupled system of Schrödinger; such solution does not exist in the integrable case for the system and for (NLS).
• l-adic,p-adic and geometric invariants in families of varieties.
  
  • Ambrosi Emiliano
  , 2019. This thesis is divided in 8 chapters. Chapter 1 is of preliminary nature: we recall the tools that we will use in the rest of the thesis and some previously known results. Chapter 2 is devoted to summarize in a uniform way the new results obtained in this thesis. The other six chapters are original. In Chapters 3 and 4, we prove the following: given a smooth proper morphism f:Y→X over a smooth geometrically connected base X over an infinite finitely generated field of positive characteristic, there are lots of closed points x∈|X| such that the rank of the N'eron-Severi group of the geometric fibre of f at x is the same of the rank of the N'eron-Severi group of the geometric generic fibre. To prove this, we first study the specialization of the ℓ-adic lisse sheaf R²f_*ℚℓ(1)(ℓ≠p), then we relate it with the specialization of the F-isocrystal R²f_{*,crys}O_{Y/K}(1) passing trough the category of overconvergent F-isocrystals. Then, the variational Tate conjecture in crystalline cohomology, allows us to deduce the result on the N'eron-Severi groups from the results on R²f_{*,crys}O_{Y/K}(1). These extend to positive characteristic results of Cadoret-Tamagawa and Andr'e in characteristic zero. Chapters 5 and 6 are devoted to the study of the monodromy groups of (over)convergent F-isocrystals. Chapter ref{chaptermarcuzzo} is a joint work with Marco D'Addezio. We study the maximal tori in the monodromy groups of (over)convergent F-isocrystals and using them we prove a special case of a conjecture of Kedlaya on homomorphism of convergent F-isocrystals. Using this special case, we prove that if A is an abelian variety without isotrivial geometric isogeny factors over a function field F over F¯_p, then the group A(F^{perf})_{tors} is finite. This may be regarded as an extension of the Lang--N'eron theorem and answer positively to a question of Esnault. In Chapter 6, we define overline Q_p-linear category of (over)convergent F-isocrystals and the monodromy groups of their objects. Using the theory of companion for overconvergent F-isocrystals and lisse sheaves, we study the specialization theory of these monodromy groups, transferring the result of Chapter 3 to this setting via the theory of companions. The last two chapters are devoted to complements and refinement of the results in the previous chapters. In Chapter 7, we show that the Tate conjecture for divisors over finitely generated fields of characteristic p above 0 follows from the Tate conjecture for divisors over finite fields of characteristic p above 0. In Chapter 8, we prove uniform boundedness results for the Brauer groups of forms of varieties in positive characteristic, satisfying the ℓ-adic Tate conjecture for divisors. This extends to positive characteristic a result of Orr-Skorobogatov in characteristic zero.
• Incompressible optimal transport : dependence to the data and entropic regularization
  
  • Baradat Aymeric
  , 2019. This thesis focuses on Incompressible Optimal Transport, a minimization problem introduced by Brenier in the late 80's, aiming at describing the evolution of an incompressible and inviscid fluid in a Lagrangian way , i.e. by prescribing the state of the fluid at the initial and final times and by minimizing some functional among the set of admissible dynamics. This text is divided into two parts.In the first part, we study the dependence of this optimization problem with respect to the data. More precisely, we analyse the dependence of the pressure field, the Lagrange multiplier corresponding to the incompressibility constraint, with respect to the endpoint conditions, described by a probability measure γ determining the state of the fluid at the initial and final times. We show in Chapter 2 by purely variational methods that the gradient of the pressure field, as an element of a space that is close to the dual of C^1, is a Hölder continuous function of γ for the Monge-Kantorovic distance. On the other hand, we prove in Chapter 4 that for all r>1 the pressure field, as an element of L^r_t L^1_x, cannot be a Lipschitz continuous function of γ for the Monge-Kantorovic distance. This last statement is linked to an ill-posedness result proved in Chapter 3 for the so-called kinetic Euler equation, a kinetic PDE interpreted as the optimality equation of the Incompressible Optimal Transport problem.In the second part, we are interested in the entropic regularization of the Incompressible Optimal Transport problem: the so-called Brödinger problem, introduced by Arnaudon, Cruzeiro, Léonard and Zambrini in 2017. On the one hand, we prove in Chapter 5 that similarly to what happens in the Incompressible Optimal Transport case, to a solution always corresponds a scalar pressure field acting as the Lagrange multiplier for the incompressibility constraint. On the other hand, we prove in Chapter 6 that when the diffusivity coefficient tends to zero, the Brödinger problem converges towards the Incompressible Optimal Transport problem in the sense of Gamma-convergence, and with convergence of the pressure fields. The results of Chapter 6 come from a joint work with L. Monsaingeon.
• Good lattices of algebraic connections
  
  • Esnault Hélène
  • Sabbah Claude
  Documenta Mathematica, Universität Bielefeld, 2019. We construct a logarithmic model of connections on smooth quasi-projective $n$-dimensional geometrically irreducible varieties defined over an algebraically closed field of characteristic $0$. It consists of a good compactification of the variety together with $(n+1)$ lattices on it which are stabilized by log differential operators, and compute algebraically de Rham cohomology. The construction is derived from the existence of good Deligne-Malgrange lattices, a theorem of Kedlaya and Mochizuki which consists first in eliminating the turning points. Moreover, we show that a logarithmic model obtained in this way, called a good model, yields a formula predicted by Michael Groechenig, computing the class of the characteristic variety of the underlying D-module in the $K$-theory group of the variety. (10.25537/dm.2019v24.271-301)
  DOI : 10.25537/dm.2019v24.271-301
• Irregular Hodge numbers of confluent hypergeometric differential equations
  
  • Sabbah Claude
  • Yu Jeng-Daw
  Épijournal de Géométrie Algébrique, EPIGA, 2019. We give a formula computing the irregular Hodge numbers for a confluent hypergeometric differential equation.
• Links of sandwiched surface singularities and self-similarity
  
  • Fantini Lorenzo
  • Favre Charles
  • Ruggiero Matteo
  Manuscripta mathematica, Springer Verlag, 2019. (10.1007/s00229-019-01126-9)
  DOI : 10.1007/s00229-019-01126-9
• Regularity of push-forward of Monge-Ampère measures
  
  • Di Nezza Eleonora
  • Favre Charles
  Annales de l'Institut Fourier, Association des Annales de l'Institut Fourier, 2019. We prove that the image under any dominant meromorphic map of the Monge-Ampère measure of a Hölder continuous quasi-psh function still possesses a Hölder potential. We also discuss the case of lower regularity. (10.5802/aif.3233)
  DOI : 10.5802/aif.3233