Publications

2017

• HYPERBOLICITY NOTIONS FOR VARIETIES DEFINED OVER A NON-ARCHIMEDEAN FIELD
  
  • Rodríguez Vázquez Rita
  , 2017. Firstly, we pursue the work of W. Cherry on the analogue of the Kobayashi semi distance dCK that he introduced for analytic spaces defined over a non-Archimedean metrized field k. We prove various characterizations of smooth projective varieties for which dCK is an actual distance. Secondly, we explore several notions of hyperbolicity for a smooth algebraic curve X defined over k. We prove a non-Archimedean analogue of the equivalence between having negative Euler characteristic and the normality of certain families of analytic maps taking values in X.
• Représentations des groupes de Lie $p$-adiques et applications globales
  
  • Schraen Benjamin
  , 2017.
• A local relative trace formula for PGL(2)
  
  • Delorme Patrick
  • Harinck Pascale
  Pacific Journal of Mathematics, Mathematical Sciences Publishers, 2017, 291 (1), pp.121 - 147. Following a scheme inspired by B. Feigon, we describe the spectral side of a local relative trace formula for $G:= PGL(2,\rm E)$ relative to the symmetric subgroup $H:=PGL(2,\rm F)$ where $\rm E/\rm F$ is an unramified quadratic extension of local non archimedean fields of characteristic $0$. This spectral side is given in terms of regularized normalized periods and normalized $C$-functions of Harish-Chandra. Using the geometric side obtained in a more general setting by P. Delorme, P. Harinck and S. Souaifi , we deduce a local relative trace formula for $G$ relative to $H$. We apply our result to invert some orbital integrals. (10.2140/pjm.2017.291.121)
  DOI : 10.2140/pjm.2017.291.121
• Degree growth of rational maps induced from algebraic structures
  
  • Favre Charles
  • Lin Jan-Li
  Conformal Geometry and Dynamics, American Mathematical Society, 2017, 21 (13), pp.353 - 368. (10.1090/ecgd/312)
  DOI : 10.1090/ecgd/312
• Positivity of valuations on convex bodies and invariant valuations by linear actions
  
  • Dang Nguyen-Bac
  • Xiao Jian
  , 2017. We define a notion of positivity on continuous and translation invariant valuations on convex bodies on a finite dimensional real vector space. We endow the valuation space generated by mixed volumes with a norm induced by the positive cone. This enables us to construct a continuous extension of the convolution operator on smooth valuations to the closure of that space. As an application, we prove a variant of Minkowski's existence theorem. Furthermore, given a linear map, we generalize a theorem of Favre-Wulcan and Lin by proving that the eigenvalues of the linear map is related to the spectral radius of the induced linear operator on the space of valuations. Finally, given a linear action and under a natural strict log-concavity assumption on certain spectral radius of the induced linear operators on valuations, we study the positivity properties of the space of invariant valuations corresponding to the spectral radius of the operator.
• Solution by convex minimization of the Cauchy problem for hyperbolic systems of conservation laws with convex entropy
  
  • Brenier Yann
  , 2017. We show that, for first-order systems of conservation laws with a strictly convex entropy, in particular for the very simple so-called "inviscid" Burgers equation, it is possible to address the Cauchy problem by a suitable convex minimization problem, quite similar to some problems arising in optimal transport or variational mean-field game theory. In the general case, we show that smooth, shock-free, solutions can be recovered on some sufficiently small interval of time. In the special situation of the Burgers equation, we further show that every "entropy solution" (in the sense of Kruzhkov) including shocks, can be recovered, for arbitrarily long time intervals.
• Birational invariants : cohomology, algebraic cycles and Hodge theory cohomologie
  
  • Mboro René
  , 2017. In this thesis, we study some birational invariants of smooth projective varieties, in view of rationality questions for these varieties. It consists of three parts, that can be read independently.In the first chapter, we study, for some families of varieties, some stable birational invariants, that vanish for projective space and that appear naturally with Manin formulas. On one hand, we show for complex cubic 5-folds that the birational invariant given by the group of torsion codimension 3 cycles annihilated by the Deligne cycle map is controlled by the group of torsion 1-cycles of its variety of lines annihilated by the Deligne cycle map. We also prove that the Griffiths group of 1-cycles for the variety of lines of a hypersurface of the projective space over an algebraically closed field of characteristic 0, is trivial when the variety is smooth and Fano of index at least 3.The two last chapters focus on different aspects of the Chow-theoretic decomposition of the diagonal, a property which is invariant under stable birational equivalence, recently introduced by Voisin. In the second chapter, we adapt in characteristic greater than 2, part of the results, obtained by Voisin over the complex numbers, on the decomposition of the diagonal of cubic threefolds.In the last chapter, we study the concept of essential CH_0-dimension introduced by Voisin and related to the decomposition of the diagonal in that having essential CH_0-dimension 0 is equivalent to admitting a Chow-theoretic decomposition of the diagonal. We give sufficient (and necessary) conditions, for a complex variety with trivial group of 0-cycles, having essential CH_0-dimension non greater than 2 to admit a Chow-theoretic decomposition of the diagonal.
• Construction of a minimal mass blow up solution of the modified Benjamin–Ono equation
  
  • Martel Yvan
  • Pilod Didier
  Mathematische Annalen, Springer Verlag, 2017, 369 (1-2), pp.153 - 245. (10.1007/s00208-016-1497-8)
  DOI : 10.1007/s00208-016-1497-8
• Optimal transport and diffusion of currents
  
  • Duan Xianglong
  , 2017. Our work concerns about the study of partial differential equations at the hinge of the continuum physics and differential geometry. The starting point is the model of non-linear electromagnetism introduced by Max Born and Leopold Infeld in 1934 as a substitute for the traditional linear Maxwell's equations. These equations are remarkable for their links with differential geometry (extremal surfaces in the Minkowski space) and have regained interest in the 90s in high-energy physics (strings and D-branches).The thesis is composed of four chapters.The theory of nonlinear degenerate parabolic systems of PDEs is not very developed because they can not apply the usual comparison principles (maximum principle), despite their omnipresence in many applications (physics, mechanics, digital imaging, geometry, etc.). In the first chapter, we show how such systems can sometimes be derived, asymptotically, from non-dissipative systems (typically non-linear hyperbolic systems), by simple non-linear change of the time variable degenerate at the origin (where the initial data are set). The advantage of this point of view is that it is possible to transfer some hyperbolic techniques to parabolic equations, which seems at first sight surprising, since parabolic equations have the reputation of being easier to treat (which is not true , in reality, in the case of degenerate systems). The chapter deals with the curve-shortening flow as a prototype, which is the simplest exemple of the mean curvature flows in co-dimension higher than 1. It is shown how this model can be derived from the two-dimensional extremal surface in the Minkowski space (corresponding to the classical relativistic strings), which can be reduced to a hyperbolic system. We obtain, almost automatically, the parabolic version of the relative entropy method and weak-strong uniqueness, which, in fact, is much simpler to establish and understand in the hyperbolic framework.In the second chapter, the same method applies to the Born-Infeld system itself, which makes it possible to obtain, in the limit, a model (not listed to our knowledge) of Magnetohydrodynamics (MHD) where we have non-linear diffusions in the magnetic induction equation and the Darcy's law for the velocity field. It is remarkable that a system of such distant appearance of the basic principles of physics can be so directly derived from a model of physics as fundamental and geometrical as that of Born-Infeld.In the third chapter, a link is established between the parabolic systems and the concept of gradient flow of differential forms with suitable transport metrics. In the case of volume forms, this concept has had an extraordinary success in the field of optimal transport theory, especially after the founding work of Felix Otto and his collaborators. This concept is really only on its beginnings: in this chapter, we study a variant of the curve-shortening flow studied in the first chapter, which has the advantage of being integrable (in a certain sense) and lead to more precise results.Finally, in the fourth chapter, we return to the domain of hyperbolic EDPs considering, in the particular case of graphs, the extremal surfaces of the Minkowski space of any dimension and co-dimension. We can show that the equations can be reformulated in the form of a symmetric first-order enlarged system (which automatically ensures the well-posedness of the equations) of a remarkably simple structure (very similar to the Burgers equation) with quadratic nonlinearities, whose calculation is not obvious.
• Remarks on approximate decompositions of the diagonal
  
  • Mboro Rene
  , 2017. In this paper, we investigate, for varieties over ℂ with trivial group of 0-cycles, the gap between essential CH_0-dimension 2 and essential CH_0-dimension 0. In particular, we present sufficient (and necessary) conditions for a variety with trivial group of 0-cycles and essential CH0-dimension ≤2 to have, in fact, essential CH_0-dimension 0.
• Haas' theorem revisited
  
  • Bertrand Benoît
  • Brugallé Erwan
  • Renaudineau Arthur
  Épijournal de Géométrie Algébrique, EPIGA, 2017. Haas' theorem describes all partchworkings of a given non-singular plane tropical curve $C$ giving rise to a maximal real algebraic curve. The space of such patchworkings is naturally a linear subspace $W_C$ of the $\mathbb{Z}/2\mathbb{Z}$-vector space $\overrightarrow \Pi_C$ generated by the bounded edges of $C$, and whose origin is the Harnack patchworking. The aim of this note is to provide an interpretation of affine subspaces of $\overrightarrow \Pi_C $ parallel to $W_C$. To this purpose, we work in the setting of abstract graphs rather than plane tropical curves. We introduce a topological surface $S_\Gamma$ above a trivalent graph $\Gamma$, and consider a suitable affine space $\Pi_\Gamma$ of real structures on $S_\Gamma$ compatible with $\Gamma$. We characterise $W_\Gamma$ as the vector subspace of $\overrightarrow \Pi_\Gamma$ whose associated involutions induce the same action on $H_1(S_\Gamma,\mathbb{Z}/2\mathbb{Z})$. We then deduce from this statement another proof of Haas' original result.
• A non-Archimedean Montel's theorem
  
  • Rodriguez Vazquez Rita
  , 2017. This thesis is devoted to the study of compactness properties of spaces of analytic maps between analytic spaces defined over a non-Archimedean metrized field k. We work in the theory of analytic spaces as developed by Berkovich to fully exploit their tame topology. One of our motivations is the strive to introduce a natural notion of Kobayashi hyperbolicity in this setting. We first prove an analogue of Montel’s theorem for analytic maps taking values in a bounded domain of the affine space. In order to do so, we parametrize the space of analytic maps from an open polydisk to a closed one by the analytic spectrum of a suitable Banach k-algebra. Our result then follows from the sequential compactness of this space.Our results naturally lead to a definition of normal families, and we subsequently introduce two notions of Fatou sets attached to an endomorphism of the projective space. We show that Fatou components behave like in the complex case and cannot contain non trivial images of the punctured affine line. Thereupon, we apply our normality notion to the study of hyperbolicity in the non-Archimedean setting. We pursue the work of W. Cherry and prove various characterizations of smooth projective varieties whose Cherry-Kobayashi semi distance on the set of rigid points defines the classical topology. We finally obtain a characterization of smooth algebraic curves X of negative Euler characteristic in terms of the normality of certain families of analytic maps taking values in X.
• Sur les sous-groupes profinis des groupes algébriques linéaires
  
  • Loisel Benoit
  , 2017. Dans cette thèse, nous nous intéressons aux sous-groupes profinis et pro-p d'un groupe algébrique linéaire connexe défini sur un corps local. Dans le premier chapitre, on résume brièvement la théorie de Bruhat-Tits et on introduit les notations nécessaires à ce travail. Dans le second chapitre, on trouve des conditions équivalentes à l'existence de sous-groupes compacts maximaux d'un groupe algébrique linéaire G connexe quelconque défini sur un corps local K. Dans le troisième chapitre, on obtient un théorème de conjugaison des sous-groupes pro-p maximaux de G(K) lorsque G est réductif. On décrit ces sous-groupes, de plus en plus précisément, en supposant successivement que G est semi-simple, puis simplement connexe, puis quasi-déployé. Dans le quatrième chapitre, on s'intéresse aux présentations d'un sous-groupe pro-p maximal du groupe des points rationnels d'un groupe algébrique G semi-simple simplement connexe quasi-déployé défini sur un corps local K. Plus spécifiquement, on calcule le nombre minimal de générateurs topologiques d'un sous-groupe pro-p maximal. On obtient une formule linéaire en le rang d'un certain système de racines, qui dépend de la ramification de l'extension minimale L=K déployant G, explicitant ainsi les contributions de la théorie de Lie et de l'arithmétique du corps de base.
• Quasineutral limit for Vlasov–Poisson via Wasserstein stability estimates in higher dimension
  
  • Han-Kwan Daniel
  • Iacobelli Mikaela
  Journal of Differential Equations, Elsevier, 2017, 263 (1), pp.1 - 25. (10.1016/j.jde.2017.01.018)
  DOI : 10.1016/j.jde.2017.01.018
• Smoothness and Classicality on eigenvarieties
  
  • Schraen Benjamin
  • Breuil Christophe
  • Hellmann Eugen
  Inventiones Mathematicae, Springer Verlag, 2017, 209 (1), pp.197--274. Let p be a prime number and f an overconvergent p-adic automorphic form on a definite unitary group which is split at p. Assume that f is of "classical weight" and that its Galois representation is crystalline at places dividing p, then f is conjectured to be a classical automorphic form. We prove new cases of this conjecture in arbitrary dimension by making crucial use of the "patched eigenvariety". (10.1007/s00222-016-0708-y)
  DOI : 10.1007/s00222-016-0708-y
• Geometric monodromy — semisimplicity and maximality
  
  • Cadoret Anna
  • Hui Chun-Yin
  • Tamagawa Akio
  Annals of Mathematics, Princeton University, Department of Mathematics, 2017, 186 (1). Let X be a connected scheme, smooth and separated over an algebraically closed field k of characteristic p ≥ 0, let f : Y → X be a smooth proper morphism and x a geometric point on X. We prove that the tensor invariants of bounded length ≤ d of π1(X, x) acting on the étale cohomology groups H * (Yx, F ) are the reduction modulo-of those of π1(X, x) acting on H * (Yx, Z ) for greater than a constant depending only on f : Y → X, d. We apply this result to show that the geometric variant with F -coefficients of the Grothendieck-Serre semisimplicity conjecture -namely that π1(X, x) acts semisimply on H * (Yx, F ) for 0 -is equivalent to the condition that the image of π1(X, x) acting on H * (Yx, Q ) is 'almost maximal' (in a precise sense; what we call 'almost hyperspecial') with respect to the group of Q -points of its Zariski closure. Ultimately, we prove the geometric variant with F -coefficients of the Grothendieck-Serre semisimplicity conjecture. (10.4007/annals.2017.186.1.5)
  DOI : 10.4007/annals.2017.186.1.5
• Kink dynamics in the $\phi ^4$ model: Asymptotic stability for odd perturbations in the energy space
  
  • Kowalczyk Michał
  • Martel Yvan
  • Muñoz Claudio
  Journal of the American Mathematical Society, American Mathematical Society, 2017, 30 (3), pp.769 - 798. (10.1090/jams/870)
  DOI : 10.1090/jams/870
• Hyperbolicity of the time-like extremal surfaces in minkowski spaces
  
  • Duan Xianglong
  , 2017. In this paper, it is established, in the case of graphs, that time-like extremal surfaces of dimension $1+n$ in the Minkowski space of dimension $1+n+m$ can be described by a symmetric hyperbolic system of PDEs with the very simple structure (reminiscent of the inviscid Burgers equation) $$ \partial_t W + \sum_{j=1}^n A_j(W)\partial_{x_j} W =0,\;\;\;W:\;(t,x)\in\mathbb{R}^{1+n} \rightarrow W(t,x)\in\mathbb{R}^{n+m+\binom{m+n}{n}}, $$ where each $A_j(W)$ is just a $\big(n+m+\binom{m+n}{n}\big)\times\big(n+m+\binom{m+n}{n}\big)$ symmetric matrix depending linearly on $W$.
• The initial value problem for the Euler equations of incompressible fluids viewed as a concave maximization problem
  
  • Brenier Yann
  , 2017. We consider the Euler equations of incompressible fluids and attempt to solve the initial value problem with the help of a concave maximization problem. We show that this problem, which shares a similar structure with the optimal transport problem with quadratic cost, in its "Benamou-Brenier" formulation, always admits a relaxed solution that can be interpreted in terms of $sub-solution$ of the Euler equations in the sense of convex integration theory. Moreover, any smooth solution of the Euler equations can be recovered from this maximization problem, at least for short times.
• C. Favre - Degeneration of measures of maximal entropy
  
  • Favre Charles
  • Magnien Jérémy
  , 2017. Consider any meromorphic family of endomorphisms of the complex projective plane parameterized by the punctured unit disk. We shall explain how to describe the behaviour of their measures of maximal entropy when one approaches the central fiber. This generalizes works by Demarco and Faber.
• Magnetohydrodynamic regime of the born-infeld electromagnetism
  
  • Duan Xianglong
  , 2017. The Born-Infeld (BI) model is a nonlinear correction of Maxwell's equations. By adding the energy and Poynting vector as additional variables, it can be augmented as a 10×10 system of hyperbolic conservation laws, called the augmented BI (ABI) equations. The author found that, through a qua-dratic change of the time variable, the ABI system gives a simple energy dissi-pation model that combines Darcy's law and magnetohydrodynamics (MHD). Using the concept of " relative entropy " (or " modulated energy "), borrowed from the theory of hyperbolic systems of conservation laws, we introduce a notion of generalized solutions, that we call dissipative solutions. For given initial conditions, the set of generalized solutions is not empty, convex, and compact. Smooth solutions to the dissipative system are always unique in this setting.
• Partial Hasse invariants, partial degrees, and the canonical subgroup
  
  • Bijakowski Stéphane
  Canadian Journal of Mathematics = Journal Canadien de Mathématiques, University of Toronto Press, 2017, pp.1 - 28. (10.4153/CJM-2016-052-8)
  DOI : 10.4153/CJM-2016-052-8
• Théorème de comparaison pour les cycles proches par un morphisme sans pente
  
  • Kochersperger Matthieu
  Journal of Singularities, Worldwide Center of Mathematics, LLC, 2017, 16, pp.52-72. Le but de cet article est de démontrer le théorème de comparaison entre les cycles proches algébriques et topologiques associés à un morphisme sans pente. Nous obtenons en particulier que dans le cas d'une famille de fonctions holomorphes sans pente, l'itération des isomorphismes de comparaison des cycles proches associés à chacune de ces fonctions ne dépend pas de l'ordre d'itération. (10.5427/jsing.2017.16b)
  DOI : 10.5427/jsing.2017.16b
• Periods of Hodge structures and special values of the gamma function
  
  • Fresán Javier
  Inventiones Mathematicae, Springer Verlag, 2017, 208 (1), pp.247 - 282. (10.1007/s00222-016-0690-4)
  DOI : 10.1007/s00222-016-0690-4
• An integrable example of gradient flows based on optimal transport of differential forms
  
  • Brenier Yann
  • Duan Xianglong
  , 2017. Optimal transport theory has been a powerful tool for the analysis of parabolic equations viewed as gradient flows of volume forms according to suitable transportation metrics. In this paper, we present an example of gradient flows for closed (d-1)-forms in the Euclidean space R^d. In spite of its apparent complexity, the resulting very degenerate parabolic system is fully integrable and can be viewed as the Eulerian version of the heat equation for curves in the Euclidean space. We analyze this system in terms of ``relative entropy" and ``dissipative solutions" and provide global existence and weak-strong uniqueness results.