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We use local symplectic Lie groupoids to construct Poisson integrators for generic Poisson structures. More precisely, recursively obtained solutions of a Hamilton-Jacobi-like equation are interpreted as Lagrangian bisections in a neighborhood of the unit manifold, that, in turn, give Poisson integrators. We also insist on the role of the Magnus formula, in the context of Poisson geometry, for the backward analysis of such integrators.

We consider a closed three-dimensional contact sub-Riemannian manifold. The objective of this note is to provide a precise description of the sub-Riemannian geodesics with large initial momenta: we prove that they "spiral around the Reeb orbits", not only in the phase space but also in the configuration space. Our analysis is based on a normal form along any Reeb orbit due to Melrose.

We show that the germ of the contact structure surrounding a certain kind of convex hypersurfaces is overtwisted. We then find such hypersurfaces close to any plastikstufe with toric core so that these imply overtwistedness. All proofs in this article are explicit, and we hope that the methods used here might hint at a deeper understanding of the size of neighborhoods in contact manifolds. In the appendix we reprove in a concise way that the Legendrian unknot is loose if the ambient manifold contains a large enough neighborhood of a 2-dimensional overtwisted disk. Additionally we prove the folklore result that the singular distribution induced on a hypersurface $\Sigma$ of a contact manifold $(M, \xi)$ determines the germ of the contact structure around $\Sigma$., Comment: 13 pages, 2 figure

We describe the coadjoint orbits of the group of volume preserving diffeomorphisms of $\mathbb{R}^3$ associated to the motion of closed vortex sheets in ideal 3D fluids. We show that these coadjoint orbits can be identified with nonlinear Grassmannians of compact surfaces enclosing a given volume and endowed with a closed 1-form describing the vorticity density. If the vorticity density has discrete period group and is nonvanishing, the vortex sheet is given by a surface of genus one fibered by its vortex lines over a circle. We determine the Hamilton equations for such vortex sheets relative to the Hamiltonian function suggested in Khesin (2012) and prove that there are no stationary solutions having rotational symmetries. These coadjoint orbits are shown to be prequantizable if the period group of the 1-form and the volume enclosed by the surface satisfy an Onsager-Feynman relation, as argued in Goldin et al. (1991) for the case of open vortex sheets (tubes/ribbons). We find a character for the prequantizable coadjoint orbits, as well as a polarization group on which the character extends, which is a first step beyond prequantization., Comment: 25 pages, 1 figure

We prove that, for closed exact embedded Lagrangian submanifolds of cotangent bundles, the homomorphism of homotopy groups induced by the stable Lagrangian Gauss map vanishes. In particular, we prove that this map is null-homotopic for all spheres. The key tool that we introduce in order to prove this is the notion of twisted generating function and we show that every closed exact Lagrangian can be described using such an object, by extending a doubling argument developed in the setting of sheaf theory. Floer theory and sheaf theory constrain the type of twisted generating functions that can appear to a class which is closely related to Waldhausen's tube space, and our main result follows by a theorem of B\"okstedt which computes the rational homotopy type of the tube space., Comment: minor modifications

In this paper, we define quantum toric varieties associated to an arbitrary fan in a finitely generated subgroup of some $\mathbb{R}^d$ generalizing the article arXiv:2002.03876 of Katzarkov, Lupercio, Meersseman and Verjovsky., 35 pages, 2 figures

Given a smooth spacelike surface $\Sigma$ of negative curvature in Anti-de Sitter space of dimension 3, invariant by a representation $\rho:\pi_1(S)\to\mathrm{PSL}_2\mathbb{R}\times\mathrm{PSL}_2\mathbb{R}$ where $S$ is a closed oriented surface of genus $\geq 2$, a canonical construction associates to $\Sigma$ a diffeomorphism $\phi_\Sigma$ of $S$. It turns out that $\phi_\Sigma$ is a symplectomorphism for the area forms of the two hyperbolic metrics $h$ and $h'$ on $S$ induced by the action of $\rho$ on $\mathbb{H}^2\times\mathbb{H}^2$. Using an algebraic construction related to the flux homomorphism, we give a new proof of the fact that $\phi_\Sigma$ is the composition of a Hamiltonian symplectomorphism of $(S,h)$ and the unique minimal Lagrangian diffeomorphism from $(S,h)$ to $(S,h')$., Comment: 20 pages

We construct quantum K-invariants in non-archimedean analytic geometry. Contrary to the classical approach in algebraic geometry via perfect obstruction theory, we build on our previous works on the foundations of derived non-archimedean geometry, the representability theorem and Gromov compactness. We obtain a list of natural geometric relations between the stacks of stable maps, directly at the derived level, with respect to elementary operations on graphs, namely, products, cutting edges, forgetting tails and contracting edges. They imply immediately the corresponding properties of quantum K-invariants. The derived approach produces highly intuitive statements and functorial proofs. The flexibility of our derived approach to quantum K-invariants allows us to impose not only simple incidence conditions for marked points, but also incidence conditions with multiplicities. This leads to a new set of enumerative invariants. For the proofs, we further develop the foundations of derived non-archimedean geometry in this paper: we study derived lci morphisms, relative analytification, and deformation to the normal bundle. Our motivations come from non-archimedean enumerative geometry and mirror symmetry., Improved exposition

We apply the canonical forms (barcodes) of Morse complexes to explore topology of loss surfaces. We present a novel algorithm for calculations of the objective function's barcodes of minima. We have conducted experiments for calculating barcodes of local minima forbenchmark functions and for loss surfaces of neural networks. Our experiments confirm two principal observations: (1) the barcodes of minima are located in a small lower part of the range of values of loss function of neural networks and (2) increase of the neural network's depth brings down the minima's barcodes. This has natural implications for the neural network learning and the ability to generalize.

There is canonical partition of set of critical values of function into pairs "birth-death" and a separate set representing basis in Betti homology, as was established in S.Barannikov "The Framed Morse complex and its invariants" Adv. in Sov. Math., vol 21, AMS transl, (1994). This partition arises from bringing the gradient (Morse) complex to so called "canonical form" by a linear transform respecting the filtration defined by the order of the critical values. These "canonical forms" are combinatorial invariants of R-filtered complexes. Starting from the beginning of 2000s these invariants became widely used in applied mathematics under the name of "Persistence diagrams" and "Persistence Bar-codes". The canonical form of an R-filtered complex is the partition of the set of critical values (indices of filtration) into "birth-death" pairs and a separate set representing homology of the complex. The "canonical form" invariants are obtained by action of upper-triangular matrices, which reduces the R-filtered complex to the simplest form. In this note I give a short introduction into the subject of these invariants. I also propose application of these invariants to artificial neural nets.

The support norm $sn(\xi)$ of a contact structure $\xi$ is the minimum of the negative Euler characteristics of the pages of the open books supporting $\xi$. In this paper we prove additivity of the support norm for tight contact structures., Comment: The paper is withdrawn due to a mistake in the proof of Proposition 3.2

The localization of the EA-matrix integrals from [B4], is calculated via determinants and tau functions of KP-type hierarchies in the case of associative algebras with even scalar product.

Given a closed symplectic 4-manifold $(X,\omega)$, we define a twisted version of the Gromov-Taubes invariants for $(X,\omega)$, where the twisting coefficients are induced by the choice of a surface bundle over $X$. Given a fibered 3-manifold $Y$, we similarly construct twisted Lefschetz zeta functions associated with surface bundles: we prove that these are essentially equivalent to the Jiang's Lefschetz zeta functions of $Y$, twisted by the representations of $\pi_1(Y)$ that are induced by monodromy homomorphisms of surface bundles over $Y$. This leads to an interpretation of the corresponding twisted Reidemeister torsions of $Y$ in terms of products of "local" commutative Reidemeister torsions. Finally we relate the two invariants by proving that, for any fixed closed surface bundle $\mathcal{B}$ over $Y$, the corresponding twisted Lefschetz zeta function coincides with the Gromov-Taubes invariant of $S^1 \times Y$ twisted by the bundle over $S^1 \times Y$ naturally induced by $\mathcal{B}$., Comment: 31 pages, 1 figure. Version 2: minor changes, typographical corrections

International audience

Line bundles of real and rational degree are constructed and their cohomology computed. This is useful in homological mirror symmetry.

Irreducible holomorphic symplectic varieties (IHSV) are the algebraic analogue of the hyperkähler Riemannian manifolds. An IHSV of dimension 2 is a K3 surface; in this case, if furthermore X is projective, any ample curve on X is linearly equivalent to a sum of rational curves (Bogomolov, Mumford). Charles, Mongardi and Pacienza proved the existence of uniruled divisors on (almost) any ample linear system on a IHSV that is deformation equivalent to an Hilbert scheme on a K3 surface, or to a generalized Kummer variety. The existence of many rational curves on X semplifies the structure of the 0-Chow group of X. In my thesis, I worked on the OG10 case, the IHSV defined by O’Grady; the variety OG10 isimportant and very actively studied. The main result of my thesis proves the existence of ample uniruled divisors on any IHSV inside three connected components of the moduli space of OG10 varieties, Les variétés holomorphes symplectiques irréductibles (VHSI) sont l'analogue algébrique des variétés Riemannienne hyperkähler. Une VHSI X avec dimension 2 est une surface K3, et dans ce cas, si de plus X est projective, chaque courbe ample sur X est linéairement équivalente à une somme de courbes rationnelles (Bogomolov, Mumford). Charles, Mongardi et Pacienza ont démontré l'existence de diviseurs uniréglés dans (presque) tous les systèmes linéaires amples sur une VHSI qui est déformation d'un schéma de Hilbert sur une surface K3 ou d'une variété de Kummer generalisée. La présence de nombreuses courbes rationnelles dans X simplifie la structure du 0-group de Chow de X. Dans ma thése, j'ai travaillé sur le cas OG10, la VHSI définie par O'Grady; la variété OG10 est importantes et très activement étudiées. Le résultat principal de ma thèse démontre l'existence de diviseurs uniréglés amples sur chaque VHSI projectives appartenant à trois composantes connexes de l'espace de modules des OG10.

Accepté le 25 mars 2015; International audience; This article provides a definition of a subdifferential for continuous functions based on homological considerations. We show that it satisfies all the requirement for a good notion of subdifferential. Moreover, we prove sublinearity, a Leibniz formula and an approximation result. This work fits in the framework of microlocal analysis of sheaves for C^0 symplectic problems and application to Aubry-Mather theory.

In this article we study the Arnold conjecture in settings where objects under consideration are no longer smooth but only continuous. The example of a Hamiltonian homeomorphism, on any closed symplectic manifold of dimension greater than 2, having only one fixed point shows that the conjecture does not admit a direct generalization to continuous settings. However, it appears that the following Arnold-type principle continues to hold in continuous settings: Suppose that X is a non-smooth object for which one can define spectral invariants. If the number of spectral invariants associated to X is smaller than the number predicted by the (homological) Arnold conjecture, then the set of fixed/intersection points of X is homologically non-trivial, hence it is infinite. We recently proved that the above principle holds for Hamiltonian homeomorphisms of closed and aspherical symplectic manifolds. In this article, we verify this principle in two new settings: $C^0$ Lagrangians in cotangent bundles and Hausdorff limits of Legendrians in 1-jet bundles which are isotopic to the zero section. An unexpected consequence of the result on Legendrians is that the classical Arnold conjecture does hold for Hausdorff limits of Legendrians in 1-jet bundles. Dedicated to Claude Viterbo on the occasion of his 60th birthday.

N the first part of the thesis, we give, under some hypotheses, a lower bound on the intersection number of a closed monotone Lagrangian submanifold L with its image by a generic Hamiltonianisotopy. For monotone Lagrangian submanifolds L which are K(pi, 1) and, in particular with negative sectional curvature, this bound is 1 + beta_1(L), where beta_1 is the first Betti number with coefficients in Z_2. Another consequence, is the non-displaceability of a monotone Lagrangian embedding of RPn x K (where K is a submanifold with negative sectional curvature such that H^1(K, Z) ≠ 0) in some symplectic manifolds. In the second part, given a closed monotone Lagrangian submanifold L, which is not displaceable, we use Floer homology defined on Lagrangians which are C^1 - close to L, to get information about it Maslov number. Besides, if L can be approached by a sequence of displaceable Lagrangians, then, under some topological assumptions on L, the displacement energy of the elements of this sequence converge to infinity., Dans la première partie de cette thèse, on donne, sous certaines hypothèses, une minoration du nombre de points d’intersections d’une sous-variété Lagrangienne monotone L avec son image par une isotopie Hamiltonienne. Dans le cas où L est un espace K(pi, 1), et en particulier à courbure sectionnelle strictement négative, le minorant est 1 + beta1(L), où beta1 est le premier nombre de Betti à coefficients dans Z2. Une autre conséquence est la non-déplaçabilité d’un plongement Lagrangien monotone de RPn × K (où K est une sous-variété à courbure sectionnelle strictement négative telle que H1(K, Z) ≠ 0) dans certaines variétés symplectiques. Dans la seconde partie, on considère une sous-variété Lagrangienne monotone L non déplaçable. En utilisant l’homologie de Floer définie pour les Lagrangiennes qui sont C-1-proches de L, on obtient des informations sur son nombre de Maslov. De plus, si L peut être approchée par une suite de Lagrangiennes déplaçables, alors, sous certaines hypothèses topologiques sur L, l’énergie de déplacement des éléments de cette suite tend vers l’infini.

The construction of cohomology classes of compactified moduli spaces of curves from asymptotic expansions of EA matrix integrals, described in the works of speaker, is presented. The construction defines a cohomological field theory, which conjecturally coincides with the all genus Gromov-Witten invariants of the mirror manifold. The construction is based on the speaker’s theorem identifying the cell complex of the compactified moduli space of curves with the Feynman transform of the operad of permutation group algebras. As a simple application a new formula for generating function for products of psi classes in the total cohomology of the compactified moduli spaces of curves is described. Relevant papers: arxiv:1803.11549 / HAL-00429963 (2009), arxiv:0912.5484 / hal-00102085(2006)

International audience

This article pursues the study of the knot state asymptotics in the large level limit initiated in "Knot sate Asymptotics I". As a main result, we prove the Witten asymptotic expansion conjecture for the Dehn fillings of the figure eight knot. The state of a knot is defined in the realm of Chern-Simons topological quantum field theory as a holomorphic section on the SU(2)-character manifold of the peripheral torus. In the previous paper, we conjectured that the knot state concentrates on the character variety of the knot with a given asymptotic behavior on the neighborhood of the abelian representations. In the present paper we study the neighborhood of irreducible representations. We conjecture that the knot state is Lagrangian with a phase and a symbol given respectively by the Chern-Simons and Reidemeister torsion invariants. We show that under some mild assumptions, these conjectures imply the Witten conjecture on the asymptotic expansion of WRT invariants of the Dehn fillings of the knot. Using microlocal techniques, we show that the figure eight knot state satisfies our conjecture starting from q-differential relations verified by the colored Jones polynomials. The proof relies on a differential equation satisfied by the Reidemeister torsion along the branches of the character variety, a phenomenon which has not been observed previously as far as we know., 45 pages, 2 figures

The notion of nondegenerate critical point in the BV formalism is studied. The analogs of the Morse and Darboux theorems in the BV formalism are proven. The theorem on the normal form of an arbitrary quadratic function on odd symplectic space is proven. This can be viewed as an analog of Jordan type decomposition for a pair of a symmetric pairing on vector space and an anti-symmetric pairing on the dual space. The last result was used for the construction of propagators in the BV formalism in the equivariant setting, see [2, 3, 4]

This thesis is divided in two parts.The first part focuses on the study of the topology of the contactomorphism group of some explicit high dimensional contact manifolds. More precisely, using constructions and results by Massot, Niederkrüger and Wendl, we construct (infinitely many) examples in all dimensions of contactomor-phisms of closed overtwisted contact manifolds that are smoothly isotopic but not contact-isotopicto the identity. We also give examples of tight high dimensional contact manifolds admitting a contactomorphism whose powers are all smoothly isotopic but not contact-isotopic to the identity ;this is a generalization of a result in dimension 3 by Ding and Geiges.In the second part, we construct examples of higher dimensional contact manifolds with specific properties. This leads us to the existence of tight virtually overtwisted closed contact manifolds in all dimensions and to the fact that every closed contact 3-manifold embeds with trivial nor-mal bundle inside a tight closed contact 5-manifold. This uses known construction procedures byBourgeois (on products with tori) and Geiges (on branched covering spaces). We pass from these procedures to definitions ; this allows to prove a uniqueness statement in the case of contact branched coverings, and to study the global properties (such as tightness and fillability) of the results of both constructions without relying on any auxiliary choice in the procedures. A second goal allowed by these definitions is to study relations between these constructions and the notions of supporting open book, due to Giroux, and of contact fiber bundle, due to Lerman. For instance,we give a definition of Bourgeois contact structures on flat contact fiber bundles which is local,(strictly) includes the results of Bourgeois’ construction, and allows to recover an isotopy class of supporting open books on the fibers. This last point relies on a reinterpretation, inspired by anidea by Giroux, of supporting open books in terms of pairs of contact vector fields., Cette thèse est subdivisée en deux parties.La première partie porte sur l’étude de la topologie de l’espace des contactomorphismes pour quelques exemples explicites de variétés de contact en grandes dimensions. Plus précisément, en utilisant des constructions et résultats dus à Massot, Niederkrüger et Wendl, on construit, en chaque dimension impaire, une infinité d’exemples de contactomorphismes de variétés de contact vrillées fermées qui sont lissement isotopes mais pas contact-isotopes à l’identité. On donne aussi,en toutes dimensions impaires, des exemples de variétés de contact tendues fermées qui admettent un contactomorphisme tel que tous ses itérées sont lissement isotopes mais pas contacto-isotopes à l’identité ; ceci généralise un résultat en dimension 3 dû à Ding et Geiges.Dans la deuxième partie, on construit des exemples de variétés de contact fermées en grandes dimensions avec des propriétés particulières. Ceci nous amène à l’existence de structures tendues virtuellement vrillées en toutes dimensions impaires, et au fait que chaque variété de contact fermée de dimension 3 se plonge dans une variété de contact tendue fermée de dimension 5 avec fibré normal trivial. Pour cela, on utilise des constructions dues à Bourgeois (sur des produits avec des tores) et à Geiges (sur des revêtements ramifiés). On passe de ces constructions à des définitions ;ceci permet de prouver un résultat d’unicité dans le cas des revêtements ramifiés de contact, et d’étudier leurs propriétés globales, en montrant qu’elles ne dépendent d’aucun choix auxiliaire fait dans les procédures. Un deuxième but permis par ces définitions est l’étude des relations entre ces constructions et les notions de livre ouvert porteur, due à Giroux, et de fibré de contact, due à Lerman. Par exemple, on donne une définition de structure de contact de Bourgeois qui est locale,inclue (strictement) les résultats de la construction de Bourgeois et permet de récupérer une classe d’isotopie de livres ouverts porteurs sur les fibres ; ceci suit d’une réinterprétation, inspirée par une idée de Giroux, des livres ouverts porteurs en termes de paires de champs de vecteurs de contact.

We study automorphisms of irreducible holomorphic symplectic manifolds of type K3^[n], i.e. manifolds which are deformation equivalent to the Hilbert scheme of n points on a K3 surface, for some n > 1. In the first part of the thesis we describe the automorphism group of the Hilbert scheme of n points on a generic projective K3 surface, i.e. a K3 surface whose Picard lattice is generated by a single ample line bundle. We show that, if it is not trivial, the automorphism group is generated by a non-symplectic involution, whose existence depends on some arithmetic conditions involving the number of points n and the polarization of the surface. We also determine necessary and sufficient conditions on the Picard lattice of the Hilbert scheme for the existence of the involution.In the second part of the thesis we study non-symplectic automorphisms of prime order on manifolds of type K3^[n]. We investigate the properties of the invariant lattice and its orthogonal complement inside the second cohomology lattice of the manifold, providing a classification of their isometry classes. We then approach the problem of constructing examples (or at least proving the existence) of manifolds of type K3^[n] with a non-symplectic automorphism inducing on cohomology each specific action in our classification. In the case of involutions, and of automorphisms of odd prime order for n=3,4, we are able to realize all possible cases. In order to do so, we present a new non-symplectic automorphism of order three on a ten-dimensional family of Lehn-Lehn-Sorger-van Straten eightfolds of type K3^[4]. Finally, for n < 6 we describe deformation families of large dimension of manifolds of type K3^[n] equipped with a non-symplectic involution., Nous allons étudier les automorphismes des variétés symplectiques holomorphes irréductibles de type K3^[n], c'est-à-dire des variétés équivalentes par déformation au schéma de Hilbert de n points sur une surface K3, pour n > 1.Dans la première partie de la thèse, nous classifions les automorphismes du schéma de Hilbert de n points sur une surface K3 projective générique, dont le réseau de Picard est engendré par un fibré ample. Nous montrons que le groupe des automorphismes est soit trivial soit engendré par une involution non-symplectique et nous déterminons des conditions numériques et géométriques pour l’existence de l’involution.Dans la deuxième partie, nous étudions les automorphismes non-symplectiques d’ordre premier des variétés de type K3^[n]. Nous déterminons les propriétés du réseau invariant de l'automorphisme et de son complément orthogonal dans le deuxième réseau de cohomologie de la variété et nous classifions leurs classes d’isométrie. Dans le cas des involutions, e des automorphismes d’ordre premier impair pour n = 3, 4, nous montrons que toutes les actions en cohomologie dans notre classification sont réalisées par un automorphism non-symplectique sur une variété de type K3^[n]. Nous construisons explicitement l’immense majorité de ces automorphismes et, en particulier, nous présentons la construction d’un nouvel automorphisme d’ordre trois sur une famille de dimension dix de variétés de Lehn-Lehn-Sorger-van Straten de type K3^[4]. Pour n < 6, nous étudions aussi les espaces de modules de dimension maximal des variétés de type K3^[n] munies d’une involution non-symplectique.

The purpose of this thesis is to construct geometric objects on a manifold C parametrizing rational curves in the twistor space of a hyperkähler manifold. We shall establish a correspondence between the complex geometry of the twistor space and some differential properties of C (differential operators and curvature of a complex riemannian structure inherited from the base hyperkähler manifold). The first chapters gather some classical results of the theory of hyperkähler manifolds and their twistor spaces. In the chapters 4, 5 and 6, we construct an equivalence of categories between bundles on the twistor space which are trivial on each line and bundles with a connexion of C satisfying certain curvature conditions. The chapter 7 extends this correspondence on the cohomological level whereas the chapter 8 explores its infinitesimal version ; it links curvature of the connexion with thickening (in the sense of LeBrun) of the bundle along the lines., L'objet de cette thèse est la construction d'objets géométriques sur une variété C paramétrant des courbes rationnelles dans l'espace des twisteurs d'une variété hyperkählérienne. On établira une correspondance entre la géométrie complexe de l'espace des twisteurs et des propriétés différentielles sur C (opérateurs différentiels et courbure de la structure riemanienne complexe héritée de la variété hyperkählérienne). Les premiers chapitres précisent le cadre et les résultats connus. Dans les chapitres 4, 5 et 6 on établit une équivalence de catégories entre fibrés triviaux en restriction à chaque droite de l'espace des twisteurs et les fibrés à connexion sur C satisfaisant une condition de courbure. Le chapitre 7 prolonge cette correspondance sur le plan cohomologique tandis que le chapitre 8 en fait l'étude infinitésimale en reliant la courbure de la connexion avec les épaississements infinitésimaux des fibrés le long des droites.

An exact Lagrangian submanifold $L$ in the symplectization of standard contact $(2n-1)$-space with Legendrian boundary $\Sigma$ can be glued to itself along $\Sigma$. This gives a Legendrian embedding $\Lambda(L,L)$ of the double of $L$ into contact $(2n+1)$-space. We show that the Legendrian isotopy class of $\Lambda(L,L)$ is determined by formal data: the manifold $L$ together with a trivialization of its complexified tangent bundle. In particular, if $L$ is a disk then $\Lambda(L,L)$ is the Legendrian unknot., Comment: 8 pages. An assumption on regularity of Lagrangian fillings was removed following a suggestion by Yang Huang. Version contains Emmy Murphy's proof of looseness of Legendrian spheres obtained by gluing two distinct disk fillings. (These spheres were originally claimed elsewhere by the second author to be non-loose.)

We interest ourselves in the spectral theory of non self-adjoint semi-classical operators in dimension one and in asymptotic expansions of eigenvalues. These expansions are written in terms of geometrical objects in a complex phase space coming from classical mechanics and correspond to a generalization of Bohr-Sommerfeld quantization conditions in the non self-adjoint case. First, we study non self-adjoint perturbations of self-adjoint pseudo-differential operators in dimension one by using techniques of analytic microlocal analysis. As a corollary, we establish for PT-symmetric perturbations of self-adjoint operators, that the spectrum is real. Then we show Bohr-Sommerfeld quantization conditions for non self-adjoint perturbations of self-adjoint Berezin-Toeplitz operators of the complex plane. In the second part, we look into quantizations of the torus, namely the Berezin-Toeplitz, the classical Weyl and the complex Weyl quantizations of the torus. We establish links between these different quantizations using Bargmann transform. We propose a conjecture, supported by numerical simulations, on Bohr-Sommerfeld quantization conditions for non self-adjoint perturbations of self-adjoint Berezin-Toeplitz operators of the torus., On s'intéresse à la théorie spectrale d'opérateurs semi-classiques non auto-adjoints en dimension un et plus précisément aux développements asymptotiques des valeurs propres. Ces derniers font intervenir des objets géométriques issus de la mécanique classique dans l'espace des phases complexifié et correspondent à une généralisation des conditions de quantification de Bohr-Sommerfeld au cadre non auto-adjoint. Plus précisément, dans un premier temps, on étudie le spectre de perturbations non auto-adjointes d'opérateurs pseudo-différentiels auto-adjoints en dimension un à l'aide de techniques d'analyse microlocale analytique et en corollaire, on établit que pour des perturbations PT-symétriques d'opérateurs auto-adjoints, le spectre est réel. Ensuite, on présente des conditions de quantification de Bohr-Sommerfeld pour des perturbations non auto-adjointes d'opérateurs de Berezin-Toeplitz du plan complexe auto-adjoints. Dans un second temps, on s'intéresse aux différentes quantifications du tore et plus précisément à la quantification de Berezin-Toeplitz du tore, à la quantification de Weyl classique du tore et à la quantification de Weyl complexe du tore. On établit des liens entre ces différentes quantifications notamment grâce à la transformée de Bargmann, puis à l'aide de simulations numériques, on met en évidence une conjecture sur des conditions de quantification de Bohr-Sommerfeld pour des perturbations non auto-adjointes d'opérateurs de Berezin-Toeplitz du tore auto-adjoints.

This thesis establishes a topological constraint on the fundamental group of some monotone Lagrangien. One useful consequence is to rule out a class of examples of Lagrangians due to L. Polterovich as monotone ones. It also leads to a discussion on the possible topologies en dimension 3., Les sous-variées isotropes maximales en géométries symplectique sont appelées lagrangiennes ; parmi celles-ci on distingue les lagrangiennes monotones. Historiquement leur définition est motivée en partie par la construction de l'homologie de Floer lagrangiennes ; elles présentent ainsi une classe plus rigide, moins étendue, de lagrangiennes. Ce manuscrit établit une contrainte sur le groupe fondamental de certaines lagrangiennes monotones, qui s'applique en particulier lorsque la variété symplectique ambiante est l'espace projectif complexe. Une des conséquences du théorème principal est d'exclure toute une classe d'exemples classiques de lagrangiennes, due à L. Polterovich, du cas monotone. Elle conduit également à une discussion sur les topologies possibles en dimension 3.

We define an index for the critical points of parametrized Hamiltonian action functionals. The expected dimension of moduli spaces of parametrized Floer trajectories equals the difference of indices of the asymptotes., Comment: 18 pages. This paper contains and extends the discussion of the index that was part of the first version of our paper arXiv:0909.4526. To appear in Geometriae Dedicata, Special Issue GESTA 2011

In this paper we introduce the concept of decorated character variety for the Riemann surfaces arising in the theory of the Painlev\'e differential equations. Since all Painlev\'e differential equations (apart from the sixth one) exhibit Stokes phenomenon, it is natural to consider Riemann spheres with holes and bordered cusps on such holes. The decorated character is defined as complexification of the bordered cusped Teichm\"uller space introduced in arXiv:1509.07044. We show that the decorated character variety of a Riemann sphere with s holes and n>1 bordered cusps is a Poisson manifold of dimension 3 s+ 2 n-6 and we explicitly compute the Poisson brackets which are naturally of cluster type. We also show how to obtain the confluence procedure of the Painlev\'e differential equations in geometric terms., Comment: Parts of this paper overlap with arXiv:1212.6723, but it presents a much deeper geometric understanding and the new notion of decorated character variety which was not included there. 19 coloured pictures, 33 pages. To appear on Int. Mat. Res. Not

The long exact sequence describes how the Floer cohomology of two Lagrangian submanifolds changes if one of them is modified by applying a Dehn twist. We give a proof in the simplest case (no bubbling). The paper contains a certain amount of material that may be of interest independently of the exact sequence: in particular, chapter 1 covers "symplectic Picard-Lefschetz theory" in some detail, and chapter 2 contains a generalization of the usual TQFT setup for Floer cohomology., 70 pages, 12 eps figures

We discuss the properties of a certain type of Dehn surgery along a Lagrangian torus in a symplectic 4-manifold, known as Luttinger's surgery, and use this construction to provide a purely topological interpretation of a non-isotopy result for symplectic plane curves with cusp and node singularities due to Moishezon., 15 pages

International audience; I show some ways of applying the meromorphic non-integability theorems of Ziglin and Morales-Ramis to the {\em real} non-integrability of some Hamiltonian systems.

To be published in Proceedings of “New Spaces in Mathematics & Physics”, 28 Septembre - 2 Octobre 2015, Institut Henri Poincaré, Paris, France.

In this thesis we study the relations between the contact topological properties of contact manifolds and the dynamics of Reeb flows. On the first part of the thesis, we establish a relation between the growth of the cylindrical contact homology of a contact manifold and the topological entropy of Reeb flows on this manifold. We build on this to show in Chapter 6 that if a contact manifold M admits a hypertight contact form A for which the cylindrical contact homology has exponential homotopical growth rate, then the Reeb flow of every contact form on M has positive topological entropy. Using this result, we exhibit in Chapter 8 and 9 numerous new examples of contact 3-manifolds on which every Reeb flow has positive topological entropy. On Chapter 10 we present a joint result with Chris Wendl that gives a dynamical obstruction for contact 3-manifold to be planar. We then use the obstruction to show that a contact 3-manifold that possesses a Reeb flow that is a transversely orientable Anosov flow, cannot be planar. On Chapter 11 we study the topological entropy for Reeb flows on spherizations. The result we obtain is a refinement of a result of Macarini and Schlenk, that states that every Reeb flow on the unit tangent bundle U of a high genus surface S has positive topological entropy. We show that for any Reeb flow on U, the omega-limit of almost every Legendrian fiber is a compact invariant set on which the dynamics has positive topological entropy., L'objectif de cette thèse est d'investiguer les relations entre les propriétés topologiques d'une variété de contact et la dynamique des flots de Reeb dans la variété de contact en question. Dans la première partie de la thèse, nous établissons une relation entre la croissance de l’homologie de contact cylindrique d'une variété de contact et l'entropie topologique des flots de Reeb dans cette variété de contact. Nous utilisons ce résultat dans les chapitres 8 et 9 pour montrer l'existence d'un grand nombre des nouvelles variétés de contact de dimension 3 dans lesquelles tous les flots de Reeb ont entropie topologique positive. Dans le chapitre 10, nous prouvons un résultat obtenu en collaboration avec Chris Wendl qui donne une obstruction dynamique pour qu'une variété de contact de dimension 3 soit planaire. Cette obstruction est utilisée pour montrer que, si une variété de contact de dimension 3 possède un flot de Reeb qui est uniformément hyperbolique (Anosov) avec variétés invariantes traversalement orientables, alors cette variété de contact n'est pas planaire. Dans le chapitre 11, nous étudions l'entropie topologique des flots de Reeb dans les fibrés unitaires des surfaces de genre plus grand que 1. Nous montrons que la restriction de chaque flot de Reeb en au ensemble limite de presque toute fibre unitaire a une entropie topologique positive.

International audience; In this paper, we give a symplectic proof of the Horn inequalities on eigenvalues of a sum of two Hermitian matrices with given spectra. Our method is a combination of tropical calculus for matrix eigenvalues, combinatorics of planar networks, and estimates for the Liouville volume. As a corollary, we give a tropical description of the Duistermaat-Heckman measure on the Horn polytope.

In a previous article, we proved that symplectic homeomorphisms preserving a coisotropic submanifold C, preserve its characteristic foliation as well. As a consequence, such symplectic homeomorphisms descend to the reduction of the coisotropic C. In this article we show that these reduced homeomorphisms continue to exhibit certain symplectic properties. In particular, in the specific setting where the symplectic manifold is a torus and the coisotropic is a standard subtorus, we prove that the reduced homeomorphism preserves spectral invariants and hence the spectral capacity. To prove our main result, we use Lagrangian Floer theory to construct a new class of spectral invariants which satisfy a non-standard triangle inequality., 39 pages, to appear in Annales scientifiques de l'\'Ecole normale sup\'erieure

Aubry-Mather is traditionally concerned with Tonelli Hamiltonian (convex and super-linear). In \cite{Vi,MVZ}, Mather's $\alpha$ function is recovered from the homogenization of symplectic capacities. This allows the authors to extend the Mather functional to non convex cases. This article shows that the relation between invariant measures and the subdifferential of Mather's functional (which is the foundational statement of Mather) is preserved in the non convex case. We give applications in the context of the classical KAM theory to the existence of invariant measures with large rotation vector after the possible disappearance of some KAM tori.

This text presents some basic notions in symplectic geometry, Poisson geometry, Hamiltonian systems, Lie algebras and Lie groups actions on symplectic or Poisson manifolds, momentum maps and their use for the reduction of Hamiltonian systems. It should be accessible to readers with a general knowledge of basic notions in differential geometry. Full proofs of many results are provided. The Marsden-Weinstein reduction procedure and the Euler-Poincar\'e equation are discussed. Symplectic cocycles involved when a momentum map is equivariant with respect to an affine action whose linear part is the coadjoint action are presented. The Hamiltonian actions of a Lie group on its cotangent bundle obtained by lifting the actions of the group on itself by translations on the left and on the right are fully discussed. We prove that there exists a two-parameter family of deformations of these actions into a pair of mutually symplectically orthogonal Hamiltonian actions whose momentum maps are equivariant with respect to an affine action involving any given Lie group symplectic cocycle. In the last section three classical examples are considered: the spherical pendulum, the motion of a rigid body around a fixed point and the Kepler problem. For each example the Euler-Poincar\'e equation is derived, the first integrals linked to symmetries are given. In this Section, the classical concepts of vector calculus on an Euclidean three-dimensional vector space (scalar, vector and mixed products) are used and their interpretation in terms of concepts such as the adjoint or coadjoint action of the group of rotations are explained., Comment: 70 pages This new version corrects a few misprints

We enumerate, via floor diagrams, complex and real curves in the projective plane blown up in $n$ points on a conic. As an application, we deduce Gromov-Witten and Welschinger invariants of Del Pezzo surfaces. These results are mainly obtained using Li's degeneration formula and its real counterpart., Comment: 49 pages, 8 figures, v3: correction of a small mistake in the computations of example 7.9 (that unfortunately remains present in the published version)

This document concerns the rigidity/flexibility phenomena related to symplectic embeddings, and their applications in symplectic topology. The two main themes developped here are the equidimensionnal symplectic embeddings in dimension 4 and the C^0 symplectic geometry., Ce mémoire concerne les phénomènes de rigidité/flexibilité liés aux plongements et leurs applications en topologie symplectique. Les deux grands thèmes abordés sont les plongements symplectiques équidimensionnels en dimension 4 et la géometrie symplectique C^0.

This PhD thesis in applied mathematics was conducted within the Electronic Systems andSensors department of the CEA-Leti (Atomic Energy and Alternative Energies Commission - Laboratory for Electronics and Information Technologies), a major organism for technological research located in Grenoble, France. This work originated from a partnership with the applied mathematics laboratory (LJK) of the Joseph Fourier university (UJF). The Leti develops embedded systems equiped with micro-sensors (magnetometers, accelerometers...) from which it is possible to retrieve informations about their spatial orientation. These systems allow for innovative applications in the field of shape acquisition and reconstruction. The problem of reconstructing surfaces from unstructured orientation data is ill-posed. However, previous work done within the Leti came up with a valid reconstruction protocol. The micro-sensors were integrated into the Morphosense ribbon : this flexible ribbon instrumented with sensor knots according to a known geometry is at the core of a number of reconstruction algorithms for the curves followed by the ribbon. When lied on a physical surface, Morphosense ribbons then allow the acquisition and reconstruction of a network of curves on the surface, that are then used for the reconstruction of the entire surface. We first propose new algorithms for curve reconstruction thanks to the Morphosense ribbon. Those new methods now integrate the orientation informations provided by the sensors in their entirety, as well as the mechanical properties of the ribbon that force it to follow geodesic curves on a surface. From this point of view, the curve reconstruction can be considered optimal, as it integrates all the information embedded in the ribbons' structure. We then study a set of methods for the reconstruction of surfaces using a network of ribbon curves. Such a network generally leads to problems linked to the closure of the network and missing data estimation. The closure of the network is essentially a numerical problem related to differential constraints. The missing data corresponds to the lack of information on the surface outside the network of curves. In order to deal with these problems and propose practical solutions for the reconstruction, hypotheses either on the surface models or the topology of the network of curves are required. Therefore, the developed methods fall within the two following approaches.– On the one hand, reconstruction methods for developable and quasi-developable surfaces, which are a good approximation for the surfaces considered in numerous applications.– On the other hand, reconstruction methods from networks of curves with specific topologies (quasi-planar curves, open network) so as to deal with the closure problem.The set of methods developed in this work allow to formulate a global process for the reconstruction of surfaces, with flexible algorithms adapting to the different practical situations, so as to propose a solution both simple and precise in each case. The validation of our results in the case of real sensors data provided by the Morphosense ribbons also led us to develop metrological device. Finally, notice that the general context of reconstruction from orientation data studied here raises original theorical problems, to which we tried to answer with innovative solutions through interpolation and optimization algorithms., Ce travail de thèse en Mathématiques Appliquées a été effectué au sein du service Capteurs et Systèmes Electroniques (SCSE) au CEA-Leti, organisme majeur de la recherche publique française implanté à Grenoble. Il s'inscrit dans le cadre d'une collaboration avec le laboratoire de mathématiques appliquées Jean Kuntzmann (LJK) de l'Université Joseph Fourier (UJF). Le Leti développe des systèmes de capteurs de données terrestres (magnétomètres, accéléromètres...) capables de se géoréférencer de manière autonome. Placés sur des objets, ces dispositifs de capteursfournissent leur orientation propre dans l'espace, et ouvrent donc un vaste champ d'applications dans le domaine de l'acquisition et la reconstruction de formes.Le problème de la reconstruction de surfaces à partir de données d'orientation non structurées est par essence un problème mal posé. Cependant, des travaux précédents effectués au Leti ont permis de dégager un protocole fournissant un cadre valide pour le processus de reconstruction. Les capteurs ont été intégrés dans les rubans Morphosense : ces rubans souples équipés de noeuds de capteurs selon unegéométrie connue permettent ainsi le développement d'algorithmes de reconstruction de la courbe suivie par le ruban. L'application de rubans Morphosense sur une surface physique permet alors d'acquérir la famille des courbes suivies par les rubans et tracées sur la surface. Il s'agit ensuite d'exploiter le réseau des courbes ainsi obtenues pour reconstruire la surface. Dans un premier temps, nous revisitons la question de la reconstruction du ruban. Nous proposons des algorithmes de reconstruction de la courbe 3D suivie par le ruban Morphosense prenant maintenant en compte l'intégralité des données fournies par les capteurs d'orientation, ainsi que les propriétés méca-niques du ruban qui le conduisent à suivre des courbes géodésiques sur une surface. De ce point de vue, la reconstruction peut être considérée comme optimale.On étudie ensuite un ensemble de méthodes pour la reconstruction de surfaces à partir d'un réseau de courbes rubans. Dans le cas général, un tel type de réseau conduit à des problèmes de fermeture et d'estimation de données manquantes. La question de la fermeture, d'ordre essentiellement numérique et liée à des contraintes différentielles, concerne le réseau des courbes et la difficulté d'obtenir des contoursfermés. La question cruciale de l'estimation des données manquantes traduit le fait qu'aucune information sur la surface n'est connue et accessible en dehors du réseau des courbes rubans.Afin de s'affranchir de ces problèmes et de proposer des solutions pratiques pour la reconstruction, il est nécessaire de faire des hypothèses sur le modèle de surfaces à reconstruire ou sur la topologie de réseau de courbes acquises. Les méthodes développées s'inscrivent donc dans l'une des deux approchessuivantes.– D'une part des méthodes de reconstruction de surfaces développables et quasi-développables, qui modélisent de manière satisfaisante les surfaces étudiées dans le cadre de nombreuses applications.– D'autre part des méthodes de reconstruction à partir d'une topologie spécifique de réseaux de courbes (courbes quasi-planaires, contour ouvert), permettant de résoudre le problème de fermeture.L'ensemble des méthodes proposées dans ces travaux permet ainsi de formuler un processus global de reconstruction de surfaces, qu'il est possible d'adapter aux problèmes étudiés en pratique, afin de proposer une solution à la fois simple et précise dans chaque cas. La validation des résultats dans le cadre des données réelles fournies par les rubans Morphosense nous a conduit à développer des dispositifs métrologiques. Enfin, notons que le contexte général des données d'orientation étudié ici soulève des problématiques peu classiques, voire nouvelles, auxquelles nous avons essayé d'apporter des solutions originales, en particulier au travers d'algorithmes d'interpolation et d'optimisation.

Le th\'eor\`eme principal de cet article est un r\'esultat de transversalit\'e quantitative relatif \`a une sous-vari\'et\'e. The main theorem of this paper is a result of estimated transversality with respect to a given submanifold., Comment: 34 pages, in French

Summary of main work 1999-2012, Comment: Habilitation memoir, Paris-Sud (Orsay) 12/12/12, (40 pages, 8 figures)

22 pages; We classify nonabelian extensions of Lie algebroids in the holomorphic or algebraic category, and introduce and study a spectral sequence that one can attach to any such extension and generalizes the Hochschild-Serre spectral sequence associated to an ideal in a Lie algebra. We compute the differentials of the spectral sequence up to $d_2$.

In this thesis, we study the solutions of Hamilton-Jacobi equations. We will compare the viscosity solution and the minmax solution, with the latter defined by a geometric method. In the literature, there are well-known cases where these two solutions coincide: if the Hamiltonian is convex or concave with respect to the momentum variable, the 8 minmax can be reduced to min or max. The minmax and viscosity solutions are different in general. We will construct "iterated minmax" by iterating the minmax step by step and prove that, as the size of steps go to zero, the iterated minmax converge to the viscosity solution. In particular, we study the equations of conservation laws in dimension one, where, by the "front tracking" method, we shall see that in the case where the initial function is convex, the viscosity solution and the minmax are equal. And as an application, we use the limiting iterated process to describe the singularities of the viscosity solution. In the end, we show that the notion of minmax is not so obvious., Dans cette thèse, nous étudions les solutions des équations Hamilton-Jacobi. Plus précisément, nous comparons la solution de viscosité, obtenue comme limite de solutions de l'équation perturbée par un petit terme de diffusion, et la solution minmax, définie géométriquement à partir d'une fonction génératrice quadratique à l'infini. Dans la littérature, il y a des cas bien connus où les deux coïncident, par exemple lorsque le hamiltonien est convexe ou concave, le minmax pouvant alors être réduit à un min ou un max. Mais les solutions minmax et de viscosité diffèrent en général. Nous construisons des "minmax itérés" en répétant pas à pas la procédure de minmax et démontrons que, quand la taille du pas tend vers zéro, les minmax itérés tendent vers la solution de viscosité. Dans une deuxième partie, nous étudions les lois de conservation en dimension un d'espace par le méthode de "front tracking". Nous montrons que dans le cas où la donnée initiale est convexe, la solution de viscosité et le minmax sont égaux. Et comme application, nous décrivons sur des exemples la manière dont sont construites les singularités de la solution de viscosité. Pour finir, nous montrons que la notion de minmax n'est pas aussi évidente qu'il y paraît.