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Let S be a compact surface - or the interior of a compact surface - and let V be the manifold of cooriented contact elements of S equiped with its canonical contact structure. A diffeomorphism of V that preserves the contact structure and its coorientation is called a contact transformation over S. We prove the following results. 1) If S is neither a sphere nor a torus then the inclusion of the diffeomorphism group of S into the contact transformation group is 0-connected. 2) If S is a sphere then the contact transformation group is connected. 3) if S is a torus then the homomorphism from the contact transformation group of S to the automorphism group of $H_1(V) \simeq Z^3$ has connected fibers and the image is (known to be) the stabilizer of $Z^2 \times \{0\}$)., 15 pages, LaTeX

The paper continues the discussion of symplectic aspects of Picard-Lefschetz theory begun in "Vanishing cycles and mutation" (this archive). There we explained how to associate to a suitable fibration over a two-dimensional disc a triangulated category, the "derived directed Fukaya category" which describes the structure of the vanishing cycles. The present second part serves two purposes. Firstly, it contains various kinds of algebro-geometric examples, including the "mirror manifold" of the projective plane. Secondly there is a (largely conjectural) discussion of more advanced topics, such as (i) Hochschild cohomology, (ii) relations between Picard-Lefschetz theory and Morse theory, (iii) a proposed "dimensional reduction" algorithm for doing certain Floer cohomology computations., 33 pages, LaTeX2e, 9 eps figures

Let M be a symplectic manifold acted on by a compact Lie group G in a Hamiltonian fashion, with proper moment map. In this situation we introduce a pushforward morphism P : H ∗ G (M)→ M −∞ ( g ∗ ) G , from the equivariant cohomology of M to the space of G-invariant distributions on g ∗ , which gives rise to symplectic invariants, in particular the pushforward of the Liouville measure. For the study of this pushforward morphism we make an intensive use of equivariant forms with generalized coefficients.

We consider an almost complex structure J on CP2, or more generally an elliptic structure E which is tamed by the standard symplectic structure. An E-curve is a surface tangent to E (this generalizes the notion of J(holomorphic)-curve), and an E-line is an E-curve of degree 1. We prove that the space of E-lines is again a CP2 with a tame elliptic structure E^*, and that each E-curve has an associated dual E^*-curve. This implies that the E-curves, and in particular the J-curves, satisfy the Pl\"ucker formulas, which restricts their possible sets of singularities., Comment: 18 pages The only difference with the first version is the mention of the thesis of Benjamin MacKay ("Duality and integrable systems of pseudoholomorphic curves", Duke University, 1999), which I did not know at the time, and which contains a large part of the results of my paper

In this paper, we study the global behaviour of contact structures on oriented manifolds V which are circle bundles over a closed orientable surface S of genus g>0. We establish in particular contact analogs of a number of classical results about foliations due to Milnor, Wood, Thurston, Matsumoto, and Ghys. In Section~1, we prove that V carries a (positive) contact structure transverse to the fibers if and only if the Euler number of the fibration is less or equal to 2g-2. In Section~2, we show that, for any contact structure $��$ on V, one of the following properties holds: either $��$ is isotopic to a contact structure transverse to the fibers or there exists, in some finite sheeted cover of V, a Legendrian curve isotopic to the fiber along which $��$ determines the same framing as the fibration $V \to S$. In Section 3, we classify contact structures that are transverse to the fibers up to isotopy and conjugation. In Section 4, we study general tight contact structures on V. We prove that virtually over-twisted contact structures form finitely many isotoy classes while isotopy classes of universally tight contact structures are in one-to-one correspondence with isotopy classes of systems of essential curves on S., 37 pages, LaTeX

International audience; PSA results are expected to be used to improve the design, the operating procedures or the safety policy of a system. It then appears logical to combine a risk analysis with an optimisation procedure, aiming at minimising a cost function to determine the most appropriate strategy. This idea is applied in this work to dynamic systems, i.e. installations for which the probabilistic risk analysis must be conducted using dynamic reliability methods. As several system parameters can be uncertain, the cost function turns out to be a random variable. Stochastic programming techniques are convenient for such calculations. The whole optimisation procedure is presented, and a first application of the algorithm on a HNO3 supply system is provided.

In this thesis we study the natural symplectic geometry of moduli spaces of meromorphic connections (with arbitrary order poles) over Riemann surfaces. The aim is to understand the symplectic geometry of the monodromy data of such connections, involving Stokes matrices. This is motivated by the appearance of Stokes matrices in the theory of Frobenius manifolds due to Dubrovin, and in the derivation of the isomonodromic deformation equations of Jimbo, Miwa and Ueno.The main results of this thesis are:1) An extension to the meromorphic case of the infinite dimensional description, due to Atiyah and Bott, of the symplectic structure on moduli spaces of flat connections. This involves using an appropriate notion of singular $C^\infty$ connections and realises the natural moduli space of monodromy data as an infinite dimensional symplectic quotient.2) An explicit finite dimensional symplectic description of moduli spaces of meromorphic connections on trivial holomorphic vector bundles over the Riemann sphere. A similar description is given of certain extended moduli spaces involving a compatible framing at each pole; these are the phase spaces of the isomonodromic deformation equations.3) A proof that the monodromy map is a symplectic map. In other words the above two symplectic structures are related by the transcendental map taking meromorphic connections to their monodromy data. The analogue of this result in inverse scattering theory is well-known and was important in developing the quantum inverse scattering method.4) A symplectic description of the full family of Jimbo-Miwa-Ueno isomonodromic deformation equations. In modern language we prove that the isomonodromic deformation equations are equivalent to a flat symplectic Ehresmann connection on a symplectic fibre bundle. This fits together, into a uniform framework, all the previous results for the six Painlevé equations and Schlesinger's equations.5) Finally we look at the simplest case involving Stokes matrices in detail. We present a conjecture relating Stokes matrices to Poisson-Lie groups (which we prove in the simplest case) and also prove directly that in low-dimensional cases the Poisson structure on the local moduli space of semisimple Frobenius manifolds does arise from a Poisson-Lie group.; Dans cette thèse, nous étudions la géométrie symplectique naturelle des espaces de modules de connexions méromorphes (avec des pôles d'ordre arbitraire) sur les surfaces de Riemann. L'objectif est de comprendre la géométrie symplectique des données de monodromie/Stokes de telles connexions. Ceci est motivé par l'apparition de matrices de Stokes dans la théorie des variétés de Frobenius due à Dubrovin et dans la dérivation des équations de déformation isomonodromique de Jimbo, Miwa et Ueno.Les principaux résultats de cette thèse sont:1) Une extension au cas méromorphe de la description de dimension infinie, due à Atiyah et Bott, de la structure symplectique sur les espaces de modules de connexions plates. Pour faire ca on utilise une notion appropriée de connexions singulières $C^\infty$ et réalise l'espace de modules naturel des données de monodromie/Stokes comme un quotient symplectique à dimensions infinies.2) Description symplectique à dimensions finies explicite d'espaces de modules de connexions méromorphes sur des fibrés vectoriels holomorphes triviaux sur la sphère de Riemann. Une description similaire est donnée de certains espaces de modules étendus impliquant un cadrage compatible à chaque pôle; ce sont les espaces de phase des équations de déformation isomonodromique.3) Une preuve que l'application monodromie est une application symplectique. En d'autres termes, les deux structures symplectiques ci-dessus sont reliées par l'application transcendantale prenant des connexions méromorphes à leurs données de monodromie. L'analogue de ce résultat dans la théorie d'inverse scattering est bien connu et a joué un rôle important dans le développement de la méthode quantique d'inverse scattering.4) Description symplectique de la famille complète des équations de déformation isomonodromique de Jimbo-Miwa-Ueno. En langage moderne, nous prouvons que les équations de déformation isomonodromique sont équivalentes à une connexion plate symplectique d'Ehresmann sur un fibrés non linéaire, de fibres symplectique. Cela généralise, dans un cadre uniforme, tous les résultats précédents pour les six équations de Painlevé et les équations de Schlesinger.5) Enfin, nous examinons en détail le cas le plus simple impliquant des matrices de Stokes. Nous présentons une conjecture reliant les matrices de Stokes à des groupes de Poisson-Lie (ce que nous prouvons dans le cas le plus simple) et prouvons directement que, dans les cas de petit dimension, la structure de Poisson sur l'espace des modules locaux des variétés semi-simples de Frobenius découle d'un groupe de Poisson-Lie.

International audience; We define a variational principle for symplectic connections of the Yang-Mills type. When the symplectic manifold is a compact surface we show that the moduli space of the connections which are extremals of the functional coincides with the Teichmüller space of the surface. We indicate that the noncompact situation is very different.

International audience; We introduce a variational principle for symplectic connections and study the corresponding field equations. For two-dimensional compact symplectic manifolds we determine all solutions of the field equations. For two-dimensional non-compact simply connected symplectic manifolds we give an essentially exhaustive list of solutions of the field equations. Finally we indicate how to construct from solutions of the field equations on $(M, \omega)$ solutions of the field equations on the cotangent bundle to M with its standard symplectic structure.

International audience; In this paper, we develop a method of localization in equivariant cohomology based on the notion of partition of unity cohomology. We apply this method in two cases. In the first case, this method gives a refinement of the localization of Atiyah-Bott and Berline-Vergne (in the frame given by Bismut). After, we consider the Hamiltonian action of a torus, and we realise, following the idea of Witten, the localization on the critical points of the square of the moment map.

Resume On construit une structure de contact sur un espace de Hurwitz de courbes hyperelliptiques de genre n . Elle est definie a l'aide d'une famille de formes meromorphes qui permet aussi de construire des coordonnees d'action pour le systeme des geodesiques de l'ellipsoide.

International audience; Using an idea of E. Witten (J. Geom. Phys. 9, 1992) we give a localization formula in equivariant cohomology in the case of a Hamiltonian torus action on a compact symplectic manifold X. The integral over X of an equivariant closed form can be written as an integral over the submanifold of critical points of the square of the moment map.

International audience

In these notes, I will describe the moduli space of flat connections on a principal bundle over a surface and its Poisson structure. I will then give examples of integrable systems on these spaces, following ideas of Goldman, Jeffrey and Weitsman, Fock and Roslyi, and Alekseev.

International audience; The invariants of filtered complexes, called "canonical forms", are introduced. The canonical form of a filtered complex describes births and deaths of cycles across the filtration. The canonical form gives a partition of the set of critical values (indices) into pairs "birth-death" and a separate set representing the homology of the complex. The canonical form is obtained by action of upper-triangular matrices, that reduce the filtered complex to its simplest form. An efficient algorithm for computation of the canonical forms of filtered complexes is described. These "canonical forms" invariants, about ten years later after the appearance of this paper, began to be widely used in applied mathematics under the name of "Persistence Diagrams" or "Persistence Bar-codes". The main result of the paper applies this canonical form invariant to solve the following problem: given a function defined on a neighborhood of boundary of an n-dimensional ball, calculate optimal estimate from below for the number of Morse critical points that any generic extension of the function inside the ball must have.

International audience

International audience

Introduction: Applications of pseudo-holomorphic curves to symplectic topology.- 1 Examples of problems and results in symplectic topology.- 2 Pseudo-holomorphic curves in almost complex manifolds.- 3 Proofs of the symplectic rigidity results.- 4 What is in the book... and what is not.- 1: Basic symplectic geometry.- I An introduction to symplectic geometry.- 1 Linear symplectic geometry.- 2 Symplectic manifolds and vector bundles.- Appendix: the Maslov class M. Audin, A. Banyaga, F. Lalonde, L. Polterovich.- II Symplectic and almost complex manifolds.- 1 Almost complex structures.- 2 Hirzebruch surfaces.- 3 Coadjoint orbits (of U(n)).- 4 Symplectic reduction.- 5 Surgery.- Appendix: The canonical almost complex structure on the manifold of 1-jets of pseudo-holomorphic mappings between two almost complexmanifolds P. Gauduchon.- 2: Riemannian geometry and linear connections.- III Some relevant Riemannian geometry.- 1 Riemannian manifolds as metric spaces.- 2 The geodesic flow and its linearisation.- 3 Minimal manifolds.- 4 Two-dimensional Riemannian manifolds.- 5 An application to pseudo-holomorphic curves.- Appendix: the isoperimetric inequality M.-P. Muller.- IV Connexions lineaires, classes de Chern, theoreme de Riemann-Roch.- 1 Connexions lineaires.- 2 Classes de Chern.- 3 Le theoreme de Riemann-Roch.- Bibliographie.- 3: Pseudo-holomorphic curves and applications.- V Some properties of holomorphic curves in almost complex manifolds.- 1 The equation $$ \bar \partial f$$ in C.- 2 Regularity of holomorphic curves.- 3 Other local properties.- 4 Properties of the area of holomorphic curves.- 5 Gromov's compactness theorem for holomorphic curves.- Appendix: Stokes' theorem for forms with differentiable coefficients.- VI Singularities and positivity of intersections of J-holomorphic curves.- 1 Elementary properties.- 2 Positivity of intersections.- 3 Local deformations.- 4 Perturbing away singularities.- Appendix: The smoothness of the dependence on ? Gang Liu.- VII Gromov's Schwarz lemma as an estimate of the gradient for holomorphic curves.- 1 Introduction.- 2 A review of some classical Schwarz lemmas.- 3 Isoperimetric inequalities for J-curves.- 4 The Schwarz and monotonicity lemmas.- 5 Continuous Lipschitz extension across a puncture.- 6 Higher derivatives.- VIII Compactness.- 1 Riemann surfaces with nodes.- 2 Cusp-curves.- 3 Proof of the compactness theorem 2.2.1.- 4 Convergence of parametrised curves.- IX Exemples de courbes pseudo-holomorphes en geometrie riemannienne.- 1 Immersions isometriques elliptiques.- 2 Courbure de Gauss prescrite.- 3 Autres exemples et constructions.- Appendice: convergence d'applications pseudo-holomorphes.- Bibliographie.- X Symplectic rigidity: Lagrangian submanifolds.- 1 Lagrangian constructions.- 2 Symplectic area and Maslov classes-rigidity in split manifolds.- 3 Soft and hard Lagrangian obstructions to Lagrangian embeddings in Cn.- 4 Rigidity in cotangent bundles and applications to mechanics.- 5 Pseudo-holomorphic curves: proof of the main rigidity theorem.- Appendix: Exotic structures on R2n.- Authors' addresses.

International audience

International audience

Resume Nous donnons une nouvelle demonstration d'une formule enumerative de Givental sur les singularites de certaines surfaces lagrangiennes. La methode est de comparer ces surfaces lagrangiennes singulieres a des courbes complexes qui se desingularisent par eclatement. On obtient aussi une formule analogue modulo 4 pour des surfaces non-orientables et on montre comment obtenir tous les exemples de plongements lagrangiens de Givental par chirurgie lagrangienne.

International audience; The aim of this paper is to present aspects of the use of Lie groups in mechanics. We start with the motion of the rigid body for which the main concepts are extracted. In a second part, we extend the theory for an arbitrary Lie group and in a thirdsection we apply these methods for the diffeomorphism group of the circle with two particular examples: the Burger equation and theCamassa-Holm equation.

International audience

Symplectic field theory (SFT) is a collection of homology theories that provide invariants for contact manifolds. We give a proof that vanishing of any one of either contact homology, rational SFT or (full) SFT are equivalent. We call a manifold for which these theories vanish "algebraically overtwisted"., Comment: 9 pages; improved description of SFT, simplified proof, added marked points in main theorem

International audience

International audience

The aim of this paper is to address the following question: given a contact manifold $(\Sigma, \xi)$, what can be said about the aspherical symplectic manifolds $(W, \omega)$ bounded by $(\Sigma, \xi)$ ? We first extend a theorem of Eliashberg, Floer and McDuff to prove that under suitable assumptions the map from $H_{*}(\Sigma)$ to $H_{*}(W)$ induced by inclusion is surjective. We then apply this method in the case of contact manifolds having a contact embedding in $ {\mathbb R}^{2n}$ or in a subcritical Stein manifold. We prove in many cases that the homology of the fillings is uniquely determined. Finally we use more recent methods of symplectic topology to prove that, if a contact hypersurface has a Stein subcritical filling, then all its weakly subcritical fillings have the same homology. A number of applications are given, from obstructions to the existence of Lagrangian or contact embeddings, to the exotic nature of some contact structures., Comment: 32 pages, one figure, one table

International audience; We describe the bifurcation diagrams of almost toric integrable Hamiltonian systems on a four dimensional symplectic manifold MM, not necessarily compact. We prove that, under a weak assumption, the connectivity of the fibers of the induced singular Lagrangian fibration M→R^2 can be detected from the bifurcation diagram alone. In this case, it is possible to give a detailed description of the image of the fibration.

In this paper we identify the cotangent to the derived stack of representations of a quiver $Q$ with the derived moduli stack of modules over the Ginzburg dg-algebra associated with $Q$. More generally, we extend this result to finite type dg-categories, to a relative setting as well, and to deformations of these. It allows us to recover and generalize some results of Yeung, and leads us to the discovery of seemingly new lagrangian subvarieties in the Hilbert scheme of points in the plane., 59 pages. v3: final version. Accepted for publication in Annales Scientifiques de l'ENS

Given an exact Lagrangian submanifold $L$ in $T^*N$, we want to construct a complex of sheaves in the derived category of sheaves on $N\times {\mathbb R} $, such that its singular support, $SS({\mathcal F}^\bullet_L)$, is equal to $\widehat L$, the cone constructed over $L$. Its existence was stated in \cite{Viterbo-ISTST} in 2011, with a sketch of proof, which however contained a gap (fixed here by the rectification). A complete proof was shortly after provided by Guillermou (\cite{Guillermou}) by a completely different method, in particular Guillermou's method does not use Floer theory. The proof provided here is, as originally planned, based on Floer homology. Besides the construction of the complex of sheaves, we prove that the filtered versions of sheaf cohomology of the quantization and of Floer cohomology coincide, that is $FH^*(N\times ]-\infty, \lambda [, {\mathcal F}^\bullet_L)\simeq FH^*(L;0_N;\lambda)$, and so do their product structures.

International audience; Soils are essential for supporting food production and providing ecosystem services but are under pressure due to population growth, higher food demand, and land use competition. Because of the effort to ensure the sustainable use of soil resources, demand for current, updatable soil information capable of supporting decisions across scales is increasing. Digital soil mapping (DSM) addresses the drawbacks of conventional soil mapping and has been increasingly used for delivering soil information in a time- and cost-efficient manner with higher spatial resolution, better map accuracy, and quantified uncertainty estimates. We reviewed 244 articles published between January 2003 and July 2021 and then summarised the progress in broad-scale (spatial extent 10,000 km2) DSM, focusing on the 12 mandatory soil properties for GlobalSoilMap. We observed that DSM publications continued to increase exponentially; however, the majority (74.6%) focused on applications rather than methodology development. China, France, Australia, and the United States were the most active countries, and Africa and South America lacked country-based DSM products. Approximately 78% of articles focused on mapping soil organic matter/carbon content and soil organic carbon stocks because of their significant role in food security and climate regulation. Half the articles focused on soil information in topsoil only (