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In this paper, we prove some convexity results associated to orbit projection of noncompact real reductive Lie groups.

In this paper we consider a punctured Riemann surface endowed with a Hermitian metric that equals the Poincaré metric near the punctures, and a holomorphic line bundle that polarizes the metric. We introduce a new method to compare the Bergman kernels of high tensor powers of the line bundle and of the Poincaré model near the singularity and show that their quotient tends to one uniformly on a neighborhood of the singularity up to arbitrary negative powers of the tensor power.

We introduce and study the barcode entropy for geodesic flows of closed Riemannian manifolds, which measures the exponential growth rate of the number of not-too-short bars in the Morse-theoretic barcode of the energy functional. We prove that the barcode entropy bounds from below the topological entropy of the geodesic flow and, conversely, bounds from above the topological entropy of any hyperbolic compact invariant set. As a consequence, for Riemannian metrics on surfaces, the barcode entropy is equal to the topological entropy. A key to the proofs and of independent interest is a crossing energy theorem for gradient flow lines of the energy functional., Comment: 41 pages

This work is at the intersection of dynamical systems and contact geometry, and it focuses on the effects of a contact surgery adapted to the study of Reeb fields and on the effects of invariance of contact homology. We show that this contact surgery produces an increased dynamical complexity for all Reeb flows compatible with the new contact structure. We study Reeb Anosov fields on closed 3manifolds that are not topologically orbit-equivalent to any algebraic flow; this includes many examples on hyperbolic 3-manifolds. Our study also includes results of exponential growth in cases where neither the flow nor the manifold obtained by surgery is hyperbolic, as well as results when the original flow is periodic. This work fully demonstrates, in this context, the relevance of contact homology to the analysis of the dynamics of Reeb fields.

Let $f$ be a Morse function on a closed surface $\Sigma$ such that zero is a regular value and such that $f$ admits neither positive minima nor negative maxima. In this expository note, we show that $\Sigma\times \mathbb{R}$ admits an $\mathbb{R}$-invariant contact form $\alpha=fdt+\beta$ whose characteristic foliation along the zero section is (negative) weakly gradient-like with respect to $f$. The proof is self-contained and gives explicit constructions of any $\mathbb{R}$-invariant contact structure in $\Sigma \times \mathbb{R}$, up to isotopy. As an application, we give an alternative geometric proof of the homotopy classification of $\mathbb{R}$-invariant contact structures in terms of their dividing set., Comment: 15 pages, 5 figures. Expository note

A b-contact structure on a b-manifold (M, Z) is a singular Jacobi structure on M satisfying a transversality condition along the hypersurface Z. We show that, in three dimensions, b-contact structures with overtwisted three-dimensional leaves satisfy an existence h-principle that allows prescribing the induced singular foliation. The existence of b-contact structures with tight leaves of maximal dimension is also established. We give a method to classify b-contact structures on a given b-manifold and use it to give a classification on S 3 with either a two-sphere or an unknotted torus as the critical surface. We also discuss generalizations to higher dimensions.

Semitoric systems are a type of four-dimensional integrable system for which one of the integrals generates a global $S^1$-action; these systems were classified by Pelayo and Vu Ngoc in terms of five symplectic invariants. We introduce and study semitoric families, which are one-parameter families of integrable systems with a fixed $S^1$-action that are semitoric for all but finitely many values of the parameter, with the goal of developing a strategy to find a semitoric system associated to a given partial list of semitoric invariants. We also enumerate the possible behaviors of such families at the parameter values for which they are not semitoric, providing examples illustrating nearly all possible behaviors, which describes the possible limits of semitoric systems with a fixed $S^1$-action. Furthermore, we introduce natural notions of blowup and blowdown in this context, investigate how semitoric families behave under these operations, and use this to prove that each Hirzebruch surface admits a semitoric family with certain desirable invariants; these families are related to the semitoric minimal model program. Finally, we give several explicit semitoric families on the first and second Hirzebruch surfaces showcasing various possible behaviors of such families which include new semitoric systems., Updated to version accepted by the Memoirs of the AMS, added references and improved exposition. 96 pages and 34 figures

We give a general overview of the influence of William Thurston on the French mathematical school and we show how some of the major problems he solved are rooted in the French mathematical tradition. At the same time, we survey some of Thurston's major results and their impact. The final version of this paper will appear in the Surveys of the European Mathematical Society.

Given a holomorphic family of Bridgeland stability conditions over a surface, we define a notion of spectral network which is an object in a Fukaya category of the surface with coefficients in a triangulated DG-category. These spectral networks are analogs of special Lagrangian submanifolds, combining a graph with additional algebraic data, and conjecturally correspond to semistable objects of a suitable stability condition on the Fukaya category with coefficients. They are closely related to the spectral networks of Gaiotto--Moore--Neitzke. One novelty of our approach is that we establish a general uniqueness results for spectral network representatives. We also verify the conjecture in the case when the surface is disk with six marked points on the boundary and the coefficients category is the derived category of representations of an $A_2$ quiver. This example is related, via homological mirror symmetry, to the stacky quotient of an elliptic curve by the cyclic group of order six., 85 pages, 24 figures. Comments welcome!

We use generating function techniques developed by Givental, Théret and ourselves to deduce a proof in CPd of the homological generalization of Franks theorem due to Shelukhin. This result proves in particular the Hofer-Zehnder conjecture in the non-degenerate case: every Hamiltonian diffeomorphism of CPd that has at least d+2 non-degenerate periodic points has infinitely many periodic points. Our proof does not appeal to Floer homology or the theory of J-holomorphic curves. An appendix written by Shelukhin contains a new proof of the Smith-type inequality for barcodes of Hamiltonian diffeomorphisms that arise from Floer theory, which lends itself to adaptation to the setting of generating functions.; Au moyen de techniques utilisant les fonctions génératrices développées par Givental, Théret et nous-même, nous donnons une preuve sur CPd de la généralisation homologique de Shelukhin du théorème de Franks. Ce résultat démontre, en particulier, la conjecture de Hofer-Zehnder dans le cas non-dégénéré : tout difféomorphisme hamiltonien de CPd ayant au moins d+2 points périodiques non-dégénérés possède une infinité de points périodiques. Notre preuve ne fait pas appel à l'homologie de Floer ou à la théorie des courbes J-holomorphes. Un appendice écrit par Shelukhin propose une nouvelle preuve de l'inégalité de type Smith pour les codes-barres de difféomorphismes hamiltoniens issus de la théorie de Floer se prêtant à l'adaptation au cadre des fonctions génératrices.

A Reeb vector field satisfies the Kupka-Smale condition when all its closed orbits are non-degenerate, and the stable and unstable manifolds of its hyperbolic closed orbits intersect transversely. We show that, on a closed 3-manifold, any Reeb vector field satisfying the Kupka-Smale condition admits a Birkhoff section. In particular, this implies that the Reeb vector field of a $C^\infty$-generic contact form on a closed 3-manifold admits a Birkhoff section, and that the geodesic vector field of a $C^\infty$-generic Riemannian metric on a closed surface admits a Birkhoff section., Comment: 25 pages, 9 figures, version 2: new title; the results are now extended beyond geodesic flows, to general Reeb flows satisfying the Kupka-Smale condition

We first study symplectically embedded curves in symplectic surfaces with high self-intersection numbers compared to their genus. We prove in two different ways that such a curve completely determines both the diffeomorphism type of the surface in which it is embedded and the embedding itself. The first proof uses Seiberg--Witten theory whereas the second one only involves pseudoholomorphic techniques. We deduce from this result that the contact $3$--manifolds naturally associated with those curves admit a unique strong symplectic filling up to diffeomorphism.We next examine symplectic sections of geometrically ruled complex surfaces over elliptic curves. We show that such a section is symplectically isotopic to a complex section.; On étudie dans un premier temps les courbes symplectiquement plongées dans les surfaces symplectiques dont les nombres d'auto-intersection sont suffisamment grands par rapport leurs genres. On montre de deux manières différentes qu'une telle courbe détermine à la fois la classe de difféomorphisme de la surface symplectique qui la contient et la manière dont elle est plongée dans cette surface. La première démonstration fait appel à la théorie de Seiberg--Witten, alors que la seconde se restreint aux techniques pseudoholomorphes. On déduit de ce résultat l'unicité à difféomorphisme près des remplissages symplectiques forts des variétés de contact de dimension $3$ naturellement associées à ce type de courbes. Dans un second temps, on s'intéresse aux sections symplectiques des surfaces complexes géométriquement réglées au-dessus de courbes elliptiques. On montre qu'une telle section est symplectiquement isotope à une section complexe.

This is an expanded version of the talk given be the first author at the conference "Topology, Geometry, and Dynamics: Rokhlin - 100". The purpose of this talk was to explain our current results on classification of rational symplectic 4-manifolds equipped with an anti-symplectic involution. Detailed exposition will appear elsewhere., Comment: 8 pages

In this thesis, we are studying topological and dynamical conditions imposing infinitely many periodic orbits for some dynamical systems. In a first part, weelaborate on theories of Givental and Théret based on generating functions in order to study the case of complex projective spaces. We find recent results back without appealing to the theory of J-holomorphic curves. Inparticular, we prove Shelukhin theorem showing a homology version of the Hofer-Zehnder conjecture. In a second part, we study the geodesic flow and show new results bringing examples of topological and dynamical conditions imposing infinitely manyclosed geodesics or geodesic chords. We give conditions under which the existence of one or two closed geodesics on a complete Riemannian plane,cylinder or Möbius band impose the existence of infinitely many closed geodesics. In particular, we show that a complete Riemannian cylinder (whose closed geodesics are isolated) has zero, one or infinitely many homologically visible closed geodesics ; it solves a conjecture of Alberto Abbondandolo. We also prove that every complete Finsler manifold with an infinite fundamental group that is not homotopy equivalent to a circle possesses infinitely many geometrically distinct geodesic chords joining any given pair of points. Results of this part are partially joint with Tobias Soethe.; Dans cette thèse, nous nous intéressons aux conditions dynamiques ou topologiques imposant l’existence d’un nombre infini de trajectoires périodiques pour certains types de systèmes hamiltoniens. Dans une première partie, nous prolongeons les théories de Givental et Théret basées sur les fonctions génératrices afin d’étudier le cas des espaces projectifs complexes ; nous retrouvons ainsi des résultats très récents sans faire appel à la théorie J-holomorphe. Nous montrons, en particulier, le théorème de Shelukhin démontrant une version homologique de la conjecture de Hofer-Zehnder. Dans une seconde partie, nous nous intéressons aux flots géodésiques et démontrons de nouveaux résultats apportant des exemples de telles conditions dynamiques ou topologiques. Nous énonçons des conditions sous lesquelles la présence d'une ou deux géodésiques fermées géométriquement distinctes sur un plan, un cylindre ou un ruban de Möbius riemannien complet impose la présence d'une infinité de géodésiques fermées géométriquement distinctes. En particulier, nous montrons qu'un cylindre riemannien complet (dont les géodésiques fermées sont isolées) admet zéro, une ou une infinité de géodésiques fermées homologiquement distinctes ; cela répond à une question d'Alberto Abbondandolo. On prouve aussi que toute variété de Finsler complète de groupe fondamental infini et non homotopiquement équivalente à un cercle possède une infinité de géodésiques géométriquement distinctes joignant n'importe quelle paire de points. Les résultats de cette seconde partie sont partiellement issus d’une collaboration avec Tobias Soethe.

We prove that the derived critical locus of a $G$-invariant function $S:X\to\mathbb{A}^1$ carries a shifted moment map, and that its derived symplectic reduction is the derived critical locus of the induced function $S_{red}:X/G\to\mathbb{A}^1$ on the orbit stack. We also provide a relative version of this result, and show that derived symplectic reduction commutes with derived lagrangian intersections., 26 Pages. Many diagrams. V2: minor corrections (typos fixed, references updated). To appear in Advances in Theoretical and Mathematical Physics (ATMP)

International audience; We develop a framework for comparing data manifolds, aimed, in particular, towards the evaluation of deep generative models. We describe a novel tool, Cross-Barcode(P,Q), that, given a pair of distributions in a high-dimensional space, tracks multiscale topology spacial discrepancies between manifolds on which the distributions are concentrated. Based on the Cross-Barcode, we introduce the Manifold Topology Divergence score (MTop-Divergence) and apply it to assess the performance of deep generative models in various domains: images, 3D-shapes, time-series, and on different datasets: MNIST, Fashion MNIST, SVHN, CIFAR10, FFHQ, chest X-ray images, market stock data, ShapeNet. We demonstrate that the MTop-Divergence accurately detects various degrees of mode-dropping, intra-mode collapse, mode invention, and image disturbance. Our algorithm scales well (essentially linearly) with the increase of the dimension of the ambient high-dimensional space. It is one of the first TDA-based practical methodologies that can be applied universally to datasets of different sizes and dimensions, including the ones on which the most recent GANs in the visual domain are trained. The proposed method is domain agnostic and does not rely on pre-trained networks.

This paper establishes an aspect of Bohr's correspondence principle, i.e. that quantum mechanics converges in the high frequency limit to classical mechanics, for commuting semiclassical unitary operators. We prove, under minimal assumptions, that the semiclassical limit of the convex hulls of the quantum spectrum of a collection of commuting semiclassical unitary operators converges to the convex hull of the classical spectrum of the principal symbols of the operators., Comment: 32 pages. Presentation substantially revised and reorganized for clarity. Main Theorem is now in in the introduction, and non central materialshave been moved to appendix 1 and appendix 2. Small mistake in the application of Cayley transform has been fixed

We prove that shifted cotangent stacks carry a canonical shifted symplectic structure. We also prove that shifted conormal stacks carry a canonical Lagrangian structure. These results were believed to be true but no written proof was available in the Artin case., 16 pages. Minor corrections. To appear in Annales de la Facult\'e des Sciences de Toulouse

International audience

International audience; This paper is dedicated to the study of the stability of multiplicities of group representations.

We prove a version of the Arnol'd conjecture for Lagrangian submanifolds of conformal symplectic manifolds: a Lagrangian $L$ which has non-zero Morse-Novikov homology for the restriction of the Lee form $\beta$ cannot be disjoined from itself by a $C^0$-small Hamiltonian isotopy. Furthermore for generic such isotopies the number of intersection points equals at least the sum of the free Betti numbers of the Morse-Novikov homology of $\beta$. We also give a short exposition of conformal symplectic geometry, aimed at readers who are familiar with (standard) symplectic or contact geometry., Comment: 15 pages. v2 fixes some attribution issues, updates bibliography and corrects some typos. v3: major strengthening of the main theorem (Theorem 1.1), plus small corrections

International audience; This is a survey paper on derived symplectic geometry, that will appear as a chapter contribution to the book "New Spaces for Mathematics and Physics", edited by Mathieu Anel and Gabriel Catren. Our goal is to explain how derived stacks can be useful for ordinary symplectic geometry, with an emphasis on examples coming from classical topological field theories. More precisely, we use classical Chern-Simons theory and moduli spaces of flat $G$-bundles and $G$-local systems as leading examples in our journey. We start in the introduction by reviewing various point-of-views on classical Chern--Simons theory and moduli of flat connections. In the main body of the Chapter we try to convince the reader how derived symplectic geometry (after Pantev-To\"en-Vaqui\'e-Vezzosi) somehow reconciles all these different point-of-views.

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 describe a novel tool, that tracks multiscale topology spacial discrepancies between manifolds

In this work, we study some convex cones associated to isotropic representations of symmetric spaces. We explain the inequalities that describe them by means of cohomological conditions. In particular, we study the singular Horn cone which is the counterpart of the classical Horn cone, where the eigenvalues of Hermitian square matrices are replaced by the singular values of rectangular matrices.

International audience; Inspired by the techniques of Givental and Théret, we provide a proof with generating functions of a recent result of Ginzburg-Gürel concerning the periodic points of Hamiltonian diffeomorphisms of CP d. For instance, we are able to prove that fixed points of pseudo-rotations are isolated as invariant sets or that a Hamiltonian diffeomorphism with a hyperbolic fixed point has infinitely many periodic points.; Inspirés des techniques de Givental et Théret, nous donnons une preuve de récents résultats de Ginzburg-Gürel concernant les points périodiques de difféomorphismes hamiltoniens de CPd utilisant les fonctions génératrices. Nous sommes par exemple en mesure de redémontrer que les points fixes des pseudo-rotations sont isolés comme ensemble invariant ou encore qu'un difféomorphisme hamiltonien ayant un point fixe hyperbolique a une infinité de points périodiques.

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

Let Z be the real part of a K{\"a}hler Hamiltonian manifold M. The O'Shea-Sjamaar's Theorem tells us that the moment polytope Delta(Z) corresponds to the anti-invariant part of the Kirwan polytope Delta(M). The purpose of the present paper is to explain how to parameterize the equations of the facets of Delta(Z) in terms of real Ressayre's pairs of Z., Comment: arXiv admin note: text overlap with arXiv:1912.10925

This thesis is devoted to the study of the algebraic integrability of the deformed 5-particle Bogoyavlenskij-Itoh systems. We rely on the method described by the Kowalevski-Painlevé analysis to establish this integrability result. In addition, we show that the nullity or not of the deformation parameters strongly affects the geometry of the divisor at infinity of the generic fiber of the system in question. This allows to show seven different curve configurations, on 2-dimensional (hyperelliptic) Jacobians.; Cette thèse a pour but l'étude de l'intégrabilité algébrique des systèmes de Bogoyavlenskij-Itoh déformés à 5 particules. Nous nous appuyons sur la méthode décrite par l'analyse de Kowalevski-Painlevé pour établir ce résultat d'intégrabilité. De plus, nous montrons que la nullité ou non des paramètres de déformation a des fortes répercussions sur la géométrie du diviseur à l'infini de la fibre générique du système en question. Ceci permet d'exhiber sept configurations de courbes différentes, sur les jacobiennes (hyperelliptiques) de dimension 2.

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

National audience; Convex Geometry is the branch of geometry studying convex sets, mainly in Euclidean spaces. Convex sets occur naturally in Geometry and in many mathematical areas: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of numbers, integral geometry, linear programming, game theory, probability theory, stochastic geometry, stereology etc.. Convex Geometry is also of interest in other scientific and engineering disciplines (e.g. in biology, chemistry, cosmology, geology, pharmaceutics, physics …) where elementary objects (cells, corpuscles, grains, particles, planets …) are often considered as convex sets. This second article deals with distances and measurements on convex sets and more broadly on star-shaped sets, as well as approximations, comparisons and symmetrizations, whose interest lies both in theory and in practice.; La géométrie convexe est la branche de la géométrie traitant des ensembles convexes, principalement dans les espaces euclidiens. Les ensembles convexes se produisent naturellement dans la géométrie et dans beaucoup de domaines mathématiques : analyse convexe, analyse fonctionnelle, géométrie calculatoire, géométrie discrète, géométrie intégrale, géométrie des nombres, géométrie stochastique, programmation linéaire, stéréologie, théorie des jeux, théorie des probabilités,etc. La géométrie convexe concerne aussi d'autres disciplines scientifiques et techniques (e.g. biologie, chimie, cosmologie, géologie, pharmacie, physique...) où les objets élémentaires (cellules, corpuscules, grains, particules, planètes...) sont souvent considérés comme des ensembles convexes. Ce second article porte sur les distances et les mesures sur les ensembles convexes et plus largement sur les ensembles étoilés, ainsi que sur les approximations, comparaisons et symétrisations, dont l’intérêt se situe à la fois en théorie et en pratique.

National audience; Convex Geometry is the branch of geometry studying convex sets, mainly in Euclidean spaces. Convex sets occur naturally in Geometry and in many mathematical areas: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of numbers, integral geometry, linear programming, game theory, probability theory, stochastic geometry, stereology etc.. Convex Geometry is also of interest in other scientific and engineering disciplines (e.g. in biology, chemistry, cosmology, geology, pharmaceutics, physics …) where elementary objects (cells, corpuscles, grains, particles, planets …) are often considered as convex sets. This first article deals with the main definitions and properties and fundamental theorems relating to convex sets and more broadly to star-shaped sets.; La géométrie convexe est la branche de la géométrie traitant des ensembles convexes, principalement dans les espaces euclidiens. Les ensembles convexes se produisent naturellement dans la géométrie et dans beaucoup de domaines mathématiques : analyse convexe, analyse fonctionnelle, géométrie calculatoire, géométrie discrète, géométrie intégrale, géométrie des nombres, géométrie stochastique, programmation linéaire, stéréologie, théorie des jeux, théorie des probabilités,etc. La géométrie convexe concerne aussi d'autres disciplines scientifiques et techniques (e.g. biologie, chimie, cosmologie, géologie, pharmacie, physique...) où les objets élémentaires (cellules, corpuscules, grains, particules, planètes...) sont souvent considérés comme des ensembles convexes. Ce premier article porte sur les principales définitions et propriétés et des théorèmes fondamentaux concernant les ensembles convexes et plus largement sur les ensembles étoilés.

We study Legendrians with boundary, in a contact manifold (V, ξ) with sutured convex boundary, and treat some examples. First we define the cylindrical and wrapped sutured Legendrian homologies of a Legendrian whose boundary is in the suture of ∂V . Moreover those homologies fit into an exact sequence, which conjecturally generalises the exact triangle arising from a Lagrangian filling.The unit conormal construction, applied to a submanifold embedded in a manifold with boundary, is a typical instance of this situation. The main illustration involves braids in a thickened surface : we prove that the conormals of two local pure 2-braids are isotopic (as Legendrians with fixed boundary) if and only if the braids are equivalent. In a second part, we apply the conormal construction to an hyperbolic knot in the 3-sphere, and study the sutured contact manifold obtained by taking the complement of the unit conormal of the knot. We show that the Legendrian contact homology of a fiber in the sutured contact manifold, with its product structure, is a complete invariant of the knot (up to mirror), which can be understood as a sutured version of a recent result.; Nous étudions des legendriennes à bord, incluses dans une variété de contact (V, ξ) à bord convexe suturé, et traitons quelques exemples. Tout d’abord on définit l’homologie cylindrique et enroulée d’une legendrienne dont le bord est inclus dans la suture de ∂V . De plus ces homologies s’inscrivent dans un suite exacte, qui généralise conjecturalement le triangle issu d’un remplissage lagrangien.La construction conormale, appliquée à un sous-variété plongée dans une variété à bord, est un exemple typique d’une telle situation. L’illustration principale concerne les tresses dans une surface épaissie : nous prouvons que les conormaux de deux 2-tresses pures et locales sont isotopes (comme legendriennes à bord fixe) si et seulement si les tresses sont équivalentes. Dans un second temps, nous appliquons la construction conormale à un noeud hyperbolique dans la 3-sphère, et étudions la variété de contact suturée obtenue en retirant un voisinage du conormal du nœud. Nous montrons que l’homologie legendrienne d’une fibre dans cette variété de contact suturée, avec sa structure produit, est un invariant complet du nœud, ce qui peut être vu comme une version suturée d’un récent résultat.

We describe the asymptotic behaviour of the quantum propagator generated by a Berezin-Toeplitz operator with real-valued principal symbol. We also give precise asymptotics for smoothed spectral projectors associated with the operator in the autonomous case; this leads us to introducting quantum states associated with immersed Lagrangian submanifolds. These descriptions involve geometric quantities of two origins, coming from lifts of the Hamiltonian flow to the prequantum bundle and the canonical bundle respectively. The latter are the main contribution of this article and are connected to the Maslov indices appearing in trace formulas, as will be explained in a forthcoming paper.

We classify rational cuspidal curves of degrees 6 and 7 in the complex projective plane, up to symplectic isotopy. The proof uses topological tools, pseudoholomorphic techniques, and birational transformations., 49 pages, many figures. We have upgraded Theorem 4.7. Comments are welcome

A closed contact manifold is called Besse when all its Reeb orbits are closed, and Zoll when they have the same minimal period. In this paper, we provide a characterization of Besse contact forms for convex contact spheres and Riemannian unit tangent bundles in terms of $S^1$-equivariant spectral invariants. Furthermore, for restricted contact type hypersurfaces of symplectic Euclidean spaces, we give a sufficient condition for the Besse property via the Ekeland-Hofer capacities., 33 pages; version 2: minor corrections

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

In this article, we obtain a recursive description of the Horn cone Horn(p,q) with respect to the integers p and q, as in the classical Horn's conjecture.

In this article, we obtain a recursive description of the Horn cone Horn(p,q) with respect to the integers p and q, as in the classical Horn's conjecture.

Any filtered complex over a field F can be brought by a linear transformation preserving the filtration to so called canonical form, a canonically defined direct sum of filtered complexes of two types: one-dimensional complexes with trivial differential d(e_{t_i})=0 and two-dimensional complexes with trivial homology d(e_{s_j})=e_{r_j}. The presentation is devoted to the proof of this theorem, that was first published in the speaker’s 1994 paper “Framed Morse complex and its invariants”, AMS, Advances in Soviet Mathematics, 21: 93–115. Starting from the early 2000s these invariants became widely popular in Applied Mathematics under the name of “persistence diagrams” and “persistence barcodes”. The mentioned classification theorem is usually referred to in Applied Mathematics as the Persistent homology Main (or Structure, or Principal) Theorem. More than 15 different software platforms, that exist actually for computation of persistence diagrams, are based on the algorithm, described in the mentioned 1994 paper, that brings filtered complexes to the canonical form.

In this announcement a diagram will be defined for each nonabelian Hodge space on the affine line., 9 pages (v2: minor improvements)

Let V be a two-dimensional vector space over a field F of characteristic not 2 or 3. We show there is a surjection ν from the set of ‘generic’ commutative algebra structures on V modulo the action ...

In this article we define intersection Floer homology for exact Lagrangian cobordisms between Legendrian submanifolds in the contactisation of a Liouville manifold, provided that the Chekanov-Eliashberg algebras of the negative ends of the cobordisms admit augmentations. From this theory we derive several exact sequences relating the Morse homology of an exact Lagrangian cobordism with the bilinearised contact homologies of its ends. These are then used to investigate the topological properties of exact Lagrangian cobordisms., Comment: 61 pages, 17 figures. Final version, accepted for publication in Journal of Differential Geometry, some missing citations have been added

In this paper, we will introduce Quantum Toric Varieties which are (non-commutative) generalizations of ordinary toric varieties where all the tori of the classical theory are replaced by quantum tori. Quantum toric geometry is the non-commutative version of the classical theory; it generalizes non-trivially most of the theorems and properties of toric geometry. By considering quantum toric varieties as (non-algebraic) stacks, we define their category and show that it is equivalent to a category of quantum fans. We develop a Quantum Geometric Invariant Theory (QGIT) type construction of Quantum Toric Varieties. Unlike classical toric varieties, quantum toric varieties admit moduli and we define their moduli spaces, prove that these spaces are orbifolds and, in favorable cases, up to homotopy, they admit a complex structure., 93 pages, 4 figures

We prove that in dimension 3 every nondegenerate contact form is carried by a broken book decomposition. As an application we get that if M is a closed irreducible oriented 3-manifold that is not a graph manifold, for example a hyperbolic manifold, then every nondegenerate Reeb vector field on M has positive topological entropy. Moreover, we obtain that on a closed 3-manifold, every nondegenerate Reeb vector field has either two or infinitely many periodic orbits, and two periodic orbits are possible only on the sphere or on a lens space.

We study relative symplectic cobordisms between contact submanifolds, and in particular relative symplectic cobordisms to the empty set, that we call hats. While we make some observations in higher dimensions, we focus on the case of transverse knots in the standard 3-sphere, and hats in blow-ups of the (punctured) complex projective planes. We apply the construction to give constraints on the algebraic topology of fillings of double covers of the 3-sphere branched over certain transverse quasipositive knots., Comment: 46 pages, 5 figures; accepted for publication by the Journal of Topology

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

A classical theorem due to Wadsley implies that, on a connected contact manifold all of whose Reeb orbits are closed, there is a common period for the Reeb orbits. In this paper we show that, for any Reeb flow on a closed connected 3-manifold, the following conditions are actually equivalent: (1) every Reeb orbit is closed; (2) all closed Reeb orbits have a common period; (3) the action spectrum has rank 1. We also show that, on a fixed closed connected 3-manifold, a contact form with an action spectrum of rank 1 is determined (up to pull-back by diffeomorphisms) by the set of minimal periods of its closed Reeb orbits., Comment: 18 pages; version 3: we specified that the contact manifolds are required to be connected. To appear in Commentarii Mathematici Helvetici