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195 results on '"[MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG]"'

1. Orbit growth of contact structures after surgery

Author

Anne Vaugon, Boris Hasselblatt, Patrick Foulon, Centre International de Rencontres Mathématiques (CIRM), Aix Marseille Université (AMU)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics, Tufts University, Tufts University, Tufts University [Medford]-Tufts University [Medford], Laboratoire de Mathématiques d'Orsay (LMO), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), Boris Hasselblatt partially supported by the Committee on Faculty Research Awards of Tufts University., and ANR-16-CE40-0017,Quantact,Topologie quantique et géométrie de contact(2016)

Subjects
Anosov flow, medicine.medical_specialty, Dynamical systems theory, Contact geometry, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], Ocean Engineering, Context (language use), Dynamical Systems (math.DS), Homology (mathematics), Reeb flow, 01 natural sciences, surgery, Intersection, 0103 physical sciences, FOS: Mathematics, medicine, Mathematics - Dynamical Systems, 0101 mathematics, Mathematics::Symplectic Geometry, 3-manifold, Mathematics, contact structure, 010102 general mathematics, contact homology, Mathematics::Geometric Topology, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], Surgery, Flow (mathematics), Mathematics - Symplectic Geometry, Orbit (dynamics), Symplectic Geometry (math.SG), 010307 mathematical physics
Abstract

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.

Published
2021

2. W. P. Thurston and French mathematics

Author

Athanase Papadopoulos, Francois Laudenbach, Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN), Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Chebyshev Laboratory, St Petersburg State University (SPbU), Université de Nantes - Faculté des Sciences et des Techniques, and Université de Nantes (UN)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Low-dimensional topology, History and Overview (math.HO), General Mathematics, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], low-dimensional topology, 01 natural sciences, 01A70, 01A60, 01A61, 57M50, 57R17, 57R30, Mathematics - Geometric Topology, contact structures, [MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT], FOS: Mathematics, 57R30 Keywords: William P Thurston: Geometric structures, 0101 mathematics, foliations, Mathematics, Final version, Mathematical society, Mathematics - History and Overview, 010102 general mathematics, Geometric Topology (math.GT), Hyperbolic structures, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], Haefliger structures, William P. Thurston: Geometric structures, history of French mathematics Part 1, [MATH.MATH-AT]Mathematics [math]/Algebraic Topology [math.AT], AMS classification: 01A70, [MATH.MATH-HO]Mathematics [math]/History and Overview [math.HO], history of French mathematics, Hyperbolic struc- tures, Mathematical economics
Abstract

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.

Published
2019

3. Anti-symplectic involutions on rational symplectic 4-manifolds

Author

Vsevolod Shevchishin, Viatcheslav Kharlamov, Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), and Faculty of Mathematics, National Research University Higher School of Economics (HSE)

Subjects
Pure mathematics, Current (mathematics), 010102 general mathematics, 01 natural sciences, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], Mathematics - Symplectic Geometry, 0103 physical sciences, FOS: Mathematics, Symplectic Geometry (math.SG), Involution (philosophy), 010307 mathematical physics, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], 0101 mathematics, Mathematics::Symplectic Geometry, Topology (chemistry), ComputingMilieux_MISCELLANEOUS, Exposition (narrative), Mathematics, Symplectic geometry
Abstract

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

Published
2021

4. Spectral asymptotics of semiclassical unitary operators

Author

Álvaro Pelayo, Yohann Le Floch, Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), University of California [San Diego] (UC San Diego), and University of California

Subjects
Convex hull, Pure mathematics, Spectral theory, Inverse, Semiclassical physics, symplectic actions, 01 natural sciences, Unitary state, Mathematics - Spectral Theory, Mathematics - Analysis of PDEs, 0103 physical sciences, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Limit (mathematics), 0101 mathematics, Spectral Theory (math.SP), Mathematics, Applied Mathematics, 010102 general mathematics, Spectrum (functional analysis), Cayley transform, spectral theory, 16. Peace & justice, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], Mathematics - Symplectic Geometry, Symplectic Geometry (math.SG), 010307 mathematical physics, Analysis, Analysis of PDEs (math.AP), semiclassical analysis, [MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP]
Abstract

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

Published
2019

5. Shifted cotangent stacks are shifted symplectic

Author

Damien Calaque, Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Institut Universitaire de France (IUF), Ministère de l'Education nationale, de l’Enseignement supérieur et de la Recherche (M.E.N.E.S.R.), Institut Universitaire de France, and ANR-14-CE25-0008,SAT,Structures supérieures en Algèbre et Topologie(2014)

Subjects
Pure mathematics, Carry (arithmetic), Structure (category theory), 01 natural sciences, Mathematics - Algebraic Geometry, symbols.namesake, 0103 physical sciences, FOS: Mathematics, Algebraic Topology (math.AT), Trigonometric functions, Mathematics - Algebraic Topology, 0101 mathematics, Algebraic Geometry (math.AG), Mathematics::Symplectic Geometry, Physics, 010102 general mathematics, General Medicine, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], Mathematics - Symplectic Geometry, [MATH.MATH-AT]Mathematics [math]/Algebraic Topology [math.AT], symbols, Symplectic Geometry (math.SG), [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], 010307 mathematical physics, Lagrangian, Symplectic geometry
Abstract

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

Published
2019

6. Stability property of multiplicities of group representations

Author

Paul-Emile Paradan, Institut Montpelliérain Alexander Grothendieck ( IMAG ), Université de Montpellier ( UM ) -Centre National de la Recherche Scientifique ( CNRS ), Institut Montpelliérain Alexander Grothendieck (IMAG), and Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)

Subjects
branching laws, Pure mathematics, Property (philosophy), [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT], multiplicities, 010102 general mathematics, [ MATH.MATH-SG ] Mathematics [math]/Symplectic Geometry [math.SG], 0102 computer and information sciences, stability, group representation, 01 natural sciences, Stability (probability), Group representation, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], [ MATH.MATH-RT ] Mathematics [math]/Representation Theory [math.RT], Mathematics - Symplectic Geometry, 010201 computation theory & mathematics, FOS: Mathematics, Symplectic Geometry (math.SG), Geometry and Topology, Representation Theory (math.RT), 0101 mathematics, Mathematics - Representation Theory, Mathematics
Abstract

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

Published
2019

7. Conformal symplectic geometry of cotangent bundles

Author

Emmy Murphy, Baptiste Chantraine, Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN), Massachusetts Institute of Technology (MIT), and ANR-13-JS01-0008,cospin,Invariants spectraux de contact(2013)

Subjects
Mathematics - Differential Geometry, Pure mathematics, Contact geometry, Conformal map, Homology (mathematics), Mathematics::Algebraic Topology, 01 natural sciences, Mathematics - Geometric Topology, symbols.namesake, Mathematics::K-Theory and Homology, [MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT], 0103 physical sciences, FOS: Mathematics, 53D, 0101 mathematics, Mathematics::Symplectic Geometry, Mathematics, Conjecture, 010102 general mathematics, Geometric Topology (math.GT), 16. Peace & justice, Mathematics::Geometric Topology, Manifold, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], Differential Geometry (math.DG), Mathematics - Symplectic Geometry, [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], symbols, Symplectic Geometry (math.SG), Novikov self-consistency principle, 010307 mathematical physics, Geometry and Topology, Hamiltonian (quantum mechanics), Symplectic geometry
Abstract

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

Published
2019

8. Derived Stacks in Symplectic Geometry

Author

Damien Calaque, Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Institut Universitaire de France (IUF), Ministère de l'Education nationale, de l’Enseignement supérieur et de la Recherche (M.E.N.E.S.R.), Mathieu Anel, Gabriel Catren, and ANR-14-CE25-0008,SAT,Structures supérieures en Algèbre et Topologie(2014)

Subjects
Field (physics), [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], 010102 general mathematics, 01 natural sciences, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], Moduli space, Moduli, Theoretical physics, [MATH.MATH-HO]Mathematics [math]/History and Overview [math.HO], 0103 physical sciences, 010307 mathematical physics, 0101 mathematics, Mathematics::Symplectic Geometry, Mathematics, Symplectic geometry
Abstract

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.

Published
2021

9. On the spectral characterization of Besse and Zoll Reeb flows

Author

Marco Mazzucchelli, Viktor L. Ginzburg, Basak Z. Gurel, Unité de Mathématiques Pures et Appliquées (UMPA-ENSL), and École normale supérieure de Lyon (ENS de Lyon)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Mathematics - Differential Geometry, Pure mathematics, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], Dynamical Systems (math.DS), 01 natural sciences, 0103 physical sciences, FOS: Mathematics, 0101 mathematics, Mathematics - Dynamical Systems, [MATH]Mathematics [math], Mathematics::Symplectic Geometry, Mathematical Physics, Mathematics, Applied Mathematics, 010102 general mathematics, High Energy Physics::Phenomenology, Regular polygon, Contact type, Tangent, Manifold, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], Differential Geometry (math.DG), Mathematics - Symplectic Geometry, Equivariant map, Symplectic Geometry (math.SG), 010307 mathematical physics, Mathematics::Differential Geometry, 53D10, 58E05, 53C22, Unit (ring theory), Analysis, Symplectic geometry, Vector space
Abstract

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

Published
2020

10. Diagrams for nonabelian Hodge spaces on the affine line

Author

Daisuke Yamakawa, Philip Boalch, Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG (UMR_7586)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), and Tokyo University of Science [Tokyo]

Subjects
Pure mathematics, General Mathematics, FOS: Physical sciences, [MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA], Space (mathematics), 01 natural sciences, High Energy Physics::Theory, Mathematics - Algebraic Geometry, Physics::Popular Physics, Mathematics::Group Theory, Mathematics::Algebraic Geometry, 0103 physical sciences, Classical Analysis and ODEs (math.CA), FOS: Mathematics, 0101 mathematics, Representation Theory (math.RT), [MATH]Mathematics [math], Algebraic Geometry (math.AG), Mathematics, Nonlinear Sciences - Exactly Solvable and Integrable Systems, 010102 general mathematics, Mathematics::History and Overview, Computer Science::Computers and Society, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], Mathematics - Classical Analysis and ODEs, [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], Line (geometry), 010307 mathematical physics, Affine transformation, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], Exactly Solvable and Integrable Systems (nlin.SI), Mathematics - Representation Theory
Abstract

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

Published
2020

11. Polynomial invariants and moduli of generic two-dimensional commutative algebras

Author

M. Rausch de Traubenberg, M. J. Slupinski, Institut de Recherche Mathématique Avancée (IRMA), and Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Discrete mathematics, Pure mathematics, Polynomial, Algebra and Number Theory, [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT], Modulo, 010102 general mathematics, Field (mathematics), 01 natural sciences, Moduli space, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], Surjective function, 0103 physical sciences, 010307 mathematical physics, 0101 mathematics, Commutative algebra, Commutative property, ComputingMilieux_MISCELLANEOUS, Mathematics, Vector space
Abstract

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 ...

Published
2020

12. Floer theory for Lagrangian cobordisms

Author

Paolo Ghiggini, Roman Golovko, Baptiste Chantraine, Georgios Dimitroglou Rizell, Laboratoire de Mathématiques Jean Leray (LMJL), Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS), University of Cambridge [UK] (CAM), Alfred Renyi Mathematical Institute, Eötvös Loránd University (ELTE), ANR-13-JS01-0008,cospin,Invariants spectraux de contact(2013), European Project: 646649,H2020,ERC-2014-CoG,SymplecticEinstein(2015), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), and Université de Nantes (UN)-Université de Nantes (UN)

Subjects
Pure mathematics, Mathematics::Algebraic Topology, 01 natural sciences, 57R58, 53D42, 53D12, symbols.namesake, Morse homology, Intersection, Mathematics::K-Theory and Homology, [MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT], FOS: Mathematics, 0101 mathematics, Mathematics::Symplectic Geometry, Mathematics, Algebra and Number Theory, 010102 general mathematics, Cobordism, Mathematics::Geometric Topology, Manifold, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], Floer homology, [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], Mathematics - Symplectic Geometry, symbols, Symplectic Geometry (math.SG), Geometry and Topology, Analysis, Lagrangian
Abstract

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

Published
2020

13. The action spectrum characterizes closed contact 3-manifolds all of whose Reeb orbits are closed

Author

Marco Mazzucchelli, Daniel Cristofaro-Gardiner, University of California [Santa Cruz] (UCSC), University of California, Unité de Mathématiques Pures et Appliquées (UMPA-ENSL), École normale supérieure - Lyon (ENS Lyon)-Centre National de la Recherche Scientifique (CNRS), University of California [Santa Cruz] (UC Santa Cruz), University of California (UC), and École normale supérieure de Lyon (ENS de Lyon)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Mathematics - Differential Geometry, Pure mathematics, General Mathematics, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], Dynamical Systems (math.DS), Rank (differential topology), 01 natural sciences, 0103 physical sciences, FOS: Mathematics, 0101 mathematics, Mathematics - Dynamical Systems, [MATH]Mathematics [math], Classical theorem, 53C22, 58E10, Mathematics::Symplectic Geometry, Mathematics, 010102 general mathematics, Manifold, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], Flow (mathematics), Differential Geometry (math.DG), Mathematics - Symplectic Geometry, Orbit (dynamics), Symplectic Geometry (math.SG), 010307 mathematical physics, Mathematics::Differential Geometry
Abstract

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

Published
2020

14. Nonexistence of global characteristics for viscosity solutions

Author

Valentine Roos, CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Unité de Mathématiques Pures et Appliquées (UMPA-ENSL), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Lyon (ENS Lyon), Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), Université Paris Dauphine-PSL, École normale supérieure de Lyon (ENS de Lyon)-Centre National de la Recherche Scientifique (CNRS), and École normale supérieure - Paris (ENS-PSL)

Subjects
Integrable system, characteristics, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], nonconvex Hamiltonian dynamics, Dynamical Systems (math.DS), 01 natural sciences, Hamilton–Jacobi equation, symbols.namesake, Mathematics - Analysis of PDEs, 0103 physical sciences, FOS: Mathematics, Initial value problem, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], wavefronts, 0101 mathematics, Mathematics - Dynamical Systems, Mathematics - Optimization and Control, Saddle, Mathematics, Wavefront, viscosity solution, Numerical Analysis, 49L25, Applied Mathematics, 010102 general mathematics, Regular polygon, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], Optimization and Control (math.OC), Mathematics - Symplectic Geometry, symbols, variational solution, Symplectic Geometry (math.SG), 010307 mathematical physics, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Viscosity solution, Hamiltonian (quantum mechanics), Analysis, Analysis of PDEs (math.AP)
Abstract

International audience; Two different types of generalized solutions, namely viscosity and variational solutions, were introduced to solve the first-order evolutionary Hamilton–Jacobi equation. They coincide if the Hamiltonian is convex in the momentum variable. In this paper we prove that there exists no other class of integrable Hamiltonians sharing this property. To do so, we build for any non-convex non-concave integrable Hamiltonian a smooth initial condition such that the graph of the viscosity solution is not contained in the wavefront associated with the Cauchy problem. The construction is based on a new example for a saddle Hamiltonian and a precise analysis of the one-dimensional case, coupled with reduction and approximation arguments.

Published
2020

15. Sufficient conditions for time optimality of systems with control on the disk

Author

Jean-Baptiste Caillau, Michael Orieux, COMUE Université Côte d'Azur (2015-2019) (COMUE UCA), Mathematics for Control, Transport and Applications (McTAO), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL), and Université Côte d'Azur (UCA)

Subjects
021103 operations research, 0211 other engineering and technologies, Field (mathematics), 010103 numerical & computational mathematics, 02 engineering and technology, Type (model theory), 01 natural sciences, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], Flow (mathematics), Optimization and Control (math.OC), FOS: Mathematics, Piecewise, Applied mathematics, Verifiable secret sharing, Point (geometry), Gravitational singularity, Minification, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], 0101 mathematics, [MATH]Mathematics [math], Mathematics - Optimization and Control, Mathematics
Abstract

International audience; The case of time minimization for affine control systems with control on the disk is studied. After recalling the standard sufficient conditions for local optimality in the smooth case, the analysis focusses on the specific type of singularities encountered when the control is prescribed to the disk. Using a suitable stratification, the regularity of the flow is analyzed, which helps to devise verifiable sufficient conditions in terms of left and right limits of Jacobi fields at a switching point. Under the appropriate assumptions, piecewise regularity of the field of extremals is obtained.

Published
2019

16. On the normality of the null-fiber of the moment map for θ - and tori representations

Author

Michaël Bulois, Algèbre, géométrie, logique (AGL), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), ANR-15-CE40-0012,GéoLie,Méthodes géométriques en théorie de Lie(2015), and ANR-10-LABX-0070,MILYON,Community of mathematics and fundamental computer science in Lyon(2010)

Subjects
Pure mathematics, torus, 01 natural sciences, moment map, Mathematics - Algebraic Geometry, commuting variety, 0101 mathematics, Moment map, Quotient, Orbifold, Mathematics, Algebra and Number Theory, Conjecture, [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT], Fiber (mathematics), 010102 general mathematics, Null (mathematics), Torus, MSC 2010 : 14L24, 17B20, 17B70, 20G05 , 53D20, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], 010101 applied mathematics, Mathematics - Symplectic Geometry, orbifold, symplectic reduction, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], theta-representations, Mathematics - Representation Theory, Symplectic geometry
Abstract

Let (G, V) be a representation with either G a torus or (G, V) a locally free stable $\theta$-representation. We study the fiber at 0 of the associated moment map, which is a commuting variety in the latter case. We characterize the cases where this fiber is normal. The quotient (i.e. the symplectic reduction) turns out to be a specific orbifold when the representation is polar. In the torus case, this confirms a conjecture stated by C. Lehn, M. Lehn, R. Terpereau and the author in a former article. In the $\theta$-case, the conjecture was already known but the present approach yield another proof., Comment: 21 pages, comments welcome

Published
2018

17. Singular fibers of the bending flows on the moduli space of 3D polygons

Author

Damien Bouloc, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), and Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)

Subjects
Pure mathematics, Polygonal modeling, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], 010102 general mathematics, Diagonal, Isotropy, Bending, Disjoint sets, 01 natural sciences, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], Moduli space, Mathematics - Symplectic Geometry, Homogeneous, Phase space, 0103 physical sciences, FOS: Mathematics, Symplectic Geometry (math.SG), 010307 mathematical physics, Geometry and Topology, 0101 mathematics, Mathematics
Abstract

In this paper, we prove that in the system of bending flows on the moduli space of polygons with fixed side lengths introduced by Kapovich and Millson, the singular fibers are isotropic homogeneous submanifolds. The proof covers the case where the system is defined by any maximal family of disjoint diagonals. We also take in account the case where the fixed side lengths are not generic. In this case, the phase space is an orbispace, and our result holds in the sense that singular fibers are isotropic orbispaces. In a last part we provide leads in favor of a similar study of the integrable systems defined by Nohara and Ueda on the Grassmannian of 2-planes in $\mathbb{C}^n$., Comment: 39 pages, 7 figures; added correspondence with the systems defined by Nohara and Ueda on the Grassmannian of 2-planes

Published
2018

18. A categorification of the Alexander polynomial in embedded contact homology

Author

Gilberto Spano, Laboratoire de Mathématiques Nicolas Oresme (LMNO), Centre National de la Recherche Scientifique (CNRS)-Université de Caen Normandie (UNICAEN), Normandie Université (NU)-Normandie Université (NU), Institut Fourier (IF ), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), Institut Fourier [1973-2019] (IF [1973-2019]), Institut Fourier (IF), and Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)

Subjects
Categorification, embedded contact homology, Alexander polynomial, 01 natural sciences, Combinatorics, Mathematics - Geometric Topology, Knot (unit), Mathematics::Quantum Algebra, [MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT], 0103 physical sciences, FOS: Mathematics, 57R58, [MATH]Mathematics [math], 0101 mathematics, Mathematics::Symplectic Geometry, ComputingMilieux_MISCELLANEOUS, 57R17, Mathematics, Conjecture, 010102 general mathematics, Geometric Topology (math.GT), Mathematics::Geometric Topology, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], categorification, Floer homology, Mathematics - Symplectic Geometry, 57M27, Symplectic Geometry (math.SG), 010307 mathematical physics, Geometry and Topology
Abstract

Given a transverse knot $K$ in a three dimensional contact manifold $(Y,\alpha)$, in [13] Colin, Ghiggini, Honda and Hutchings define a hat version of embedded contact homology for $K$, that we call $\widehat{ECK}(K,Y,\alpha)$, and conjecture that it is isomorphic to the knot Floer homology $\widehat{HFK}(K,Y)$. We define here a full version $ECK(K,Y,\alpha)$ and generalise the definitions to the case of links. We prove then that, if $Y = S^3$, $ECK$ and $\widehat{ECK}$ categorify the (multivariable) Alexander polynomial of knots and links, obtaining expressions analogue to that for knot and link Floer homologies in the plus and, respectively, hat versions., Comment: 48 pages, 4 figures

Published
2017

19. GIVENTAL'S NON-LINEAR MASLOV INDEX ON LENS SPACES

Author

Sheila Sandon, Yael Karshon, Gustavo Granja, Milena Pabiniak, Instituto Superior Técnico, Universidade Técnica de Lisboa (IST), Department of Mathematics [University of Toronto], University of Toronto, Mathematisch-Naturwissenschaftliche Fakultät [Köln], Universität zu Köln, Institut de Recherche Mathématique Avancée (IRMA), and Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Pure mathematics, Covering space, General Mathematics, 010102 general mathematics, 02 engineering and technology, 021001 nanoscience & nanotechnology, 01 natural sciences, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], Nonlinear system, Discriminant, Mathematics - Symplectic Geometry, FOS: Mathematics, Symplectic Geometry (math.SG), 53D35 53D10 57R17, Identity component, 0101 mathematics, Invariant (mathematics), [MATH]Mathematics [math], 0210 nano-technology, Symplectomorphism, Mathematics::Symplectic Geometry, Real projective space, Mathematics
Abstract

Givental's non-linear Maslov index, constructed in 1990, is a quasimorphism on the universal cover of the identity component of the contactomorphism group of real projective space. This invariant was used by several authors to prove contact rigidity phenomena such as orderability, unboundedness of the discriminant and oscillation metrics, and a contact geometric version of the Arnold conjecture. In this article we give an analogue for lens spaces of Givental's construction and its applications., 44 pages; v3: minor changes; v2: besides minor changes, we corrected a mistake in Corollary 1.3(iv)

Published
2019

20. Correction to: Inverse spectral theory for semiclassical Jaynes–Cummings systems

Author

San Vũ Ngọc, Yohann Le Floch, Álvaro Pelayo, Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Universidad Complutense de Madrid = Complutense University of Madrid [Madrid] (UCM), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), and Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)

Subjects
Conjecture, Spectral theory, General Mathematics, 010102 general mathematics, Inverse, Semiclassical physics, 01 natural sciences, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], 0103 physical sciences, 010307 mathematical physics, 0101 mathematics, ComputingMilieux_MISCELLANEOUS, Mathematical physics, Mathematics, [MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP]
Abstract

We explain why Theorem B in the original article does not follow from the main result of this paper (Theorem A). While we conjecture that Theorem B should nevertheless be true, in this erratum we prove a slightly weaker version of it.

Published
2019

21. On the invariance of Welschinger invariants

Author

Erwan Brugallé, Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), and Université de Nantes (UN)-Université de Nantes (UN)

Subjects
Pure mathematics, Computation, 02 engineering and technology, Mathematical proof, 01 natural sciences, symbols.namesake, Mathematics - Algebraic Geometry, Algebraic surface, 0202 electrical engineering, electronic engineering, information engineering, FOS: Mathematics, 0101 mathematics, [MATH]Mathematics [math], Algebraic Geometry (math.AG), Mathematics::Symplectic Geometry, Mathematics, Statement (computer science), Algebra and Number Theory, Applied Mathematics, 010102 general mathematics, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], Mathematics - Symplectic Geometry, symbols, Symplectic Geometry (math.SG), 020201 artificial intelligence & image processing, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], Analysis, Lagrangian, Symplectic geometry, Sign (mathematics)
Abstract

We collect in this note some observations about original Welschinger invariants of real symplectic fourfolds. None of their proofs is difficult, nevertheless these remarks do not seem to have been made before. Our main result is that when $X$ is a real rational algebraic surface, Welschinger invariants only depend on the number of real interpolated points, and some homological data associated to $X$. This strengthened the invariance statement initially proved by Welschinger. This main result follows easily from a formula relating Welschinger invariants of two real symplectic manifolds differing by a surgery along a real Lagrangian sphere. In its turn, once one believes that such formula may hold, its proof is a mild adaptation of the proof of analogous formulas previously obtained by the author on the one hand, and by Itenberg, Kharlamov and Shustin on the other hand. We apply the two aforementioned results to complete the computation of Welschinger invariants of real rational algebraic surfaces, and to obtain vanishing, sign, and sharpness results for these invariants that generalize previously known statements. We also discuss some hypothetical relations of our work with tropical refined invariants defined by Block-G\"ottsche and G\"ottsche-Schroeter., Comment: 15 pages, 2 figures; V2: minor edition

Published
2019

22. A Novikov fundamental group

Author

R Golovko, J-F Barraud, Hông Vân Lê, Agnès Gadbled, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Uppsala Universitet [Uppsala], Département de mathématiques Université Libre de Bruxelles, Université libre de Bruxelles (ULB), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), and Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)

Subjects
Mathematics - Differential Geometry, Pure mathematics, Fundamental group, Closed manifold, General Mathematics, 57R19 (Primary) 57R70, 57R17 (Secondary), Homology (mathematics), Morse code, 01 natural sciences, Mathematics::Algebraic Topology, law.invention, Mathematics - Geometric Topology, Morse homology, law, Mathematics::K-Theory and Homology, FOS: Mathematics, 0101 mathematics, Mathematics::Symplectic Geometry, Mathematics, 010102 general mathematics, Geometric Topology (math.GT), [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], Differential Geometry (math.DG), Mathematics - Symplectic Geometry, Symplectic Geometry (math.SG), Novikov self-consistency principle
Abstract

Given a $1$-cohomology class $u$ on a closed manifold $M$, we define a Novikov fundamental group associated to $u$, generalizing the usual fundamental group in the same spirit as Novikov homology generalizes Morse homology to the case of non exact $1$-forms. As an application, lower bounds for the minimal number of index $1$ and $2$ critical points of Morse closed $1$-forms are obtained, that are different in nature from those derived from the Novikov homology., 39 pages, 4 drawings (inkscape) Added a discussion of an Hurewicz morphism and examples

Published
2019

23. Taut foliations

Author

Colin, Vincent, Kazez, William, Roberts, Rachel, Colin, Vincent, Topologie quantique et géométrie de contact - - Quantact2016 - ANR-16-CE40-0017 - AAPG2016 - VALID, Geometry and dynamics via contact topology - GEODYCON - - EC:FP7:ERC2012-01-01 - 2016-12-31 - 278246 - VALID, Université de Nantes (UN), University of Georgia [USA], Washington University in Saint Louis (WUSTL), ANR-16-CE40-0017,Quantact,Topologie quantique et géométrie de contact(2016), and European Project: 278246,EC:FP7:ERC,ERC-2011-StG_20101014,GEODYCON(2012)

Subjects
Statistics and Probability, 010102 general mathematics, 01 natural sciences, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], [MATH.MATH-SG] Mathematics [math]/Symplectic Geometry [math.SG], [MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT], 0103 physical sciences, 010307 mathematical physics, Geometry and Topology, 0101 mathematics, Statistics, Probability and Uncertainty, Analysis, ComputingMilieux_MISCELLANEOUS, [MATH.MATH-GT] Mathematics [math]/Geometric Topology [math.GT]
Abstract

International audience

Published
2019

24. Symplectic geometry and spectral properties of classical and quantum coupled angular momenta

Author

Yohann Le Floch, Álvaro Pelayo, Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), University of California [San Diego] (UC San Diego), and University of California

Subjects
Integrable system, Computation, semitoric systems, FOS: Physical sciences, 01 natural sciences, 010305 fluids & plasmas, Mathematics - Spectral Theory, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, FOS: Mathematics, 0101 mathematics, Invariant (mathematics), Spectral Theory (math.SP), Quantum, Mathematics::Symplectic Geometry, Mathematical Physics, Mathematical physics, Physics, Applied Mathematics, Spectral properties, General Engineering, Mathematical Physics (math-ph), [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], 010101 applied mathematics, Mathematics - Symplectic Geometry, symplectic geometry, Modeling and Simulation, Integrable systems, Symplectic Geometry (math.SG), Symplectic geometry, semiclassical analysis, [MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP]
Abstract

We give a detailed study of the symplectic geometry of a family of integrable systems obtained by coupling two angular momenta in a non trivial way. These systems depend on a parameter t $\in$ [0, 1] and exhibit different behaviors according to its value. For a certain range of values, the system is semitoric, and we compute some of its symplectic invariants. Even though these invariants have been known for almost a decade, this is to our knowledge the first example of their computation in the case of a non-toric semitoric system on a compact manifold (the only invariant of toric systems is the image of the momentum map). In the second part of the paper we quantize this system, compute its joint spectrum, and describe how to use this joint spectrum to recover information about the symplectic invariants., Comment: Exposition revised. A problem which affected the computation of one invariant has been fixed

Published
2019

25. Variational Discretization Framework for Geophysical Flow Models

Author

François Gay-Balmaz, Werner Bauer, Fluid Flow Analysis, Description and Control from Image Sequences (FLUMINANCE), Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-AGROCAMPUS OUEST, Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA)-Inria Rennes – Bretagne Atlantique, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Centre National de la Recherche Scientifique (CNRS), Laboratoire de Météorologie Dynamique (UMR 8539) (LMD), Institut national des sciences de l'Univers (INSU - CNRS)-École polytechnique (X)-École des Ponts ParisTech (ENPC)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Département des Géosciences - ENS Paris, École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA)-Inria Rennes – Bretagne Atlantique, École normale supérieure - Paris (ENS-PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-École normale supérieure - Paris (ENS-PSL), ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011), Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-AGROCAMPUS OUEST-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-Centre National de la Recherche Scientifique (CNRS)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Centre National de la Recherche Scientifique (CNRS)-Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA)-Inria Rennes – Bretagne Atlantique, and École normale supérieure - Paris (ENS Paris)-École normale supérieure - Paris (ENS Paris)

Subjects
Discretization, Lie group, 010103 numerical & computational mathematics, 01 natural sciences, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], 010101 applied mathematics, Variational method, Geophysical fluid dynamics, [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], Variational principle, Fluid dynamics, Applied mathematics, Diffeomorphism, 0101 mathematics, Variational integrator, ComputingMilieux_MISCELLANEOUS, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

We introduce a geometric variational discretization framework for geophysical flow models. The numerical scheme is obtained by discretizing, in a structure-preserving way, the Lie group formulation of fluid dynamics on diffeomorphism groups and the associated variational principles. Being based on a discrete version of the Euler-Poincare variational method, this discretization approach is widely applicable. We present an overview of structure-preserving variational discretizations of various equations of geophysical fluid dynamics, such as the Boussinesq, anelastic, pseudo-incompressible, and rotating shallow-water equations. We verify the structure-preserving nature of the resulting variational integrators for test cases of geophysical relevance. Our framework applies to irregular mesh discretizations in 2D and 3D in planar and spherical geometry and produces schemes that preserve invariants of the equations such as mass and potential vorticity. Descending from variational principles, the discussed variational schemes exhibit a discrete version of Kelvin circulation theorem and show excellent long term energy behavior.

Published
2019

26. POSITIVE LEGENDRIAN ISOTOPIES AND FLOER THEORY

Author

Baptiste Chantraine, Vincent Colin, Georgios Dimitroglou Rizell, Laboratoire de Mathématiques Jean Leray (LMJL), Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics [Uppsala], Uppsala University, ANR-13-JS01-0008,cospin,Invariants spectraux de contact(2013), European Project: 278246,EC:FP7:ERC,ERC-2011-StG_20101014,GEODYCON(2012), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), and Université de Nantes (UN)-Université de Nantes (UN)

Subjects
Pure mathematics, Geometry, Mathematics::Algebraic Topology, 01 natural sciences, symbols.namesake, Mathematics::K-Theory and Homology, 0103 physical sciences, FOS: Mathematics, Geometri, Point (geometry), 0101 mathematics, Mathematics::Symplectic Geometry, Mathematics, Algebra and Number Theory, 010102 general mathematics, Mathematics::Geometric Topology, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], Loop (topology), Floer homology, Mathematics - Symplectic Geometry, Floer, symbols, Symplectic Geometry (math.SG), Legendrian, 010307 mathematical physics, Geometry and Topology, Positive isotopies, Lagrangian
Abstract

Positive loops of Legendrian embeddings are examined from the point of view of Floer homology of Lagrangian cobordisms. This leads to new obstructions to the existence of a positive loop containing a given Legendrian, expressed in terms of the Legendrian contact homology of the Legendrian submanifold. As applications, old and new examples of orderable contact manifolds are obtained and discussed. We also show that contact manifolds admitting a filling of a Liouville domain with non-zero symplectic homology is strongly orderable in the sense of Liu., 43 pages. v2: more details, mainly in Section 5. Changes in introduction, added some references and Theorem 1.19. v3: minor corrections, v4: update in the bibliography and change the references accordingly. To appear at "Annales de l'Institut Fourier"

Published
2019

27. Minimal boundaries in Tonelli Lagrangian systems

Author

Luca Asselle, Marco Mazzucchelli, Gabriele Benedetti, Ruhr-Universitat Bochum, Fakultat fur Mathematik, Ruhr-Universität Bochum [Bochum], Mathematisches Institut der Universität Leipzig, Universität Leipzig, Unité de Mathématiques Pures et Appliquées (UMPA-ENSL), and École normale supérieure de Lyon (ENS de Lyon)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Mathematics - Differential Geometry, Surface (mathematics), Tangent bundle, Pure mathematics, Geodesic, General Mathematics, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], Dynamical Systems (math.DS), 01 natural sciences, Lagrangian system, FOS: Mathematics, Mathematics - Dynamical Systems, 0101 mathematics, Abelian group, [MATH]Mathematics [math], 37J45, 58E05, Mathematics, 010102 general mathematics, Base (topology), Action (physics), [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], 010101 applied mathematics, Cover (topology), Differential Geometry (math.DG), Mathematics - Symplectic Geometry, Symplectic Geometry (math.SG)
Abstract

We prove several new results concerning action minimizing periodic orbits of Tonelli Lagrangian systems on an oriented closed surface $M$. More specifically, we show that for every energy larger than the maximal energy of a constant orbit and smaller than or equal to the Ma\~n\'e critical value of the universal abelian cover, the Lagrangian system admits a minimal boundary, i.e. a global minimizer of the Lagrangian action on the space of smooth boundaries of open sets of $M$. We also extend the celebrated graph theorem of Mather in this context: in the tangent bundle $TM$, the union of the supports of all lifted minimal boundaries with a given energy projects injectively to the base $M$. Finally, we prove the existence of action minimizing simple periodic orbits on energies just above the Ma\~n\'e critical value of the universal abelian cover. This provides in particular a class of non-reversible Finsler metrics on the 2-sphere possessing infinitely many closed geodesics., Comment: 31 pages, 4 figures. This version also incorporates the results of arXiv:1702.08815 (the preprint arXiv:1702.08815 has been withdrawn from the arXiv, and will not be submitted for publication)

Published
2019

28. The fundamental group of a rigid Lagrangian cobordism

Author

Lara Simone Suárez, Jean-François Barraud, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Fundamental group, General Mathematics, 010102 general mathematics, Mathematical analysis, Dimension (graph theory), Cobordism, 01 natural sciences, Omega, Mathematics::Algebraic Topology, Injective function, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], Surjective function, Combinatorics, Monotone polygon, Number theory, Mathematics - Symplectic Geometry, 0103 physical sciences, FOS: Mathematics, Symplectic Geometry (math.SG), 010307 mathematical physics, 0101 mathematics, Mathematics::Symplectic Geometry, Mathematics
Abstract

In this article we extend the construction of the Floer fundamental group to the monotone Lagrangian setting (for weakly exact or monotone Lagrangians with large minimal Maslov number) and use it to study the fundamental group of a Lagrangian cobordism $$W\subset (\mathbb {C}\times M, \omega _{st}\oplus \omega )$$ between two Lagrangian submanifolds $$L, L'\subset ( M, \omega )$$ . We show that under natural conditions the inclusions $$L,L'\hookrightarrow W$$ induce surjective maps $$\pi _{1}(L)\twoheadrightarrow \pi _{1}(W)$$ , $$\pi _{1}(L')\twoheadrightarrow \pi _{1}(W)$$ and when the previous maps are injective then W is an h-cobordism. In particular, in dimension at least 6, W is topologically trivial in this case.

Published
2018

29. Lie rackoids integrating Courant algebroids

Author

Camille Laurent-Gengoux, Friedrich Wagemann, Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN), and Université de Nantes (UN)

Subjects
Pure mathematics, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], Standard Courant algebroid, 01 natural sciences, Courant algebroid, Dorfman bracket, Leibniz algebroid, Poisson manifold, 0103 physical sciences, FOS: Mathematics, Equivalence relation, Algebraic Topology (math.AT), Mathematics - Algebraic Topology, 0101 mathematics, Mathematics::Symplectic Geometry, Quotient, Integration of Courant algebroids, Mathematics, Lie rackoid, Weinstein groupoid, 010102 general mathematics, Automorphism, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], Lie groupoid, Bracket (mathematics), Extended tangent bundle, Mathematics - Symplectic Geometry, Product (mathematics), [MATH.MATH-AT]Mathematics [math]/Algebraic Topology [math.AT], Symplectic Lie groupoid, Symplectic Geometry (math.SG), 010307 mathematical physics, Geometry and Topology, Analysis
Abstract

We construct an infinite-dimensional Lie rackoid Y which hosts an integration of the standard Courant algebroid. As a set, $$Y={{\mathcal {C}}}^{\infty }([0,1],T^*M)$$ for a compact manifold M. The rackoid product is by automorphisms of the Dorfman bracket. The first part of the article is a study of the Lie rackoid Y and its tangent Leibniz algebroid, a quotient of which is the standard Courant algebroid. In the second part, we study the equivalence relation related to the quotient on the rackoid level and restrict then to an integrable Dirac structure. We show how our integrating object contains the corresponding integrating Weinstein Lie groupoid in the case where the Dirac structure is given by a Poisson structure.

Published
2018

30. Toric aspects of the first eigenvalue

Author

Eveline Legendre, Rosa Sena-Dias, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Mathematics - Differential Geometry, Polytope, Lambda, 01 natural sciences, Upper and lower bounds, Combinatorics, 0103 physical sciences, FOS: Mathematics, 0101 mathematics, Mathematics::Symplectic Geometry, [MATH.MATH-MG]Mathematics [math]/Metric Geometry [math.MG], Eigenvalues and eigenvectors, Mathematics, 010102 general mathematics, Mathematics::Spectral Theory, 16. Peace & justice, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], Differential Geometry (math.DG), Differential geometry, Mathematics - Symplectic Geometry, Bounded function, Symplectic Geometry (math.SG), Equivariant map, 010307 mathematical physics, Geometry and Topology, Mathematics::Differential Geometry, Laplace operator
Abstract

In this paper we study the smallest non-zero eigenvalue $\lambda_1$ of the Laplacian on toric K\"ahler manifolds. We find an explicit upper bound for $\lambda_1$ in terms of moment polytope data. We show that this bound can only be attained for $\mathbb{CP}^n$ endowed with the Fubini-Study metric and therefore $\mathbb{CP}^n$ endowed with the Fubini-Study metric is spectrally determined among all toric K\"ahler metrics. We also study the equivariant counterpart of $\lambda_1$ which we denote by $\lambda_1^T$. It is the the smallest non-zero eigenvalue of the Laplacian restricted to torus-invariant functions. We prove that $\lambda_1^T$ is not bounded among toric K\"ahler metrics thus generalizing a result of Abreu-Freitas on $S^2$. In particular, $\lambda_1^T$ and $\lambda_1$ do not coincide in general., Comment: 24 pages. Added some more details and corrected an estimate. Fixed some typos

Published
2018

31. Symplectomorphisms of exotic discs

Author

Ivan Smith, Sylvain Courte, Ailsa Keating, Roger Casals, Keating, Ailsa [0000-0002-1288-3117], Apollo - University of Cambridge Repository, Instituto de Ciencias Matematicas CSIC-UAM-UC3M-UCM, Consejo Superior de Investigaciones Cientficas, Consejo Superior de Investigaciones Científicas [Madrid] (CSIC), Institut Fourier (IF ), Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), Consejo Superior de Investigaciones Científicas [Spain] (CSIC), Institut Fourier (IF), and Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)

Subjects
Engineering, ComputingMilieux_THECOMPUTINGPROFESSION, business.industry, General Mathematics, 010102 general mathematics, Foundation (engineering), Astrophysics::Instrumentation and Methods for Astrophysics, Library science, 01 natural sciences, Computer Science::Digital Libraries, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], 0103 physical sciences, ComputingMethodologies_DOCUMENTANDTEXTPROCESSING, 010307 mathematical physics, 0101 mathematics, business, GeneralLiterature_REFERENCE(e.g.,dictionaries,encyclopedias,glossaries), Mathematics::Symplectic Geometry, ComputingMilieux_MISCELLANEOUS
Abstract

We construct a symplectic structure on a disc that admits a compactly supported symplectomorphism which is not smoothly isotopic to the identity. The symplectic structure has an overtwisted concave end; the construction of the symplectomorphism is based on a unitary version of the Milnor--Munkres pairing. En route, we introduce a symplectic analogue of the Gromoll filtration. The Appendix by S. Courte shows that for our symplectic structure the map from compactly supported symplectic mapping classes to compactly supported smooth mapping classes is in fact surjective.

Published
2018
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32. Notes on open book decompositions for Engel structures

Author

Thomas Vogel, Vincent Colin, Francisco Presas, Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN), Instituto de Ciencias Matematicas CSIC-UAM-UC3M-UCM, Consejo Superior de Investigaciones Cientficas, Consejo Superior de Investigaciones Científicas [Madrid] (CSIC), Mathematisches Institut [München] (LMU), Ludwig-Maximilians-Universität München (LMU), ANR-16-CE40-0017,Quantact,Topologie quantique et géométrie de contact(2016), European Project: 278246,EC:FP7:ERC,ERC-2011-StG_20101014,GEODYCON(2012), Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), and Université de Nantes (UN)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Pure mathematics, contact structure, 58A30, Parallelizable manifold, 010102 general mathematics, Mathematics::Rings and Algebras, Tangent, Engel structures, 16. Peace & justice, 01 natural sciences, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], Mathematics::Group Theory, Monodromy, Mathematics - Symplectic Geometry, Open book decomposition, 0103 physical sciences, open book decomposition, FOS: Mathematics, Symplectic Geometry (math.SG), 010307 mathematical physics, Geometry and Topology, Mathematics::Differential Geometry, 0101 mathematics, Mathematics::Symplectic Geometry, Mathematics
Abstract

Algebr. Geom. Topol. 18 (2018) 4275-4303; International audience; We relate open book decompositions of a 4-manifold M with its Engel structures. Our main result is, given an open book decomposition of M whose binding is a collection of 2-tori and whose monodromy preserves a framing of a page, the construction of an En-gel structure whose isotropic foliation is transverse to the interior of the pages and tangent to the binding. In particular the pages are contact man-ifolds and the monodromy is a contactomorphism. As a consequence, on a parallelizable closed 4-manifold, every open book with toric binding carries in the previous sense an Engel structure. Moreover, we show that amongst the supported Engel structures we construct, there is a class of loose Engel structures.

Published
2018

33. Noncommutative geometry and diffeology: The case of orbifolds

Author

Patrick Iglesias-Zemmour, Jean-Pierre Laffineur, Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG (UMR_7586)), and Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Pure mathematics, Algebra and Number Theory, Mathematics::Operator Algebras, 58B34 (57R18), 010102 general mathematics, Noncommutative geometry, 0102 computer and information sciences, 01 natural sciences, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], 010201 computation theory & mathematics, Mathematics::K-Theory and Homology, diffeology, Mathematics::Category Theory, Diffeology, orbifolds, Geometry and Topology, 0101 mathematics, Mathematics::Symplectic Geometry, Mathematical Physics, Mathematics
Abstract

International audience; We establish a first structural link between noncommutative geometry and diffeology, in the particular case of orbifolds. Precisely, we associate a structure groupoid with every atlas of a diffeological orbifold. We show that different atlases give equivalent groupoids, that generate strongly Morita equivalent C*-algebras, according to standards. Thus, diffeomorphisms translate naturally into Morita equivalences.

Published
2018

34. A Brief Introduction to Berezin–Toeplitz Operators on Compact Kähler Manifolds

Author

Yohann Le Floch, Institut de Recherche Mathématique Avancée (IRMA), and Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
010102 general mathematics, 0103 physical sciences, 010307 mathematical physics, 0101 mathematics, 01 natural sciences, ComputingMilieux_MISCELLANEOUS, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], [MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP]
Abstract

International audience

Published
2018

35. Surgery of real symplectic fourfolds and Welschinger invariants

Author

Erwan Brugallé, Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), and Université de Nantes (UN)-Université de Nantes (UN)

Subjects
medicine.medical_specialty, Computation, 02 engineering and technology, Real structure, 01 natural sciences, symbols.namesake, Mathematics - Algebraic Geometry, Mathematics::Algebraic Geometry, 0203 mechanical engineering, Genus (mathematics), FOS: Mathematics, medicine, Enumeration, 0101 mathematics, [MATH]Mathematics [math], Algebraic Geometry (math.AG), Mathematics::Symplectic Geometry, Symplectic manifold, Mathematics, Applied Mathematics, 010102 general mathematics, 16. Peace & justice, Mathematics::Geometric Topology, Surgery, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], 020303 mechanical engineering & transports, Conic section, Mathematics - Symplectic Geometry, symbols, Symplectic Geometry (math.SG), Geometry and Topology, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], Lagrangian, Symplectic geometry
Abstract

A surgery of a real symplectic manifold $X_{\mathbb R}$ along a real Lagrangian sphere $S$ is a modification of the symplectic and real structure on $X_{\mathbb R}$ in a neigborhood of $S$. Genus 0 Welschinger invariants of two real symplectic $4$-manifolds differing by such a surgery have been related in a previous work in collaboration with N. Puignau. In the present paper, we explore some particular situations where these general formulas greatly simplify. As an application, we complete the computation of genus 0 Welschinger invariants of all del~Pezzo surfaces, and of all $\mathbb R$-minimal real conic bundles. As a by-product, we establish the existence of some new relative Welschinger invariants. We also generalize our results to the enumeration of curves of higher genus, and give relations between hypothetical invariants defined in the same vein as a previous work by Shustin., 28 pages, 2 figures. V2: Major edition (hopefully simplifications) of the first version, references precised. V3: Minor editions

Published
2018

36. A Floer fundamental group

Author

Jean-François Barraud, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Fundamental group, Pure mathematics, General Mathematics, 010102 general mathematics, Mathematical analysis, Degenerate energy levels, Multiplicity (mathematics), Geometric Topology (math.GT), Fixed point, 16. Peace & justice, 01 natural sciences, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], Mathematics - Geometric Topology, Floer homology, Mathematics - Symplectic Geometry, 57R17, 53D40, 14F35, 0103 physical sciences, FOS: Mathematics, Symplectic Geometry (math.SG), 010307 mathematical physics, 0101 mathematics, Mathematics::Symplectic Geometry, Symplectic manifold, Mathematics
Abstract

The main purpose of this paper is to provide a description of the fundamental group of a symplectic manifold in terms of Floer theoretic objects. As an application, we show that when counted with a suitable notion of multiplicity, non degenerate Hamiltonian diffeomorphisms have enough fixed points to generate the fundamental group., 38 pages. (added a quick discussion of the Morse-stable case)

Published
2018

37. The equivariant index of twisted dirac operators and semi-classical limits

Author

Mich`ele Vergne, Paul-`Emile Paradan, Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Jussieu (IMJ), and Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)

Subjects
010102 general mathematics, Lie group, Geometry, (g,K)-module, Dirac operator, 01 natural sciences, spin structure, moment map, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], symbols.namesake, Line bundle, [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], Irreducible representation, semi-classical limit, 0103 physical sciences, Lie algebra, symbols, Equivariant map, 010307 mathematical physics, 0101 mathematics, geometric quantization, Moment map, Mathematics, Mathematical physics
Abstract

Let G be a compact connected Lie group with Lie algebra g. Let M be a compact spin manifold with a G-action, and ℒ be a G-equivariant line bundle on M. Consider an integer k, and let \( \vartheta _G^{{\rm{spin}}} \) (M, ℒk) be the equivariant index of the Dirac operator on M twisted by ℒk. Let mG(λ, k) be the multiplicity in \( \vartheta _G^{{\rm{spin}}} \) (M, ℒk) of the irreducible representation of G attached to the admissible coadjoint orbit Gλ. We prove that the distribution ⟨Θk,φ⟩ = kdim(G/T)/2 Σλ mG(λ,k)⟨βλ/k,φ⟩ has an asymptotic expansion when k tends to infinity of the form ⟨Θk,φ⟨ ≡ kdimM/2\( \sum\nolimits_{n = 0}^\infty {{k^{ - n}}} \left\langle {{\theta _n},\phi } \right\rangle \). Here φ is a test function on g* and ⟨βξ,φ⟩ is the integral of φ on the coadjoint orbit Gφ with respect to the canonical Liouville measure. We compute explicitly the distribution θn in terms of the graded  class of M and the equivariant curvature of ℒ.

Published
2018

38. Symplectic capacities from positive S 1 -equivariant symplectic homology

Author

Michael Hutchings, Jean Gutt, Institut national universitaire Champollion [Albi] (INUC), Université de Toulouse (UT), Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), University of California [Berkeley] (UC Berkeley), and University of California (UC)

Subjects
Pure mathematics, 53D40, Homology (mathematics), 01 natural sciences, 0103 physical sciences, FOS: Mathematics, 0101 mathematics, Mathematics::Symplectic Geometry, 57R17, Axiom, Computer Science::Information Theory, Mathematics, Sequence, 2010 MSC: 53D05, 53D40, 57R17, 010102 general mathematics, Regular polygon, equivariant symplectic homology, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], symplectic capacities, 53D05, Mathematics - Symplectic Geometry, Domain (ring theory), Symplectic Geometry (math.SG), Equivariant map, 010307 mathematical physics, Geometry and Topology, Symplectic geometry, cube capacity
Abstract

We use positive S^1-equivariant symplectic homology to define a sequence of symplectic capacities c_k for star-shaped domains in R^{2n}. These capacities are conjecturally equal to the Ekeland-Hofer capacities, but they satisfy axioms which allow them to be computed in many more examples. In particular, we give combinatorial formulas for the capacities c_k of any "convex toric domain" or "concave toric domain". As an application, we determine optimal symplectic embeddings of a cube into any convex or concave toric domain. We also extend the capacities c_k to functions of Liouville domains which are almost but not quite symplectic capacities., Comment: 63 pages

Published
2018

39. Admissible coadjoint orbits for compact Lie groups

Author

Michèle Vergne, Paul-Emile Paradan, Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Jussieu (IMJ), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Institut Montpelliérain Alexander Grothendieck ( IMAG ), Université de Montpellier ( UM ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de Mathématiques de Jussieu ( IMJ ), and Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Université Paris Diderot - Paris 7 ( UPD7 ) -Centre National de la Recherche Scientifique ( CNRS )

Subjects
Pure mathematics, [ MATH.MATH-SG ] Mathematics [math]/Symplectic Geometry [math.SG], 01 natural sciences, Coadjoint orbit, Mathematics::Quantum Algebra, 0103 physical sciences, Quantization, FOS: Mathematics, Representation Theory (math.RT), 0101 mathematics, Algebra over a field, Mathematics::Representation Theory, Spin^c structures, Mathematics::Symplectic Geometry, Mathematics, Algebra and Number Theory, [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT], Quantization (signal processing), 010102 general mathematics, Lie group, 16. Peace & justice, [ MATH.MATH-RT ] Mathematics [math]/Representation Theory [math.RT], [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], Condensed Matter::Strongly Correlated Electrons, Astrophysics::Earth and Planetary Astrophysics, 010307 mathematical physics, Geometry and Topology, Mathematics - Representation Theory
Abstract

We describe the admissible coadjoint orbits of a compact connected Lie group and their spin-c quantization., Comment: The authors have decided to divide the preprint "Multiplicities of equivariant Spin-c Dirac operators" into two separate publications. The present paper is one of them, the other part is entitled "Equivariant Dirac operators and differentiable geometric invariant theory" (arXiv:1411.7772, version 3)

Published
2018

40. Symplectic homology and the Eilenberg–Steenrod axioms

Author

Kai Cieliebak, Alexandru Oancea, Universität Augsburg [Augsburg], Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG (UMR_7586)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Cieliebak, Kai, Oancea, Alexandru, and Albers, Peter

Subjects
Mathematics - Differential Geometry, Pure mathematics, 53D40, Eilenberg–Steenrod axioms for a homology theory, Homology (mathematics), 01 natural sciences, Mathematics::Algebraic Topology, Floer homology, Rabinowitz–Floer homology, Liouville cobordisms, Mathematics::K-Theory and Homology, 0103 physical sciences, FOS: Mathematics, Algebraic Topology (math.AT), Mathematics - Algebraic Topology, Eilenberg–Steenrod axioms, ddc:510, 0101 mathematics, [MATH]Mathematics [math], Mathematics::Symplectic Geometry, 57R17, Axiom, Mathematics, 53D40, 55N40, 57R17, 57R90, Differential Geometry, Exact sequence, 010102 general mathematics, contact homology, Symplectic Geometry, 57R90, Mathematics::Geometric Topology, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], 55N40, Differential Geometry (math.DG), Algebraic Topology, Mathematics - Symplectic Geometry, [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], symplectic homology, [MATH.MATH-AT]Mathematics [math]/Algebraic Topology [math.AT], Symplectic Geometry (math.SG), 010307 mathematical physics, Geometry and Topology, Symplectic geometry
Abstract

We give a definition of symplectic homology for pairs of filled Liouville cobordisms, and show that it satisfies analogues of the Eilenberg-Steenrod axioms except for the dimension axiom. The resulting long exact sequence of a pair generalizes various earlier long exact sequences such as the handle attaching sequence, the Legendrian duality sequence, and the exact sequence relating symplectic homology and Rabinowitz Floer homology. New consequences of this framework include a Mayer-Vietoris exact sequence for symplectic homology, invariance of Rabinowitz Floer homology under subcritical handle attachment, and a new product on Rabinowitz Floer homology unifying the pair-of-pants product on symplectic homology with a secondary coproduct on positive symplectic homology. In the appendix, joint with Peter Albers, we discuss obstructions to the existence of certain Liouville cobordisms., Comment: v3: corrected Lemma 7.11. Various other minor modifications and reformatting. Final version to be published in Algebraic and Geometric Topology

Published
2018

41. Representations, sheaves, and Legendrian $(2,m)$ torus links

Author

Lenhard Ng, Baptiste Chantraine, Steven Sivek, Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN), Duke University [Durham], Imperial College London, ANR-13-JS01-0008,cospin,Invariants spectraux de contact(2013), and ANR-16-CE40-0017,Quantact,Topologie quantique et géométrie de contact(2016)

Subjects
Pure mathematics, General Mathematics, 01 natural sciences, 0101 Pure Mathematics, Mathematics - Geometric Topology, Differential graded algebra, [MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT], Mathematics::Category Theory, 0103 physical sciences, FOS: Mathematics, math.GT, Rank (graph theory), 0101 mathematics, Link (knot theory), Representation (mathematics), Mathematics::Symplectic Geometry, Mathematics, Conjecture, math.SG, 010102 general mathematics, Torus, Geometric Topology (math.GT), 16. Peace & justice, Mathematics::Geometric Topology, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], Mathematics - Symplectic Geometry, [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], Symplectic Geometry (math.SG), 010307 mathematical physics, 57R17, 53D42 (primary), 14F05, 53D37 (secondary)
Abstract

We study an $A_\infty$ category associated to Legendrian links in $\mathbb{R}^3$ whose objects are $n$-dimensional representations of the Chekanov-Eliashberg differential graded algebra of the link. This representation category generalizes the positive augmentation category and we conjecture that it is equivalent to a category of sheaves of microlocal rank $n$ constructed by Shende, Treumann, and Zaslow. We establish the cohomological version of this conjecture for a family of Legendrian $(2,m)$ torus links., Comment: v3: 50 pages, small changes, to appear in Journal of the London Mathematical Society

Published
2018
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42. Integrable systems, symmetries and quantization

Author

Daniele Sepe, San Vu Ngoc, Instituto de Computação - Universidade Federal Fluminense ( IC-UFF ), Universidade Federal Fluminense [Rio de Janeiro] ( UFF ), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Instituto de Computação [Niteroi-Rio de Janeiro] (IC-UFF), Universidade Federal Fluminense [Rio de Janeiro] (UFF), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), and Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)

Subjects
Spectral theory, Integrable system, moment(um) maps, Complex system, Semiclassical physics, FOS: Physical sciences, [ MATH.MATH-SG ] Mathematics [math]/Symplectic Geometry [math.SG], 01 natural sciences, Mathematics - Spectral Theory, Quantization (physics), Theoretical physics, FOS: Mathematics, [NLIN.NLIN-SI]Nonlinear Sciences [physics]/Exactly Solvable and Integrable Systems [nlin.SI], 0101 mathematics, Quantum, Spectral Theory (math.SP), Mathematical Physics, Physics, classical and quantum integrable systems, [ MATH.MATH-SP ] Mathematics [math]/Spectral Theory [math.SP], Nonlinear Sciences - Exactly Solvable and Integrable Systems, 010102 general mathematics, Statistical and Nonlinear Physics, spectral theory, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], 37J35, 53D05, 70H06, 81Q20, 010101 applied mathematics, Mathematics - Symplectic Geometry, Homogeneous space, Symplectic Geometry (math.SG), Gravitational singularity, quantization, Exactly Solvable and Integrable Systems (nlin.SI), [ NLIN.NLIN-SI ] Nonlinear Sciences [physics]/Exactly Solvable and Integrable Systems [nlin.SI], [MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP]
Abstract

These notes correspond to a mini-course given at the Poisson 2016 conference in Geneva. Starting from classical integrable systems in the sense of Liouville, we explore the notion of non-degenerate singularity and expose recent research in connection with semi-toric systems. The quantum and semiclassical counterpart will also be presented, in the viewpoint of the inverse question: from the quantum mechanical spectrum, can you recover the classical system?, 91 pages, 4 figures

Published
2018

43. On para-Kähler and hyper-para-Kähler Lie algebras

Author

Mohamed Boucetta, Saïd Benayadi, Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Faculte des Sciences et techniques Gueliz (FSTG), Université Cadi Ayyad [Marrakech] (UCA), and Action concertée CNRST(Maroc)-CNRS (France)Project SPM04/13.

Subjects
Non-associative algebra, 01 natural sciences, [MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR], Representation of a Lie group, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, MSC:53C25, 53D05: 17B30, 0101 mathematics, [MATH.MATH-MG]Mathematics [math]/Metric Geometry [math.MG], Mathematics::Symplectic Geometry, Mathematics, Hyper-para-Kähler Lie algebra Left symmetric algebra Para-Kähler Lie algebra S-matrix Symplectic Lie algebra, Algebra and Number Theory, [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT], [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], 010102 general mathematics, Killing form, Kac–Moody algebra, Affine Lie algebra, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], Lie conformal algebra, Algebra, Adjoint representation of a Lie algebra, [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], Freudenthal magic square, Mathematics::Differential Geometry, 010307 mathematical physics
Abstract

International audience; In this study, we consider Lie algebras that admit para-Kählerand hyper-para-Kähler structures. We provide new charac-terizations of these Lie algebras and develop many methodsfor building large classes of examples. Previously, Bai consid-ered para-Kähler Lie algebras as left symmetric bialgebras.We reconsider this viewpoint and make improvements in or-der to obtain some new results. The study of para-Kähler andhyper-para-Kähler is intimately linked to the study of leftsymmetric algebras, particularly those that admit invariantsymplectic forms. In this study, we provide many new classesof left symmetric algebras and complete descriptions of all theassociative algebras that admit an invariant symplectic form.We also describe all four-dimensional hyper-para-Kähler Liealgebras.

Published
2015

44. EA-Matrix integrals of associative algebras and equivariant localization

Author

Serguei Barannikov, Institut de Mathématiques de Jussieu (IMJ), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG (UMR_7586)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Centre National de la Recherche Scientifique (CNRS), Université Paris Cité (UPCité), and Sorbonne Université (SU)

Subjects
Pure mathematics, Differential form, Mirror symmetry, General Mathematics, Scalar (mathematics), Gromov-Witten invariants, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], 01 natural sciences, Noncommutative varieties, Mathematics - Algebraic Geometry, 0103 physical sciences, Lie algebra, Mathematics - Quantum Algebra, FOS: Mathematics, Quantum Algebra (math.QA), 0101 mathematics, Algebraic Geometry (math.AG), Associative property, Batalin-Vilkovisky formalism, Mathematics, 11G42, 14J33, 53D37, Mathematics::History and Overview, 010102 general mathematics, 16. Peace & justice, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], Mathematics - Symplectic Geometry, Equivariant map, Symplectic Geometry (math.SG), 010307 mathematical physics, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG]
Abstract

Preprint HAL-00507788 (2010) from the CNRS open online arxive HAL. The equivariantly closed matrix integrals introduced in [B06], are studied in the case of the graded associative algebras with odd or even scalar product.I prove that the EA-matrix integrals for associative algebras with scalar product are integrals of equivariantly closed differential forms with respect to the Lie algebra $gl_N(A)$., Preprint HAL-00507788 (2010) from the CNRS open online arxive HAL

Published
2017

45. Toric contact geometry in arbitrary codimension

Author

Eveline Legendre, David M. J. Calderbank, Vestislav Apostolov, Paul Gauduchon, Centre interuniversitaire de recherches en géométrie et topologie [Montréal] (CIRGET), Centre de Recherches Mathématiques [Montréal] (CRM), Université de Montréal (UdeM)-Université de Montréal (UdeM), Centre de Mathématiques Laurent Schwartz (CMLS), Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X), Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Mathematics - Differential Geometry, Pure mathematics, 53C15, 53D10, 53D20, Mathematics::Commutative Algebra, General Mathematics, Contact geometry, 010102 general mathematics, Polytope, Codimension, 01 natural sciences, Mathematics::Geometric Topology, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], Mathematics::Algebraic Geometry, Differential Geometry (math.DG), Mathematics - Symplectic Geometry, Grassmannian, FOS: Mathematics, Symplectic Geometry (math.SG), Mathematics::Differential Geometry, 0101 mathematics, [MATH.MATH-MG]Mathematics [math]/Metric Geometry [math.MG], Mathematics::Symplectic Geometry, Symplectic manifold, Mathematics
Abstract

We define toric contact manifolds in arbitrary codimension and give a description of such manifolds in terms of a kind of labelled polytope embedded into a grassmannian, analogous to the Delzant polytope of a toric symplectic manifold., 22 pages (v2) added example and references

Published
2017

46. Surface singularities and planar contact structures

Author

Paolo Ghiggini, Olga Plamenevskaya, Marco Golla, Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN), and Centre National de la Recherche Scientifique (CNRS)

Subjects
Surface (mathematics), Pure mathematics, Fundamental group, Fibered knot, 01 natural sciences, Mathematics - Geometric Topology, [MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT], Genus (mathematics), 0103 physical sciences, FOS: Mathematics, 0101 mathematics, Mathematics::Symplectic Geometry, Mathematics, Algebra and Number Theory, 010102 general mathematics, Fibration, Geometric Topology (math.GT), Mathematics::Geometric Topology, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], Mathematics - Symplectic Geometry, Open book decomposition, Symplectic Geometry (math.SG), [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], 010307 mathematical physics, Geometry and Topology, Normal surface, Symplectic geometry
Abstract

We prove that if a contact 3-manifold admits an open book decomposition of genus 0, a certain intersection pattern cannot appear in the homology of any of its minimal symplectic fillings, and moreover, fillings cannot contain symplectic surfaces of positive genus. Applying these obstructions to canonical contact structures on links of normal surface singularities, we show that links of isolated singularities of surfaces in the complex 3-space are planar only in the case of $A_n$-singularities. In general, we characterize completely planar links of normal surface singularities (in terms of their resolution graphs); these singularities are precisely rational singularities with reduced fundamental cycle (also known as minimal singularities). We also establish non-planarity of tight contact structures on certain small Seifert fibered L-spaces and of contact structures arising from the Boothby--Wang construction applied to surfaces of positive genus. Additionally, we prove that every finitely presented group is the fundamental group of a Lefschetz fibration with planar fibers., 26 pages, 8 figures; corrected a reference, this is the final version, to appear on the Annales de l'Institut Fourier

Published
2017

47. Topological entropy for Reeb vector fields in dimension three via open book decompositions

Author

Ko Honda, Vincent Colin, Marcelo R. R. Alves, Université de Neuchâtel (UNINE), Université de Nantes (UN), Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN), University of California [Los Angeles] (UCLA), University of California, Swiss National FoundationNSF Grant DMS-1406564, ANR-16-CE40-0017,Quantact,Topologie quantique et géométrie de contact(2016), European Project: 278246,EC:FP7:ERC,ERC-2011-StG_20101014,GEODYCON(2012), Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS), and University of California (UC)

Subjects
General Mathematics, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], Topological entropy, Dynamical Systems (math.DS), Homology (mathematics), 01 natural sciences, Combinatorics, 0502 economics and business, FOS: Mathematics, 0101 mathematics, Mathematics - Dynamical Systems, Mathematics::Symplectic Geometry, Mathematics, 010102 general mathematics, 05 social sciences, Weinstein conjecture, 16. Peace & justice, Mathematics::Geometric Topology, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], Dehn twist, Monodromy, Mathematics - Symplectic Geometry, Open book decomposition, Symplectic Geometry (math.SG), topological entropy, contact structure, open book decomposition,mapping class group, Reeb dynamics, pseudo-Anosov, contact homology, Vector field, 050203 business & management, Knot (mathematics)
Abstract

Given an open book decomposition of a contact three man-ifold (M, $\xi$) with pseudo-Anosov monodromy and fractional Dehn twist coefficient c = k n, we construct a Legendrian knot $\Lambda$ close to the stable foliation of a page, together with a small Legendrian pushoff $\Lambda$. When k $\ge$ 5, we apply the techniques of [CH2] to show that the strip Legen-drian contact homology of $\Lambda$ $\rightarrow$ $\Lambda$ is well-defined and has an exponential growth property. The work [Al2] then implies that all Reeb vector fields for $\xi$ have positive topological entropy.

Published
2017

48. Crepant resolutions and open strings

Author

Dustin Ross, Renzo Cavalieri, Andrea Brini, GTA, Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), and Colorado State University [Fort Collins] (CSU)

Subjects
High Energy Physics - Theory, Pure mathematics, Class (set theory), QUANTIZATION, General Mathematics, FOS: Physical sciences, QUANTUM COHOMOLOGY, 01 natural sciences, 0101 Pure Mathematics, Mathematics - Algebraic Geometry, Symplectic vector space, Morphism, Mathematics::Algebraic Geometry, INTEGRALS, 0103 physical sciences, MIRROR SYMMETRY, FOS: Mathematics, Brane cosmology, 0101 mathematics, Algebraic Geometry (math.AG), Mathematics::Symplectic Geometry, Mathematics, Conjecture, Science & Technology, [PHYS.HTHE]Physics [physics]/High Energy Physics - Theory [hep-th], 010308 nuclear & particles physics, Applied Mathematics, 010102 general mathematics, LOCALIZATION, GROMOV-WITTEN THEORY, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], High Energy Physics - Theory (hep-th), Mathematics - Symplectic Geometry, Physical Sciences, Crepant resolution, Symplectic Geometry (math.SG), Equivariant map, Gravitational singularity, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG]
Abstract

We formulate a Crepant Resolution Correspondence for open Gromov-Witten invariants (OCRC) of toric Lagrangian branes inside Calabi-Yau 3-orbifolds by encoding the open theories into sections of Givental's symplectic vector space. The correspondence can be phrased as the identification of these sections via a linear morphism of Givental spaces. We deduce from this a Bryan-Graber-type statement for disk invariants, which we extend to arbitrary topologies in the Hard Lefschetz case. Motivated by ideas of Iritani, Coates-Corti-Iritani-Tseng and Ruan, we furthermore propose 1) a general form of the morphism entering the OCRC, which arises from a geometric correspondence between equivariant K-groups, and 2) an all-genus version of the OCRC for Hard Lefschetz targets. We provide a complete proof of both statements in the case of minimal resolutions of threefold An singularities; as a necessary step of the proof we establish the all-genus closed Crepant Resolution Conjecture with descendents in its strongest form for this class of examples. Our methods rely on a new description of the quantum D-modules underlying the equivariant Gromov-Witten theory of this family of targets., This paper supersedes arXiv:1303.0723 by the same authors, which will be withdrawn. v2: minor changes, references added. v3: arguments strengthened in Section 6.1 with reference to Teleman's theorem, statements about analytic continuation of flat sections of the Dubrovin connection have been clarified in Section 5.3 (Lemma 5.8); version accepted for publication in Crelle. 48 pages, 6 figures

Published
2017

49. Towards a symplectic version of the Chevalley restriction theorem

Author

Ronan Terpereau, Manfred Lehn, Christian Lehn, Michael Bulois, Algèbre, géométrie, logique ( AGL ), Institut Camille Jordan [Villeurbanne] ( ICJ ), École Centrale de Lyon ( ECL ), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 ( UCBL ), Université de Lyon-Institut National des Sciences Appliquées de Lyon ( INSA Lyon ), Université de Lyon-Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Université Jean Monnet [Saint-Étienne] ( UJM ) -Centre National de la Recherche Scientifique ( CNRS ) -École Centrale de Lyon ( ECL ), Université de Lyon-Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Université Jean Monnet [Saint-Étienne] ( UJM ) -Centre National de la Recherche Scientifique ( CNRS ), Fakultät für Mathematik der Technischen Universität Chemnitz, Fakultaet fuer Mathematik der Technischen Universitaet Chemnitz, Institut für Mathematik ( MI ), Johannes Gutenberg - Universität Mainz ( JGU ), Institut de Mathématiques de Bourgogne [Dijon] ( IMB ), Université de Bourgogne ( UB ) -Centre National de la Recherche Scientifique ( CNRS ), Max Planck Institute for Mathematics ( Bonn ), The second-named author gratefully acknowledges the support by the DFG through the research grant Le 3093/2-1. The thirdnamed author is supported by the SFB Transregio 45 'Periods, Moduli Spaces and Arithmetic of Algebraic Varieties'. The fourth-named author is grateful to the Max-Planck-Institut für Mathematik of Bonn for the warm hospitality and support provided during the writing of this paper. Part ofthe genesis of this paper occurred during the half semester Algebraic Groups and Representations supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program 'Investissements d'Avenir' (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR)., ANR-11-IDEX-0007-02/10-LABX-0070,MILYON,Community of mathematics and fundamental computer science in Lyon ( 2011 ), ANR-15-CE40-0012,GeoLie,GeoLie, Algèbre, géométrie, logique (AGL), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet [Saint-Étienne] (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet [Saint-Étienne] (UJM)-Centre National de la Recherche Scientifique (CNRS), Chemnitz University of Technology / Technische Universität Chemnitz, Institut für Mathematik (MI), Johannes Gutenberg - Universität Mainz (JGU), Institut de Mathématiques de Bourgogne [Dijon] (IMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB), Max Planck Institute for Mathematics (MPIM), Max-Planck-Gesellschaft, ANR-10-LABX-0070,MILYON,Community of mathematics and fundamental computer science in Lyon(2010), and ANR-15-CE40-0012,GéoLie,Méthodes géométriques en théorie de Lie(2015)

Subjects
Polar representation, Symplectic variety, [ MATH.MATH-SG ] Mathematics [math]/Symplectic Geometry [math.SG], 01 natural sciences, Combinatorics, Mathematics - Algebraic Geometry, symbols.namesake, Morphism, 0103 physical sciences, FOS: Mathematics, 14L30, 53D20, 20G05, 20G20, 13A50, Representation Theory (math.RT), 0101 mathematics, MSC: 14L30, 53D20, 20G05, 20G20, 13A50, Mathematics::Representation Theory, Symplectic reduction, Algebraic Geometry (math.AG), Moment map, Mathematics, Weyl group, Algebra and Number Theory, Conjecture, [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT], Reductive group, 010102 general mathematics, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], [ MATH.MATH-RT ] Mathematics [math]/Representation Theory [math.RT], [ MATH.MATH-AG ] Mathematics [math]/Algebraic Geometry [math.AG], Mathematics - Symplectic Geometry, symbols, Symplectic Geometry (math.SG), 010307 mathematical physics, Isomorphism, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], Mathematics - Representation Theory, Symplectic geometry
Abstract

If $(G,V)$ is a polar representation with Cartan subspace $\mathfrak c$ and Weyl group $W$, it is shown that there is a natural morphism of Poisson schemes $\mathfrak c \oplus {\mathfrak c}^*/W \to V\oplus V^*/\!\!/\!\!/ G$. This morphism is conjectured to be an isomorphism of the underlying reduced varieties if $(G,V)$ is visible. The conjecture is proved for visible stable locally free polar representations and certain further examples., Comment: 18 pages, final version

Published
2017

50. Equivariant Dirac operators and differentiable geometric invariant theory

Author

Michèle Vergne, Paul-Emile Paradan, Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Jussieu (IMJ), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques et de Modélisation de Montpellier ( I3M ), Université Montpellier 2 - Sciences et Techniques ( UM2 ) -Université de Montpellier ( UM ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de Mathématiques de Jussieu ( IMJ ), Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Université Paris Diderot - Paris 7 ( UPD7 ) -Centre National de la Recherche Scientifique ( CNRS ), and Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Mathematics - Differential Geometry, Pure mathematics, General Mathematics, [ MATH.MATH-SG ] Mathematics [math]/Symplectic Geometry [math.SG], Dirac operator, 01 natural sciences, Mathematics::Algebraic Topology, symbols.namesake, Mathematics::K-Theory and Homology, 0103 physical sciences, [ MATH.MATH-KT ] Mathematics [math]/K-Theory and Homology [math.KT], FOS: Mathematics, Differentiable function, 0101 mathematics, Spin^c structures, Mathematics::Symplectic Geometry, Mathematics, 010102 general mathematics, Multiplicity (mathematics), K-Theory and Homology (math.KT), [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], [ MATH.MATH-DG ] Mathematics [math]/Differential Geometry [math.DG], Equivariant index, Transversally elliptic symbol, Differential Geometry (math.DG), Multiplicity, Mathematics - Symplectic Geometry, [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], Mathematics - K-Theory and Homology, symbols, [MATH.MATH-KT]Mathematics [math]/K-Theory and Homology [math.KT], Equivariant map, Symplectic Geometry (math.SG), Condensed Matter::Strongly Correlated Electrons, 010307 mathematical physics, Geometric invariant theory
Abstract

In this paper, we give a geometric expression for the multiplicities of the equivariant index of a spin-c Dirac operator., Comment: The authors have decided to divide the preprint "Multiplicities of equivariant Spin-c Dirac operators" into two separate publications. The present paper is the first of them. The other part is the preprint arXiv:1512.02367 entitled "Admissible orbits for compact Lie group"

Published
2017

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