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63 results

1. A note on hypocoercivity for kinetic equations with heavy-tailed equilibrium

Author: Isabelle Tristani, Hélène Hivert, Maxime Herda, Nathalie Ayi, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Reliable numerical approximations of dissipative systems (RAPSODI ), Laboratoire Paul Painlevé - UMR 8524 (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université de Lille-Centre National de la Recherche Scientifique (CNRS), Institut Camille Jordan [Villeurbanne] (ICJ), Centre National de la Recherche Scientifique (CNRS)-Université Jean Monnet [Saint-Étienne] (UJM)-École Centrale de Lyon (ECL), Université de Lyon-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 Claude Bernard Lyon 1 (UCBL), Université de Lyon, Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS Paris)-Centre National de la Recherche Scientifique (CNRS), 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), and 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)
Subjects: Fokker-Planck operator, General Mathematics, 010102 general mathematics, fractional diffusion, Kinetic energy, 82C40, 35K65, 35Q84, 60G22, 01 natural sciences, Hypocoercivity, Mathematics - Analysis of PDEs, Exponential growth, Heavy-tailed distribution, Kinetic equations, Regularization (physics), 0103 physical sciences, Fractional diffusion, FOS: Mathematics, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 010307 mathematical physics, 0101 mathematics, heavy-tailed distribution, linear kinetic equations, Mathematics, Analysis of PDEs (math.AP)
Abstract: In this paper we are interested in the large time behavior of linear kinetic equations with heavy-tailed local equilibria. Our main contribution concerns the kinetic Lévy-Fokker-Planck equation, for which we adapt hypocoercivity techniques in order to show that solutions converge exponentially fast to the global equilibrium. Compared to the classical kinetic Fokker-Planck equation, the issues here concern the lack of symmetry of the non-local Lévy-Fokker-Planck operator and the understanding of its regularization properties. As a complementary related result, we also treat the case of the heavy-tailed BGK equation.; In this paper we are interested in the large time behavior of linear kinetic equations with heavy-tailed local equilib-ria. Our main contribution concerns the kinetic Lévy-Fokker-Planck equation, for which we adapt hypocoercivity techniques in order to show that solutions converge exponentially fast to the global equilibrium. Compared to the classical kinetic Fokker-Planck equation, the issues here concern the lack of symmetry of the non-local Lévy-Fokker-Planck operator and the understanding of its regularization properties. As a complementary related result, we also treat the case of the heavy-tailed BGK equation. Résumé Une note sur l'hypocoercivité pour leséquations cinétiques avecéquilibresà queue lourde. Dans cet article, on s'intéresse au comportement en temps long d'équations cinétiques linéaires dont leséquilibres locaux sontà queue lourde. Notre contribution principale concerne l'équation de Lévy-Fokker-Planck cinétique, pour laquelle nous adaptons des techniques d'hypocoercivité afin de démontrer la convergence exponentielle des solutions vers unéquilibre global. En comparant au cas de l'équation de Fokker-Planck cinétique classique, les enjeux ici sont liés au manque de symétrie de l'opérateur non-local de Lévy-Fokker-Planck età la compréhension de ses propriétés de régularisation. En complément de notre analyse, nous traitonségalement le cas de l'équation de BGKà queue lourde.
Published: 2020

2. HYPERTRANSCENDENCE OF SOLUTIONS OF MAHLER EQUATIONS

Author: Thomas Dreyfus, Charlotte Hardouin, Julien Roques, Institut de Recherche Mathématique Avancée (IRMA), Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA), 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), Institut Fourier (IF ), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Institut Fourier (IF), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA), 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: Class (set theory), Pure mathematics, Series (mathematics), Applied Mathematics, General Mathematics, 010102 general mathematics, Galois theory, Context (language use), 39A06, 12H10, Mathematics - Commutative Algebra, Commutative Algebra (math.AC), 01 natural sciences, Set (abstract data type), [MATH.MATH-GM]Mathematics [math]/General Mathematics [math.GM], Algebraic group, 0103 physical sciences, FOS: Mathematics, 010307 mathematical physics, 0101 mathematics, U-1, Differential (mathematics), Mathematics
Abstract: International audience; The last years have seen a growing interest from mathematicians in Mahler functions. This class of functions includes the generating series of the automatic sequences. The present paper is concerned with the following problem, which is rather frequently encountered in combinatorics: a set of Mahler functions u 1 , ..., un being given, are u 1 , ..., un and their successive derivatives algebraically independent? In this paper, we give general criteria ensuring an affirmative answer to this question. We apply our main results to the generating series attached to the so-called Baum-Sweet and Rudin-Shapiro automatic sequences. In particular, we show that these series are hyperalge-braically independent, i.e., that these series and their successive derivatives are algebraically independent. Our approach relies of the parametrized difference Galois theory (in this context, the algebro-differential relations between the solutions of a given Mahler equation are reflected by a linear differential algebraic group).
Published: 2018

3. KANTOROVICH POTENTIALS AND CONTINUITY OF TOTAL COST FOR RELATIVISTIC COST FUNCTIONS

Author: Marjolaine Puel, Jérôme Bertrand, Aldo Pratelli, 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), Department Mathematik [Erlangen], Friedrich-Alexander Universität Erlangen-Nürnberg (FAU), 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: Kantorovich potential, Critical time, Mass transport, Optimal transportation, Relativistic heat equation, Relativistic cost, Total cost, Applied Mathematics, General Mathematics, 010102 general mathematics, Optimal map, 01 natural sciences, 0103 physical sciences, Mathematics (all), Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Chain lemma, 010307 mathematical physics, Uniqueness, Ball (mathematics), 0101 mathematics, Convex function, [MATH.MATH-MG]Mathematics [math]/Metric Geometry [math.MG], Mathematics
Abstract: In this paper we consider the optimal mass transport problem for relativistic cost functions, introduced in [12] as a generalization of the relativistic heat cost. A typical example of such a cost function is c t ( x , y ) = h ( y − x t ) , h being a strictly convex function when the variable lies on a given ball, and infinite otherwise. It has been already proved that, for every t larger than some critical time T > 0 , existence and uniqueness of optimal maps hold; nonetheless, the existence of a Kantorovich potential is known only under quite restrictive assumptions. Moreover, the total cost corresponding to time t has been only proved to be a decreasing right-continuous function of t. In this paper, we extend the existence of Kantorovich potentials to a much broader setting, and we show that the total cost is a continuous function. To obtain both results the two main crucial steps are a refined “chain lemma” and the result that, for t > T , the points moving at maximal distance are negligible for the optimal plan.
Published: 2017

4. Symplectic theory of completely integrable Hamiltonian systems

Author: San Vũ Ngọc, Álvaro Pelayo, Department of Mathematics [Berkeley], University of California [Berkeley], University of California-University of California, 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), Mathematics Department, 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 ), University of California [Berkeley] (UC Berkeley), University of California (UC)-University of California (UC), 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, and 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)
Subjects: Pure mathematics, Integrable system, General Mathematics, Dynamical Systems (math.DS), 01 natural sciences, Hamiltonian system, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], 0103 physical sciences, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Covariant Hamiltonian field theory, Mathematics - Dynamical Systems, 0101 mathematics, Symplectomorphism, Mathematics::Symplectic Geometry, Moment map, ComputingMilieux_MISCELLANEOUS, Mathematics, Symplectic manifold, Applied Mathematics, 010102 general mathematics, Symplectic representation, Mathematics - Symplectic Geometry, Symplectic Geometry (math.SG), 010307 mathematical physics, 37J35, 14H70, 37J05, 37J15, 53D20, Symplectic geometry
Abstract: This paper explains the recent developments on the symplectic theory of Hamiltonian completely integrable systems on symplectic 4-manifolds, compact or not. One fundamental ingredient of these developments has been the understanding of singular affine structures. These developments make use of results obtained by many authors in the second half of the twentieth century, notably Arnold, Duistermaat and Eliasson, of which we also give a concise survey. As a motivation, we present a collection of remarkable results proven in the early and mid 1980s in the theory of Hamiltonian Lie group actions by Atiyah, Guillemin-Sternberg and Delzant among others, and which inspired many people, including the authors, to work on more general Hamiltonian systems. The paper concludes discussing a spectral conjecture for quantum integrable systems., 40 pages
Published: 2011

5. Deformation theory of representation of prop(erad)s I

Author: Bruno Vallette, Sergei Merkulov, Laboratoire Jean Alexandre Dieudonné (JAD), Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), and ANR-06-JCJC-0042,OBTH,Opérades Bigèbres et Théories d'Homotopie(2006)
Subjects: Koszul duality, Model category, formal manifolds, General Mathematics, Minimal models, Commutative ring, Homology (mathematics), 01 natural sciences, Mathematics::Algebraic Topology, Morphism, Mathematics::K-Theory and Homology, deformation theory, Mathematics::Quantum Algebra, Mathematics::Category Theory, 0103 physical sciences, Lie algebra, Operads, 0101 mathematics, homotopy Lie algebras, Mathematics, Applied Mathematics, Homotopy, 010102 general mathematics, 16. Peace & justice, Algebra, [MATH.MATH-KT]Mathematics [math]/K-Theory and Homology [math.KT], [MATH.MATH-QA]Mathematics [math]/Quantum Algebra [math.QA], 010307 mathematical physics
Abstract: 55 pages; International audience; In this paper and its follow-up, we study the deformation theory of morphisms of properads and props thereby extending Quillen's deformation theory for commutative rings to a non-linear framework. The associated chain complex is endowed with an L_infinity-algebra structure. Its Maurer-Cartan elements correspond to deformed structures, which allows us to give a geometric interpretation of these results. To do so, we endow the category of prop(erad)s with a model category structure. We provide a complete study of models for prop(erad)s. A new effective method to make minimal models explicit, that extends the Koszul duality theory, is introduced and the associated notion is called homotopy Koszul. As a corollary, we obtain the (co)homology theories of (al)gebras over a prop(erad) and of homotopy (al)gebras as well. Their underlying chain complex is endowed with an L_infinity-algebra structure in general and a Lie algebra structure only in the Koszul case. In particular, we make the deformation complex of morphisms from the properad of associative bialgebras explicit. For any minimal model of this properad, the boundary map of this chain complex is shown to be the one defined by Gerstenhaber and Schack. As a corollary, this paper provides a complete proof of the existence of an L_infinity-algebra structure on the Gerstenhaber-Schack bicomplex associated to the deformations of associative bialgebras.
Published: 2009

6. SCATTERING DIAGRAMS, SHEAVES, AND CURVES

Author: Pierrick Bousseau, Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques d'Orsay (LMO), and Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)
Subjects: Pure mathematics, Conjecture, Applied Mathematics, General Mathematics, 010102 general mathematics, 01 natural sciences, Cubic plane curve, Moduli, Moduli space, Coherent sheaf, Mathematics - Algebraic Geometry, Mathematics::Algebraic Geometry, Mathematics - Symplectic Geometry, Genus (mathematics), 0103 physical sciences, FOS: Mathematics, Symplectic Geometry (math.SG), 010307 mathematical physics, 0101 mathematics, Algebraic number, [MATH]Mathematics [math], Algebraic Geometry (math.AG), Mathematics::Symplectic Geometry, Complex projective plane, Mathematics
Abstract: We review the recent proof of the N.Takahashi's conjecture on genus $0$ Gromov-Witten invariants of $(\mathbb{P}^2, E)$, where $E$ is a smooth cubic curve in the complex projective plane $\mathbb{P}^2$. The main idea is the use of the algebraic notion of scattering diagram as a bridge between the world of Gromov-Witten invariants of $(\mathbb{P}^2, E)$ and the world of moduli spaces of coherent sheaves on $\mathbb{P}^2$. Using this bridge, the N.Takahashi's conjecture can be translated into a manageable question about moduli spaces of coherent sheaves on $\mathbb{P}^2$. This survey is based on a three hours lecture series given as part of the Beijing-Zurich moduli workshop in Beijing, 9-12 September 2019., Comment: Expository paper. 18 pages, 1 figure
Published: 2021

7. Integral representation of unbounded variational functionals on Sobolev spaces

Author: Omar Anza Hafsa, Jean-Philippe Mandallena, Mathématiques, Informatique, Physique, et Applications / Université de Nîmes (MIPA), and Université de Nîmes (UNIMES)
Subjects: Pure mathematics, Integral representation, Applied Mathematics, General Mathematics, Numerical analysis, 010102 general mathematics, Dimension (graph theory), [MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA], 01 natural sciences, Homogenization (chemistry), Sobolev space, Nonlinear system, 0103 physical sciences, Convergence (routing), 010307 mathematical physics, Relaxation (approximation), 0101 mathematics, [MATH]Mathematics [math], Mathematics
Abstract: In this paper we establish an unbounded version of the integral representation theorem by Buttazzo and Dal Maso [see Buttazzo Dal Maso (Nonlinear Anal 9(6):515–532, 1985) and also Bouchitte et al. (Arch Ration Mech Anal 165(3), 187–242, 2002)]. More precisely, we prove an integral representation theorem (with a formula for the integrand) for functionals defined on $$W^{1,p}$$ with $$p>N$$ (N being the dimension) that do not satisfy a standard p-growth condition from above and can take infinite values. Applications to $$\Gamma $$ -convergence, relaxation and homogenization are also developed.
Published: 2021

8. Geometry of the moduli of parabolic bundles on elliptic curves

Author: Néstor Fernández Vargas, 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), ANR-11-LABX-0020-01, Agence Nationale de la Recherche, ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011), 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 ), ANR-11-LABX-0020/11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation ( 2011 ), 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, and 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)
Subjects: Del Pezzo surface, General Mathematics, Vector bundle, Geometry, Rank (differential topology), Space (mathematics), 01 natural sciences, Moduli space, Moduli, Mathematics - Algebraic Geometry, Mathematics::Algebraic Geometry, 0103 physical sciences, Elliptic curve, 0101 mathematics, Parabolic structure, Mathematics::Symplectic Geometry, Mathematics, Primary 14H60, Secondary 14D20, 14H52, 14Q10, 14H60, 14D20, 14H52, 14Q10, parabolic vector bundle, Applied Mathematics, 010102 general mathematics, [ MATH.MATH-AG ] Mathematics [math]/Algebraic Geometry [math.AG], Cover (topology), 010307 mathematical physics, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG]
Abstract: The goal of this paper is the study of simple rank 2 parabolic vector bundles over a 2 2 -punctured elliptic curve C C . We show that the moduli space of these bundles is a non-separated gluing of two charts isomorphic to P 1 × P 1 \mathbb {P}^1 \times \mathbb {P}^1 . We also showcase a special curve Γ \Gamma isomorphic to C C embedded in this space, and this way we prove a Torelli theorem. This moduli space is related to the moduli space of semistable parabolic bundles over P 1 \mathbb {P}^1 via a modular map which turns out to be the 2:1 cover ramified in Γ \Gamma . We recover the geometry of del Pezzo surfaces of degree 4 and we reconstruct all their automorphisms via elementary transformations of parabolic vector bundles.
Published: 2021

9. TOPOLOGICAL MIXING OF WEYL CHAMBER FLOWS

Author: Olivier Glorieux, Nguyen-Thi Dang, 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), Université du Luxembourg (Uni.lu), 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, and 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)
Subjects: medicine.medical_specialty, Physics::Instrumentation and Detectors, General Mathematics, Nonpositive curvature, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], Topological dynamics, Space (mathematics), Translation (geometry), Topology, 01 natural sciences, 0103 physical sciences, medicine, 0101 mathematics, Mathematics::Representation Theory, Mixing (physics), Mathematics, Applied Mathematics, 010102 general mathematics, Infinite volume, 54H20, 37B99, 53C30, 58E40, Action (physics), Topological Dynamics, Weyl chamber flow, Flow (mathematics), 010307 mathematical physics, Semisimple Lie groups
Abstract: International audience; In this paper, we study topological properties of the right action by translation of the Weyl Chamber flow on the space of Weyl chambers. We obtain a necessary and sufficient condition for topological mixing. (1)
Published: 2021

10. THE HÖLDER CONTINUOUS SUBSOLUTION THEOREM FOR COMPLEX HESSIAN EQUATIONS

Author: Amel Benali, Ahmed Zeriahi, 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: Hessian equation, Mathematics - Complex Variables, General Mathematics, 010102 general mathematics, Hölder condition, [MATH.MATH-CV]Mathematics [math]/Complex Variables [math.CV], 01 natural sciences, Mathematics - Analysis of PDEs, 31C45, 32U15, 32U40, 32W20, 35J96, 0103 physical sciences, Applied mathematics, 010307 mathematical physics, 0101 mathematics, Mathematics
Abstract: Let $\Omega \Subset \mathbb C^n$ be a bounded strongly $m$-pseudoconvex domain ($1\leq m\leq n$) and $\mu$ a positive Borel measure with finite mass on $\Omega$. Then we solve the H\"older continuous subsolution problem for the complex Hessian equation $(dd^c u)^m \wedge \beta^{n - m} = \mu$ on $\Omega$. Namely, we show that this equation admits a unique H\"older continuous solution on $\Omega$ with a given H\"older continuous boundary values if it admits a H\"older continuous subsolution on $\Omega$. The main step in solving the problem is to establish a new capacity estimate showing that the $m$-Hessian measure of a H\"older continuous $m$-subharmonic function on $\Omega$ with zero boundary values is dominated by the $m$-Hessian capacity with respect to $\Omega$ with an (explicit) exponent $\tau > 1$., Comment: This is a corrected version of a published paper where a new correct versions of Theorem B and Lemma 4.2 was provided
Published: 2020

11. Generalized traveling waves for time-dependent reaction–diffusion systems

Author: Arnaud Ducrot, B. Ambrosio, Shigui Ruan, Laboratoire de Mathématiques Appliquées du Havre (LMAH), Université Le Havre Normandie (ULH), Normandie Université (NU)-Normandie Université (NU), Department of Mathematics [Miami], and University of Miami [Coral Gables]
Subjects: education.field_of_study, General Mathematics, 010102 general mathematics, Population, Context (language use), System of linear equations, 01 natural sciences, 0103 physical sciences, Reaction–diffusion system, Traveling wave, Quantitative Biology::Populations and Evolution, Carrying capacity, Applied mathematics, 010307 mathematical physics, 0101 mathematics, [MATH]Mathematics [math], Epidemic model, education, Mathematics
Abstract: Traveling wave solutions in general time-dependent (including time-periodic) reaction–diffusion equations and systems of equations have attracted great attention in the last two decades. The aim of this paper is to study the propagation phenomenon in a general time-heterogeneous environment. More specifically, we investigate generalized traveling wave solutions for a two-component time-dependent non-cooperative reaction–diffusion system which has applications in epidemiology and ecology. Sufficient conditions on the existence and nonexistence of generalized traveling wave solutions are established. In the susceptible-infectious epidemic model setting, generalized traveling waves describe the spatio-temporal invasion of a disease into a totally susceptible population. In the context of predator–prey systems, the generalized traveling waves describe the spatial invasion of predators introduced into a new environment where the prey population is at its carrying capacity.
Published: 2020

12. Unique ergodicity of the horocycle flow on Riemannnian foliations

Author: F. Alcalde Cuesta, Alberto Verjovsky, Matilde Martinez, Françoise Dal'Bo, Universidade de Santiago de Compostela [Spain] (USC ), 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), Instituto de Matemática y Estadística Rafael Laguardia [Montevideo] (IMERL), Universidad de la República [Montevideo] (UCUR), Instituto de Matematicas (UNAM), Universidad Nacional Autónoma de México (UNAM), 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), Universidad de la República [Montevideo] (UDELAR), and Universidad Nacional Autónoma de México = National Autonomous University of Mexico (UNAM)
Subjects: Pure mathematics, Mathematics::Dynamical Systems, General Mathematics, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], Dynamical Systems (math.DS), 01 natural sciences, Group action, group actions, 0103 physical sciences, Affine group, FOS: Mathematics, 0101 mathematics, Mathematics - Dynamical Systems, classical ergodic theory, Mathematics, foliations, Conjecture, Applied Mathematics, 010102 general mathematics, Ergodicity, 37C85, 37D40, 57R30, 16. Peace & justice, Surface (topology), Compact space, Horocycle, Flow (mathematics), horocycle flow, 010307 mathematical physics, Mathematics::Differential Geometry
Abstract: A classic result due to Furstenberg is the strict ergodicity of the horocycle flow for a compact hyperbolic surface. Strict ergodicity is unique ergodicity with respect to a measure of full support, and therefore implies minimality. The horocycle flow has been previously studied on minimal foliations by hyperbolic surfaces on closed manifolds, where it is known not to be minimal in general. In this paper, we prove that for the special case of Riemannian foliations, strict ergodicity of the horocycle flow still holds. This in particular proves that this flow is minimal, which establishes a conjecture proposed by Matsumoto. The main tool is a theorem due to Coud\`ene, which he presented as an alternative proof for the surface case. It applies to two continuous flows defining a measure-preserving action of the affine group of the line on a compact metric space, precisely matching the foliated setting. In addition, we briefly discuss the application of Coud\`ene's theorem to other kinds of foliations., Comment: 18 pages, 1 figure, to appear in Ergodic Theory and Dynamical Systems
Published: 2020

13. Modulo $p$ representations of reductive $p$-adic groups: functorial properties

Author: M.-F. Vignéras, G. Henniart, Noriyuki Abe, Hokkaido University [Sapporo, Japan], Laboratoire de Mathématiques d'Orsay (LM-Orsay), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11), 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), Hokkaido University, and Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Pure mathematics, Mathematics - Number Theory, [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT], Applied Mathematics, General Mathematics, Modulo, 010102 general mathematics, 01 natural sciences, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], 0103 physical sciences, FOS: Mathematics, primary 20C08, secondary 11F70, 010307 mathematical physics, Number Theory (math.NT), 0101 mathematics, Representation Theory (math.RT), Mathematics::Representation Theory, Mathematics - Representation Theory, Mathematics
Abstract: Let $F$ be a local field with residue characteristic $p$, let $C$ be an algebraically closed field of characteristic $p$, and let $\mathbf{G}$ be a connected reductive $F$-group. In a previous paper, Florian Herzig and the authors classified irreducible admissible $C$-representations of $G=\mathbf{G}(F)$ in terms of supercuspidal representations of Levi subgroups of $G$. Here, for a parabolic subgroup $P$ of $G$ with Levi subgroup $M$ and an irreducible admissible $C$-representation $\tau$ of $M$, we determine the lattice of subrepresentations of $\mathrm{Ind}_P^G \tau$ and we show that $\mathrm{Ind}_P^G \chi \tau$ is irreducible for a general unramified character $\chi$ of $M$. In the reverse direction, we compute the image by the two adjoints of $\mathrm{Ind}_P^G$ of an irreducible admissible representation $\pi$ of $G$. On the way, we prove that the right adjoint of $\mathrm{Ind}_P^G $ respects admissibility, hence coincides with Emerton's ordinary part functor $\mathrm{Ord}_{\overline{P}}^G$ on admissible representations., Comment: 39 pages
Published: 2018

14. Moser’s theorem on manifolds with corners

Author: Peter W. Michor, Adam Parusinski, Armin Rainer, Martins Bruveris, Brunel University London [Uxbridge], University of Vienna [Vienna], Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), and Universität Wien
Subjects: Mathematics - Differential Geometry, Pure mathematics, Group (mathematics), Differential form, Applied Mathematics, General Mathematics, Operator (physics), 010102 general mathematics, Boundary (topology), 16. Peace & justice, Space (mathematics), 53C65, 58A10, 01 natural sciences, Manifold, Differential Geometry (math.DG), 0103 physical sciences, Lefschetz duality, FOS: Mathematics, 010307 mathematical physics, Diffeomorphism, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], 0101 mathematics, Mathematics::Symplectic Geometry, ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract: Moser's theorem (1965) states that the diffeomorphism group of a compact manifold acts transitively on the space of all smooth positive densities with fixed volume. Here we describe the extension of this result to manifolds with corners. In particular we obtain Moser's theorem on simplices. The proof is based on Banyaga's paper (1974), where Moser's theorem is proven for manifolds with boundary. A cohomological interpretation of Banyaga's operator is given, which allows a proof of Lefschetz duality using differential forms., 9 pages; mistakes corrected, final accepted version
Published: 2018

15. An introduction to pressure metrics for higher Teichmüller spaces

Author: Richard D. Canary, Martin Bridgeman, Andrés Sambarino, Boston College (BC), Department of Mathematics - University of Michigan, University of Michigan [Ann Arbor], University of Michigan System-University of Michigan System, and Université Paris-Sud - Paris 11 (UP11)
Subjects: Teichmüller space, Pure mathematics, Applied Mathematics, General Mathematics, 010102 general mathematics, Lie group, Bridgeman, 01 natural sciences, Formalism (philosophy of mathematics), 0103 physical sciences, 010307 mathematical physics, 0101 mathematics, [MATH]Mathematics [math], ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract: We discuss how one uses the thermodynamic formalism to produce metrics on higher Teichmüller spaces. Our higher Teichmüller spaces will be spaces of Anosov representations of a word-hyperbolic group into a semi-simple Lie group. We begin by discussing our construction in the classical setting of the Teichmüller space of a closed orientable surface of genus at least 2, then we explain the construction for Hitchin components and finally we treat the general case. This paper surveys results of Bridgeman, Canary, Labourie and Sambarino, The pressure metric for Anosov representations, and discusses questions and open problems which arise.
Published: 2018

16. UNIFORM K-STABILITY AND ASYMPTOTICS OF ENERGY FUNCTIONALS IN KÄHLER GEOMETRY

Author: Tomoyuki Hisamoto, Sébastien Boucksom, Mattias Jonsson, Centre de Mathématiques Laurent Schwartz (CMLS), Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X), Department of Mathematics, University of Michigan [Ann Arbor], University of Michigan System-University of Michigan System, 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), Graduate School of Mathematics [Nagoya], Nagoya University, Department of Ecology [Warsaw], Institute of Zoology [Warsaw], Faculty of Biology [Warsaw], University of Warsaw (UW)-University of Warsaw (UW)-Faculty of Biology [Warsaw], University of Warsaw (UW)-University of Warsaw (UW), Department of Mathematics - University of Michigan, and Boucksom, Sebastien
Subjects: Applied Mathematics, General Mathematics, 010102 general mathematics, Mathematical analysis, [MATH.MATH-CV]Mathematics [math]/Complex Variables [math.CV], [MATH] Mathematics [math], 01 natural sciences, Stability (probability), 0103 physical sciences, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], 010307 mathematical physics, 0101 mathematics, [MATH]Mathematics [math], ComputingMilieux_MISCELLANEOUS, Energy (signal processing), Mathematics
Abstract: Consider a polarized complex manifold (X, L) and a ray of positive metrics on L defined by a positive metric on a test configuration for (X, L). For many common functionals in Kähler geometry, we prove that the slope at infinity along the ray is given by evaluating the non-Archimedean version of the functional (as defined in our earlier paper [BHJ15]) at the non-Archimedean metric on L defined by the test configuration. Using this asymptotic result, we show that coercivity of the Mabuchi functional implies uniform K-stability, as defined in [Der15, BHJ15]. As a partial converse, we show that uniform K-stability implies coercivity of the Mabuchi functional when restricted to Bergman metrics.
Published: 2018

17. Definable equivalence relations and zeta functions of groups (with an appendix by Raf Cluckers)

Author: Ben Martin, Raf Cluckers, Silvain Rideau, Ehud Hrushovski, Laboratoire de Mathématiques d'Orsay (LM-Orsay), and Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS)
Subjects: European community, Hebrew, Applied Mathematics, General Mathematics, European research, 010102 general mathematics, Representation (arts), 01 natural sciences, language.human_language, Algebra, 0103 physical sciences, language, Equivalence relation, media_common.cataloged_instance, 010307 mathematical physics, Cell decomposition, 0101 mathematics, European union, [MATH]Mathematics [math], Classics, ComputingMilieux_MISCELLANEOUS, Mathematics, media_common, Exposition (narrative)
Abstract: The authors wish to thank Thomas Rohwer, Deirdre Haskell, Dugald Macpherson and Elisabeth Bouscaren for their comments on earlier drafts of this work, Martin Hils for suggesting that the proof could be adapted to finite extensions and Zo´e Chatzidakis for pointing out an error in how constants were handled in earlier versions. The second author is grateful to Jamshid Derakhshan, Marcus du Sautoy, Andrei Jaikin-Zapirain, Angus Macintyre, Dugald Macpherson, Mark Ryten, Christopher Voll and Michele Zordan for helpful conversations. We are grateful to Alex Lubotzky for suggesting studying representation growth; several of the ideas in Section 8 are due to him. The first author was supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement no. 291111/ MODAG, the second author was supported by a Golda Meir Postdoctoral Fellowship at the Hebrew University of Jerusalem and the third author was partly supported by ANR MODIG (ANR-09-BLAN-0047) Model Theory and Interactions with Geometry. The author of the appendix would like to thank M. du Sautoy, C. Voll, and Kien Huu Nguyen for interesting discussions on this and related subjects. He was partially supported by the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) with ERC Grant Agreement nr. 615722 MOTMELSUM and he thanks the Labex CEMPI (ANR-11-LABX-0007-01). We are grateful to the referee for their careful reading of the paper and for their many comments, corrections and suggestions for improving the exposition. In memory of Fritz Grunewald.
Published: 2018

18. Reduction of dimension as a consequence of norm-resolvent convergence and applications

Author: Nicolas Raymond, David Krejčiřík, Julien Royer, Petr Siegl, Czech Technical University in Prague (CTU), 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), 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), Institute of Mathematical Statistics and Actuarial Science [Bern] (IMSV), University of Bern, Centre International de Mathématiques et Informatique de Toulouse, Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung, PZ00P2_154786, Statens Naturvidenskabelige Forskningsrad, PTDC/MAT‐CAL/4334/2014, Fundação para a Ciência e a Tecnologia, 18‐088355, Grantová Agentura České Republiky, Czech Technical University in Prague ( CTU ), 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 ), Institut de Mathématiques de Toulouse UMR5219 ( IMT ), 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 ) -PRES Université de Toulouse-Université Paul Sabatier - Toulouse 3 ( UPS ) -Université Toulouse - Jean Jaurès ( UT2J ) -Université Toulouse 1 Capitole ( UT1 ), Institute of Mathematical Statistics and Actuarial Science [Bern] ( IMSV ), 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 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: [ MATH ] Mathematics [math], Quantum Physics, General Mathematics, 010102 general mathematics, Mathematical analysis, FOS: Physical sciences, Mathematical Physics (math-ph), 01 natural sciences, Mathematics - Spectral Theory, Mathematics - Analysis of PDEs, 510 Mathematics, 0103 physical sciences, FOS: Mathematics, Applied mathematics, 010307 mathematical physics, Norm (social), 0101 mathematics, [MATH]Mathematics [math], Quantum Physics (quant-ph), Spectral Theory (math.SP), Mathematical Physics, Resolvent, Mathematics, Analysis of PDEs (math.AP)
Abstract: This paper is devoted to dimensional reductions via the norm resolvent convergence. We derive explicit bounds on the resolvent difference as well as spectral asymptotics. The efficiency of our abstract tool is demonstrated by its application on seemingly different PDE problems from various areas of mathematical physics; all are analysed in a unified manner now, known results are recovered and new ones established., Comment: 15 pages
Published: 2018

19. Entropy of $C^1$ diffeomorphisms without a dominated splitting

Author: Todd Fisher, Jérôme Buzzi, Sylvain Crovisier, Laboratoire de Mathématiques d'Orsay (LMO), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), BYU Mathematics Department (BYU Math), and Brigham Young University (BYU)
Subjects: Pure mathematics, Mathematics::Dynamical Systems, General Mathematics, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], Lyapunov exponent, Topological entropy, Dynamical Systems (math.DS), 01 natural sciences, symbols.namesake, 0103 physical sciences, FOS: Mathematics, Homoclinic orbit, 0101 mathematics, Mathematics - Dynamical Systems, Mathematics, Applied Mathematics, Principle of maximum entropy, 010102 general mathematics, 16. Peace & justice, Homoclinic connection, 37C15, 37B40, 37D05, 37D30, Bounded function, symbols, 010307 mathematical physics, Diffeomorphism, Symplectic geometry
Abstract: A classical construction due to Newhouse creates horseshoes from hyperbolic periodic orbits with large period and weak domination through local $C^1$-perturbations. Our main theorem shows that, when one works in the $C^1$ topology, the entropy of such horseshoes can be made arbitrarily close to an upper bound deriving from Ruelle's inequality, i.e., the sum of the positive Lyapunov exponents (or the same for the inverse diffeomorphism, whichever is smaller). Adapting classical techniques, we use perturbations that are local and can be chosen to preserve volume or symplectic form or a homoclinic connection. This optimal entropy creation yields a number of consequences for $C^1$-generic diffeomorphisms, especially in the absence of a dominated splitting. For instance, in the conservative settings, we find formulas for the topological entropy, deduce that the topological entropy is continuous but not locally constant at the generic diffeomorphism and we prove that these generic diffeomorphisms have no measure of maximum entropy. In the dissipative setting, we show the locally generic existence of infinitely many homoclinic classes with entropy bounded away from zero., 56 pages, 2 figures. Proofs of the local perturbative tools to appear in a separate paper. Accepted by Trans. AMS. This is a revised version with corrected proof of Theorem 7.6
Published: 2018

20. Exotic tilting sheaves, parity sheaves on affine Grassmannians, and the Mirkovic-Vilonen conjecture

Author: Simon Riche, Carl Mautner, Department of Mathematics (UC Riverside), University of California [Riverside] (UC Riverside), University of California (UC)-University of California (UC), Institut National des Sciences Mathématiques et de leurs Interactions (INSMI), Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Blaise Pascal - Clermont-Ferrand 2 (UBP)-Centre National de la Recherche Scientifique (CNRS), University of California [Riverside] (UCR), and University of California-University of California
Subjects: Pure mathematics, Derived category, Equivalence of categories, Applied Mathematics, General Mathematics, 010102 general mathematics, Langlands dual group, Reductive group, 16. Peace & justice, 01 natural sciences, Representation theory, Coherent sheaf, 0103 physical sciences, FOS: Mathematics, Equivariant map, 010307 mathematical physics, Representation Theory (math.RT), 0101 mathematics, Algebraically closed field, [MATH]Mathematics [math], Mathematics::Representation Theory, Mathematics - Representation Theory, Mathematics
Abstract: Let $\mathbf{G}$ be a connected reductive group over an algebraically closed field $\mathbb{F}$ of good characteristic, satisfying some mild conditions. In this paper we relate tilting objects in the heart of Bezrukavnikov's exotic t-structure on the derived category of equivariant coherent sheaves on the Springer resolution of $\mathbf{G}$, and Iwahori-constructible $\mathbb{F}$-parity sheaves on the affine Grassmannian of the Langlands dual group. As applications we deduce in particular the missing piece for the proof of the Mirkovic-Vilonen conjecture in full generality (i.e. for good characteristic), a modular version of an equivalence of categories due to Arkhipov-Bezrukavnikov-Ginzburg, and an extension of this equivalence., Comment: v1: 66 pages; v2: 67 pages, minor corrections; v3: 65 pages, final version, to appear in JEMS
Published: 2018

21. Reducibility in Sasakian Geometry

Author: Charles P. Boyer, Christina W. Tønnesen-Friedman, Eveline Legendre, Hongnian Huang, 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), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), and Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)
Subjects: Mathematics - Differential Geometry, General Mathematics, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], Geometry, Type (model theory), 01 natural sciences, 53C25, symbols.namesake, Genus (mathematics), 0103 physical sciences, FOS: Mathematics, 0101 mathematics, Mathematics::Symplectic Geometry, decomposable, Mathematics, Applied Mathematics, Riemann surface, Sasakian, 010102 general mathematics, Zero (complex analysis), Algebraic variety, Join (topology), join, Cone (topology), Differential Geometry (math.DG), reducible, Isotopy, symbols, 010307 mathematical physics, Mathematics::Differential Geometry
Abstract: The purpose of this paper is to study reducibility properties in Sasakian geometry. First we give the Sasaki version of the de Rham Decomposition Theorem; however, we need a mild technical assumption on the Sasaki automorphism group which includes the toric case. Next we introduce the concept of {\it cone reducible} and consider $S^3$ bundles over a smooth projective algebraic variety where we give a classification result concerning contact structures admitting the action of a 2-torus of Reeb type. In particular, we can classify all such Sasakian structures up to contact isotopy on $S^3$ bundles over a Riemann surface of genus greater than zero. Finally, we show that in the toric case an extremal Sasaki metric on a Sasaki join always splits., Comment: 58 pages, minor corrections made in latest version; a reference added and references updated; to appear in the Transactions of the American Mathematical Society
Published: 2018

22. On ergodic averages for parabolic product flows

Author: Boris Solomyak, Alexander I. Bufetov, Chebyshev Laboratory, St Petersburg State University (SPbU), Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Steklov Mathematical Institute [Moscow] (SMI), Russian Academy of Sciences [Moscow] (RAS), Institute for Information Transmission Problems, Vysšaja škola èkonomiki = National Research University Higher School of Economics [Moscow] (HSE), Bar-Ilan University [Israël], Russian Federation MD 5991.2016.1, RFBR 18-31-20031, Gabriel Lame Chair at the Chebyshev Laboratory of the SPbSU, a joint initiative of the French Embassy in the Russian Federation and Saint-Petersburg State University, Israel Science Foundation 396/15, European Project: 647133,H2020,ERC-2014-CoG,IChaos(2016), and National Research University Higher School of Economics [Moscow] (HSE)
Subjects: Dynamical systems theory, substitution dynamical system, General Mathematics, 010102 general mathematics, Substitution (logic), [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], spectral measure, Dynamical Systems (math.DS), Dynamical system, 01 natural sciences, Suspension (topology), Hoelder continuity, Flow (mathematics), Product (mathematics), 0103 physical sciences, FOS: Mathematics, 37A30, 37B10, Applied mathematics, Ergodic theory, 010307 mathematical physics, 0101 mathematics, Mathematics - Dynamical Systems, Direct product, Mathematics
Abstract: We consider a direct product of a suspension flow over a substitution dynamical system and an arbitrary ergodic flow and give quantitative estimates for the speed of convergence for ergodic integrals of such systems. Our argument relies on new uniform estimates of the spectral measure for suspension flows over substitution dynamical systems. The paper answers a question by Jon Chaika., 14 pages, minor corrections, to appear in Bulletin de la SMF. arXiv admin note: text overlap with arXiv:1305.7373
Published: 2018

23. On foliations with nef anti-canonical bundle

Author: Stéphane Druel, Institut Fourier (IF ), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), Institut Fourier (IF), and Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)
Subjects: Pure mathematics, Applied Mathematics, General Mathematics, 010102 general mathematics, Holomorphic function, Codimension, 01 natural sciences, Manifold, Canonical bundle, Mathematics - Algebraic Geometry, Mathematics::Algebraic Geometry, Bundle, 0103 physical sciences, FOS: Mathematics, Foliation (geology), Kodaira dimension, 010307 mathematical physics, Mathematics::Differential Geometry, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], 0101 mathematics, Algebraic Geometry (math.AG), Mathematics::Symplectic Geometry, ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract: In this paper we prove that the anti-canonical bundle of a holomorphic foliation $\mathcal{F}$ on a complex projective manifold cannot be nef and big if either $\mathcal{F}$ is regular, or $\mathcal{F}$ has a compact leaf. Then we address codimension one regular foliations whose anti-canonical bundle is nef with maximal Kodaira dimension., References were added and a few typos were fixed
Published: 2017

24. Direct and inverse scattering at fixed energy for massless charged Dirac fields by Kerr-Newman-de Sitter black holes

Author: François Nicoleau, Thierry Daudé, Analyse, Géométrie et Modélisation (AGM - UMR 8088), Centre National de la Recherche Scientifique (CNRS)-CY Cergy Paris Université (CY), 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), Nicoleau, François, and Laboratoires d'excellence - Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation - - LEBESGUE2011 - ANR-11-LABX-0020 - LABX - VALID
Subjects: Angular momentum, Primaries 81U40, 35P25, Secondary 58J50, General Mathematics, FOS: Physical sciences, 01 natural sciences, symbols.namesake, General Relativity and Quantum Cosmology, De Sitter universe, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, black hole, Primaries 81U40, 35P25, Secondary 58J50, Dirac equation, [MATH.MATH-MP] Mathematics [math]/Mathematical Physics [math-ph], 0101 mathematics, Chandrasekhar limit, Mathematical Physics, Mathematics, Mathematical physics, Scattering, Applied Mathematics, 010102 general mathematics, Mathematical Physics (math-ph), inverse scattering, Black hole, Ordinary differential equation, Inverse scattering problem, symbols, 010307 mathematical physics
Abstract: In this paper, we study the direct and inverse scattering theory at fixed energy for massless charged Dirac fields evolving in the exterior region of a Kerr-Newman-de Sitter black hole. In the first part, we establish the existence and asymptotic completeness of time-dependent wave operators associated to our Dirac fields. This leads to the definition of the time-dependent scattering operator that encodes the far-field behavior (with respect to a stationary observer) in the asymptotic regions of the black hole: the event and cosmological horizons. We also use the miraculous property (quoting Chandrasekhar) - that the Dirac equation can be separated into radial and angular ordinary differential equations - to make the link between the time-dependent scattering operator and its stationary counterpart. This leads to a nice expression of the scattering matrix at fixed energy in terms of stationary solutions of the system of separated equations. In a second part, we use this expression of the scattering matrix to study the uniqueness property in the associated inverse scattering problem at fixed energy. Using essentially the particular form of the angular equation (that can be solved explicitely by Frobenius method) and the Complex Angular Momentum technique on the radial equation, we are finally able to determine uniquely the metric of the black hole from the knowledge of the scattering matrix at a fixed energy., Comment: 115 pages
Published: 2017

25. Besicovitch Covering Property for homogeneous distances in the Heisenberg groups

Author: Séverine Rigot, Enrico Le Donne, Department of Mathematics and Statistics [Jyväskylä Univ] (JYU), University of Jyväskylä (JYU), Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), and ANR-12-BS01-0014,GEOMETRYA,Théorie géométrique de la mesure et applications(2012)
Subjects: Unit sphere, Pure mathematics, Property (philosophy), General Mathematics, Computer Science::Computational Geometry, Mathematical proof, 01 natural sciences, Perspective (geometry), Completeness (order theory), 0103 physical sciences, Mathematics (all), 0101 mathematics, [MATH]Mathematics [math], 10. No inequality, ComputingMilieux_MISCELLANEOUS, Mathematics, Degree (graph theory), Heisenberg groups, Applied Mathematics, 010102 general mathematics, Radius, Metric space, Covering theorems, 010307 mathematical physics, Homogeneous distances
Abstract: The main result of the present paper is a positive answer to the question whether one can nd homogeneous distances on the Heisenberg groups that have the Besicovitch Covering Property (BCP). We prove that BCP is satised by the homogeneous distances whose unit ball centered at the origin coincides with an Euclidean ball with a small enough radius. By homogeneous distance we mean a left-invariant distance that is homogeneous with degree one with respect to intrinsic dilations. To put our result in perspective, we also prove two geometric criteria that imply the non-validity of BCP, showing that in some sense our example is sharp. Commonly used homogeneous distances on the Heisenberg groups, such as the Cygan-Kor anyi (also called Kor anyi) and Carnot- Carath eodory distances, are known not to satisfy BCP. As a matter of fact, our rst criterion applies to these distances, thus giving also new geometric proofs of the failure of BCP for these distances. To put a dierent perspective on these results and for sake of completeness, we also give a proof of the fact, noticed by D. Preiss, that in a general metric space, one can always construct a bi-Lipschitz equivalent distance that does not satisfy BCP.
Published: 2017

26. A GROUPOID APPROACH TO PSEUDODIFFERENTIAL CALCULI

Author: Robert Yuncken, Erik van Erp, Department of Mathematics [Dartmouth], Dartmouth College [Hanover], Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Blaise Pascal - Clermont-Ferrand 2 (UBP)-Centre National de la Recherche Scientifique (CNRS), ANR-14-CE25-0012,SINGSTAR,Analyse sur les espaces singuliers et non compacts: une approche par les C*-algèbres(2014), Yuncken, Robert, and Appel à projets générique - Analyse sur les espaces singuliers et non compacts: une approche par les C*-algèbres - - SINGSTAR2014 - ANR-14-CE25-0012 - Appel à projets générique - VALID
Subjects: Tangent bundle, Pure mathematics, General Mathematics, hypoelliptic operators, [MATH.MATH-OA]Mathematics [math]/Operator Algebras [math.OA], [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], 01 natural sciences, Operator (computer programming), 0103 physical sciences, Lie groupoids, Double groupoid, 0101 mathematics, Algebraic number, Pseudodifferential operators, Mathematics, Applied Mathematics, 010102 general mathematics, Mathematical analysis, [MATH.MATH-FA] Mathematics [math]/Functional Analysis [math.FA], Tangent, 2010 Mathematics Subject Classification. Primary: 58J40, Secondary: 35S05, 47G30, 58H05, 22A22, Manifold, Kernel (algebra), filtered manifolds, [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], 010307 mathematical physics, [MATH.MATH-DG] Mathematics [math]/Differential Geometry [math.DG], [MATH.MATH-OA] Mathematics [math]/Operator Algebras [math.OA], Distribution (differential geometry)
Abstract: In this paper we give an algebraic/geometric characterization of the classical pseudodifferential operators on a smooth manifold in terms of the tangent groupoid and its natural ℝ + × {\mathbb{R}^{\times}_{+}} -action. Specifically, a properly supported semiregular distribution on M × M {M\times M} is the Schwartz kernel of a classical pseudodifferential operator if and only if it extends to a smooth family of distributions on the range fibers of the tangent groupoid that is homogeneous for the ℝ + × {\mathbb{R}^{\times}_{+}} -action modulo smooth functions. Moreover, we show that the basic properties of pseudodifferential operators can be proven directly from this characterization. Further, with the appropriate generalization of the tangent bundle, the same definition applies without change to define pseudodifferential calculi on arbitrary filtered manifolds, in particular the Heisenberg calculus.
Published: 2016

27. Around supersymmetry for semiclassical second order differential operators

Author: Laurent Michel, Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), and COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Physics, Applied Mathematics, General Mathematics, 010102 general mathematics, Scalar (mathematics), Semiclassical physics, Supersymmetry, Differential operator, 01 natural sciences, Mathematics - Spectral Theory, Mathematics - Analysis of PDEs, 0103 physical sciences, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 010307 mathematical physics, 0101 mathematics, Spectral Theory (math.SP), Analysis of PDEs (math.AP), Mathematical physics, [MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP]
Abstract: Let P ( h ) , h ∈ ] 0 , 1 ] P(h),h\in ]0,1] be a semiclassical scalar differential operator of order 2 2 . The existence of a supersymmetric structure given by a matrix G ( x ; h ) G(x;h) was exhibited by Hérau, Hitrik, and Sjöstrand (2011) under rather general assumptions. In this paper we give a sufficient condition on the coefficients of P ( h ) P(h) so that the matrix G ( x ; h ) G(x;h) enjoys some nice estimates with respect to the semiclassical parameter.
Published: 2016

28. Imaginaries and invariant types in existentially closed valued differential fields

Author: Silvain Rideau, Laboratoire de Mathématiques d'Orsay (LM-Orsay), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11), Département de Mathématiques et Applications - ENS Paris (DMA), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Department of Mathematics [Berkeley], University of California [Berkeley], University of California-University of California, ANR-13-BS01-0006,ValCoMo,Valuations, Combinatoire et Théorie des Modèles(2013), École normale supérieure - Paris (ENS Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Model theory, Pure mathematics, Applied Mathematics, General Mathematics, 010102 general mathematics, Mathematics - Logic, 16. Peace & justice, 01 natural sciences, [MATH.MATH-LO]Mathematics [math]/Logic [math.LO], 0103 physical sciences, FOS: Mathematics, Differential algebra, Haskell, 010307 mathematical physics, 0101 mathematics, Invariant (mathematics), Algebraically closed field, Logic (math.LO), computer, ComputingMilieux_MISCELLANEOUS, Mathematics, computer.programming_language
Abstract: We answer two open questions about the model theory of valued differential fields introduced by Scanlon. We show that they eliminate imaginaries in the geometric language introduced by Haskell, Hrushovski and Macpherson and that they have the invariant extension property. These two result follow from an abstract criterion for the density of definable types in enrichments of algebraically closed valued fields. Finally, we show that this theory is metastable., 41 pages. The paper was significantly shortened for publication
Published: 2015

29. Weighted L^2 -invariants and applications to knot theory

Author: Jérôme Dubois, Christian Wegner, Laboratoire de Mathématiques Blaise Pascal (LMBP), and Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS)
Subjects: Knot complement, Pure mathematics, Applied Mathematics, General Mathematics, Quantum invariant, 010102 general mathematics, Skein relation, Knot polynomial, Tricolorability, Mathematics::Geometric Topology, 01 natural sciences, Combinatorics, Knot invariant, Knot group, 0103 physical sciences, 010307 mathematical physics, 0101 mathematics, [MATH]Mathematics [math], ComputingMilieux_MISCELLANEOUS, Knot (mathematics), Mathematics
Abstract: This paper deals with the study of a new family of knot invariants: the L2-Alexander invariant. A main result is to give a method of computation of the L2-Alexander invariant of a knot complement using any presentation of default 1 of the knot group.
Published: 2015

30. Intertwining operators of irreducible representations for exponential solvable Lie groups

Author: Ali Baklouti, Jean Ludwig, Hidenori Fujiwara, Faculté des Sciences de Sfax, Université de Sfax - University of Sfax, Institut Élie Cartan de Lorraine (IECL), and Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Pure mathematics, Applied Mathematics, General Mathematics, Simple Lie group, 010102 general mathematics, Lie group, (g,K)-module, 01 natural sciences, Representation of a Lie group, Representation theory of SU, 0103 physical sciences, Homogeneous space, Lie algebra, Fundamental representation, 010307 mathematical physics, 0101 mathematics, Arithmetic, [MATH]Mathematics [math], ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract: Let G be a real solvable exponential Lie group with Lie algebra 𝔤 and let f ∈ 𝔤 * ${ f\in \mathfrak {g}^* }$ . We take two polarizations 𝔭 j ${ \mathfrak {p}_j}$ , j = 1,2, at f which meet the Pukanszky condition. Let P j : = exp ( 𝔭 j ) ${ P_j:=\exp ( \mathfrak {p}_j)}$ , j = 1,2, be the associated subgroups in G. The linear functional f defines unitary characters χ f ( exp X ) : = e i 〈 f , X 〉 ${ \chi _f(\exp X):=e^{i \langle { f},{X}\rangle }}$ , X ∈ 𝔭 j ${X\in \mathfrak {p}_j }$ , of Pj . Let π j : = ind P j G χ f ${\pi _j :=\hbox{ind}_{ P_j }^{G} \chi _f}$ , j = 1,2, be the corresponding induced representations, which are unitary and irreducible. It is well known that τ 1 ${ \tau _1 }$ and τ 2 ${ \tau _2 }$ are unitarily equivalent. The description of the intertwining operator of such an equivalence is given via an abstract integral I 𝔭 2 , 𝔭 1 ${ I_{\mathfrak {p}_2,\mathfrak {p}_1} }$ . In this paper, we show that this formal integral gives us a concrete intertwining operator provided that it converges absolutely on a G- and 𝒰 ( 𝔤 ) ${\mathcal {U}(\mathfrak {g})}$ -invariant dense subspace of the space ℋ τ 1 ∞ ${\mathcal {H}_{\tau _1}^\infty }$ of C ∞ vectors of π1 (which is the case when the simple product P 1 P 2 ${P_1P_2}$ is closed). Finally, given three Pukanszky polarizations P i = exp ( 𝔭 i ) ${ P_i=\exp ( \mathfrak {p}_i)}$ , i = 1,2,3, at f of that kind, we accurately determine the composition formula of I 𝔭 3 , 𝔭 2 ∘ I 𝔭 2 , 𝔭 1 ∘ I 𝔭 1 , 𝔭 3 ${ I_{\mathfrak {p}_3,\mathfrak {p}_2}\circ I_{\mathfrak {p}_2,\mathfrak {p}_1}\circ I_{\mathfrak {p}_1,\mathfrak {p}_3}}$ using Maslov's index.
Published: 2015

31. Semiclassical hypoelliptic estimates with a loss of many derivatives

Author: Karel Pravda-Starov, Alberto Parmeggiani, Dipartimento di Matematica [Bologna], Università di Bologna [Bologna] ( UNIBO ), 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 ), Alma Mater Studiorum Università di Bologna [Bologna] (UNIBO), 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), Alberto Parmeggiani, Karel Pravda-Starov, 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: Grushin-reduction method, Pseudodifferential operators, General Mathematics, resolvent estimates, 010102 general mathematics, Mathematical analysis, Resolvent estimate, Mathematics::Analysis of PDEs, Semiclassical physics, Mathematics::Spectral Theory, 01 natural sciences, hypoellipticity with a loss of many derivatives, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Hypoelliptic operator, 0103 physical sciences, Hypoellipticity with a loss of many derivative, A priori and a posteriori, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], doubly characteristic pseudodifferential operators, 010307 mathematical physics, 0101 mathematics, Characteristic point, Mathematics, Doubly characteristic pseudodifferential operator
Abstract: We study the pseudospectral properties of general pseudodifferential operators around a doubly characteristic point and provide necessary and sufficient conditions for semiclassical hypoelliptic a priori estimates with a loss of many derivatives. In the case of the first author, the research that led to the present paper was partially supported by a grant of the group GNAMPA of INdAM.
Published: 2015

32. On the structure of smooth components of Springer fibers

Author: Lucas Fresse, Sammar Sakas-Obeid, Anna Melnikov, Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), and University of Haifa [Haifa]
Subjects: Pure mathematics, Betti number, General Mathematics, iterated bundles, Structure (category theory), Grassmannian varieties, 01 natural sciences, [MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR], Grassmannian, 0103 physical sciences, Young tableau, Betti numbers, 0101 mathematics, Mathematics::Representation Theory, Mathematics, Components of a Springer fiber, flag manifolds, [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT], MSC (2010): 14M15, 05E10, 17B08, Applied Mathematics, 010102 general mathematics, Base (topology), Algebra, Nilpotent, Iterated function, Bundle, 010307 mathematical physics, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG]
Abstract: International audience; The aim of this paper is to study the structure of the smooth irreducible components of the Springer fibers associated to nilpotent endo-morphisms of nilpotency order 2. Relying on its combinatorial interpretation in terms of standard Young tableaux, we show that each smooth component has a structure of iterated bundle of Grassmannian varieties, with explicit base. Using this description, we then classify the smooth components according to their Poincaré polynomials.
Published: 2015

33. Deformation of the O'Grady moduli spaces

Author: Arvid Perego, Antonio Rapagnetta, Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), and Università degli Studi di Roma Tor Vergata [Roma]
Subjects: Pure mathematics, General Mathematics, Abelian surface, Algebraic Geometry, Moduli spaces of sheaves, Deformation theory, Hyperkahler manifolds, K3 surfaces, 01 natural sciences, K3 surface, symplectic resolutions, Mathematics - Algebraic Geometry, Mathematics::Algebraic Geometry, 0103 physical sciences, FOS: Mathematics, 0101 mathematics, Abelian group, Algebraic Geometry (math.AG), Mathematics::Symplectic Geometry, Moduli spaces of sheaves, Symplectic manifold, Mathematics, deformations, Applied Mathematics, 010102 general mathematics, Moduli space, Deformation theory, Hyperkahler manifolds, abelian surfaces, Settore MAT/03 - Geometria, 010307 mathematical physics, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], Algebraic Geometry, Symplectic geometry
Abstract: In this paper we study moduli spaces of sheaves on an abelian or projective K3 surface. If $S$ is a K3, $v=2w$ is a Mukai vector on $S$, where $w$ is primitive and $w^{2}=2$, and $H$ is a $v-$generic polarization on $S$, then the moduli space $M_{v}$ of $H-$semistable sheaves on $S$ whose Mukai vector is $v$ admits a symplectic resolution $\widetilde{M}_{v}$. A particular case is the $10-$dimensional O'Grady example $\widetilde{M}_{10}$ of irreducible symplectic manifold. We show that $\widetilde{M}_{v}$ is an irreducible symplectic manifold which is deformation equivalent to $\widetilde{M}_{10}$ and that $H^{2}(M_{v},\mathbb{Z})$ is Hodge isometric to the sublattice $v^{\perp}$ of the Mukai lattice of $S$. Similar results are shown when $S$ is an abelian surface., 29 pages
Published: 2013

34. Berezin-Toeplitz quantization on Kaehler manifolds

Author: George Marinescu, Xiaonan Ma, 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), Mathematisches Institut, Universität zu Köln, and Universität zu Köln = University of Cologne
Subjects: Mathematics - Differential Geometry, Pure mathematics, General Mathematics, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], FOS: Physical sciences, 01 natural sciences, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, Mathematics - Quantum Algebra, FOS: Mathematics, Quantum Algebra (math.QA), 0101 mathematics, Complex Variables (math.CV), Mathematics::Symplectic Geometry, Mathematical Physics, Mathematics, Mathematics::Functional Analysis, Mathematics - Complex Variables, Mathematics::Operator Algebras, Applied Mathematics, Quantization (signal processing), 010102 general mathematics, [MATH.MATH-CV]Mathematics [math]/Complex Variables [math.CV], Mathematical Physics (math-ph), Composition (combinatorics), Mathematics::Spectral Theory, 53D50, 53C21, 32Q15, Toeplitz matrix, Kernel (algebra), Differential Geometry (math.DG), [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], [MATH.MATH-QA]Mathematics [math]/Quantum Algebra [math.QA], 010307 mathematical physics, Mathematics::Differential Geometry
Abstract: We study the Berezin-Toeplitz quantization on Kaehler manifolds. We explain first how to compute various associated asymptotic expansions, then we compute explicitly the first terms of the expansion of the kernel of the Berezin-Toeplitz operators, and of the composition of two Berezin-Toeplitz operators. As application we estimate the norm of Donaldson's Q-operator., Comment: 45 pages, footnote at page 3 and Remark 0.5 added; v.3 is a final update to agree with the published paper
Published: 2012

35. Stacks of trigonal curves

Author: Michele Bolognesi, Angelo Vistoli, 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), Scuola Normale Superiore di Pisa (SNS), 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 ), Scuola Normale Superiore di Pisa ( SNS ), Scuola Normale Superiore di Pisa, Bolognesi, M., Vistoli, Angelo, 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), and Bolognesi, Michele
Subjects: Pure mathematics, General Mathematics, Picard group, MathematicsofComputing_GENERAL, 14H10, 14C22, 14D20, 01 natural sciences, Mathematics - Algebraic Geometry, Mathematics::Algebraic Geometry, 0103 physical sciences, ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION, FOS: Mathematics, Computer Science::Symbolic Computation, 0101 mathematics, Physics::Chemical Physics, Algebraic Geometry (math.AG), Quotient, Mathematics, Applied Mathematics, 010102 general mathematics, [MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG], Algebraic variety, Quotient stack, [ MATH.MATH-AG ] Mathematics [math]/Algebraic Geometry [math.AG], Algebra, Algebraic group, Homogeneous space, Condensed Matter::Strongly Correlated Electrons, 010307 mathematical physics, Geometric invariant theory, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], Stack (mathematics)
Abstract: In this paper we give a presentation of the stack of trigonal curves as a quotient stack, and we compute its Picard group., Comment: Corrected typos, added references and improved the exposition, following the suggestions of the referees. To appear in Transactions of the American Mathematical Society"
Published: 2012

36. Locally nilpotent derivations with a PID ring of constants

Author: M’hammed El Kahoui, Moulay A. Barkatou, DMI (XLIM-DMI), XLIM (XLIM), Université de Limoges (UNILIM)-Centre National de la Recherche Scientifique (CNRS)-Université de Limoges (UNILIM)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics, Faculty of Sciences Semlalia, and Université Cadi Ayyad [Marrakech] (UCA)
Subjects: Discrete mathematics, Reduced ring, Ring (mathematics), Pure mathematics, Applied Mathematics, General Mathematics, 010102 general mathematics, Locally nilpotent, Field (mathematics), Locally nilpotent derivation, Plinth ideal, Unipotent, Fixed point, 01 natural sciences, Nilpotent, 0103 physical sciences, affine modification, 010307 mathematical physics, 0101 mathematics, Nilpotent group, 14R20, Mathematics
Abstract: Let K \mathcal {K} be a commutative field of characteristic zero, A \mathcal {A} be a domain containing K \mathcal {K} and ∂ \partial be a locally nilpotent K \mathcal {K} -derivation of A \mathcal {A} . We give in this paper a description of the differential K \mathcal {K} -algebra ( A , ∂ ) (\mathcal {A},\partial ) under the assumptions that the ring of constants A ∂ \mathcal {A}^{\partial } of ∂ \partial is a PID, ∂ \partial is fixed point free and its special fibers are reduced.
Published: 2012

37. Discrete spectrum of a model Schrödinger operator on the half-plane with Neumann conditions

Author: Monique Dauge, Virginie Bonnaillie-Noël, Nicolas Raymond, Nicolas Popoff, 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 ), 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: 35P20,35Q40, General Mathematics, Essential spectrum, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], General Physics and Astronomy, Boundary (topology), 01 natural sciences, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, Neumann boundary condition, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Born-Oppenheimer approximation, Eigenvalues and eigenvectors, Mathematics, Schrödinger operator, Applied Mathematics, Operator (physics), 010102 general mathematics, Degenerate energy levels, Mathematical analysis, [ MATH.MATH-MP ] Mathematics [math]/Mathematical Physics [math-ph], Mathematics::Spectral Theory, Eigenfunction, 16. Peace & justice, Agmon estimates, [ PHYS.MPHY ] Physics [physics]/Mathematical Physics [math-ph], 010307 mathematical physics, Semi classical limit, Asymptotic expansion
Abstract: International audience; We study the eigenpairs of a model Schrödinger operator with a quadratic potential and Neumann boundary conditions on a half-plane. The potential is degenerate in the sense that it reaches its minimum all along a line which makes the angle \theta with the boundary of the half-plane. We show that the first eigenfunctions satisfy localization properties related to the distance to the minimum line of the potential. We investigate the densification of the eigenvalues below the essential spectrum in the limit \theta \to 0 and we prove full asymptotic expansion for these eigenvalues and their associated eigenvectors. We conclude the paper by numerical experiments obtained by a finite element method. The numerical results confirm and enlighten the theoretical approach.
Published: 2012

38. The Roquette category of finite p-groups

Author: Serge Bouc, Laboratoire Amiénois de Mathématique Fondamentale et Appliquée (LAMFA), and Université de Picardie Jules Verne (UPJV)-Centre National de la Recherche Scientifique (CNRS)
Subjects: p-group, Pure mathematics, Derived functor, General Mathematics, Functor category, Group Theory (math.GR), 01 natural sciences, [MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR], Mathematics::Category Theory, 0103 physical sciences, rational, biset, FOS: Mathematics, Category Theory (math.CT), Representation Theory (math.RT), 0101 mathematics, Abelian group, Adjoint functors, Mathematics, [MATH.MATH-CT]Mathematics [math]/Category Theory [math.CT], Functor, [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT], Direct sum, Applied Mathematics, 010102 general mathematics, Mathematics - Category Theory, 18B99, 19A22, 20C99, 20J15, Roquette, Natural transformation, 010307 mathematical physics, genetic, Mathematics - Group Theory, Mathematics - Representation Theory
Abstract: Let p be a prime number. This paper introduces the Roquette category R_p of finite p-groups, which is an additive tensor category containing all finite p-groups among its objects. In R_p, every finite p-group P admits a canonical direct summand, called the edge of P. Moreover P splits uniquely as a direct sum of edges of Roquette p-groups, and the tensor structure of R_p can be described in terms of such edges. The main motivation for considering this category is that the additive functors from R_p to abelian groups are exactly the rational p-biset functors. This yields in particular very efficient ways of computing such functors on arbitrary p-groups : this applies to the representation functors R_K, where K is any field of characteristic 0, but also to the functor of units of Burnside rings, or to the torsion part of the Dade group.
Published: 2011

39. Higher order Lagrange-Poincaré and Hamilton-Poincaré reductions

Author: Tudor S. Ratiu, François Gay-Balmaz, Darryl D. Holm, Laboratoire de Météorologie Dynamique (UMR 8539) (LMD), 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)-Centre National de la Recherche Scientifique (CNRS)-École des Ponts ParisTech (ENPC)-École polytechnique (X)-Institut national des sciences de l'Univers (INSU - CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC), Department of Mathematics [Imperial College London], Imperial College London, Section de Mathématiques and Bernoulli Center, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland, Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-École polytechnique (X)-École des Ponts ParisTech (ENPC)-Centre National de la Recherche Scientifique (CNRS)-Département des Géosciences - ENS Paris, École normale supérieure - Paris (ENS-PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-École normale supérieure - Paris (ENS-PSL), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)
Subjects: Sequence, General Mathematics, 010102 general mathematics, Mathematical analysis, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], 70H50, 37J15, 70H25, 70H30, 01 natural sciences, Symmetry (physics), Reduction (complexity), Poisson bracket, symbols.namesake, [MATH.MATH-GM]Mathematics [math]/General Mathematics [math.GM], Geometric mechanics, Variational principle, 0103 physical sciences, Poincaré conjecture, symbols, Applied mathematics, 010307 mathematical physics, 0101 mathematics, Hamiltonian (control theory), Mathematical Physics, Mathematics
Abstract: Motivated by the problem of longitudinal data assimilation, e.g., in the registration of a sequence of images, we develop the higher-order framework for Lagrangian and Hamiltonian reduction by symmetry in geometric mechanics. In particular, we obtain the reduced variational principles and the associated Poisson brackets. The special case of higher order Euler-Poincar\'e and Lie-Poisson reduction is also studied in detail., Comment: This paper has appeared in 2011 in the Journal of the Brazilian Mathematical Society 42(4), 579--606
Published: 2011

40. The holomorphy conjecture for ideals in dimension two

Author: Lise Van Proeyen, Ann Lemahieu, Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA), and Catholic University of Leuven - Katholieke Universiteit Leuven (KU Leuven)
Subjects: Polynomial, General Mathematics, Holomorphic function, 01 natural sciences, Combinatorics, symbols.namesake, Mathematics - Algebraic Geometry, Mathematics::Algebraic Geometry, Integer, 0103 physical sciences, Domain of holomorphy, FOS: Mathematics, Order (group theory), 0101 mathematics, [MATH]Mathematics [math], Algebraic Geometry (math.AG), Mathematics, Conjecture, 14B05, Mathematics::Commutative Algebra, Mathematics::Complex Variables, Applied Mathematics, 010102 general mathematics, Riemann zeta function, Algebra, Monodromy, symbols, 010307 mathematical physics
Abstract: The holomorphy conjecture predicts that the topological zeta function associated to a polynomial f ∈ C [ x 1 , … , x n ] f \in \mathbb {C}[x_1,\ldots ,x_n] and an integer d > 0 d > 0 is holomorphic unless d d divides the order of an eigenvalue of local monodromy of f f . In this paper, we generalise the holomorphy conjecture to the setting of arbitrary ideals in C [ x 1 , … , x n ] \mathbb {C}[x_1,\ldots ,x_n] , and we prove it when n = 2 n=2 .
Published: 2011

41. Generic super-exponential stability of invariant tori in Hamiltonian systems

Author: Abed Bounemoura, Laboratoire de Mathématiques d'Orsay (LM-Orsay), Université Paris-Sud - Paris 11 (UP11)-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 Centre de Recerca Matemàtica
Subjects: Pure mathematics, Dynamical systems theory, General Mathematics, Nekhoroshev theory, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], Dynamical Systems (math.DS), Fixed point, 37J25, 37J40, 01 natural sciences, Hamiltonian system, Exponential stability, Stability theory, 0103 physical sciences, FOS: Mathematics, 0101 mathematics, Mathematics - Dynamical Systems, Mathematics, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Torus, Invariant (physics), Dynamical Systems, Hamiltonian Systems, Hamilton, Sistemes de, 010307 mathematical physics, 515.1 - Topologia
Abstract: In this article, we consider solutions that start close to some linearly stable invariant tori in an analytic Hamiltonian system, and we prove results of stability for a super-exponentially long interval of time, under generic conditions. The proof combines classical Birkhoff normal forms with a new method for obtaining generic Nekhoroshev estimates developed by the author and L. Niederman in another paper. We will focus mainly on the neighbourhood of elliptic fixed points, since with our approach the other cases can be treated in a very similar way.
Published: 2011

42. Iterative q-difference Galois theory

Author: Charlotte Hardouin, 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: Pure mathematics, Root of unity, Mathematics::Number Theory, General Mathematics, Galois theory, Field (mathematics), Group Theory (math.GR), Translation (geometry), 01 natural sciences, 510 Mathematics, Operator (computer programming), 12H10, 12H05, Mathematics - Quantum Algebra, 0103 physical sciences, FOS: Mathematics, Quantum Algebra (math.QA), 0101 mathematics, ComputingMilieux_MISCELLANEOUS, Mathematics, Applied Mathematics, 010102 general mathematics, Order (ring theory), 16. Peace & justice, Differential Galois theory, 12H10 (39A13), 010307 mathematical physics, Transcendental number, Mathematics - Group Theory
Abstract: We propose in this paper a Galois theory of $q$-difference equations where q is a root of unity. This theory is the q difference analogue of the Galois theory of iterative differential equations, that is differential equations over fields of positive characteristic. This theory contains and generalizes the Galois theory of q difference equations developed by Singer and van der Put., to appear in "Journal fur die reine und angewandte Mathematik (Crelle's Journal)"
Published: 2010

43. The solar Julia sets of basic quadratic Cremer polynomials

Author: Alexander Blokh, Lex Oversteegen, Xavier Buff, Arnaud Chéritat, Department of Mathematics, University of Alabama at Birmingham [ Birmingham] (UAB), 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: Mathematics::Dynamical Systems, General Mathematics, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], Structure (category theory), Dynamical Systems (math.DS), 01 natural sciences, Combinatorics, Set (abstract data type), Quadratic equation, [MATH.MATH-GN]Mathematics [math]/General Topology [math.GN], 0103 physical sciences, FOS: Mathematics, Point (geometry), Mathematics - Dynamical Systems, 0101 mathematics, Mathematics - General Topology, Mathematics, Lebesgue measure, Mathematics::Complex Variables, Applied Mathematics, 010102 general mathematics, Degenerate energy levels, General Topology (math.GN), 16. Peace & justice, 37F10 (Primary), 37B45, 37C25 (Secondary), Julia set, 010307 mathematical physics
Abstract: In general, little is known about the exact topological structure of Julia sets containing a Cremer point. In this paper we show that there exist quadratic Cremer Julia sets of positive area such that for a full Lebesgue measure set of angles the impressions are degenerate, the Julia set is connected im kleinen at the landing points of these rays, and these points are contained in no other impression., 20 pages, 3 figures, to appear in Ergodic Theory and Dynamical Systems
Published: 2010

44. On t -conformal measures and Hausdorff dimension for a family of non-uniformly hyperbolic horseshoes

Author: Isabel Rios, Renaud Leplaideur, Laboratoire de Mathématiques de Bretagne Atlantique (LMBA), Université de Bretagne Sud (UBS)-Université de Brest (UBO)-Centre National de la Recherche Scientifique (CNRS), Institut de sciences exactes et appliquées (ISEA), and Université de la Nouvelle-Calédonie (UNC)
Subjects: Pure mathematics, Applied Mathematics, General Mathematics, 010102 general mathematics, Mathematical analysis, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], Minkowski–Bouligand dimension, Dimension function, Effective dimension, 01 natural sciences, Hausdorff distance, Packing dimension, Hausdorff dimension, 0103 physical sciences, Hausdorff measure, 010307 mathematical physics, Homoclinic orbit, 0101 mathematics, Mathematics
Abstract: In this paper we consider horseshoes with homoclinic tangencies inside the limit set. For a class of such maps, we prove the existence of a unique equilibrium state μt, associated to the (non-continuous) potential −tlog Ju. We also prove that the Hausdorff dimension of the limit set, in any open piece of unstable manifold, is the unique number t0 such that the pressure of μt0 is zero. To deal with the discontinuity of the jacobian, we introduce a countable Markov partition adapted to the dynamics, and work with the first return map defined in a rectangle of it.
Published: 2009

45. Deformation theory of representation of prop(erad)s II

Author: Sergei Merkulov, Bruno Vallette, Laboratoire Jean Alexandre Dieudonné (JAD), Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), and ANR-06-JCJC-0042,OBTH,Opérades Bigèbres et Théories d'Homotopie(2006)
Subjects: Pure mathematics, Koszul duality, Model category, formal manifolds, General Mathematics, Minimal models, Commutative ring, Homology (mathematics), Mathematics::Algebraic Topology, 01 natural sciences, Morphism, Mathematics::K-Theory and Homology, deformation theory, Mathematics::Quantum Algebra, Mathematics::Category Theory, Mathematics - Quantum Algebra, 0103 physical sciences, Lie algebra, FOS: Mathematics, Quantum Algebra (math.QA), Algebraic Topology (math.AT), Operads, Mathematics - Algebraic Topology, 0101 mathematics, homotopy Lie algebras, Mathematics, Applied Mathematics, Homotopy, 010102 general mathematics, 16. Peace & justice, Gerstenhaber-Schack bicomplex, [MATH.MATH-KT]Mathematics [math]/K-Theory and Homology [math.KT], [MATH.MATH-QA]Mathematics [math]/Quantum Algebra [math.QA], 010307 mathematical physics
Abstract: We study the deformation theory of morphisms of properads and props thereby extending to a non-linear framework Quillen's deformation theory for commutative rings. The associated chain complex is endowed with a Lie algebra up to homotopy structure. Its Maurer-Cartan elements correspond to deformed structures, which allows us to give a geometric interpretation of these results. To do so, we endow the category of prop(erad)s with a model category structure. We provide a complete study of models for prop(erad)s. A new effective method to make minimal models explicit, that extends Koszul duality theory, is introduced and the associated notion is called homotopy Koszul. As a corollary, we obtain the (co)homology theories of (al)gebras over a prop(erad) and of homotopy (al)gebras as well. Their underlying chain complex is endowed with a canonical Lie algebra up to homotopy structure in general and a Lie algebra structure only in the Koszul case. In particular, we explicit the deformation complex of morphisms from the properad of associative bialgebras. For any minimal model of this properad, the boundary map of this chain complex is shown to be the one defined by Gerstenhaber and Schack. As a corollary, this paper provides a complete proof of the existence of a Lie algebra up to homotopy structure on the Gerstenhaber-Schack bicomplex associated to the deformations of associative bialgebras., Comment: Version 4 : Statement about the properad of (non-commutative) Frobenius bialgebras fixed in Section 4. [82 pages]
Published: 2009

46. On the moduli space of certain smooth codimension-one foliations of the 5-sphere by complex surfaces

Author: Laurent Meersseman, Alberto Verjovsky, Laboratoire Angevin de Recherche en Mathématiques (LAREMA), Université d'Angers (UA)-Centre National de la Recherche Scientifique (CNRS), Instituto de Matematicas de la UNAM unidad Cuernavaca, and UNAM
Subjects: Pure mathematics, Integrable system, Applied Mathematics, General Mathematics, 010102 general mathematics, Mathematical analysis, Structure (category theory), [MATH.MATH-CV]Mathematics [math]/Complex Variables [math.CV], Codimension, 16. Peace & justice, 01 natural sciences, Moduli space, Set (abstract data type), Moduli of algebraic curves, 0103 physical sciences, Foliation (geology), Mathematics::Differential Geometry, 010307 mathematical physics, 0101 mathematics, Mathematics::Symplectic Geometry, ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract: In this paper we first determine the set of all possible integrable almost CR-structures on the smooth foliation of S 5 constructed in (M-V). We give a specific concrete model of each of these structures. We show that this set can be naturally identified with C × C × C. We then adapt the classical notions of coarse and fine moduli space to the case of a foliation by complex manifolds. We prove that the previous set, identified with C 3 , defines a coarse moduli space for the foliation of (M-V), but that it does not have a fine moduli space. Finally, using the same ideas we prove that the standard Lawson foliation on the 5-sphere can be endowed with almost CR-structures but none of these is integrable. This is a foliated analogue to the examples of almost complex manifolds without complex structure.
Published: 2009

47. Optimal length estimates for stable CMC surfaces in $3$-space forms

Author: Laurent Mazet, Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA), Université Paris-Est Marne-la-Vallée (UPEM)-Fédération de Recherche Bézout-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS), and Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Fédération de Recherche Bézout-Université Paris-Est Marne-la-Vallée (UPEM)
Subjects: Surface (mathematics), Mathematics - Differential Geometry, Mean curvature, Applied Mathematics, General Mathematics, 010102 general mathematics, Mathematical analysis, Boundary (topology), 53A10, Geometry, Space (mathematics), 01 natural sciences, Differential Geometry (math.DG), [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], 0103 physical sciences, Pi, FOS: Mathematics, Point (geometry), 010307 mathematical physics, 0101 mathematics, Constant (mathematics), Mathematics
Abstract: In this paper, we study stable constant mean curvature $H$ surfaces in $\R^3$. We prove that, in such a surface, the distance from a point to the boundary is less that $\pi/(2H)$. This upper-bound is optimal and is extended to stable constant mean curvature surfaces in space forms., Comment: 7 pages
Published: 2009

48. Corps de nombres peu ramifies et formes automorphes autoduales

Author: Laurent Clozel, Gaëtan Chenevier, Centre de Mathématiques Laurent Schwartz (CMLS), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques d'Orsay (LM-Orsay), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), and Chenevier, Gaetan
Subjects: Trace (linear algebra), General Mathematics, media_common.quotation_subject, Mathematics::Number Theory, 01 natural sciences, Combinatorics, 11R32, 11F70, 0103 physical sciences, FOS: Mathematics, Number Theory (math.NT), 0101 mathematics, Representation Theory (math.RT), [MATH.MATH-RT] Mathematics [math]/Representation Theory [math.RT], Mathematics::Representation Theory, Finite set, media_common, Mathematics, Mathematics - Number Theory, [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT], Applied Mathematics, 010102 general mathematics, Algebraic extension, 16. Peace & justice, Infinity, Injective function, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], Algebra, 010307 mathematical physics, Mathematics - Representation Theory, Symplectic geometry, [MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT]
Abstract: Let S be a finite set of primes, p in S, and Q_S a maximal algebraic extension of Q unramified outside S and infinity. Assume that |S|>=2. We show that the natural maps Gal(Q_p^bar/Q_p) --> Gal(Q_S/Q) are injective. Much of the paper is devoted to the problem of constructing selfdual automorphic cuspidal representations of GL(2n,A_Q) with prescribed properties at all places, that we study via the twisted trace formula of J. Arthur. The techniques we develop shed also some lights on the orthogonal/symplectic alternative for selfdual representations of GL(2n)., 50 pages, french. Section 4.18 has been extended : let F be a totally real field and \pi a selfdual cuspidal automorphic representation of GL(2n,A_F) which is cohomological at all the archimedean places and discrete at a finite place at least, we show that for each place v the L-parameter of \pi_v preserves a non-degenerate symplectic pairing
Published: 2009

49. Stable self-similar blow up dynamics for the three dimensional relativistic gravitational Vlasov-Poisson system

Author: Mohammed Lemou, Florian Méhats, Pierre Raphaël, Mathématiques pour l'Industrie et la Physique (MIP), 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 III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), 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), Laboratoire de Mathématiques d'Orsay (LM-Orsay), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), 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), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11), Mathématiques pour l'Industrie et la Physique ( MIP ), Université Toulouse 1 Capitole ( UT1 ) -Institut National des Sciences Appliquées - Toulouse ( INSA Toulouse ), Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Université Paul Sabatier - Toulouse 3 ( UPS ) -Centre National de la Recherche Scientifique ( CNRS ), 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 ), Laboratoire de Mathématiques d'Orsay ( LM-Orsay ), and Université Paris-Sud - Paris 11 ( UP11 ) -Centre National de la Recherche Scientifique ( CNRS )
Subjects: Gravity (chemistry), 2 SPACE DIMENSIONS, MAXWELL SYSTEM, General Mathematics, 01 natural sciences, Stability (probability), Gravitation, symbols.namesake, Singularity, Stellar dynamics, 0103 physical sciences, 0101 mathematics, GLOBAL SYMMETRIC-SOLUTIONS, Nonlinear Schrödinger equation, ComputingMilieux_MISCELLANEOUS, Mathematics, Mathematical physics, INITIAL DATA, STABILITY, Applied Mathematics, 010102 general mathematics, STEADY-STATES, [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], State (functional analysis), EXISTENCE, 35Q55, STELLAR DYNAMICS, GALACTIC DYNAMICS, symbols, 010307 mathematical physics, Relativistic quantum chemistry, NONLINEAR SCHRODINGER-EQUATION, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: The three dimensional gravitational Vlasov-Poisson system ∂ t f + v ⋅ ∇ x f − E f ⋅ ∇ v f = 0 \partial _tf+v\cdot \nabla _x f-E_f\cdot \nabla _vf=0 , where E f ( x ) = ∇ x ϕ f ( x ) E_f(x)=\nabla _x \phi _f(x) , Δ x ϕ f = ρ f ( x ) \Delta _x\phi _f=\rho _f(x) , ρ f ( x ) = ∫ R N f ( x , v ) d v \rho _f(x)=\int _{\mathbb {R}^N} f(x,v)dv , describes the mechanical state of a stellar system subject to its own gravity. Smooth initial data yield unique global in time solutions from a celebrated result by Pfaffelmoser. There exists a hierarchy of physical models which aim at taking into account further relativistic effects. The simplest one is the three dimensional relativistic gravitational Vlasov-Poisson system ∂ t f + v 1 + | v | 2 ⋅ ∇ x f − E f ⋅ ∇ v f = 0 \partial _tf+\frac {v}{\sqrt {1+|v|^2}}\cdot \nabla _x f-E_f\cdot \nabla _vf=0 which we study here. A striking feature as observed by Glassey and Schaeffer is that this system now admits finite blow up solutions. Nevertheless, the existence argument is purely obstructive and provides no insight into the description of the singularity formation. We exhibit in this paper a family of finite time blow up self-similar solutions and prove that their blow up dynamic is stable with respect to radially symmetric perturbations. Our analysis applies to the four dimensional gravitational Vlasov-Poisson system as well.
Published: 2008

50. Lie Symmetries and CR Geometry

Author: Joel Merker, Laboratoire de Mathématiques d'Orsay (LMO), and Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11)
Subjects: Statistics and Probability, Maple, Pure mathematics, Integrable system, Jet (mathematics), Applied Mathematics, General Mathematics, 010102 general mathematics, Order (ring theory), Geometry, engineering.material, 01 natural sciences, Part iii, Algebra, 0103 physical sciences, Homogeneous space, engineering, Vector field, 010307 mathematical physics, 0101 mathematics, [MATH]Mathematics [math], Flatness (mathematics), Mathematics
Abstract: International audience; This paper is divided into three parts 1. Part I develops a general, new theory (inspired by modern CR geometry) of Lie symmetries of completely integrable pde systems, viewed from their associated submanifolds of solutions. Part II constructs general combinatorial formulas for the prolongations of vector fields to jet spaces. Part III explicitly characterizes the flatness of some systems of the second order. The results presented here are original and were not published elsewhere; most formulas of Parts II and III were verified by means of Maple Release 7.
Published: 2008

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