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3. Shellable tilings on relative simplicial complexes and their h-vectors

Author

Jean-Yves Welschinger, Algèbre, géométrie, logique (AGL), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), and ANR-15-CE40-0007,MICROLOCAL,Topologie symplectique, théorie microlocal des faisceaux et quantification(2015)

Subjects
Shellable complex, Barycentric subdivision, 55U10, 52C22, 57Q70, Geometric Topology (math.GT), K-Theory and Homology (math.KT), Mathematics - Geometric Topology, Tilings, Stellar subdivision, [MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT], [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], Mathematics - K-Theory and Homology, FOS: Mathematics, [MATH.MATH-KT]Mathematics [math]/K-Theory and Homology [math.KT], Mathematics - Combinatorics, Geometry and Topology, Combinatorics (math.CO), Simplicial complex, Discrete Morse theory
Abstract

An h-tiling on a finite simplicial complex is a partition of its geometric realization by maximal simplices deprived of several codimension one faces together with possibly their remaining face of highest codimension. In this last case, the tiles are said to be critical. An h-tiling thus induces a partitioning of its face poset by closed or semi-open intervals. We prove the existence of h-tilings on every finite simplicial complex after finitely many stellar subdivisions at maximal simplices. These tilings are moreover shellable. We also prove that the number of tiles of each type used by a tiling, encoded by its h-vector, is determined by the number of critical tiles of each index it uses, encoded by its critical vector. In the case of closed triangulated manifolds, these vectors satisfy some palindromic property. We finally study the behavior of tilings under any stellar subdivision., 24 pages, 3 figures

Published
2020

4. Asymptotic Topology of Random Subcomplexes in a Finite Simplicial Complex

Author

Jean-Yves Welschinger, Nermin Salepci, Algèbre, géométrie, logique (AGL), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), and ANR-15-CE40-0007,MICROLOCAL,Topologie symplectique, théorie microlocal des faisceaux et quantification(2015)

Subjects
Betti, Betti number, General Mathematics, 0102 computer and information sciences, Barycentric coordinate system, Morse code, Topology, 01 natural sciences, Upper and lower bounds, Mathematics::Algebraic Topology, law.invention, Simplicial complex, symbols.namesake, Mathematics - Algebraic Geometry, Mathematics - Geometric Topology, law, Euler characteristic, [MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT], [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], FOS: Mathematics, Mathematics - Combinatorics, barycentric subdivisions, 0101 mathematics, Algebraic Geometry (math.AG), Topology (chemistry), Mathematics, 010102 general mathematics, Probability (math.PR), triangulations, Geometric Topology (math.GT), [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], random variable, 010201 computation theory & mathematics, numbers, symbols, simplicial complex, MSC 52C99, 60C05, 60B05, Combinatorics (math.CO), [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], Random variable, Mathematics - Probability
Abstract

International audience; We consider a finite simplicial complex $K$ together with its successive barycentric subdivisions $Sd^d(K), d\geq0,$ and study the expected topology of a random subcomplex in $Sd^d(K), d\gg0$. We get asymptotic upper and lower bounds for the expected Betti numbers of those subcomplexes, together with the average Morse inequalities and expected Euler characteristic.

Published
2019
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5. Remote sensing of methane with OSAS-lidar on the 2ν3 band Q-branch: Experimental proof

Author

Patrick Rairoux, Jean-Pierre Cariou, J.F. Sivignon, Sandrine Galtier, Christophe Anselmo, Alain Miffre, Jean-Yves Welschinger, Institut Lumière Matière [Villeurbanne] (ILM), Centre National de la Recherche Scientifique (CNRS)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon, Algèbre, géométrie, logique (AGL), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), LEOSPHERE France, LEOSPHERE, Université Claude Bernard Lyon 1 (UCBL), and Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS)

Subjects
[PHYS.PHYS.PHYS-OPTICS]Physics [physics]/Physics [physics]/Optics [physics.optics], 010504 meteorology & atmospheric sciences, Absorption spectroscopy, Flux, 01 natural sciences, 7. Clean energy, Atomic and Molecular Physics, and Optics, Methane, Trace gas, 010309 optics, Atmosphere, Wavelength, chemistry.chemical_compound, Lidar, chemistry, 13. Climate action, 0103 physical sciences, Environmental science, Physical and Theoretical Chemistry, Spectroscopy, ComputingMilieux_MISCELLANEOUS, 0105 earth and related environmental sciences, Remote sensing
Abstract

Optical sensors based on absorption spectroscopy play a central role in the detection and monitoring of atmospheric trace gases. We here present for the first time the experimental demonstration of OSAS-Lidar on the remote sensing of CH4 in the atmosphere. This new methodology, the OSAS-Lidar, couples the Optical Similitude Absorption Spectroscopy (OSAS) methodology with a light detection and ranging device. It is based on the differential absorption of spectrally integrated signals following Beer Lambert-Bouguer law, which are range-resolved. Its novelty originates from the use of broadband laser spectroscopy and from the mathematical approach used to retrieve the trace gas concentration. We previously applied the OSAS methodology in laboratory on the 2ν3 methane absorption band, centered at the 1665 nm wavelength and demonstrated that the OSAS-methodology is almost independent from atmospheric temperature and pressure. In this paper, we achieve an OSAS-Lidar device capable of observing large concentrations of CH4 released from a methane source directly into the atmosphere. Comparison with a standard in-situ measurement device shows that the path-integrated concentrations retrieved from OSAS-Lidar methodology exhibit sufficient sensitivity (2 000 ppm m) and observational time resolution (1 s) to remotely sense methane leaks in the atmosphere. The coupling of OSAS-lidar with a wind measurement device opens the way to monitor time-resolved methane flux emissions, which is important in regards to future climate mitigation involving regional reduction of CH4 flux emissions.

Published
2018
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6. Topology of Random Real Hypersurfaces

Author

Jean-Yves Welschinger

Subjects
Discrete mathematics, Combinatorics, Hypersurface, Degree (graph theory), Betti number, General Mathematics, Expected value, Projective test, Mathematics
Abstract

Las siguientes son las notas de un mini curso que dí durante la escuela de verano CIMPA en Villa de Leyva, Colombia, en julio de 2014. El tema fue el trabajo que en conjunto se desarrolló con Damien Gayet sobre la topología de las hipersuperficies reales aleatorias, restringiéndonos al caso de los espacios proyectivos y enfocándonos en nuestras estimaciones inferiores. Particularmente, estimamos (por arriba y) por abajo la esperanza matemática de todos los números de Betti de las hipersuperficies reales proyectivas aleato- rias de grado d. De hecho, para cualquier hipersuperficie cerrada y conexa ∑ de Rn, estimamos por abajo la esperanza del número de componentes conexas de éstas hipersuperficies reales proyectivas aleatorias de grado d, las cuales son difeomorfas a ∑.

Published
2015
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7. Remote sensing of methane emissions by combining optical similitude absorption spectroscopy (OSAS) and lidar

Author

Christophe Anselmo, Sandrine Galtier, Jean-Yves Welschinger, Jean-Pierre Cariou, Alain Miffre, Jean-François Sivignon, and Patrick Rairoux

Subjects
Methane emissions, Differential absorption, 010504 meteorology & atmospheric sciences, Absorption spectroscopy, Astrophysics::High Energy Astrophysical Phenomena, Physics, QC1-999, Gas concentration, 01 natural sciences, Similitude, 010309 optics, Lidar, Remote sensing (archaeology), Proof of concept, 0103 physical sciences, Astrophysics::Galaxy Astrophysics, 0105 earth and related environmental sciences, Remote sensing
Abstract

Monitoring the emission of gases is difficult to achieve in industrial sites and in environments presenting poor infrastructures. Hence, robust methodologies should be developed and coupled to Lidar technology to allow remote sensing of gas emission. OSAS is a new methodology to evaluate gas concentration emission from spectrally integrated differential absorption measurements. Proof of concept of OSAS-Lidar for CH4 emission monitoring is here presented.

Published
2018

8. Open Gromov–Witten invariants in dimension four

Author

Jean-Yves Welschinger, Algèbre, géométrie, logique (AGL), Institut Camille Jordan (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), ANR-08-BLAN-0291,Floer Power,Applications des courbes holomorphes en géométrie symplectique et de contact(2008), European Project: 258204,EC:FP7:ERC,ERC-2010-StG_20091028,REALUMAN(2010), Institut Camille Jordan [Villeurbanne] (ICJ), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Surface (mathematics), Gromov-Witten invariants, Dimension (graph theory), Structure (category theory), Boundary (topology), 16. Peace & justice, 53D45, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], Combinatorics, Integer, holomorphic discs, Geometry and Topology, Algebraic number, Mathematics::Symplectic Geometry, Symplectic geometry, Mathematics, Relative homology
Abstract

26 pages; Given a closed orientable Lagrangian surface L in a closed symplectic four-manifold X together with a relative homology class d in H_2 (X , L ; Z) with vanishing boundary in H_1 (L ; Z), we prove that the algebraic number of J-holomorphic discs with boundary on L, homologous to d and passing through the adequate number of points neither depends on the choice of the points nor on the generic choice of the almost-complex structure J. We furthermore get analogous open Gromov-Witten invariants by counting, for every non-negative integer k, unions of k discs instead of single discs.

Published
2015
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9. Betti numbers of random nodal sets of elliptic pseudo-differential operators

Author

Jean-Yves Welschinger, Damien Gayet, Institut Fourier (IF ), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), Algèbre, géométrie, logique (AGL), Institut Camille Jordan (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), European Project: 258204,EC:FP7:ERC,ERC-2010-StG_20091028,REALUMAN(2010), Institut Fourier (IF), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA), Institut Camille Jordan [Villeurbanne] (ICJ), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet [Saint-Étienne] (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet [Saint-Étienne] (UJM)-Centre National de la Recherche Scientifique (CNRS), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Pure mathematics, Closed manifold, Betti number, General Mathematics, 01 natural sciences, Mathematics - Spectral Theory, Line bundle, 0103 physical sciences, FOS: Mathematics, 0101 mathematics, Spectral Theory (math.SP), Morse theory, Mathematics, Lebesgue measure, random nodal set, Applied Mathematics, 010102 general mathematics, Probability (math.PR), random matrix, Mathematics::Spectral Theory, Differential operator, Pseudo-differential operator, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 34L20, 58J40, 60D05, 60B20, Bounded function, 010307 mathematical physics, Mathematics - Probability, [MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP]
Abstract

Given an elliptic self-adjoint pseudo-differential operator $P$ bounded from below, acting on the sections of a Riemannian line bundle over a smooth closed manifold $M$ equipped with some Lebesgue measure, we estimate from above, as $L$ grows to infinity, the Betti numbers of the vanishing locus of a random section taken in the direct sum of the eigenspaces of $P$ with eigenvalues below $L$. These upper estimates follow from some equidistribution of the critical points of the restriction of a fixed Morse function to this vanishing locus. We then consider the examples of the Laplace-Beltrami and the Dirichlet-to-Neumann operators associated to some Riemannian metric on $M$., Comment: 34

Published
2017
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10. Asymptotic measures and links in simplicial complexes

Author

Nermin Salepci, Jean-Yves Welschinger, Algèbre, géométrie, logique (AGL), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), and ANR-15-CE40-0007,MICROLOCAL,Topologie symplectique, théorie microlocal des faisceaux et quantification(2015)

Subjects
link of a simplex, Dimension (graph theory), Block (permutation group theory), measure, 0102 computer and information sciences, Barycentric coordinate system, 01 natural sciences, Measure (mathematics), Theoretical Computer Science, Combinatorics, Simplicial complex, Mathematics - Geometric Topology, [MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT], [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], FOS: Mathematics, barycentric subdivisions, Discrete Mathematics and Combinatorics, Mathematics::Metric Geometry, Mathematics - Combinatorics, 0101 mathematics, Mathematics, Lebesgue measure, face vector, 010102 general mathematics, Geometric Topology (math.GT), dual block, Link (geometry), face polynomial, Computer Science::Graphics, Computational Theory and Mathematics, 010201 computation theory & mathematics, simplicial complex, Combinatorics (math.CO), MSC 52C99, 28C15, 28A33, Geometry and Topology
Abstract

We introduce canonical measures on a locally finite simplicial complex $K$ and study their asymptotic behavior under infinitely many barycentric subdivisions. We also compute the face polynomial of the asymptotic link and dual block of a simplex in the $d^{th}$ barycentric subdivision $Sd^d(K)$ of $K$, $d\gg0$. It is almost everywhere constant. When $K$ is finite, we study the limit face polynomial of $Sd^d(K)$ after F.Brenti-V.Welker and E.Delucchi-A.Pixton-L.Sabalka., 15 pages, 1 figure

Published
2017
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11. Universal components of random nodal sets

Author

Damien Gayet, Jean-Yves Welschinger, Institut Fourier (IF ), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), Algèbre, géométrie, logique (AGL), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), ANR-08-BLAN-0291,Floer Power,Applications des courbes holomorphes en géométrie symplectique et de contact(2008), European Project: 258204,EC:FP7:ERC,ERC-2010-StG_20091028,REALUMAN(2010), Institut Fourier (IF), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet [Saint-Étienne] (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet [Saint-Étienne] (UJM)-Centre National de la Recherche Scientifique (CNRS), ANR Floer Power,ANR Floer Power, Institut Camille Jordan (ICJ), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), 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
Betti number, 01 natural sciences, Upper and lower bounds, Combinatorics, Mathematics - Spectral Theory, Mathematics - Geometric Topology, Line bundle, [MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT], 0103 physical sciences, FOS: Mathematics, Betti numbers, Ball (mathematics), 0101 mathematics, Spectral Theory (math.SP), Mathematical Physics, Physics, Lebesgue measure, Probability (math.PR), 010102 general mathematics, Geometric Topology (math.GT), Statistical and Nonlinear Physics, random nodal sets, Pseudo-differential operator, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Hypersurface, Bounded function, 010307 mathematical physics, Locus (mathematics), Mathematics - Probability, [MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP]
Abstract

We give, as L grows to infinity, an explicit lower bound of order \({L^{\frac{n}{m}}}\) for the expected Betti numbers of the vanishing locus of a random linear combination of eigenvectors of P with eigenvalues below L. Here, P denotes an elliptic self-adjoint pseudo-differential operator of order \({m > 0}\), bounded from below and acting on the sections of a Riemannian line bundle over a smooth closed n-dimensional manifold M equipped with some Lebesgue measure. In fact, for every closed hypersurface \({\Sigma}\) of \({\mathbb{R}^n}\), we prove that there exists a positive constant \({p_\Sigma}\) depending only on \({\Sigma}\), such that for every large enough L and every \({x \in M}\), a component diffeomorphic to \({\Sigma}\) appears with probability at least \({p_\Sigma}\) in the vanishing locus of a random section and in the ball of radius \({L^{-\frac{1}{m}}}\) centered at x. These results apply in particular to Laplace–Beltrami and Dirichlet-to-Neumann operators.

Published
2016
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12. Gas concentration measurement by optical similitude absorption spectroscopy: methodology and experimental demonstration

Author

Christophe Anselmo, Patrick Rairoux, Jean-Yves Welschinger, Jean-Pierre Cariou, Alain Miffre, Institut Lumière Matière [Villeurbanne] (ILM), Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS), Algèbre, géométrie, logique (AGL), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), LEOSPHERE France, and LEOSPHERE

Subjects
[PHYS.PHYS.PHYS-OPTICS]Physics [physics]/Physics [physics]/Optics [physics.optics], Materials science, Absorption spectroscopy, business.industry, Spectral density, 02 engineering and technology, 021001 nanoscience & nanotechnology, 01 natural sciences, Atomic and Molecular Physics, and Optics, Similitude, Spectral line, 010309 optics, Optics, 13. Climate action, Greenhouse gas, 0103 physical sciences, Spectral width, Allan variance, 0210 nano-technology, Spectroscopy, business, ComputingMilieux_MISCELLANEOUS
Abstract

We propose a new methodology to measure gas concentration by light-absorption spectroscopy when the light source spectrum is larger than the spectral width of one or several molecular gas absorption lines. We named it optical similitude absorption spectroscopy (OSAS), as the gas concentration is derived from a similitude between the light source and the target gas spectra. The main OSAS-novelty lies in the development of a robust inversion methodology, based on the Newton-Raphson algorithm, which allows retrieving the target gas concentration from spectrally-integrated differential light-absorption measurements. As a proof, OSAS is applied in laboratory to the 2ν3 methane absorption band at 1.66 µm with uncertainties revealed by the Allan variance. OSAS has also been applied to non-dispersive infra-red and the optical correlation spectroscopy arrangements. This all-optics gas concentration retrieval does not require the use of a gas calibration cell and opens new tracks to atmospheric gas pollution and greenhouse gases sources monitoring.

Published
2016
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14. Betti numbers of random real hypersurfaces and determinants of random symmetric matrices

Author

Damien Gayet, Jean-Yves Welschinger, Algèbre, géométrie, logique (AGL), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), European Project: 258204,EC:FP7:ERC,ERC-2010-StG_20091028,REALUMAN(2010), Institut Camille Jordan (ICJ), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), 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
Ample line bundle, Pure mathematics, Betti number, General Mathematics, Real projective manifold, 010103 numerical & computational mathematics, 01 natural sciences, Mathematics - Algebraic Geometry, Square root, FOS: Mathematics, Symmetric matrix, random polynomial, Complex Variables (math.CV), 0101 mathematics, Algebraic Geometry (math.AG), Morse theory, Mathematics, Mathematics - Complex Variables, Applied Mathematics, Probability (math.PR), 010102 general mathematics, random matrix, [MATH.MATH-CV]Mathematics [math]/Complex Variables [math.CV], Manifold, MSC 2010: 14P25, 32Q15, 60B20, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Mathematics::Differential Geometry, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], Locus (mathematics), Random matrix, Mathematics - Probability, ample line bundle
Abstract

We asymptotically estimate from above the expected Betti numbers of random real hypersurfaces in smooth real projective manifolds. Our upper bounds grow as the square root of the degree of the hypersurfaces as the latter grows to infinity, with a coefficient involving the K\"ahlerian volume of the real locus of the manifold as well as the expected determinant of random real symmetric matrices of given index. In particular, for large dimensions, these coefficients get exponentially small away from mid-dimensional Betti numbers. In order to get these results, we first establish the equidistribution of the critical points of a given Morse function restricted to the ran- dom real hypersurfaces., Comment: 34 pages

Published
2016
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15. Invariant count of holomorphic disks in the cotangent bundles of the two-sphere and real projective plane

Author

Jean-Yves Welschinger

Subjects
Pure mathematics, Real projective plane, Holomorphic function, Cotangent bundle, General Medicine, Projective plane, Invariant (mathematics), Mathematics::Symplectic Geometry, Symplectic manifold, Mathematics, Symplectic geometry, Enumerative geometry
Abstract

We first define enumerative invariants of the cotangent bundles of the two-sphere and real projective plane. These invariants are obtained in the framework of symplectic field theory by counting with respect to some sign holomorphic disks with punctures sitting on the zero section. Then, we relate these invariants with the ones of closed real symplectic four-manifolds which have been constructed earlier. This relation provides some congruences and recursive formulas for the latter as well as sharpness results for the associated lower bounds in real enumerative geometry. To cite this article: J.-Y. Welschinger, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

Published
2007
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16. Towards relative invariants of real symplectic four-manifolds

Author

Jean-Yves Welschinger

Subjects
Discrete mathematics, Conic section, Tangent, Geometry and Topology, Homology (mathematics), Invariant (mathematics), Analysis, Symplectic geometry, Mathematics, Enumerative geometry, Real plane
Abstract

Let (X, ω, c X ) be a real symplectic four-manifold with real part $$\mathbb{R}X$$ . Let $$L \subset \mathbb{R}X$$ be a smooth curve such that $$[L] = 0 \in H_1 (\mathbb{R}X;\mathbb{Z}/2\mathbb{Z}).$$ We construct invariants under deformation of the quadruple (X, ω, c X , L) by counting the number of real rational J-holomorphic curves which realize a given homology class d, pass through an appropriate number of points and are tangent to L. As an application, we prove a relation between the count of real rational J-holomorphic curves done in [W2] and the count of reducible real rational curves done in [W3]. Finally, we show how these techniques also allow us to extract an integer valued invariant from a classical problem of real enumerative geometry, namely about counting the number of real plane conics tangent to five given generic real conics.

Published
2006
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17. Invariants of real symplectic four-manifolds out of reducible and cuspidal curves

Author

Jean-Yves Welschinger

Subjects
Combinatorics, General Mathematics, Mathematics, Symplectic geometry
Abstract

Nous construisons des invariants par deformation des varietes symplectiques reelles de dimension quatre. Ces invariants sont obtenus en comptant trois differents types de courbes J-holomorphes rationnelles reelles qui realisent une classe d'homologie donnee et passent par une configuration reelle donnee d'un nombre (adequat) de points. Ces courbes sont des courbes cuspidales, reductibles et des courbes ayant une tangente prescrite en l'un des points de la configuration. Elles sont comptees en fonction d'un signe qui depend de la parite du nombre de leurs points doubles reels isoles et, dans le cas des courbes reductibles, en fonction d'une multiplicite. Dans le cas du plan projectif complexe muni de ses formes symplectiques et structures reelles standards, ces invariants coincident avec ceux precedemment construits dans [11]. Ceci mene a une relation entre le comptage de courbes J-holomorphes rationnelles reelles realise dans [11] et le comptage de courbes J-holomorphes rationnelles reductibles reelles presente ici.

Published
2006
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18. Invariants of real rational symplectic 4-manifolds and lower bounds in real enumerative geometry

Author

Jean-Yves Welschinger

Subjects
Pure mathematics, Holomorphic function, General Medicine, Algebraic geometry, Homology (mathematics), Invariant (mathematics), Mathematics::Symplectic Geometry, Mathematics, Enumerative geometry, Symplectic manifold, Symplectic geometry
Abstract

Following the approach of Gromov and Witten, we construct invariants under deformation of real rational symplectic 4-manifolds. These invariants provide lower bounds for the number of real rational J -holomorphic curves in a given homology class passing through a given real configuration of points. To cite this article: J.-Y. Welschinger, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Published
2003
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19. Expected topology of random real algebraic submanifolds

Author

Jean-Yves Welschinger, Damien Gayet, Institut Fourier (IF ), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), Algèbre, géométrie, logique (AGL), Institut Camille Jordan (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), European Project: 258204,EC:FP7:ERC,ERC-2010-StG_20091028,REALUMAN(2010), Institut Fourier (IF), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA), Institut Camille Jordan [Villeurbanne] (ICJ), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet [Saint-Étienne] (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet [Saint-Étienne] (UJM)-Centre National de la Recherche Scientifique (CNRS), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Ample line bundle, Betti number, General Mathematics, Holomorphic function, Codimension, Submanifold, Combinatorics, Section (fiber bundle), Mathematics - Algebraic Geometry, Mathematics::Algebraic Geometry, Normal bundle, FOS: Mathematics, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], Algebraic Geometry (math.AG), Mathematics::Symplectic Geometry, 14P25, 32Q15, 60D05, Holomorphic vector bundle, Mathematics
Abstract

Let X be a smooth complex projective manifold of dimension n equipped with an ample line bundle L and a rank k holomorphic vector bundle E. We assume that 0< k, 32 pages

Published
2014
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20. Open Gromov-Witten invariants in dimension six

Author

Jean-Yves Welschinger, Algèbre, géométrie, logique (AGL), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), The research leading to these results has received funding from the European Community's Seventh Framework Progamme ([FP7/2007-2013] [FP7/2007-2011]) under grant agreement $\text{n}\textsuperscript{o}$ [258204]., European Project: 258204,EC:FP7:ERC,ERC-2010-StG_20091028,REALUMAN(2010), Institut Camille Jordan (ICJ), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), 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
Ring (mathematics), Pure mathematics, 010308 nuclear & particles physics, Group (mathematics), General Mathematics, 010102 general mathematics, Gromov-Witten invariants, Boundary (topology), Commutative ring, Submanifold, 01 natural sciences, 53D45, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], Mathematics - Symplectic Geometry, Holomorphic discs, 0103 physical sciences, FOS: Mathematics, Symplectic Geometry (math.SG), 0101 mathematics, Mathematics::Symplectic Geometry, Incidence (geometry), Relative homology, Symplectic geometry, Mathematics
Abstract

Let $L$ be a closed orientable Lagrangian submanifold of a closed symplectic six-manifold $(X, \omega)$. We assume that the first homology group $H_1 (L ; A)$ with coefficients in a commutative ring $A$ injects into the group $H_1 (X ; A)$ and that $X$ contains no Maslov zero pseudo-holomorphic disc with boundary on $L$. Then, we prove that for every generic choice of a tame almost-complex structure $J$ on $X$, every relative homology class $d \in H_2 (X, L ; \Z)$ and adequate number of incidence conditions in $L$ or $X$, the weighted number of $J$-holomorphic discs with boundary on $L$, homologous to $d$, and either irreducible or reducible disconnected, which satisfy the conditions, does not depend on the generic choice of $J$, provided that at least one incidence condition lies in $L$. These numbers thus define open Gromov-Witten invariants in dimension six, taking values in the ring $A$., Comment: 19 pages, 1 figure

Published
2013
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21. Lower estimates for the expected Betti numbers of random real hypersurfaces

Author

Jean-Yves Welschinger, Damien Gayet, Algèbre, géométrie, logique (AGL), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), European Project: 258204,EC:FP7:ERC,ERC-2010-StG_20091028,REALUMAN(2010), Institut Camille Jordan (ICJ), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), 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
Ample line bundle, Betti number, General Mathematics, Real projective manifold, Curvature, 01 natural sciences, Combinatorics, 010104 statistics & probability, Mathematics::Algebraic Geometry, FOS: Mathematics, Hermitian manifold, random polynomial, Betti numbers, Ball (mathematics), 0101 mathematics, Algebraic number, Mathematics, 14P25, 32Q15, 60D05, 010102 general mathematics, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], Hypersurface, Mathematics - Symplectic Geometry, Symplectic Geometry (math.SG), [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], Mathematics::Differential Geometry, Locus (mathematics), ample line bundle
Abstract

We estimate from below the expected Betti numbers of real hypersurfaces taken at random in a smooth real projective n-dimensional manifold. These random hypersurfaces are chosen in the linear system of a large d-th power of a real ample line bundle equipped with a Hermitian metric of positive curvature. As for the upper bounds that we recently established, these lower bounds read as a product of a constant which only depends on the dimension n of the manifold with the K\"ahlerian volume of its real locus RX and d^{n/2}. Actually, any closed affine real algebraic hypersurface appears with positive probability as part of such random real hypersurfaces in any ball of RX of radius O(d^{-1/2})., Comment: 19 pages

Published
2013
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22. What is the total Betti number of a random real hypersurface?

Author

Jean-Yves Welschinger, Damien Gayet, Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Algèbre, géométrie, logique (AGL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), European Project: 258204,EC:FP7:ERC,ERC-2010-StG_20091028,REALUMAN(2010), Institut Camille Jordan (ICJ), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), 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)-École Centrale de Lyon (ECL)

Subjects
Pure mathematics, Degree (graph theory), Betti number, Mathematics::Complex Variables, Applied Mathematics, General Mathematics, 010102 general mathematics, 14P25, 32U40, 60F10, 16. Peace & justice, Mathematical proof, 01 natural sciences, Upper and lower bounds, Domain (mathematical analysis), Manifold, Lefschetz pencil, Mathematics - Algebraic Geometry, Hypersurface, Mathematics::Algebraic Geometry, 0103 physical sciences, FOS: Mathematics, 010307 mathematical physics, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], 0101 mathematics, Algebraic Geometry (math.AG), Mathematics
Abstract

We bound from above the expected total Betti number of a high degree random real hypersurface in a smooth real projective manifold. This upper bound is deduced from the equidistribution of critical points of a real Lefschetz pencil restricted to the complex domain of such a random hypersurface, equidistribution which we first establish. Our proofs involve Hörmander's theory of peak sections as well as the formula of Poincaré–Martinelli.

Published
2012
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23. Do uniruled six-manifolds contain Sol Lagrangian submanifolds?

Author

Jean-Yves Welschinger, Frédéric Mangolte, Laboratoire Angevin de Recherche en Mathématiques (LAREMA), Université d'Angers (UA)-Centre National de la Recherche Scientifique (CNRS), Algèbre, géométrie, logique (AGL), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), ANR-08-BLAN-0291,Floer Power,Applications des courbes holomorphes en géométrie symplectique et de contact(2008), Institut National des Sciences Appliquées (INSA)-Université de Lyon-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), and Institut National des Sciences Appliquées (INSA)-Université de Lyon-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Pure mathematics, General Mathematics, Fano plane, 01 natural sciences, 53D12, 53D25, 14J30, 14P25, Mathematics - Algebraic Geometry, Mathematics::Algebraic Geometry, 0103 physical sciences, 0101 mathematics, Mathematics::Symplectic Geometry, Mathematics, 010102 general mathematics, Fibration, Submanifold, Mathematics::Geometric Topology, Manifold, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], Monotone polygon, Mathematics - Symplectic Geometry, 010307 mathematical physics, Diffeomorphism, Mathematics::Differential Geometry, MSC 2000 53D12, 53D25, 14J30, 14P25, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], Locus (mathematics), Symplectic geometry
Abstract

We prove using symplectic field theory that if the suspension of a hyperbolic diffeomorphism of the two-torus Lagrangian embeds in a closed uniruled symplectic six-manifold, then its image contains the boundary of a symplectic disc with vanishing Maslov index. This prevents such a Lagrangian submanifold to be monotone, for instance the real locus of a smooth real Fano manifold. It also prevents any Sol manifold to be in the real locus of an orientable real Del Pezzo fibration over a curve, confirming an expectation of J. Koll\'ar. Finally, it constraints Hamiltonian diffeomorphisms of uniruled symplectic four-manifolds., Comment: 23 pages, 2 figures

Published
2012
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24. Exponential rarefaction of real curves with many components

Author

Jean-Yves Welschinger, Damien Gayet, Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Algèbre, géométrie, logique (AGL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), and ANR-08-BLAN-0291,Floer Power,Applications des courbes holomorphes en géométrie symplectique et de contact(2008)

Subjects
Real analysis, General Mathematics, 010102 general mathematics, Mathematical analysis, Holomorphic function, 14P25, 32U40, 60F10, Gaussian measure, 01 natural sciences, Hermitian matrix, large deviations, Mathematics - Algebraic Geometry, 010104 statistics & probability, Real projective line, real maximal curves, Line bundle, FOS: Mathematics, Projective space, positive currents, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], 0101 mathematics, Algebraic Geometry (math.AG), Mathematics::Symplectic Geometry, Real projective space, Mathematics
Abstract

Given a positive real Hermitian holomorphic line bundle L over a smooth real projective manifold X, the space of real holomorphic sections of the bundle L^d inherits for every positive integer d a L^2 scalar product which induces a Gaussian measure. When X is a curve or a surface, we estimate the volume of the cone of real sections whose vanishing locus contains many real components. In particular, the volume of the cone of maximal real sections decreases exponentially as d grows to infinity., Comment: 21 pages

Published
2011
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25. Invariants entiers en géométrie énumérative réelle

Author

Jean-Yves Welschinger, Algèbre, géométrie, logique (AGL), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Hindustan Book Agency, New Delhi, and Welschinger, Jean-Yves

Subjects
010102 general mathematics, [MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG], 01 natural sciences, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], Mathematics - Algebraic Geometry, [MATH.MATH-SG] Mathematics [math]/Symplectic Geometry [math.SG], 53D45, 14N35, Mathematics - Symplectic Geometry, 0103 physical sciences, 010307 mathematical physics, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], 0101 mathematics, Mathematics::Symplectic Geometry, Mathematics
Abstract

I first recall the various problems of real enumerative geometry out of which I could extract some integer valued invariants, providing some real counterpart to Gromov-Witten invariants. I then discuss sharpness of the lower bounds given by these invariants and some of their arithmetical properties. Finally, I present further results which guaranty the presence or absence of pseudo-holomorphic discs with boundary on a Lagrangian submanifold of a given symplectic manifold., Comment: 26 pages, 2 figures

Published
2010

26. Effective classes and Lagrangian tori in symplectic four-manifolds

Author

Jean-Yves Welschinger, Unité de Mathématiques Pures et Appliquées (UMPA-ENSL), École normale supérieure de Lyon (ENS de Lyon)-Centre National de la Recherche Scientifique (CNRS), and École normale supérieure - Lyon (ENS Lyon)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Pure mathematics, Homology (mathematics), 01 natural sciences, 0103 physical sciences, FOS: Mathematics, Intersection form, Lagrangian torus, 0101 mathematics, Symplectomorphism, Moment map, Mathematics::Symplectic Geometry, Mathematics, 010102 general mathematics, Mathematical analysis, Pseudo-holomorphic curves, Symplectic representation, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], Hypersurface, 53D05, Mathematics - Symplectic Geometry, Symplectic field theory, Cotangent bundle, Symplectic Geometry (math.SG), 010307 mathematical physics, Geometry and Topology, Symplectic geometry
Abstract

An effective class in a closed symplectic four-manifold $(X, \omega)$ is a two-dimensional homology class which is realized by a $J$-holomorphic cycle for every tamed almost complex structure $J$. We prove that effective classes are orthogonal to Lagrangian tori in $H_2 (X ; \Bbb{Z})$., Comment: 5 pages

Published
2007
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27. Spinor states of real rational curves in real algebraic convex 3-manifolds and enumerative invariants

Author

Jean-Yves Welschinger, Arxiv, Import, Unité de Mathématiques Pures et Appliquées (UMPA-ENSL), École normale supérieure de Lyon (ENS de Lyon)-Centre National de la Recherche Scientifique (CNRS), and École normale supérieure - Lyon (ENS Lyon)-Centre National de la Recherche Scientifique (CNRS)

Subjects
General Mathematics, Structure (category theory), 01 natural sciences, 53D45, Mathematics - Algebraic Geometry, Real data type, Rational point, ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION, 0103 physical sciences, Real algebraic geometry, FOS: Mathematics, 0101 mathematics, Algebraic number, 14N35, Equivalence class, Algebraic Geometry (math.AG), 14P25, Mathematics, Discrete mathematics, Spinor, 010102 general mathematics, Regular polygon, [MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG], [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], 010307 mathematical physics
Abstract

Let $X$ be a real algebraic convex 3-manifold whose real part is equipped with a $Pin^-$ structure. We show that every irreducible real rational curve with non-empty real part has a canonical spinor state belonging to $\{\pm 1\}$. The main result is then that the algebraic count of the number of real irreducible rational curves in a given numerical equivalence class passing through the appropriate number of points does not depend on the choice of the real configuration of points, provided that these curves are counted with respect to their spinor states. These invariants provide lower bounds for the total number of such real rational curves independantly of the choice of the real configuration of points., Comment: 24 pages, 5 figures

Published
2005

28. Deformation classes of real ruled manifolds

Author

Jean-Yves Welschinger, Unité de Mathématiques Pures et Appliquées (UMPA-ENSL), École normale supérieure - Lyon (ENS Lyon)-Centre National de la Recherche Scientifique (CNRS), Arxiv, Import, and École normale supérieure de Lyon (ENS de Lyon)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Class (set theory), Pure mathematics, 14J10, Applied Mathematics, General Mathematics, 010102 general mathematics, [MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG], ComputingMethodologies_IMAGEPROCESSINGANDCOMPUTERVISION, Real structure, Deformation (meteorology), 14M99, 14P25, 01 natural sciences, GeneralLiterature_MISCELLANEOUS, Mathematics - Algebraic Geometry, 0103 physical sciences, FOS: Mathematics, 010307 mathematical physics, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], 0101 mathematics, Algebraic Geometry (math.AG), Topology (chemistry), Mathematics, ComputingMethodologies_COMPUTERGRAPHICS
Abstract

A complete description of the deformation classes of real ruled manifolds is given. In particular, we prove that once the complex deformation class is fixed, the real deformation class is prescribed by the topology of the real structure., Comment: 18 pages, 1 figure

Published
2004

29. Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry

Author

Jean-Yves Welschinger, Unité de Mathématiques Pures et Appliquées (UMPA-ENSL), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Lyon (ENS Lyon), Arxiv, Import, and École normale supérieure de Lyon (ENS de Lyon)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Pure mathematics, General Mathematics, Pseudoholomorphic curve, 01 natural sciences, 53D45, Mathematics - Algebraic Geometry, 0103 physical sciences, FOS: Mathematics, Gromov–Witten invariant, 0101 mathematics, Symplectomorphism, Algebraic Geometry (math.AG), Moment map, Mathematics::Symplectic Geometry, 14P25, Symplectic manifold, Mathematics, Symplectic group, 010102 general mathematics, [MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG], 16. Peace & justice, Symplectic matrix, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], Algebra, [MATH.MATH-SG] Mathematics [math]/Symplectic Geometry [math.SG], 53D05, Mathematics - Symplectic Geometry, Symplectic Geometry (math.SG), 14N10, 010307 mathematical physics, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], Symplectic geometry
Abstract

We first present the construction of the moduli space of real pseudo-holomorphic curves in a given real symplectic manifold. Then, following the approach of Gromov and Witten, we construct invariants under deformation of real rational symplectic 4-manifolds. These invariants provide lower bounds for the number of real rational $J$-holomorphic curves in a given homology class passing through a given real configuration of points., 31 pages, 8 figures

Published
2003

30. Real structures on minimal ruled surfaces

Author

Jean-Yves Welschinger, Arxiv, Import, Unité de Mathématiques Pures et Appliquées (UMPA-ENSL), École normale supérieure - Lyon (ENS Lyon)-Centre National de la Recherche Scientifique (CNRS), and École normale supérieure de Lyon (ENS de Lyon)-Centre National de la Recherche Scientifique (CNRS)

Subjects
General Mathematics, 010102 general mathematics, [MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG], Real structure, Deformation (meteorology), Topology, 14J26, 14P25, 01 natural sciences, Mathematics - Algebraic Geometry, 0103 physical sciences, FOS: Mathematics, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], 010307 mathematical physics, 0101 mathematics, Algebraic Geometry (math.AG), Topology (chemistry), Mathematics, ComputingMethodologies_COMPUTERGRAPHICS
Abstract

In this paper, we give a complete description of the deformation classes of real structures on minimal ruled surfaces. In particular, we show that these classes are determined by the topology of the real structure, which means that real minimal ruled surfaces are quasi-simple. As an intermediate result, we obtain the classification, up to conjugation, of real structures on decomposable ruled surfaces., Comment: 24 pages, 3 figures

Published
2003

31. Courbes algébriques réelles et courbes flexibles sur les surfaces réglées de base C P 1

Author

Jean-Yves Welschinger, Unité de Mathématiques Pures et Appliquées (UMPA-ENSL), École normale supérieure - Lyon (ENS Lyon)-Centre National de la Recherche Scientifique (CNRS), and École normale supérieure de Lyon (ENS de Lyon)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Surface (mathematics), Comparison theorem, Pure mathematics, General Mathematics, 010102 general mathematics, Base (topology), 01 natural sciences, Discriminant, 0103 physical sciences, Isotopy, Real algebraic geometry, Equivariant map, 010307 mathematical physics, Algebraic curve, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], 0101 mathematics, ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract

A first aim of this paper is to answer, in the case of real ruled surfaces $X_l$ of base $\mathbb{C}P^1$, with $l \geq 2$, a question of V. A. Rokhlin: is it true that the equivariant isotopy class does not suffice to distinguish the connected components of the space of smooth real algebraic curves of $X_l$? A second aim is to prove that there exist in these surfaces some real schemes realized by real flexible curves but not by smooth real algebraic curves. These two results of real algebraic geometry are deduced from the following comparison theorem: when $m = l + 2k$, with $k > 0$, the discriminants of the surface $X_m$ are deduced from those of the surface $X_l$ via weighted homotheties. All these results are obtained from a study of a deformation of ruled surfaces.The paper is written in French.2000 Mathematical Subject Classification: 14H10, 14J26, 14P25.

Published
2002
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32. Real automorphisms of a bundle, Cauchy-Riemann operators and orientability of moduli spaces

Author

Crétois , Rémi, Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Université Claude Bernard - Lyon I, Jean-Yves Welschinger, Institut Camille Jordan [Villeurbanne] ( ICJ ), École Centrale de Lyon ( ECL ), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 ( UCBL ), Université de Lyon-Institut National des Sciences Appliquées de Lyon ( INSA Lyon ), Université de Lyon-Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Université Jean Monnet [Saint-Étienne] ( UJM ) -Centre National de la Recherche Scientifique ( CNRS ), and Crétois, Rémi

Subjects
invariants de Gromov-Witten, [MATH.MATH-SG] Mathematics [math]/Symplectic Geometry [math.SG], Cauchy-Riemann operators, Gromov-Witten invariants, courbes réelles, moduli spaces, [ MATH.MATH-SG ] Mathematics [math]/Symplectic Geometry [math.SG], espaces de modules, fibrés vectoriels, opérateurs de Cauchy-Riemann, vector bundles, real curves, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG]
Abstract

We consider a complex vector bundle N equipped with a real structure cN over a real curve. The set of all real Cauchy-Riemann operators on (N, cN) is an infinite dimensional affine space. The union of all the determinant lines of such operators is a real line bundle Det(N) over this space. The aim of this work is to study the action of the the automorphism group of (N, cN) on the orientations of Det(N) and to deduct some results concerning the orientability of the moduli spaces of real curves in a real symplectic manifold. We first compute the action of the automorphisms which induce the identity on the complex line bundle det(N) in terms of Pin± structures on the real part of (N, cN). We then note that an automorphism of (N, cN) which lifts the identity of the curve acts on the bordism classes of real Spin structures on the curve. We use this action to give a topological description of the action on the orientations of Det(N). Finally, we introduce the notion of a lift of a diffeomorphism of the curve associated to a divisor compatible with (N, cN). Those are automorphisms of (N, cN) for which we can compute explicitly the action on the orientations of Det(N). The last part is devoted to some applications to the study of the orientability of moduli spaces of real curves in a real symplectic manifold. We compute for example, the first Stiefel-Whitney class of such moduli spaces in the case of the complex projective space of dimension three., L'ensemble des opérateurs de Cauchy-Riemann réels sur un fibré vectoriel complexe N muni d'une structure réelle cN au-dessus d'une courbe réelle est un espace affine de dimension infinie. L'union des déterminants de ces opérateurs est un fibré en droites réelles au-dessus de cet espace. L'objet de cette thèse est l'étude de l'action des automorphismes du fibré (N, cN) sur les orientations de ce fibré déterminant ainsi que de ses conséquences sur l'orientabilité des espaces de modules de courbes réelles dans une variété symplectique réelle. Nous commençons par interpréter l'action des automorphismes qui induisent l'identité sur le fibré en droites complexes det(N) en termes d'action sur les structures Pin± de la partie réelle de N. Nous remarquons ensuite qu'un automorphisme au-dessus de l'identité agit sur les classes de bordisme de structures Spin réelles de la courbe et nous utilisons cette action afin d'obtenir une description en termes topologiques de l'action sur les orientations du fibré déterminant. Enfin, pour comprendre l'action des automorphismes de (N, cN) qui ne relèvent pas l'identité, nous introduisons la notion de relevé d'un difféomorphisme de la courbe associé à un diviseur compatible avec (N, cN) et nous calculons le signe de l'action d'un tel relevé sur les orientations du fibré déterminant. Dans une dernière partie, nous appliquons les résultats obtenus à l'étude de l'orientabilité des espaces de modules de courbes réelles dans des variétés symplectiques réelles. Nous calculons en particulier la première classe de Stiefel-Whitney de l'espace de modules des courbes réelles dans l'espace projectif complexe de dimension trois.

Published
2011

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