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H 1 Noncommutative geometry and time-frequency analysis

Auteurs : Luef, Franz (Auteur de la Conférence)
CIRM (Editeur )

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    Résumé : In my talk I am presenting a link between time-frequency analysis and noncommutative geometry. In particular, a connection between the Moyal plane, noncommutative tori and time-frequency analysis. After a brief description of a dictionary between these two areas I am going to explain some consequences for time-frequency analysis and noncommutative geometry such as the construction of projections in the mentioned operator algebras and Gabor frames.

    Keywords: modulation spaces - Banach-Gelfand triples - noncommutative tori - Moyal plane - noncommutative geometry - deformation quantization

    Codes MSC :
    46Fxx - Distributions, generalized functions, distribution spaces [See also 46T30]
    46Kxx - Topological (rings and) algebras with an involution [See also 16W10]
    81S05 - Canonical quantization, commutation relations and statistics
    81S10 - Geometric quantization, symplectic methods (quantum theory)
    81S30 - Phase space methods including Wigner distributions, etc.
    46S60 - Functional analysis on superspaces (supermanifolds) or graded spaces [See also 58A50 and 58C50]

      Informations sur la Vidéo

      Langue : Anglais
      Date de publication : 10/03/15
      Date de captation : 23/01/15
      Collection : Special events ; 30 Years of Wavelets ; Analysis and its Applications ; Mathematical Physics ; Mathematics in Science and Technology
      Format : quicktime ; audio/x-aac
      Durée : 00:28:45
      Domaine : Analysis and its Applications ; Mathematical Physics ; Mathematics in Science & Technology
      Audience : Chercheurs ; Doctorants , Post - Doctorants
      Download : https://videos.cirm-math.fr/2015-01-23_Luef.mp4

    Informations sur la rencontre

    Nom de la rencontre : 30 years of wavelets / 30 ans des ondelettes
    Organisateurs de la rencontre : Feichtinger, Hans G. ; Torrésani, Bruno
    Dates : 23/01/15 - 24/01/15
    Année de la rencontre : 2015
    URL Congrès : http://feichtingertorresani.weebly.com/3...

    Citation Data

    DOI : 10.24350/CIRM.V.18713603
    Cite this video as: Luef, Franz (2015). Noncommutative geometry and time-frequency analysis. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.18713603
    URI : http://dx.doi.org/10.24350/CIRM.V.18713603


    Bibliographie

    1. [1] de Gosson, M., Luef, F. (2014). Metaplectic group, symplectic Cayley transform, and fractional Fourier transforms. Journal of Mathematical Analysis and Applications, 416(2), 947-968 - http://dx.doi.org/10.1016/j.jmaa.2014.03.013

    2. [2] de Gosson, M., Luef, F. (2010). Spectral and regularity properties of a pseudo-differential calculus related to Landau quantization. Journal of Pseudo-Differential Operators and Applications, 1(1), 3-34 - http://dx.doi.org/10.1007/s11868-010-0001-6

    3. [3] de Gosson, M., Luef, F. (2009). Symplectic capacities and the geometry of uncertainty: the irruption of symplectic topology in classical and quantum mechanics. Physics Reports, 484(5), 131-179 - http://dx.doi.org/10.1016/j.physrep.2009.08.001

    4. [4] de Gosson, M., Luef, F. (2009). On the usefulness of modulation spaces in deformation quantization. Journal of Physics A: Mathematical and Theoretical, 42(31), 315205 - http://dx.doi.org/10.1088/1751-8113/42/31/315205

    5. [5] de Gosson, M., Luef, F. (2007). Quantum states and Hardy's formulation of the uncertainty principle: a symplectic approach. Letters in Mathematical Physics, 80(1), 69-82 - http://dx.doi.org/10.1007/s11005-007-0150-6

    6. [6] de Gosson, M., Luef, F. (2007). Remarks on the fact that the uncertainty principle does not determine the quantum state. Physics Letters A, 364(6), 453-457 - http://dx.doi.org/10.1016/j.physleta.2006.12.024

    7. [7] de Gosson, M., Luef, F. (2008). A new approach to the ${\ast}$-genvalue equation. Letters in Mathematical Physics, 85(2-3), 173-183 - http://dx.doi.org/10.1007/s11005-008-0261-8

    8. [8] Dias, N.C., de Gosson, M., Luef, F., Prata, J.N. (2012). Quantum mechanics in phase space: the Schrödinger and the Moyal representations. Journal of Pseudo-Differential Operators and Applications, 3(4), 367-398 - http://dx.doi.org/10.1007/s11868-012-0054-9

    9. [9] Dias, N.C., de Gosson, M., Luef, F., Prata, J.N. (2011). A pseudo-differential calculus on non-standard symplectic space; spectral and regularity results in modulation spaces. Journal de Mathématiques Pures et Appliquées, 96(5), 423-445 - http://dx.doi.org/10.1016/j.matpur.2011.07.006

    10. [10] Dias, N.C., de Gosson, M., Luef, F., Prata, J.N. (2010). A deformation quantization theory for noncommutative quantum mechanics. Journal of Mathematical Physics, 51(7), 072101 - http://dx.doi.org/10.1063/1.3436581

    11. [11] Feichtinger, H.G., Kozek, W., Luef, F. (2009). Gabor analysis over finite abelian groups. Applied and Computational Harmonic Analysis, 26(2), 230-248 - http://dx.doi.org/10.1016/j.acha.2008.04.006

    12. [12] Feichtinger, H., Luef, F., Cordero, E. (2008). Banach Gelfand triples for Gabor analysis. In L. Rodino, & M.W. Wong (Eds.), Pseudo-differential operators (1-33). Berlin: Springer. (Lecture Notes in Mathematics, 1949) - http://dx.doi.org/10.1007/978-3-540-68268-4_1

    13. [13] Feichtinger, H.G., Luef, F., Werther, T. (2007). A guided tour from linear algebra to the foundations of Gabor analysis. In S.S. Goh, A. Ron, & Z. Shen (Eds.), Gabor and wavelet frames (1-49). Hackensack, NJ: World Scientific. (Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore, 10) - http://dx.doi.org/10.1142/9789812709080_0001

    14. [14] Luef, F. (2011). Preferred quantization rules: Born-Jordan versus Weyl. The pseudo-differential point of view. Journal of Pseudo-Differential Operators and Applications, 2(1), 115-139 - http://dx.doi.org/10.1007/s11868-011-0025-6

    15. [15] Luef, F. (2011). Projections in noncommutative tori and Gabor frames. Proceedings of the American Mathematical Society, 139(2), 571-582 - http://dx.doi.org/10.1090/s0002-9939-2010-10489-6

    16. [16] Luef, F., Rahbani, Z. (2011). On pseudodifferential operators with symbols in generalized Shubin classes and an application to Landau-Weyl operators. Banach Journal of Mathematical Analysis, 5(2), 59-72 - http://dx.doi.org/10.15352/bjma/1313363002

    17. [17] Luef, F. (2009). Projective modules over noncommutative tori are multi-window Gabor frames for modulation spaces. Journal of Functional Analysis, 257(6), 1921-1946 - http://dx.doi.org/10.1016/j.jfa.2009.06.001

    18. [18] Luef, F., Manin, Y.I. (2009). Quantum theta functions and Gabor frames for modulation spaces. Letters in Mathematical Physics, 88(1-3), 131-161 - http://dx.doi.org/10.1007/s11005-009-0306-7

    19. [19] Luef, F. (2007). Gabor analysis, noncommutative tori and Feichtinger's algebra. In S.S. Goh, A. Ron, & Z. Shen (Eds.), Gabor and wavelet frames (77-106). Hackensack, NJ: World Scientific. (Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore, 10) - http://dx.doi.org/10.1142/9789812709080_0003

    20. [20] Luef, F. (2006). On spectral invariance of non-commutative tori. In D. Han, P.E.T. Jorgensen, & D.R. Larson (Eds.), Operator theory, operator algebras, and applications (131-146). Providence, RI: American Mathematical Society. (Contemporary Mathematics, 414) - http://dx.doi.org/10.1090/conm/414/07805

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