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Multi angle
Integrable systems and spectral curves
Auteurs : Eynard, Bertrand (Auteur de la Conférence)
CIRM (Editeur )
CIRM (Editeur )
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Résumé :
Usually one defines a Tau function Tau(t_1,t_2,...) as a function of a family of times having to obey some equations, like Miwa-Jimbo equations, or Hirota equations.
Here we shall view times as local coordinates in the moduli-space of spectral curves, and define the Tau-function of a spectral curve Tau(S), in an intrinsic way, independent of a choice of coordinates. Deformations are tangent vectors, and the tangent space is isomorphic to the space of cycles (cf Goldman bracket), so that Hamiltonians can be represented by cycles.
All the integrable system formalism can then be represented geometrically in the space of cycles: the Poisson bracket is the intersection, the conserved quantities are periods, Miwa-Jimbo equations and Seiberg-Witten equations are a mere consequence of the definition, Hirota equation is a vanishing monodromy condition, and Virasoro-W constraint are automatically satisfied by our definition, showing that our Tau-function is also a conformal block. Our definition contains KdV, KP multicomponent KP, Hitchin systems, and probably all known classical integrable systems.
Codes MSC : Here we shall view times as local coordinates in the moduli-space of spectral curves, and define the Tau-function of a spectral curve Tau(S), in an intrinsic way, independent of a choice of coordinates. Deformations are tangent vectors, and the tangent space is isomorphic to the space of cycles (cf Goldman bracket), so that Hamiltonians can be represented by cycles.
All the integrable system formalism can then be represented geometrically in the space of cycles: the Poisson bracket is the intersection, the conserved quantities are periods, Miwa-Jimbo equations and Seiberg-Witten equations are a mere consequence of the definition, Hirota equation is a vanishing monodromy condition, and Virasoro-W constraint are automatically satisfied by our definition, showing that our Tau-function is also a conformal block. Our definition contains KdV, KP multicomponent KP, Hitchin systems, and probably all known classical integrable systems.
37K20 - Relations with algebraic geometry, complex analysis, special functions
60B20 - Random matrices (probabilistic aspects)
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Informations sur la Vidéo
Langue : AnglaisDate de publication : 09/05/2019 Date de captation : 09/04/2019 Sous collection : Research talks Format : MP4 arXiv category : Mathematical Physics ; Analysis of PDEs Domaine : Mathematical Physics ; Probability & Statistics ; Analysis and its Applications Durée : 00:53:10 Audience : Chercheurs ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2019-04-09_Eynard.mp4 
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Informations sur la rencontre
Nom de la rencontre : Chaire Jean-Morlet : Equation intégrable aux données initiales aléatoires / Jean-Morlet Chair : Integrable Equation with Random Initial DataOrganisateurs de la rencontre : Basor, Estelle ; Bufetov, Alexander ; Cafasso, Mattia ; Grava, Tamara ; McLaughlin, Ken Dates : 08/04/2019 - 12/04/2019 Année de la rencontre : 2019 URL Congrès : https://www.chairejeanmorlet.com/2104.html Citation Data
DOI : 10.24350/CIRM.V.19515203Cite this video as: Eynard, Bertrand (2019). Integrable systems and spectral curves. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19515203 URI : http://dx.doi.org/10.24350/CIRM.V.19515203 |
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