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VIDEO LIBRARY |
Algebraic Structures in Integrable Systems
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Integrability of generalized pentagram maps and cluster algebra (Lecture 1) M. Z. Shapiro Michigan State University |
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Abstract: The pentagram map that associates to a projective polygon a new one formed by intersections of short diagonals was introduced by R. Schwartz and was shown to be integrable by V. Ovsienko, R. Schwartz and S. Tabachnikov. Recently, M. Glick demonstrated that the pentagram map can be put into the framework of the theory of cluster algebras. We extend and generalize Glick's work by including the pentagram map into a family of discrete completely integrable systems.Our main tool is Poisson geometry of weighted directed networks on surfaces.. The ingredients necessary for complete integrability – invariant Poisson brackets, integrals of motion in involution, Lax representation – are recovered from combinatorics of the networks. Our integrable systems depend on one discrete parameter This is a joint work with M. Gekhtman, S. Tabachnikov, and A. Vainshtein. Language: English
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