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Michael Shapiro

  1. Teichm\"uller spaces of Riemann surfaces with orbifold points of arbitrary order and cluster variables.

    Fri, 11/18/2011 - 15:01 — SomNambul
    Authors: Michael Shapiro, Leonid Chekhov
    Subjects: Mathematical Physics
    Abstract

    We generalize a new class of cluster type mutations for which exchange
    transformations are given by reciprocal polynomials. In the case of
    second-order polynomials of the form $x+2\cos{\pi/n_o}+x^{-1}$ these
    transformations are related to triangulations of Riemann surfaces of arbitrary
    genus with at least one hole/puncture and with an arbitrary number of orbifold
    points of arbitrary integer orders $n_o$.

  2. Poisson structures compatible with the cluster algebra structure in Grassmannians.

    Thu, 09/03/2009 - 17:38 — SomNambul
    Authors: Michael Gekhtman, Michael Shapiro, Alexander Stolin, Alek Vainshtein
    Subjects: Quantum Algebra
    Abstract

    We describe all Poisson brackets compatible with the natural cluster algebra
    structure in the open Schubert cell of the Grassmannian $G_k(n)$ and show that
    any such bracket endows $G_k(n)$ with a structure of a Poisson homogeneous
    space with respect to the natural action of $SL_n$ equipped with an R-matrix
    Poisson-Lie structure. The corresponding R-matrices belong to the simplest
    class in the Belavin-Drinfeld classification. Moreover, every compatible
    Poisson structure can be obtained this way.

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