MS Seminar (Mathematics - String Theory)

Speaker: Irina Davydenkova (U of Geneva)
Title: Inequalities from Poisson brackets
Date: Mon, Mar 23, 2015, 15:30 - 17:00
Place: Seminar Room A
Abstract: We introduce the notion of tropicalization for Poisson structures on R^n with coefficients in Laurent polynomials. To such a Poisson structure we associate a polyhedral cone and a constant Poisson bracket on this cone. There is a version of this formalism applicable to C^n viewed as a real Poisson manifold. In this case, the tropicalization gives rise to a completely integrable system with action variables taking values in a polyhedral cone and angle variables spanning a torus.

As an example, we consider the canonical Poisson bracket on the dual Poisson-Lie group G^* for G=U(n) in the cluster coordinates of Fomin-Zelevinsky defined by a certain choice of solid minors. We prove that the corresponding integrable system is isomorphic to the Gelfand-Zeiltin completely integrable system.