Vladimir Fock | Goncharov–Kenyon integrable systems 2/3
Channel: EIMI, PDMI RAS and Chebyshev Laboratory
Date: 2021-11-25

School "Moduli Spaces, Combinatorics and Integrable Systems", November 23, 2021
Vladimir Fock (IRMA)
Goncharov–Kenyon integrable systems (2/3)

Starting from any integral convex polygon on a plane one can construct an integrable system with commuting continuous and discrete flows. The simplest example of such system is the Poncelet porism, but this scheme include a very large class of integrable systems (so far except Hitchin and Calogero systems). The phase space of this system can be viewed either as a configuration space of flags in an infinite dimensional space, or as weight space of a dimer model on a torus or as a symplectic leaf of an affine Lie group or as a bundle over the space of planar curves with Jacobians as fibers.

These four points of view permit to describe these systems very explicitly in (cluster) coordinates (in terms of tau-functions), give their solution (in terms of theta-functions) and establish relations to several other elementary problems.