Friday, October 9, 2020
10/09/2020 - 1:00pm
An alternative approach to the cluster integrable systems of Goncharov and Kenyon is to consider paths on a directed graph instead of dimer covers. To a network embedded in a disc or on a cylinder, one can associate a "boundary measurement matrix", as defined by Postnikov. It was proved by Gekhtman, Shapiro, and Vainshtein that there is a combinatorially defined Poisson structure in which the boundary measurements satisfy classical R-matrix Poisson relations, and hence spectral invariants of the matrix form an integrable system. In recent work with Shapiro and Arthamonov, we give a non-commutative analogue of this Poisson structure for networks, and prove an analogous R-matrix formulation. Time permitting, I will discuss an application of this result to the dynamical system known as the "Pentagram Map".