Friday, October 9, 2020
Time  Items 

All day 

1:00pm 
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 Rmatrix Poisson relations, and hence spectral invariants of the matrix form an integrable system. In recent work with Shapiro and Arthamonov, we give a noncommutative analogue of this Poisson structure for networks, and prove an analogous Rmatrix formulation. Time permitting, I will discuss an application of this result to the dynamical system known as the "Pentagram Map". Location: 