KP soliton theory, dimer models in the disc and totally non-negative Grassmannians

Totally non-negative Grassmannians are a special case of G. Lusztig extension of the classical notion of total positivity, and have been combinatorially characterized in a seminal paper by A. Postnikov. They also appear in many relevant problems of mathematical and theoretical physics. The Kadomtsev-Petviashvili (KP) equation is the first non-trivial flow of the most relevant classical integrable hierarchy, and was originally introduced to study the stability of soliton solutions of another integrable system, the Kortweg-de Vries equation. Kasteleyn theorem represents the number of dimer configurations in planar graphs as determinants of sign matrices. In this talk I shall explain the role of totally non-negative Grassmannians in the characterization of the asymptotic behavior in space-time of a class of solutions of the Kadomtsev-Petviashvili equation, in the solution of a spectral problem for the same equation and in counting dimer configurations in planar bipartite graphs in the disc. The presentation will be elementary and self-contained.