Abstract or Additional Information
Let \(Gr(N,M)\) be the real Grassmann manifold defined by the set of all \(N\)-dimensional subspaces of \(\mathbb{R}^M\). Each point on \(Gr(N,M)\) can be represented by an \(N\times M\) matrix \(A\) of rank \(N\). If all the \(N\times N\) minors of \(A\) are nonnegative, the set of all points associated with those matrices forms the totally nonnegative part of the Grassmannian, denoted by \(Gr(N,M)^+\).
In this talk, I start to give a realization of \(Gr(N,M)^+\) in terms of the (regular) soliton solutions of the KP (Kadomtsev-Petviashvili) equation which is a two-dimensional extension of the KdV equation. The KP equation describes small amplitude and long waves on a surface of shallow water. I then construct a cellular decomposition of \(Gr(N,M)^+\) with the asymptotic form of the soliton solutions. This leads to a classification theorem of all solitons solutions of the KP equation, showing that each soliton solution is uniquely parametrized by a derrangement of the symmetric group \(S_M\). Each derangement defines a combinatorial object called the Le-diagram (a Young diagram with zeros in particular boxes). Then I show that the Le-diagram provides a complete classification of the "entire" spatial patterns of the soliton solutions coming from the \(Gr(N,M)^+\) for asymptotic values of the time. I will also present some movies of real experiments of shallow water waves which represent some of those solutions obtained in the classification problem. If time permits, I will discuss an application of those results to analyze the Tohoku-tsunami of March 2011. The talk is elementary, and shows interesting connections among combinatorics, geometry and integrable systems.