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Chords and Solitons: Combinatorics and Geometry in Integrable Systems

4:00 pm Thursday, September 16, 2010
Yuji Kodama (Ohio State University)

Abstract: Let Gr$(N,M)$ be the real Grassmannian defined by the set of all $N$-dimenaional 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 will give a realization of Gr$^+(N,M)$ in terms of the 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 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 $S_M$. Expressing each derrangement by a unique chord diagram, I will show that the chord diagrams can be used to analyze the asymptotic behavior of certain initial value problems of the KP equation. I will also present some movies of real experiments of shallow water waves which represent some of new solutions obtained in the classification problem. The talk is elementary, and shows interesting connections among combinatorics, geometry and integrable systems.

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