Title: Totally nonnegative Grassmann cells and soliton solutions of
       the KP equation 


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 equation, and 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 permutation 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.