Institute of Mathematics for Industry

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Real Grassmannian and KP solitons


Hold Date 2012-10-04 16:00~2012-10-04 17:00

Place Seminar Room 2, Faculty of Mathematics building, Ito Campus

Object person  

Speaker Yuji KODAMA (Ohio State University)

Date:Thursday, October 4th, 2012
        3:30PM-4:00PM (teatime in the lounge)
        4:00PM-5:00PM (lectures in the seminar room 2)

summary:
Let  Gr(k,n)  be the real Grassmann manifold defined by the set of all  k-dimensional
subspaces of R^n.  Each point on  Gr(k,n) can be represented by a kxn matrix A of rank k.
If all the kxk 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(k,n)^+.

In this talk, I show how one can construct a cellular decomposition of Gr(k,n)^+   using
the "asymptotic" spatial patterns of certain "regular" solutions of the KP (Kadomtsev-Petviashvili) equation.
This provides 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_n.
Each derangement defines a combinatorial object called the Le-diagram (a Young diagram with zeros in
particular boxes). The Le-diagram then provides a classification of the ''entire'' spatial patterns
of the KP solitons coming from the Gr(k,n)^+ for asymptotic values of the time.

If time permits, I will also explain how one can compute the integral cohomology of the real Grassmannian using certain "singular" KP solitons.