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## Colloquiums (FM & IMI)

### Real Grassmannian and KP solitons [FM]

- 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.