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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.