Seminar
Real Grassmannian and KP solitons

Hold Date 
20121004 16:00～20121004 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
3:30PM4:00PM (teatime in the lounge)
4:00PM5:00PM (lectures in the seminar room 2)
summary:
Let Gr(k,n) be the real Grassmann manifold defined by the set of all kdimensional
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 (KadomtsevPetviashvili) 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 Lediagram (a Young diagram with zeros in
particular boxes). The Lediagram 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.