数理談話会

日時：2012年10月4日 (木)
午後3時30分～4時 (ティータイム)
午後4時～5時 (講演)

場所：数理棟 談話室 (ティータイム)
図書館3階 中セミナー室2 (講演)

講師：児玉 裕治 氏 (オハイオ州立大学)

題目：Real Grassmannian and KP solitons

概要：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.