Mathematics Graduate Students Organization

Seminar:

Date: October 30

Flag varieties (aka flag manifolds) are ubiquitous. Examples include the grassmannians Gr(k,n) of k-dimensional subspaces in complex n-space. Schubert subvarieties play an indispensable role in the geometry, representation theory and combinatorics associated with flag varieties. I will discuss an instance of this phenomenon in which the Schubert subvarieties tell us a great deal about a differential system on the flag variety.

Each flag variety is a homogeneous manifold G/P. The differential system we will consider may be characterized as the unique, minimal, bracket-generating G-homogeneous distribution on G/P. (All these words mean that the system is a very natural object of study. For example, this system is the differential constraint governing variations of Hodge structure.)

I will introduce flag varieties, their Schubert subvarieties and this canonical system; discuss several illuminating examples; and indicate how the Schubert varieties help us understand the solutions of the system.