Promotion Colloquium for Dr. Colleen Robles

Date: September 11, 2014

Time: 4:00PM - 5:00PM

Location: Bloc 220

Speaker: Professor Colleen Robles, Texas A&M University

Description:

Title: Homogeneous spaces and Hodge theory

Abstract:
A flag variety X is a compact, algebraic manifold admitting a transitive, holomorphic action of a complex, semisimple Lie group G. (An example is the Grassmannian X=Gr(k,n) of k-planes in complex n-space, upon which G=SL(n) acts transitively.) Every flag variety X admits a distinguished sub-bundle H of the (holomorphic) tangent bundle: H is the the unique, bracket-generating, G-homogeneous distribution on X. In Hodge theory, H is known as the infinitesimal period relation, or Griffiths's transversality condition: it is the system of differential equations constraining a variation of Hodge structure.

The Schubert subvarieties of X play a central role in the geometry and representation theory associated with X, so it is natural to ask: which Schubert varieties are integrals of H? I will answer this question, and explain what these Schubert solutions can tell us about the space of all H-integrals. We obtain, as a corollary, sharp bounds on the dimension of a variation of Hodge structure, answering a long standing question in Hodge theory.