Third Year Major Oral: Smoothing Schubert classes by homogeneous subvarieties

Abstract: In a homogeneous variety G/P, classes [X_w] of Schubert varieties X_w generate the homology of G/P. In 1961, Borel and Haefliger posed the question: when does there exist a subvariety Y in G/P that is homologous to X_w? Many authors have studied this question, particularly in cominiscule homogeneous varieties and the special case of finding smooth subvarieties that represent Schubert classes. 

This leads to my question: In G/P, which combinations of Schubert classes can be represented by a homogeneous subvariety? I will discuss a strategy for answering this in SO(7)/P_2 and my current progress. This uses work by Bernstein-Gelfand-Gelfand that gives a nice way to compute the duals of the Schubert classes as polynomials, under Borel's presentation of the cohomology ring of G/P.

Advisors: Laura Escobar Vega and Matt Kerr