Eric Berry (MSU) "The cohomology of real Grassmannians via Schubert stratifications"

The cohomology of real Grassmannians via Schubert stratifications

Consider the collection of all k-dimensional subvector spaces of n-dimensional Euclidean space, e.g. all lines through the origin in R^3. This collection is called the Grassmannian of k-planes in R^n. Grassmannians play an important role in geometry and topology. For instance, they classify vector bundles. One can show that Grassmannians are smooth manifolds. Even better, one can show they possess a natural CW structure. Better still, Grassmannians possess the structure of a stratified space. In this talk, I will introduce the Schubert stratification of Grassmannians and discuss how this extra structure enables us to give an explicit description of the Schubert CW (co)chain complex. Using these methods, we can, for instance, give significance to the number 30,525: It is the number of summands of Z/2Z in the 72nd homology group of the Grassmannian of 12-planes in 24-dimensional space. I will show some neat visualizations of the (co)chain complex and discuss some observations and their implications.