This course discusses two families of classical manifolds which have a central place in mathematics: the Grassmann manifold of k-planes in \mathbb{F}^n and the Stiefel manifold of orthonormal k-frames in \mathbb{F}^n. Here, the underlying field may be real, complex, or even quaternionic: we’ll see that each choice adds interesting properties to the manifolds.

There is so much known about these manifolds, and from so many different perspectives, that the really hard part about designing the course is figuring out what to leave out. My ambition (and it may prove to be too much!) is to restrict our attention to three basic perspectives: topological, in which these are classical examples of principal bundles, geometric, in which they are studied by the Schubert calculus (in algebraic geometry) and as homogeneous spaces (in Riemannian geometry), and numerical, in which they are seen as ambient spaces for methods in numerical linear algebra and statistics. The hope is that by intertwining these various views of the manifolds, we will be able to make some new observations in each field!

Course Notes

I intend to scan and post my lecture notes before class and encourage you to read them over in advance. I also encourage you to try to read some of the source material that’s posted on the webpage. ”Courses like this are hard, and you shouldn’t expect to get everything out of the lectures.”

Part 1. Matrices and Differential Geometry

Part 2. Topology, Homotopy, and Cohomology Groups, Schubert Calculus

Part 3. Numerical Applications

Other course materials

Course syllabus

Details

The course meets 1:20-2:15 in Boyd 326 on Mondays, Wednesdays, and Fridays.