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

- 1. Introduction, Polar Decomposition, and Frames
- 2. Grassmann and Stiefel manifolds as quotients, tangent spaces, dimension
- 3. Lie groups, Lie Algebras, the Matrix Exponential and Geodesics
- 4. More on Lie Groups, Geodesics on Grassmann and Stiefel manifolds
- 5. Geodesics and the Matrix Exponential (Practical Demo)
- 6. Geodesics in (nxk) coordinates on the Stiefel Manifold
- 7. Grassmannian geodesics in (nxk) coordinates
- 8. Distances and Angles between Subspaces
- Edelman, Arias, Smith,
*The Geometry of Algorithms with Orthogonality Constraints* - Paige, Wei,
*History and Generality of the CS Decomposition* - Sutton,
*Computing the Complete CS Decomposition* - James, Wilkinson,
*Factorization of the Residual Operator and Canonical Decomposition of Nonorthogonal Factors in the Analysis of Variance*

- Edelman, Arias, Smith,

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

- 9. Fibre Bundle Structure and Homotopy Groups
- 10. Homotopy and Homology Groups
- 11. Review of Morse Theory
- 12. A User’s Guide to Morse-Bott Theory
- 13. Morse-Bott Theory on Complex Grassmannians
- 14. Schubert Cells and the Schubert Calculus
- 15. Schubert Cells and the Schubert Calculus II
- 16. The Plucker embedding and Plucker Relations I
- 17. The Plucker embedding and Plucker Relations II
- 18. The Plucker embedding and Plucker Relations III: Plucker Relations for Polygons
- 19. Schubert Relations and Subspace Intersections I
- 20. The Cohomology Ring of (Complex) Grassmannians and Schubert Cycles
- 21. The Determinental Relation, Pieri’s Formula, and Explicit Schubert Calculus Calculations
- 22. Infinite-Dimensional Grassmannians and Curve Spaces

Part 3. Numerical Applications

- 23. Blind Source Separation I
- 23. Blind Source Separation II
- Cruces-Alvarez, Cichocki, Amari,
*From Blind Signal Extraction to Blind Instantaneous Signal Separation: Criteria, Algorithms, and Stability* - Paninski,
*Estimation of Entropy and Mutual Information* - Kraskov, Stogbauer, Grassberger,
*Estimating Mutual Information* - Hyvarinen, Oja
*Independent Component Analysis: Algorithms and Applications* - Bell, Sejnowski
*An Information Maximization Approach to Blind Separation and Blind Deconvolution* - Ikeda,
*Audio Examples of Blind Source Separation*(link) - Wikipedia,
*Kullback-Leibler Divergence*

- Cruces-Alvarez, Cichocki, Amari,
- 24. Circular Statistics I
- 25. Circular Statistics II – Asymptotics and Testing
- 26. Grassmann and Stiefel Statistics

## Other course materials

## Details

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