Geometry and topology of homogeneous spaces

Time/Place: MWF 10:30am--11:20am, C-401 Padelford.

The plan of the course is roughly as follows:

1. Semisimple algebraic groups.
2. Equivariant cohomology, especially of flag varieties.
3. Topics on Schubert varieties: toric degenerations, singularities, explicit realizations.

References

There is no single book covering everything we need. I plan to supply some notes as the course goes, but for now here are some references by topic.

For algebraic groups and root systems:

• My notes for a crash course on linear algebraic groups, in this PDF.
• A. Borel, Linear Algebraic Groups, Springer.
• J. E. Humphreys, Linear Algebraic Groups, Springer.
• T. A. Springer, Linear Algebraic Groups, Birkhauser.
• N. Bourbaki, Lie Groups and Lie Algebras, Ch IV-VI.
• J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer.
• J. E. Humphreys, Reflection groups and Coxeter groups, Cambridge.

For equivariant cohomology:

• My lectures from IMPANGA, in this PDF.
• Notes from Bill Fulton's lectures at Columbia, here.
• Steve Mitchell's course.

For Schubert calculus and geometry of flag varieties:

• W. Fulton, Young tableaux, Cambridge.
• S. Kumar, Kac-Moody groups, their flag varieties and representation theory, Birkhauser.
• M. Brion, "Lectures on the geometry of flag varieties", arXiv.