Emily Dumas
Fall 2024
Course number | Math 569 (CRN 38308) |
---|---|
Meetings | MWF 12:00pm in Taft 316 |
(Zoom meeting info on Blackboard) | |
Required texts | None |
Optional texts | See full list below; no purchases are recommended. |
Instructor office hours | Mon 1pm and Fri 10am in SEO 722 |
This is a graduate topics course on the generalized flag varieties associated to semisimple complex or real Lie groups. While these objects are often approached from the perspective of algebraic geometry, in this course we will mainly focus on topological, differential-geometric, and dynamical aspects.
The course will begin with a discussion of the classical partial and full flag varieties as generalizations of projective spaces and Grassmannian manifolds. We will then introduce the generalized flag varieties associated to semisimple Lie groups. We will study smooth and complex structures on these compact manifolds, describe their topology in terms of a natural CW structure (the Schubert cells), and compute topological invariants such as the homology groups and cohomology ring.
Next we will place flag varieties into a larger context by relating them to boundaries (or parts of boundaries) of symmetric spaces of noncompact type. Here a central theme will be the interplay between the G-invariant geometries of these two spaces: The Riemannian geometry of the nonpositively curved, complete symmetric space and the non-Riemannian, higher-order geometric structure of the flag variety.
Finally we will turn to studying dynamics on flag varieties, first for individual automorphisms and then for discrete groups of automorphisms. We will emphasize the concept of proximality and the way it gives rise to a well-behaved notion of "limit set". We will introduce the notion of Anosov subgroups, which are an important and dynamically well-behaved class of discrete subgroups of Lie groups introduced by Francois Labourie^{1, 2}. Finally we will discuss a characterization of the Anosov property in terms of dynamics on the flag variety that was established by Kapovich, Leeb, and Porti^{3,4}.
The lecture video recordings and zoom meeting details are only available through UIC's Blackboard Learn LMS (netid login required).
These contain a lot of material about flag varieties.
Books that may be mentioned or cited at some point in the course.