Flag Varieties: Geometry, Topology, and Dynamics

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

Course description

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.

Documents

• Course syllabus
• Lecture 1 (Aug 26) - Projective space, Grassmannian, Plücker embedding, flag variety
• Lecture 2 (Aug 28) - Tangent spaces, full and partial flags, model by G/P
• Lecture 3 (Aug 30) - Flag stabilizers; dimension, compactness, connectedness of G/P
• Lecture 4 (Sep 4) - Fibrations, fundamental group
• Lecture 5 (Sep 6) - Maps between flag varieties; simplicity and semisimplicity
• Lecture 6 (Sep 9) - Roots of sl_nC
• Lecture 7 (Sep 11) - Cartan subalgebras, roots of a semisimple Lie algebra
• Lecture 8 (Sep 13) - Dynkin diagrams and the classification
• Lecture 9 (Sep 16) - Borel and parabolic subalgebras
• Lecture 10 (Sep 18) - Complex semisimple Lie groups, normalizers
• Lecture 11 (Sep 20) - Parabolics and general flag varieties
• Lecture 12 (Sep 23) - Real forms and compactness of G/P
• Lecture 13 (Sep 25) - G/P and G/K for SL₂C; Statement of Bruhat decomposition
• Lecture 14 (Sep 27) - Proof of Bruhat decomposition; relation to Gaussian elimination
• Lecture 15 (Sep 30) - Schubert cells and varieties
• Lecture 16 (Oct 2) - Bott-Samelson resolution, CW structure of G/B, length and Bruhat order on W
• Lecture 17 (Oct 4) - Bruhat order and Schubert cells for SL₃C
• Lecture 18 (Oct 7) - Bruhat order and Schubert cells for SL_nC
• Lecture 19 (Oct 9) - More homology calculations (Sp(4,C) and grassmannians)
• Lecture 20 (Oct 11) - Cohomology of G/P: Fundamental classes of Schubert varieties
• Lecture 21 (Oct 14) - Cohomology of G/P: Intersection pairing

Blackboard

The lecture video recordings and zoom meeting details are only available through UIC's Blackboard Learn LMS (netid login required).

Reference Materials

Books — Main

These contain a lot of material about flag varieties.

[BE1989]

Baston, Robert J. and Eastwood, Michael G. (1989). The Penrose Transform: Its Interaction with Representation Theory. Oxford Mathematical Monographs, Clarendon Press. ISBN-13: 978-0198535652

[Brion2005]

Brion, Michel (2005). Lectures on the Geometry of Flag Varieties. In: Pragacz, P. (eds) Topics in Cohomological Studies of Algebraic Varieties. Trends in Mathematics. Birkhäuser. doi:10.1007/3-7643-7342-3_2

[LB2018]

Lakshmibai, V. and Brown, Justin (2018). Flag Varieties. Texts and Readings in Mathematics, vol 53. Springer. doi:10.1007/978-981-13-1393-6

[OV1994]

Onishchik, A. and Vinberg, E. (1994). Lie Groups and Lie Algebras III: Structure of Lie Groups and Lie Algebras. Encyclopaedia of Mathematical Sciences, Springer. ISBN-13: 978-3540546832

Books — Other

Books that may be mentioned or cited at some point in the course.

[BB2005]

Bjorner, Anders and Brenti, Francesco (2005). Combinatorics of Coxeter Groups. Graduate Texts in Mathematics, Springer. doi:10.1007/3-540-27596-7

[GH1978]

Griffiths, Philip and Harris, Joseph (1978). Principles of Algebraic Geometry. Wiley. doi:10.1002/9781118032527

[Knapp13]

Knapp, Anthony W. (2013). Lie Groups Beyond an Introduction. Progress in Mathematics, Birkhäser. doi:10.1007/978-1-4757-2453-0

[Lee2012]

Lee, John M. (2012). Introduction to Smooth Manifolds. Graduate Texts in Mathematics, Springer. doi:10.1007/978-1-4419-9982-5

[Serre2001]

Serre, Jean-Pierre (2001). Complex Semisimple Lie Algebras. Springer Monographs in Mathematics. doi:10.1007/978-3-642-56884-8

Papers

Coming “soon”

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