title: Three lectures on the geometry of generalized flag varieties
Speaker: Prof. William Haboush (University of Illinois at Urbana-Champaign)
I. TITS SYSTEMS AND THE BRUHAT DECOMPOSITION
-16:00-18:00, August 11 2014
After a brief review of root systems with particular attention to the exchange condition, I will discuss Tits systems and the Bruhat decomposition. After a short discussion of the structure of parabolic groups and a small result on certain pairs of parabolic subgroups I will describe the Chevalley-Bruhat ordering. I will end with some results concerning intersections of Schubert cells.
II. THE COHOMOLOGY OF LINE BUNDLES ON G/B
-16:00-18:00, August 12 2014
I will begin with Demazure’s proof of Bott’s theorem and his proof of the Weyl character formula. I will discuss the fail- ure of Bott vanishing in positive characteristic and I will explain how Kempf vanishing establishes the result in positive characteris- tic. I will discuss proofs of Kempf vanishing in positive character- istic. I will discuss the Steinberg tensor product theorem and the Steinberg module and give a proof of Kempf vanishing in positve characteristic.
Ⅲ. FROBENIUS SPLITTING: THE WORK OF MEHTA AND RAMANATHAN
-10:00-12:00, August 13 2014
I will define Frobenius splitting and give criteria for splitting. I will prove that Schubert cells are compatibly split and I will describe the geometric consequences of this fact.
Place: Room 404, Math Science bldg.
Date: August 11-13 2014