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.