I will talk on "Frobenius splittings and Vanishing Theorems." The basic idea is the following: let X be a variety over a finite field, and F: X --> X the Frobenius endomorphism. A splitting of the natural map O_X --> F_* O_X induced by F, if it exists, is called a Frobenius splitting.
Varieties with Frobenius splittings are special (in a sense to be made at least somewhat precise at the talk) -- examples include toric varieties and Schubert varieties (over finite fields). Using Frobenius splittings in char p, one can prove strong vanishing theorems for toric varieties and Schubert varieties over arbitrary fields -- for instance, if L is an ample line bundle on a toric variety or Schubert variety, then H^i (X,L) = 0 for i > 0.