Title: "Regularity, K-regularity and negative K-theory"

Abstract:

 We report on recent progress on two conjectures from the 1970's
 in K-theory.

 The conjecture of Weibel states that for a noetherian ring
 of dimension d, the K-theory vanishes below degree -d.
 The conjecture of Vorst states that if a noetherian ring of dimension d is
 K_{d+1}-regular (i.e. K_{d+1} (R) agrees with  K_{d+1}(R[t_1,...,t_s])
 for all ?s), then R is regular.

 Recently, both conjectures have been proven over a field
 of characteristic 0. We discuss the situation in
 characteristic p.