Cohomology theories for algebraic varieties

Abstract

In 1949 Andre Weil proposed a series of conjectures about varieties over finite fields and showed that they would follow from the existence of a ‘good’ cohomology theory. Twenty years later Alexander Grothendieck constructed such a ‘good’ cohomology theory and proved all but one of Weil's conjectures. However, in doing so he constructed infinitely many and so began our embarrassment of riches. Today we have even more cohomology theories and for fifty years one of the central problems of arithmetic geometry has been to try to understand how they are related and, just maybe, to find one to rule them all. In this talk I will discuss these cohomology theories, how they are related and recent progress towards their unification.