|Location:||MSRI: Simons Auditorium|
Grothendieck's theory of (homological) motives is grounded on the conjectural existence of certain algebraic cycles, among others the Künneth projectors: This is the standard conjecture of Künneth type. The standard sign conjecture is a weakening of this, and is strong enough to imply that the homological motives form a Tannakian category.
After reviewing the conjecture, its origin and its consequences, we sketch some of the ideas involved in our proof of the sign conjecture in the case of certain Shimura varieties. This is joint work with Sophie Morel.