Arithmetic Study Group: Motives: A colloquial introduction
29 November 2007 14:15 in E102
In this talk, we present a bird's-eye view of the theory of motives. We begin with an overview of the theory of pure motives based on correspondences on algebraic cycles (as envisioned by Grothendieck in the 1960s in-order to prove the so-called "standard conjectures".). We then introduce mixed Hodge structures and using the definition of pure (Tate) motives and mixed Hodge structures over the rationals, we explain what mixed Tate motives are, and list the desirable properties of the conjectural abelian category of mixed motives of which mixed Tate motives are a subcategory. (All through this we treat Voevodsky's construction of a derived triangulated category of mixed motives as a "black-box"- in a later talk, we will return to Voevodsky's theory.) We finish by mentioning a few applications of mixed Tate motives to questions about special values of zeta and multizeta functions, especially with a teaser on the recent work of Bloch-Esnault-Kreimer.