John Rognes: Motivic complexes and the rank filtration

The advances on the Milnor- and Bloch-Kato conjectures have led to a good  understanding of motivic cohomology and algebraic K-theory with finite  coefficients.  However, important questions remain about rational motivic  cohomology and algebraic K-theory, including the Beilinson-Soulé vanishing  conjecture.  We discuss how the speaker's "connectivity conjecture" for  the stable rank filtration of algebraic K-theory leads to the construction  of chain complexes whose cohomology groups may compute rational motivic  cohomology, and simultaneously satisfy the vanishing conjecture.  These  "rank complexes" serve a similar purpose as Goncharov's candidates for  motivic complexes, but have the advantage that they have a precise  relation to rational algebraic K-theory.