$A^1$ Homotopy Theory and Its Applications
$A^1$-homotopy theory is the homotopy theory for algebraic varieties and more generally for schemes which is based on the analogy between the affine line and the unit interval. During this year we will concentrate on two topics. One is the extension of the existing theory of triangulated motives from varieties over fields to general schemes. The main remaining problem there can be reformulated in terms of the $A^1$-homotopy theory as the problem of finding a good recognition principle for T-loop spaces. Another one is ordering and reevaluation of many concrete computations obtained for the proof of the Milnor and Bloch-Kato conjectures.
Better understanding of both of these topics seems to be necessary for the future construction of the theory of motivic homotopy type.