Alexander Vishik (Nottingham): Isotropic motives
I will discuss the "isotropic motivic category". This "local" version of Voevodsky motivic category (with finite coefficients), obtained from the "global" one by, roughly speaking, annihilating the motives of anisotropic varieties, has many remarkable properties. Considering such "local" versions for all finitely generated extensions of a ground field, permits to read global information in a rather simple form. For appropriate (so-called, "flexible") fields, "isotropic motives" are more reminiscent of their topological counterparts. In particular, "isotropic Chow groups" hypothetically coincide with Chow groups modulo numerical equivalence (with finite coefficients) and so should be finite-dimensional (checked in various cases). On the other hand, the "isotropic motivic cohomology" ring of a point doesn't depend on a field and encodes Milnor's operations.