Alexander Vishik--Isotropic Motives

Algebraic Geometry is a more complicated version of Topology. The linearisation of algebraic varieties and topological spaces is given by motives. In topology, the motive of a topological space is its chain complex which encodes homology and cohomology of the space. In Algebraic Geometry, the respective category of motives DM(k) was constructed by Voevodsky. The motive of a variety keeps the homological information of it. But the category of Voevodsky is rather large and complex, and although we have the topological realization functor from DM(k) to its (much simpler) topological counterpart D(Ab), it is far from conservative. Isotropic realisations permit to supplement this functor with many similar functors of a comparable level of complexity which should provide the points of the Balmer’s tenzor triangulated spectrum of DM(k). I will discuss some features of the respective category of isotropic motives. Among other things, numerical equivalence of cycles (with finite coefficients) and Milnor’s operations naturally appear in this picture.