Isotropic motives and affine quadrics

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The subject area of this proposal is the new motivic invariants of algebraic varieties introduced by the principal investigator. These methods enhance and extend the groundbreaking development in algebra associated with the names of V.Voevodsky (Fields Medal 2002), M.Levine, A.Merkurjev, F.Morel, M.Rost, A.Suslin.The proposed research lies at the boundary of algebraic geometry, topology and algebra. These areas use different sets of tools, and any connection between them permits to study the same mathematical object from different perspectives. Topology is a "flexible version" of geometry, and the simplest topological object is a point. In algebraic geometry there are also points, but these depend on the choice of a field, and so, substantially vary in shape. This is the source of richness of the algebro-geometric world. In fact, as was demonstrated by the spectacular work of Morel-Voevodsky, the topological world is just a "toy version" of it. The basic information on a topological object is contained in its homology, while the homological information on an algebraic variety is encoded in its motive. The theory of such motives was developed by Voevodsky.The principal investigator has introduced the new approach to the study of motives based on the, so called, "isotropic realization functors", which assign to a motive a family of its "shadows". These "shadows" are parameterized by the extensions of the ground field (or, all the algebro-geometric points, if you want) and are similar in complexity to "topological motives" (singular complexes). Thus, a complicated object is substituted by an array of simple ones.The principal aim of the proposal is to study these functors and extend them to a complete motivic invariant. To establish the connection to the numerical equivalence of algebraic cycles with finite and rational coefficients. To apply them to the computation of invariants of quadrics and the Picard group of the Voevodsky category, as well as to the Rost Nilpotence Conjecture and the Standard Conjectures on algebraic cycles. The second aim of the proposal is to generalize these invariants to the homotopic context, and study "isotropic" versions of classical topological objects.The research will be undertaken at the School of Mathematical Sciences, University of Nottingham.