Isotropic motives and affine quadrics

这个建议的主题领域是新的motivic不变量的代数品种介绍的首席研究员。这些方法增强和扩展了与V.Voevodsky（Fields Medal 2002），M.Levine，A.Merkurjev，F.Morel，M.Rost，A.Suslin的名字相关的代数的突破性发展。这些领域使用不同的工具集，它们之间的任何联系都允许从不同的角度研究同一个数学对象。拓扑是几何的“灵活版本”，最简单的拓扑对象是点。在代数几何中也有点，但这些点取决于场的选择，因此形状上有很大的不同。这是代数几何世界丰富性的源泉。事实上，正如Morel-Voevodsky的惊人工作所证明的那样，拓扑世界只是它的“玩具版”，拓扑对象的基本信息包含在它的同调中，而代数簇的同调信息则编码在它的动机中。这种动机的理论是由Voevodsky发展的。主要研究者介绍了新的方法来研究动机的基础上，所谓的“各向同性实现函子”，它分配给一个动机的家庭，它的“阴影”。这些“阴影”由基场的扩张（或者，如果你愿意，所有的代数几何点）参数化，并且在复杂性上类似于“拓扑动机”（奇异复形）。因此，一个复杂的对象被一个简单的数组所取代。该建议的主要目的是研究这些函子，并将其扩展到一个完整的motivic不变量。建立与有限有理系数代数圈的数值等价的联系。应用于二次曲面和Voevodsky范畴的Picard群的不变量的计算，以及Rost Nilerian猜想和代数圈上的标准猜想。该提案的第二个目的是将这些不变量推广到同伦上下文中，并研究经典拓扑对象的“各向同性”版本。该研究将在诺丁汉大学数学科学学院进行。

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.