Quadratic Forms and Algebraic Cobordism

该提案的主题领域是现代理论的二次形式使用新的几何方法建议的主要研究者。这些方法的基础上，并进一步扩大了新的战略重要发展，代数和代数几何有关的名称V。Rost，A.Merkurjev，A.作者声明：A.该方案的研究处于代数、代数几何和拓扑学的边界。这些领域中的每一个都有自己的世界，由特定于它的定律支配，但以一种特殊的方式相互作用;代数抓住了数学对象的最基本的本质，代数几何将其翻译成人们可以使用几何推理的语言，拓扑学提供了几何对象的粗略阴影，精确的形状丢失了，但最重要的“拓扑”不变量被保留了下来。在某种意义上，拓扑只是更丰富的代数几何拓扑的一个“玩具模型”。人们在拓扑学中遇到的所有现象也出现在代数几何世界中，但以无比多变的形式出现。最简单的对象的拓扑是一个点，其中的帮助下，暂停操作之一产生其衍生物-领域。计算这些对象之间的态射的问题，即所谓的球面同伦群，是拓扑学的中心问题，也是整个数学的主要问题之一。尽管多次尝试，这个问题仍然无法解决。在代数几何中，也有一个点，但现在它依赖于一个基场，还有一个球，但现在它们有两个维度--“圆”和"方“--因为有两种不同的悬挂运算。特别是，同伦群现在不是由一个自然数参数化，而是由两个。这些群给出拓扑对应物作为某种退化，但除此之外它们要丰富得多。这些群以压缩的形式存储关于拓扑世界和代数几何世界之间差异的信息。而其中的一个中心可以用二次型来描述（由F.Morel的结果）。因此，研究二次型，我们在现实中，研究同伦群的领域，并在这里得到的结果可以适用于中心问题的拓扑，因为这两个（二次型和拓扑同伦群）相互作用的一部分，同一个对象。本文的主要目的是研究二次型的不变量（包括PI引入的新不变量和经典不变量）。第二个目的是研究代数配边中的新的上同调运算（与第一个目的有关，并且是独立的）。第三个目标是发展关于二次型理论的新同伦理论观点（从而扩展Morel-Voevodsky的理论）。这项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 modern theory of quadratic forms using new geometric methods suggested by the principal investigator. These methods are based on and further extend the new strategically important developments in algebra and algebraic geometry related to the names of V. Voevodsky (Fields Medal 2002), M. Rost, A.Merkurjev, A. Suslin, M.Levine, F. Morel.The research of the proposal lies at the boundary between algebra, algebraic geometry and topology. Each of these fields has a world of its own, governed by laws specific to it but interacting in a peculiar way; with algebra catching the most basic essence of mathematical objects, algebraic geometry translating this into the language where one can use geometric reasoning, and topology providing the rough shade of the geometric object, where exact shape is lost, but the most important ``topological'' invariants are kept. In a sense, topology is just a ``toy model'' of the much richer field algebro-geometric topology. All the phenomena one encounters in topology appear also in the algebro-geometric world, but in incomparably more variable shapes. The simplest object of topology is a point, out of which with the help of suspension operation one produces its derivatives - the spheres. The problem of computing morphisms between these objects, the so-called homotopy groups of spheres, is the central question of topology, and one of the main problems of mathematics as a whole. Despite many attempts, this problem resists breaking. In algebraic geometry one also has a point, but now it depends on a base-field, and one has spheres, but now they have two dimensions - the ``round'' and the ``square'' one - because there are two different suspension operations. In particular, the homotopy groups are now parametrized not by one natural number, but by two. These groups give the topological counterparts as a certain degeneration, but otherwise they are much richer. These groups store in a compressed form the information about the difference between topological and algebro-geometric worlds. And the central one of them can be described in terms of quadratic forms (by the result of F.Morel). Thus, studying quadratic forms we, in reality, study homotopy groups of spheres,and the results obtained here can be applied to the central question of topology, since both (quadratic forms and topological homotopy groups) interact as parts of the same object. And the properties of quadratic forms are well seen through their invariants.The principal aim of this proposal is the study of invariants of quadratic forms (both, new ones introduced by the PI, and the classical ones as well). The second aim is the study of the new cohomological operations in algebraic cobordism (in connection with the first aim, and independently). The third aim is the development of the new homotopy-theoretic viewpoint on quadratic form theory (thus, extending the theory of Morel-Voevodsky).The research will be undertaken at the School of Mathematical Sciences, University of Nottingham.