Applying motivic filtrations

经典代数拓扑学借助于代数不变量对拓扑流形和空间进行分类。特别有趣的是可表示的代数不变量，也被称为上同调理论，其表示对象(称为谱)可以被视为具有特别好的加法定律的拓扑空间。所有谱的集合形成了稳定的同伦范畴，这一观点在原来的分类问题上取得了相当惊人的进展。与代数拓扑学不同的是，代数几何处理的是相当不灵活的代数流形，局部给出的是多项式方程的解。20世纪90年代，Fabien Morel和Vladimir Voevodsky通过广义上同调理论将拓扑学方法扩展到代数几何的背景下，开启了这一方向的大量新研究。这一研究为代数流形上许多非常有趣的上同调理论的构建和研究提供了一个框架，现在被称为域上或更广泛地说在基模式上的Motivic稳定同伦范畴。对于这些代数不变量的研究，一个技术工具--过滤的使用是必不可少的。在拓扑环境中，为了更好地从理论上理解基本的研究对象，以及进行具体的计算，使用了许多不同的过滤器或塔，并相互竞争。动机稳定同伦范畴已经产生了许多这样的过滤，一些是拓扑学的直接推广，另一些是相当新的，它们的研究在过去十年里带来了令人震惊的结果。这个项目的目标是研究几个这样的动机过滤，并利用它们的特性来实现具体的结果。特别重要的研究例子是Voevodsky的切片过滤和连通性过滤。这些滤子及其相互之间的关系应该有助于理解代数K-理论、模动机上同调的合作、球谱的同伦层以及推广经典Todd亏格的动机定向。

Classical algebraic topology pursues the classification of topological manifolds and spaces with the aid of algebraic invariants. Especially interesting ones are the representable algebraic invariants, also known as cohomology theories, whose representing object (known as a spectrum) may be seen as a topological space endowed with a particularly nice type of addition law. The collection of all spectra forms the stable homotopy category and this viewpoint has led to quite amazing progress in the original classification problem. In contrast to algebraic topology, algebraic geometry deals with the considerably more inflexible algebraic manifolds, given locally as the solutions of polynomial equations. In the 1990s, Fabien Morel and Vladimir Voevodsky extended the topological approach via generalised cohomology theories to the setting of algebraic geometry, initiating a great deal of new research in this direction. This research furnished a framework for the construction and study of many very interesting cohomology theories on algebraic manifolds, now known as motivic stable homotopy category over a field or more generally over a base-scheme. For the study of these algebraic invariants, a technical tool, the use of filtrations, is essential. In the topological setting, the use of many different filtrations, or towers, have been used and played off against each other in order to gain a better theoretic understanding of basic objects of study, as well as for making concrete computations. The motivic stable homotopy category has given rise to many such filtrations, some direct generalisations of the topological ones, others quite new, whose study in the last decade has brought astounding results. The goal of this project is to study several of these motivic filtrations and to use their properties to achieve concrete results. Especially important examples for study are Voevodsky’s slice filtration and the filtrations by connectivity. These filtrations and their relation with one another should be helpful in understanding algebraic K-theory, cooperations for mod p motivic cohomology, homotopy sheaves of the sphere-spectrum, as well as motivic orientations, generalising the classical Todd genus.