Applying motivic filtrations

经典代数拓扑学借助代数不变量对拓扑流形和空间进行分类。特别有趣的是可表征代数不变量，也被称为上同调理论，它的表征对象（称为谱）可以被看作是一个具有特别好的加法律的拓扑空间。所有谱的集合构成了稳定同伦范畴，这一观点使原来的分类问题取得了相当惊人的进展。与代数拓扑相反，代数几何处理相当不灵活的代数流形，局部给出多项式方程的解。20世纪90年代，Fabien Morel和Vladimir Voevodsky通过广义上同调理论将拓扑方法扩展到代数几何的设置中，开启了这一方向的大量新研究。这一研究为构造和研究代数流形上的许多有趣的上同调理论提供了一个框架，这些理论现在被称为域上或更一般地称为基方案上的动机稳定同调范畴。对于这些代数不变量的研究，一个技术工具，使用过滤，是必不可少的。在拓扑设置中，为了更好地从理论上理解研究的基本对象，以及进行具体计算，使用了许多不同的过滤或塔，并相互比较。动机稳定同伦范畴已经产生了许多这样的过滤，一些是拓扑过滤的直接推广，另一些是相当新的过滤，它们的研究在过去十年中带来了惊人的结果。这个项目的目标是研究几种这些动机过滤，并利用它们的性质来实现具体的结果。特别值得研究的例子是Voevodsky的切片滤波和连通性滤波。这些过滤及其相互之间的关系有助于理解代数k理论、模p动机上同调的合作、球谱的同伦束以及动机取向，推广经典Todd格。

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.