Applying motivic filtrations

经典代数拓扑学借助于代数不变量来研究拓扑流形和拓扑空间的分类。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律类型的拓扑空间。所有谱的集合形成稳定同伦范畴，这一观点在最初的分类问题上取得了令人惊讶的进展。相对于代数拓扑学，代数几何学处理的是相当不灵活的代数流形，局部地作为多项式方程的解给出。在20世纪90年代，法比安莫雷尔和弗拉基米尔Voevodsky扩大了拓扑方法通过广义上同调理论的设置代数几何，开始了大量的新的研究在这一方向。这项研究提供了一个框架，许多非常有趣的代数流形上的上同调理论的建设和研究，现在被称为motivic稳定同伦范畴在一个领域或更普遍的基础计划。为了研究这些代数不变量，一个技术工具，使用过滤，是必不可少的。在拓扑设置中，许多不同的过滤或塔的使用已经被使用，并且相互抵消，以便对基本研究对象获得更好的理论理解，以及进行具体计算。动机稳定同伦范畴产生了许多这样的过滤，一些直接概括的拓扑，其他相当新，其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其中的几个动机过滤，并利用它们的属性来实现具体的结果。特别重要的例子是Voevodsky的切片过滤和过滤的连接。这些filtrations和他们之间的关系应该有助于理解代数K-理论，合作模p motivic上同调，同伦层的球谱，以及motivic方向，概括的经典托德属。

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.