Motivic Cohomology and K-Theory

这项研究项目将试图解决关于K-理论的结构和动机的旧猜想，并澄清最近发现的代数、代数几何和代数拓扑不同领域之间的关系。代数簇的动机建立在几何关系上，并被设计为上同调不变量的通用主题(因此而得名)，因此它的结构揭示了几何的各个方面。相反，环或簇的K-理论是由涉及线性代数的关系使用代数拓扑建立的，因此它的结构揭示了代数和拓扑的各个方面。澄清这些结构之间的关系将有助于我们理解许多目前知之甚少的现象。该项目的一个部分是为最近证实的Bloch-Kato猜想的主要证明找到干净的证据。这包括尝试：根据代数余边理论，找到一个结合了Rost的各种次数公式的通用度公式；在正规簇的对称幂的模型之间建立等价；以及更好地理解Motivic上同调运算。这些都在证明中发挥了作用。该项目的这一部分将对培训世界各地的年轻研究人员具有不可估量的价值。该项目的一个相关部分是了解沃沃茨基的切片过滤是如何与代数余弦论和其他动机谱相关的。其目的是利用同调代数和代数Thom空间来验证几个切片猜想，特别是在有限特征下。这其中的一部分将是与研究生共同努力，为他们的培训做出贡献。该项目还将利用最近发展的上同调技术，研究一个簇的奇点之间的关系，它的K-理论和它的cdh-上同调，以及与循环同调和de Rham-Witt理论的关系。这应该会对与杜波依斯奇点有关的奇点类别有一些启发。本项目的最后部分是建立数域上动机的阿基米德上同调，并将其与Serre的L函数的局部因素联系起来。这推广了Connes和Consani的工作，并对方案使用了循环同调。

The research project will try to settle old conjectures about the structure of K-theory and Motives, and clarify recently discovered relationships between disparate areas of algebra, algebraic geometry, and algebraic topology. The motive of an algebraic variety is built from geometric relationships, and is designed to be the universal motif for cohomological invariants (hence the name), so its structure reveals aspects of geometry. In contrast, the K-theory of a ring or variety is built from relationships involving linear algebra using algebraic topology, so its structure reveals aspects of algebra and topology. Clarifying the relationships between these constructs will help our understanding of many phenomena that are currently understand only poorly.One part of the project is to find clean proofs of major pieces of the proof of the recently-verified Bloch-Kato conjecture. This includes trying to: find a general degree formula which combines various degree formulas due to Rost, in terms of algebraic cobordism theory; establish an equivalence between models of the symmetric powers of a normal variety; and understand motivic cohomology operations better. These all play a role in the proof. This part of the project will be invaluable in training young researchers around the world. A related part of the project is to understand how Voevodsky's slice filtration is related to algebraic cobordism and other motivic spectra. The goal is to verify several of the slice conjectures, especially in finite characteristic, using homological algebra and algebraic Thom spaces. Part of this will be joint work with graduate students, contributing to their training. The project will also study the relationship between the singularities of a variety, its K-theory and its cdh-cohomology, using recently developed cohomological techniques and the relationship to cyclic homology and de Rham-Witt theory. This should shed some light on classes of singularities related to du Bois singularities. The final part of the project is to formulate an archimedean cohomology for a motive over a number field and relate it to Serre's local factors for the L-functions of the motive. This generalizes work of Connes and Consani, and uses cyclic homology for schemes.