Algebraic K-theory and Motivic Cohomology

韦贝尔建议在代数K理论和动机理论的相关领域进行研究。首先，他建议完成一个定理的证明，该定理描述了K-理论和动机上同调与代数几何和数论等相关领域的关系。该项目的第二部分是将新的动机技术应用于处理奇点的代数几何中长期存在的问题。项目的第三部分将研究动机的内部结构，试图解释作为代数变量的经典不变量的周群的行为。具体地说，魏贝尔提出了三个研究方向。他将试图证明罗斯特变种的存在，从而完成对布洛赫-加藤猜想的验证。这个猜想将K-理论与数论和上同调联系起来。他还将使用最近发展的上同调技术，研究簇的奇点之间的关系，它的K-理论和它的cdh-上同调。这涉及到关于整环的K-理论的几个经典问题，在过去的35年里，特别是在最近的十年里，K-理论一直是K-理论研究的焦点。此外，他还将研究动机的结构，特别是切片过滤的结构，以此来破解关于周氏集团和更高周氏集团消失的猜测。

Weibel proposes to do research in the related fields of algebraic K-theory and motivic theory. First, he proposes to complete the proof of a theorem which describes the relationship that K-theory and motivic cohomology have with the related fields of algebraic geometry and number theory. A second part of the project is to apply new motivic techniques to long-standing problems in algebraic geometry dealing with singularities. The third part of the project will be to study the internal structure of motives in an attempt to explain the behavior of Chow groups, which are classical invariants of algebraic varieties.Specifically, Weibel proposes three strands of research. He will try to establish the existence of Rost varieties, completing the verification of the Bloch-Kato Conjecture. This conjecture links K-theory to number theory and etale cohomology. He will also study the relationship between the singularities of a variety, its K-theory and its cdh-cohomology, using recently developed cohomological techniques. This involves several classical questions about the K-theory of rings of integers in global fields which have been a focal point of research in K-theory for the last 35 years, especially in the last decade. In addition, he will study the structure of Motives, especially the structure of slice filtration, as a way of attacking conjectures about vanishing of Chow groups and higher Chow groups.