Mathematical Sciences: K-Theory & Motivic Cohomology

首席研究员将继续他的动机范畴和其他相关结构的研究。他将研究整体动机上同调与Bloch的高周群之间的关系，构建动机范畴的权结构和对偶性，并将动机Tate范畴与Bloch和Kriz构建的动机李代数联系起来。他打算将霍奇理论中的各种构念提升到理据设定上，研究理据上同的模式。主要研究者还将利用Hoobler的一些新思想来攻击Kato的猜想，并给出一个权值为2的上同调的候选构造。本研究涉及代数k理论。从广义上讲，代数k理论涉及线性代数概念的演变，如基和向量空间。这项工作对数论和代数几何具有重要意义，并有望在数学的许多不同领域之间建立令人兴奋的联系。＊＊＊

Levine The principal investigator will continue his study of the motivic category and other related constructions. He will investigate the relationship between the integral motivic cohomology and Bloch's higher Chow groups, to construct a weight structure and a duality on the motivic category, and to relate the motivic Tate category with the motivic Lie algebra constructed by Bloch and Kriz. He plans to lift various constructions in Hodge theory to the motivic setting and to study mod n motivic cohomology. The principal investigator will also use some new ideas of Hoobler to attack Kato's conjecture and will give a construction of a candidate for weight two cohomology. This research is concerned with algebraic K-theory. In a broad sense algebraic K-theory concerns the evolution of concepts from linear algebra such as basis and vector space. This work has significant implications for number theory and algebraic geometry, and promises to make exciting connections between a number of different areas in mathematics. ***