Motivic Cohomology, Motivic Homotopy Theory and K-theory

该奖项支持首席研究员对代数，代数几何和拓扑之间的联系的研究。目标是更好地理解代数几何中的结构如何被拓扑结构所反映，使用代数和拓扑方法的混合。几个古老的堡垒将受到攻击，并有望得到解决。此外，该项目还为研究生提供了研究培训的机会。该项目的一部分是为最近验证的Bloch-Kato猜想的部分证明找到干净的证明。该项目的一个相关部分涉及动机同伦理论;一个目标是了解Voevodsky的切片过滤与代数配边和其他动机谱的关系，并验证几个切片结构。PI还将使用最近开发的上同调技术研究一个品种的奇点之间的关系，其K-理论和cdh-上同调。该项目的最后一部分是要表明，代数维特群的品种表现良好，就多项式和洛朗多项式extensions.This奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

This award supports the principal investigator's research on the connections between Algebra, Algebraic Geometry, and Topology. The goal is to gain a better understanding of the way that structures in algebraic geometry are reflected by structures in topology, using a blend of algebraic and topological methods. Several old conjectures will be attacked, and hopefully settled. In addition the project provides research training opportunities for graduate students.One part of the project is to find clean proofs of parts of the proof of the recently-verified Bloch-Kato conjecture. A related part of the project involves motivic homotopy theory; one goal is to understand how Voevodsky's slice filtration is related to algebraic cobordism and other motivic spectra and to verify several of the slice conjectures. The PI will also study the relationship between the singularities of a variety, its K-theory and its cdh-cohomology, using recently developed cohomological techniques. A final part of this project is to show that the algebraic Witt group of varieties behaves well with respect to polynomial and Laurent polynomial extensions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.