Motivic Cohomology, Motivic Homotopy Theory and K-theory

动机上同调、动机同伦理论和 K 理论

• 批准号: 2001417
• 负责人: Charles Weibel
• 金额: $ 38.95万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-08-01 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2001417&HistoricalAwards=false
• 关键词: Motivic; Cohomology; Homotopy; Theory; theory

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.

该奖项支持首席研究员对代数、代数几何和拓扑之间联系的研究。目标是更好地理解代数几何中的结构是如何通过拓扑结构来反映的，使用代数和拓扑方法的混合。几个老的猜想将受到攻击，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清晰的证据。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的切片过滤是如何与代数共坐标和其他动力谱相关联的，并验证几个切片猜想。PI也将研究奇点之间的关系，它的k理论和它的cdh-上同调，使用最近发展的上同调技术。本课题的最后一部分是证明了代数Witt群的变体在多项式和Laurent多项式扩展方面表现良好。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Localization, monoid sets and K-theory

定位、幺半群集和 K 理论

• DOI: 10.1016/j.jalgebra.2022.09.009
• 发表时间: 2023
• 期刊: Journal of Algebra
• 影响因子: 0.9
• 作者: Coley, Ian;Weibel, Charles
• 通讯作者: Weibel, Charles

Persistent homology with non-contractible preimages

与不可收缩原像的持久同源性

• DOI: 10.4310/hha.2022.v24.n2.a16
• 发表时间: 2022
• 期刊: Homotopy and Applications
• 作者: Mischaikow, Konstantin;Weibel, Charles
• 通讯作者: Weibel, Charles

The $K’$–theory of monoid sets

幺半群集合的 $K–$– 理论

• DOI: 10.1090/proc/15517
• 发表时间: 2021
• 期刊: Proceedings of the American Mathematical Society
• 影响因子: 1
• 作者: Haesemeyer, Christian;Weibel, Charles A.
• 通讯作者: Weibel, Charles A.

Grothendieck–Witt groups of some singular schemes

一些奇异方案的 Grothendieck-Witt 群

• 发表时间: 2020
• 期刊: Proceedings of the London Mathematical Society
• 影响因子: 1.8
• 作者: M. Karoubi;M. Schlichting;C. Weibel
• 通讯作者: C. Weibel