Topics in Commutative Algebra, Singularities, and Valuations

交换代数、奇点和估值主题

• 批准号: 1700046
• 负责人: Steven Cutkosky
• 金额: $ 25万
• 依托单位: University of Missouri-Columbia
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2022-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1700046&HistoricalAwards=false
• 关键词: Topics; Commutative; Algebra; Singularities; Valuations

This project will investigate foundational problems in algebraic geometry and related areas of mathematics. Algebraic geometry is the study of solutions to polynomial equations; it has many applications in other areas of mathematics, the sciences, and engineering. The project will focus on the structure of algebraic transformations over arbitrary fields and real and complex analytic transformations. The results of this research have potential for practical application to computer science.The research in this project is in the areas of commutative algebra, algebraic geometry, valuation theory, and singularity theory. The local structure of algebraic and analytic mappings will be investigated, especially with regard to ramification and good stable forms of mappings after enough blowups along a valuation or an étoile. Motivating problems are local uniformization, resolution of singularities and equisingularity in positive characteristic, and the global structure of algebraic and analytic maps.

本专题将探讨代数几何及相关数学领域的基础问题。代数几何是研究多项式方程的解的学科;它在数学、科学和工程的其他领域有许多应用。该项目将侧重于任意域上的代数变换的结构以及真实的和复杂的解析变换。本计画的研究领域包括交换代数、代数几何、赋值理论、奇异性理论等，其研究成果具有实际应用的潜力。代数映射和解析映射的局部结构将被研究，特别是关于映射的分支和好的稳定形式，在沿着一个赋值或一个顶点足够的爆破之后。激发问题是局部一致化，解决奇异性和正特征的equisingularity，以及代数和解析映射的整体结构。

项目成果

Defect and local uniformization

缺陷和局部均匀化

• DOI: 10.1007/s13398-019-00717-1
• 发表时间: 2019
• 期刊: Físicas y Naturales. Serie A. Matemáticas
• 作者: Cutkosky, Steven Dale;Mourtada, Hussein
• 通讯作者: Mourtada, Hussein

Mixed multiplicities of filtrations

混合多重过滤

• DOI: 10.1090/tran/7745
• 发表时间: 2019
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: Cutkosky, Steven Dale;Sarkar, Parangama;Srinivasan, Hema
• 通讯作者: Srinivasan, Hema

Extensions of Valuations to the Henselization and Completion

估值扩展到亨塞尔化和完成

• DOI: 10.1007/s40306-018-0267-y
• 发表时间: 2019
• 期刊: Acta Mathematica Vietnamica
• 影响因子: 0.5
• 作者: Cutkosky, Steven Dale

Essentially finite generation of valuation rings in terms of classical invariants

根据经典不变量，估值环的生成本质上是有限的

• DOI: 10.1002/mana.201900287
• 发表时间: 2020
• 期刊: Mathematische Nachrichten
• 影响因子: 1
• 作者: Cutkosky, Steven Dale;Novacoski, Josnei
• 通讯作者: Novacoski, Josnei

Positivity of mixed multiplicities of filtrations

混合多重过滤的积极性

• DOI: 10.1112/blms.12328
• 发表时间: 2020
• 期刊: Bulletin of the London Mathematical Society
• 影响因子: 0.9
• 作者: Cutkosky, Steven Dale;Srinivasan, Hema;Verma, Jugal
• 通讯作者: Verma, Jugal