Commutative Algebra: Singularities in All Characteristics with Geometric Applications

交换代数：所有特征中的奇点及其几何应用

• 批准号: 1801849
• 负责人: Karl Schwede
• 金额: $ 18.2万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-09-01 至 2022-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1801849&HistoricalAwards=false
• 关键词: Commutative; Algebra; Singularities; Characteristics; Geometric

This research project explores questions in commutative algebra. Commutative algebra is the study of solutions of polynomial equations, which are ubiquitous throughout science. One way to study polynomial functions is to view them instead in a simpler clock-arithmetic-like setting (where for example, 6+7 = 1, in other words, 7 hours after 6 o'clock is 1 o'clock). Translating between these two number systems has been a fruitful strategy for centuries, with applications on both sides; for example, this sort of mathematics is essential in modern secure communications systems. This project aims to develop tools to help work directly in what is called a "mixed characteristic" setting, which sits between the classical numerical world and the clock-arithmetic-like setting. These tools will help unify the classical and clock arithmetic settings. The investigator plans to develop packages for open source software and to write a book on these topics. The project includes training of graduate students through involvement in the research.This project aims to develop a theory of singularities in mixed characteristic commutative algebra, with an eye towards geometric applications. This theory will unify and generalize the singularities coming from the minimal model program in algebraic geometry with the singularities coming out of tight closure theory in commutative algebra. Recent advances in commutative algebra utilizing the theory of perfectoid algebras and spaces has made now the right time to develop this theory. These tools should be able to replace Kodaira-type vanishing theorems in some applications where the latter do not apply. Inspired by these connections, the investigator will also study characteristic 0 and characteristic p 0 commutative algebra and algebraic geometry.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.

该研究项目探讨了交换代数中的问题。 交换代数是对多项式方程的解决方案的研究，在整个科学中无处不在。 研究多项式函数的一种方法是在更简单的时钟示例设置中查看它们（例如，6+7 = 1，换句话说，6点钟后7小时是1点钟）。 几个世纪以来，在这两个数字系统之间翻译一直是一种富有成果的策略，双方都有应用。例如，这种数学在现代安全通信系统中至关重要。 该项目旨在开发工具，以直接在所谓的“混合特征”设置中工作，该设置位于经典数值世界和类似钟度的设置之间。 这些工具将有助于统一经典和时钟算术设置。研究人员计划为开源软件开发包裹，并撰写有关这些主题的书。 该项目包括通过参与研究对研究生进行培训。该项目旨在发展混合特征交换代数的奇异性理论，并着眼于几何应用。 该理论将统一和概括来自代数几何模型计划的奇异性，而奇异性是从通勤代数中紧密封闭理论出现的。 利用完美代数和空间的理论的交换代数的最新进展使现在成为发展这一理论的合适时机。 这些工具应该能够在某些不适用的应用程序中替换Kodaira型消失的定理。 受这些联系的启发，研究人员还将研究特征0和特征P 0的交换代数和代数几何。该奖项反映了NSF的法定任务，并认为通过基金会的智力优点和更广泛影响的评估标准，认为值得通过评估来获得支持。

项目成果

RationalMaps, a package for Macaulay2

RationalMaps，Macaulay2 的软件包

• DOI: 10.2140/jsag.2022.12.17
• 发表时间: 2022
• 期刊: Journal of Software for Algebra and Geometry
• 作者: Bott, C. J.;Hassanzadeh, Seyed Hamid;Schwede, Karl;Smolkin, Daniel
• 通讯作者: Smolkin, Daniel

An analogue of adjoint ideals and PLT singularities in mixed characteristic

混合特征中伴随理想和PLT奇点的类比

• DOI: 10.1090/jag/797
• 发表时间: 2022
• 期刊: Journal of Algebraic Geometry
• 影响因子: 1.8
• 作者: Ma, Linquan;Schwede, Karl;Tucker, Kevin;Waldron, Joe;Witaszek, Jakub
• 通讯作者: Witaszek, Jakub

The TestIdeals package for Macaulay2

Macaulay2 的 TestIdeals 包

• DOI: 10.2140/jsag.2019.9.89
• 发表时间: 2019
• 期刊: Journal of Software for Algebra and Geometry
• 作者: F. Boix, Alberto;Hernández, Daniel;Kadyrsizova, Zhibek;Katzman, Mordechai;Malec, Sara;Robinson, Marcus;Schwede, Karl;Smolkin, Daniel;Teixeira, Pedro;Witt, Emily
• 通讯作者: Witt, Emily

Covers of rational double points in mixed characteristic

混合特征中有理双点的覆盖

• DOI: 10.5427/jsing.2021.23h
• 发表时间: 2021
• 期刊: Journal of Singularities
• 影响因子: 0.4
• 作者: Carvajal-Rojas, Javier;Ma, Linquan;Polstra, Thomas;Schwede, Karl;Tucker, Kevin
• 通讯作者: Tucker, Kevin

The FrobeniusThresholds package for Macaulay2

Macaulay2 的 FrobeniusThresholds 包

• DOI: 10.2140/jsag.2021.11.25
• 发表时间: 2021
• 期刊: The journal of software for algebra and geometry
• 作者: Hernández, Daniel;Schwede, Karl;Teixeira, Pedro;Witt, Emily
• 通讯作者: Witt, Emily