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.

本研究计划探讨交换代数中的问题。交换代数是对多项式方程解的研究，它在科学中无处不在。研究多项式函数的一种方法是在更简单的类似时钟算术的设置中查看它们（例如，6+7 = 1，换句话说，6点后7小时是1点）。几个世纪以来，这两种数字系统之间的转换一直是一种卓有成效的策略，在双方都有应用；例如，这种数学在现代安全通信系统中是必不可少的。该项目旨在开发工具，以帮助直接在所谓的“混合特征”设置中工作，该设置位于经典数值世界和时钟算术设置之间。这些工具将有助于统一经典和时钟算法设置。研究者计划为开源软件开发软件包，并写一本关于这些主题的书。该项目包括通过参与研究来培训研究生。本项目旨在发展混合特征交换代数中的奇点理论，并着眼于几何应用。该理论将代数几何中极小模型规划的奇异性与交换代数中紧闭理论的奇异性统一和推广。利用完美代数和空间理论的交换代数的最新进展使得现在是发展这一理论的最佳时机。这些工具应该能够在某些应用中取代kodaira型消失定理，后者不适用。受这些联系的启发，研究者还将研究特征0和特征p 0交换代数和代数几何。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

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

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

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 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

A Kunz-type characterization of regular rings via alterations

通过改变对规则环进行 Kunz 型表征

• DOI: 10.1016/j.jpaa.2019.07.008
• 发表时间: 2020
• 期刊: Journal of Pure and Applied Algebra
• 影响因子: 0.8
• 作者: Ma, Linquan;Schwede, Karl
• 通讯作者: Schwede, Karl