Commutative Algebra: Singularities in All Characteristics with Geometric Applications

本研究计画探讨交换代数中的问题。 交换代数是研究多项式方程的解的学科，这在整个科学中是普遍存在的。 学习多项式函数的一种方法是在一个更简单的时钟算术设置中查看它们（例如，6+7 = 1，换句话说，6点之后的7小时是1点）。 几个世纪以来，在这两种数字系统之间进行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“混合特性”设置中工作，该设置位于经典数字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。研究人员计划开发开源软件包，并就这些主题撰写一本书。 该项目包括通过参与研究培训研究生，该项目旨在发展混合特征交换代数中的奇点理论，着眼于几何应用。 这一理论将代数几何中的极小模型规划中的奇异性与交换代数中的紧闭包理论中的奇异性统一和推广。 最近的进展，在交换代数利用理论的完美代数和空间，现在已经取得了正确的时间来发展这一理论。 这些工具应该能够取代Kodaira型消失定理在某些应用中，后者不适用。 受这些联系的启发，研究人员还将研究特征0和特征p 0交换代数和代数几何。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.