Commutative Algebra: Singularities in All Characteristics with Geometric Applications

这项研究项目探索交换代数中的问题。交换代数是研究在整个科学中普遍存在的多项式方程的解的学科。学习多项式函数的一种方法是在更简单的时钟算术设置中查看多项式函数(例如，6+7=1，换句话说，6点过7小时就是1点)。几个世纪以来，在这两个数字系统之间进行转换一直是一种卓有成效的策略，在两边都有应用；例如，这种数学在现代安全通信系统中是必不可少的。这个项目的目标是开发工具，帮助直接在所谓的“混合特征”环境中工作，这种环境介于经典的数字世界和钟表算术般的环境之间。这些工具将有助于统一经典和时钟算术设置。这位研究人员计划为开源软件开发包，并就这些主题写一本书。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中的奇点理论，着眼于几何应用。该理论将代数几何中最小模型程序产生的奇点与交换代数中紧闭包理论产生的奇点统一和推广。利用完备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合适时机。这些工具应该能够在某些不适用Kodaira型消失定理的应用中取代后者。在这些联系的启发下，研究人员还将学习特征0和特征p0交换代数和代数几何。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.