A Unified Perspective on Singularities in Commutative Algebra and Algebraic Geometry

这个研究项目汇集了交换代数和代数几何中奇点的不同观点。代数几何和交换代数都涉及由多项式方程定义的形状，例如抛物线是方程y = x^2的图形。并非所有形状都如此光滑；有些图形，如方程y^2 = x^3的图形，有尖锐的点或其他奇点。对几何奇点的研究在数学中是很重要的，并广泛应用于重要的物理、社会和经济系统的数学模型。研究人员早就知道，当研究模（或时钟）算法中的几何形状时，经典几何设置中的重要奇点类型也会自然出现。事实上，时钟算法设置中的代数几何是现代通信基础设施的关键组成部分。该项目旨在发展一种混合特征的奇点理论，这是一种介于经典和时钟算术世界之间的中间地带（具有两者的各个方面）。该项目还旨在开发这一理论的几何应用，并开发开源软件来研究交换代数和代数几何中的奇点。数论和算术几何中发展的技术在混合特征交换代数中产生了许多突破性的结果。本项目旨在利用这些方法发展一种适合于研究混合特征的高维双元代数几何的奇点理论，这也将有助于将许多正特征交换代数的成功转化为这种设置。该项目还旨在开发混合特征中的f纯或对数正则奇点（及其中心）的模拟，研究奇点的基本群，并研究这种情况下的边界除数。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。

This research project brings together different perspectives on singularities in commutative algebra and algebraic geometry. Both algebraic geometry and commutative algebra concern shapes defined by polynomial equations, such as the parabola that is the graph of the equation y = x^2. Not every shape is so smooth; some, such as the graph of the equation y^2 = x^3, have sharp points or other singularities. Study of geometric singularities is important in mathematics and has application to mathematical models for a wide range of important physical, social, and economic systems. Researchers have long known that important types of singularities in classical geometric settings also appear naturally when studying geometric shapes in modular (or clock) arithmetic. In fact, algebraic geometry in the clock arithmetic setting is a key component of modern communications infrastructure. This project aims to develop a theory of singularities in mixed characteristic, a middle ground between the classical and clock arithmetic worlds (that possesses aspects of both). The project also aims to develop geometric applications of this theory and to develop open-source software to study singularities in commutative algebra and algebraic geometry. Techniques developed in number theory and arithmetic geometry have given rise to numerous recent breakthrough results in mixed characteristic commutative algebra. This project aims to use these methods to develop a singularity theory suitable for studying higher dimensional birational algebraic geometry in mixed characteristic, which would also facilitate translating many of the successes of positive characteristic commutative algebra to this setting. The project additionally aims to develop an analog of F-pure or log canonical singularities (and their centers) in mixed characteristic, to study fundamental groups of singularities, and to study boundary divisors in this setting.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.