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关注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。并不是每个形状都是如此光滑的;有些形状，比如方程y^2 = x^3的图形，有尖锐的点或其他奇点。几何奇点的研究在数学中很重要，并应用于广泛的重要物理，社会和经济系统的数学模型。研究人员早就知道，在模（或时钟）算术中研究几何形状时，经典几何环境中重要类型的奇点也会自然出现。事实上，代数几何中的时钟运算设置是现代通信基础设施的关键组成部分。这个项目的目的是发展一个混合特性的奇点理论，一个古典和时钟算术世界之间的中间地带（拥有两者的方面）。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。在数论和算术几何中发展起来的技术在混合特征交换代数中产生了许多最近的突破性结果。这个项目的目的是使用这些方法来开发一个奇点理论适合于研究高维双有理代数几何的混合特征，这也将有助于翻译的许多成功的正特征交换代数这一设置。该项目的另一个目的是开发一个类似的F-纯或对数正则奇点（及其中心）的混合特性，研究基本群体的奇点，并研究在此设置的边界因子。该奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

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.