FRG: Collaborative Research: Algebraic Geometry and Singularities in Positive and Mixed Characteristic

代数几何研究的是由多项式方程定义的几何对象——代数变量。这一领域最自然的问题之一是理解在考虑代数变体时自然出现的奇点以及这些奇点如何影响代数变体的整体几何。近年来已经有了一些突破，特别是在我们考虑复数解的情况下。同时出现了新的技术和方法来研究正和混合特性的解。这个合作项目的主要目标是推进和统一这些想法，以进一步理解和解决局部和全球代数几何中一些最具挑战性的项目。此外，该项目还为研究生提供了研究培训机会。pi将通过使用各种技术来研究正特征和混合特征中的奇点，这些技术包括来自最小模型程序，f -奇点理论以及Scholze关于完美代数和空间的工作。pi还将组织讲习班、暑期学校和会议，旨在培训这一领域的年轻研究人员，传播最近的成果并促进进一步的进展和突破。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。

Algebraic Geometry studies algebraic varieties which are geometric objects defined by polynomial equations. One of the most natural problems in this area is to understand the singularities that naturally occur when considering algebraic varieties and how these singularities influence the global geometry of algebraic varieties. In recent years there have been a number of breakthroughs, especially in the case where we consider solutions over the complex numbers. At the same time new techniques and approaches have emerged for studying solutions in positive and mixed characteristics. The primary goal of this collaborative project is to advance and unify these ideas to further understand and solve some of the most challenging programs in both local and global algebraic geometry. In addition the project provides research training opportunities for graduate students. The PIs will investigate singularities in positive and mixed characteristics by using a variety of techniques including those arising from the minimal model program, from the theory of F-singularities, and from Scholze's work on perfectoid algebras and spaces. The PIs will also organize workshops, a summer school and a conference, aimed at training young researchers in this area, disseminating recent results and facilitating further advances and breakthroughs.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.