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

代数几何研究由多项式方程定义的几何对象的代数变体。这一领域最自然的问题之一是理解在考虑代数簇时自然出现的奇点，以及这些奇点如何影响代数簇的整体几何。近年来，已经有了一些突破，特别是在我们考虑复数的解决方案的情况下。与此同时，出现了研究具有正性和混合性的解决方案的新技术和新方法。这个合作项目的主要目标是推进和统一这些想法，以进一步了解和解决局部和全局代数几何中一些最具挑战性的程序。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，这些技术包括最小模型程序、F-奇点理论以及肖尔茨关于完美拟态代数和空间的工作。PIS还将组织研讨会、暑期学校和会议，旨在培训这一领域的年轻研究人员，传播最新成果，促进进一步的进步和突破。该奖项反映了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。

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.