Singularities and Duality with Applications to Moduli Theory

这个项目是在代数几何领域进行的，这是现代数学中最古老的领域之一，也是一个蓬勃发展到解决了数百年历史的问题的领域。在其最简单的形式中，它处理由多项式定义的在空间中的图形，例如球体或(无限)锥体。今天，这个领域不仅使用代数的方法，而且还使用分析和拓扑学的方法，相反，它在这些领域被广泛使用。此外，它已被证明适用于物理、理论计算机科学、密码学、编码理论和机器人学等多个领域。代数几何中的一个中心问题是对所有几何对象的分类。反过来，分类理论的一个重要部分就是模理论。后者的核心思想是，人们不仅想要了解这些物体，而且还想了解它们可以变形的方式。模空间在理论物理中扮演着重要的角色：研究模空间上的曲线提供了关于物体如何在时空中运动的信息。研究人员还参与了指导下一代研究人员的工作。这个项目支持几个研究生，他们在研究人员的指导下正在攻读博士学位。这个项目涉及高维代数几何的几个主题，特别是奇点及其在模理论中的应用。中心是奇点及其相互联系的研究，特别是有理奇点、Du Bois奇点，以及与高维代数簇的极小模型程序和模理论有关的其他奇点，这反过来又是高维簇分类理论的两个主要支柱。该项目的主要目标之一是进一步发展具有任意特征的有理奇点理论。特别是，研究人员正在努力获得比当前定义更容易检查的标准，以及建立具有有理奇点的部分解的存在。另一个主要目的是将Du Bois奇点的定义推广到任意特征，并研究它们的形变理论性质及其与有理奇点的联系。该项目的另一个目标是更好地理解“可提升局部上同调”这一相对较新的概念及其与变形的关系。关于奇点和可提升局部上同调的结果将用于模理论的应用。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。

This project is in the field of algebraic geometry, one of the oldest fields in modern mathematics, and one that blossomed to the point where it has solved centuries-old problems. In its simplest form it treats figures defined in space by polynomials, such as a sphere or an (infinite) cone. Today, the field uses methods not only from algebra, but also from analysis and topology, and conversely it is extensively used in those fields. Moreover, it has proved applicable in fields as diverse as physics, theoretical computer science, cryptography, coding theory, and robotics. A central problem in algebraic geometry is the classification of all geometric objects. In turn, an important part of classification theory is the theory of moduli. The latter's core idea is that one not only wants to understand these objects, but also understand the way they can be deformed. Moduli spaces play an important role in theoretical physics: studying curves on moduli spaces provides information on how an object is moving in space-time. The investigator is also involved in mentoring the next generation of researchers. This project supports several graduate students, who are working toward their doctoral degree on related projects under the direction of the investigator.The project concerns several topics in higher dimensional algebraic geometry, especially singularities and their applications to moduli theory. At the center is the study of singularities and their interconnections, especially rational, Du Bois, and other singularities related to the minimal model program and moduli theory of higher dimensional algebraic varieties, which in turn are the two main pillars of classification theory of higher dimensional varieties. One of the main goals of the project is to further develop the theory of rational singularities in arbitrary characteristic. In particular, the investigator is working toward obtaining criteria which are easier to check than the current definition as well as toward establishing the existence of partial resolutions with rational singularities. Another of the main goals is to extend the definition of Du Bois singularities to arbitrary characteristics and study their deformation theoretic properties and their connections to rational singularities. Yet another goal of the project is to better understand the relatively new notion of 'liftable local cohomology' and its relations to deformations. The results obtained with regard to singularities and liftable local cohomology will be used in applications to moduli theory.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.