Singularities and Duality with Applications to Moduli Theory

这个项目是在代数几何领域，在现代数学中最古老的领域之一，一个蓬勃发展的点，它已经解决了几个世纪的老问题。在其最简单的形式，它对待的图形在空间中定义的多项式，如一个球或一个（无限）锥。今天，该领域使用的方法不仅来自代数，而且来自分析和拓扑学，相反，它被广泛用于这些领域。此外，它已被证明适用于物理学、理论计算机科学、密码学、编码理论和机器人技术等不同领域。代数几何中的一个中心问题是所有几何对象的分类。反过来，分类理论的一个重要部分是模理论。后者的核心思想是，人们不仅要了解这些物体，还要了解它们可以变形的方式。模空间在理论物理学中起着重要的作用：研究模空间上的曲线可以提供关于物体如何在时空中运动的信息。研究人员还参与指导下一代研究人员。该项目支持几名研究生，他们正在研究员的指导下进行相关项目的博士学位。该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。 在中心是研究奇点及其相互联系，特别是合理的，杜波依斯，和其他奇点有关的最小模型计划和模理论的高维代数簇，这反过来又是两个主要支柱分类理论的高维品种。 该项目的主要目标之一是进一步发展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目前的定义更容易检查的标准，以及建立合理的奇点的部分决议的存在。另一个主要目标是将杜波依斯奇点的定义扩展到任意特征，并研究它们的变形理论性质及其与理性奇点的联系。 该项目的另一个目标是更好地理解相对较新的“可提升局部上同调”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量理论的应用。该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.