Singularities and Moduli Theory

这个研究项目是在代数几何领域进行的，这是现代数学中最古老的部分之一，但它发展到了解决几个世纪以来一直存在的问题的地步。在其最简单的形式中，它处理由多项式在平面中定义的图形。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，它已被证明在物理、理论计算机科学、密码学、编码理论和机器人学等领域都很有用。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非常重要的角色：研究模空间上的曲线提供了关于物体在时空中如何变化的信息。这个项目的焦点之一是紧模空间，它给出了关于奇异变形的额外信息，这些变形本质上与其他变形不同。这位研究人员还参与了向高中生推广数学的工作；他和华盛顿大学夏季数学研究所的其他负责人特别努力让该项目所有级别的女性参与进来，从学生到助教再到教师，以加强她们在数学中的领导作用。这个项目涉及高维代数几何的几个主题，特别是模理论和奇点。研究的主要主题是稳定原木品种的紧模空间，这是一个仍处于发展阶段的重要领域。具体地说，即使是正确的模函子也需要确定，大多数研究都是基于对这些模空间的基本性质的理解。这涉及到理解在稳定的原木品种上可能发生的奇点。在这方面，重要的奇点类别是有理奇点和最小模型程序的其他奇点。研究人员将致力于推进我们目前对任意特征中的这些奇点的有限理解。该项目还旨在开发上同调方法来处理任意特征的有理对和节约解。特别是，研究人员将研究Hodge上同调的对数版本和Grothendieck基本类的对数版本。同样源于模项目，研究人员计划研究杜波依斯奇点和杜波伊斯对的性质。该项目这一部分的主要目标是建立一个在任意特征下有意义的奇点的定义，并证明有理和Du Bois对的一个次伴随定理。

This research project is in the field of algebraic geometry, one of the oldest parts of modern mathematics, but one that blossomed to the point where it has solved problems that have stood for centuries. In its simplest form it treats figures defined in the plane by polynomials. 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 itself useful 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 does not only want to understand these objects, but also understand the way they can be deformed. Moduli spaces play a very important role in theoretical physics: studying curves on moduli spaces provides information on how an object is changing in space-time. One of the foci of this project is on compact moduli spaces, which give additional information about singular deformations, ones that are essentially different from others. The investigator is also involved in promoting mathematics to high school students; he and fellow directors of the Summer Institute for Mathematics at the University of Washington make special efforts to involve women at all levels of the program, from students through teaching assistants to instructors, to reinforce their leadership roles in mathematics.This project concerns several topics in higher dimensional algebraic geometry, especially moduli theory and singularities. The overarching theme of the research is centered on compact moduli spaces of stable log varieties, an important area that is still in the developmental stage. In particular, even the correct moduli functor needs to be identified, and most of the research is motivated by understanding the basic properties of these moduli spaces. This involves understanding the singularities that can occur on stable log varieties. Important classes of singularities in this regard are that of rational singularities and other singularities of the minimal model program. The investigator will work on advancing our currently very limited understanding of these singularities in arbitrary characteristic. The project also aims to develop cohomological methods to deal with rational pairs and thrifty resolutions in arbitrary characteristic. In particular, the investigator will study logarithmic versions of Hodge cohomology and of Grothendieck's fundamental class. Also stemming from the moduli project, the investigator plans to study properties of Du Bois singularities and Du Bois pairs. The main objectives of this part of the project are to develop a definition of these singularities that makes sense in arbitrary characteristic and to prove a subadjunction theorem for rational and Du Bois pairs.