CAREER: Theory of Moduli

调查研究问题的代数几何有关的家庭的高维品种。该项目的主要目标是找到紧模函子的良好定义，特别是关于曲面的模理论。作为模理论项目的一部分，特别努力致力于自反层及其关于态射的行为，以努力发展一种理论，该理论包括秩为1的自反层的许多结果，这些结果类似于已知的线丛结果。这是非常重要的，以发展一个模理论的奇异品种，这反过来是必不可少的几何意义紧模空间。另一个问题的调查研究是概括Shafarevich猜想及其应用的情况下，高维基地，特别是问题的有界性和刚性的家庭品种的一般类型。 这项研究是在代数几何领域，现代数学的最古老的部分之一，但一个蓬勃发展的地步，它已经解决了几个世纪以来的问题。最初，仍然在其最简单的形式，它对待的数字定义在平面上的多项式。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，它已被证明在物理学、理论计算机科学、密码学、编码理论和机器人学等不同领域都很有用。代数几何中的一个中心问题是所有几何对象的分类。分类理论的一个重要组成部分是模理论。后者的核心思想是，人们不仅要理解这些物体，还要理解它们变形的方式。模空间在理论物理中起着非常重要的作用。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。这个项目的重点是紧模空间。这些是模空间的一般扩展，它们提供了关于奇异变形的额外信息，这些信息与其他信息有本质的不同。目前，对于低维问题，紧模空间的存在性是已知的。调查员正在研究更高维的情况。作为项目的一部分，但有点独立于上述，调查员和他的研究生正在运行一个网站，作为一个论坛的研究生和年轻的研究人员在代数几何。本网站的用户可以提出专家可能知道但文献中没有的问题。该项目的主要目标是为更多的学生开辟一个新的场所，以获得该领域专家的知识，帮助实现一个机会更加平等的环境。此外，收集的数据库提供了参考结果，这些结果太“小”，无法占据整篇文章，但太重要，不能忽视，以及为研究生收集潜在的研究问题。

The investigator studies problems in algebraic geometry related to families of higher dimensional varieties. The main goal of the project is to find good definitions of compact moduli functors, with special regard to the moduli theory of surfaces. As part of the moduli theory project special efforts are devoted to reflexive sheaves and their behavior with respect to morphisms in an effort to develop a theory that includes many results for reflexive sheaves of rank 1 that are similar to results already known for line bundles. This is very important in order to develop a moduli theory of singular varieties, which in turn is essential for geometrically meaningful compact moduli spaces. Another problem the investigator is studying is generalizations of Shafarevich's conjecture and its applications to the case of higher dimensional bases, in particular the problem of boundedness and rigidity for families of varieties of general type. This research 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. Originally, and still 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. This project focuses on compact moduli spaces. Those are extensions of moduli spaces in general and they give additional information about singular deformations, ones that are essentially different from others. Presently the existence of compact moduli spaces is known for low dimensional problems. The investigator is studying the higher dimensional case. As part of the project, but somewhat independent of the above, the investigator and his graduate students are running a website that works as a forum for graduate students and young researchers in algebraic geometry. Users of this website can pose questions that are likely to be known to experts but not available in the literature. The main goal of this project is to open a new venue for a larger number of students to access the knowledge of experts in the field helping to achieve a more equal opportunity environment. In addition, the collected database provides references to results that are too "small" to occupy an entire article, but too important to ignore, as well as a collection of potential research problems for graduate students.