CAREER: Theory of Moduli

职业：模数理论

• 批准号: 0092165
• 负责人: Sandor Kovacs
• 金额: $ 25万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0092165&HistoricalAwards=false
• 关键词: CAREER; Theory; Moduli

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.

研究者研究代数几何中与高维变量族有关的问题。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，我们致力于研究自反束及其在态射方面的行为，以发展一个包含秩1自反束的许多结果的理论，这些结果与已知的线束的结果相似。这对于发展奇异变元的模理论是非常重要的，而奇异变元的模理论又对几何上有意义的紧模空间至关重要。研究者正在研究的另一个问题是Shafarevich猜想的推广及其在高维基情况下的应用，特别是一般类型的变种族的有界性和刚性问题。这项研究属于代数几何领域，这是现代数学中最古老的部分之一，但它已经发展到解决了几个世纪以来一直存在的问题的地步。最初，它仍然是最简单的形式，它用多项式来处理平面上的图形。今天，该领域不仅使用代数的方法，而且还使用分析和拓扑的方法，相反，它在这些领域中被广泛使用。此外，它已经证明自己在物理学、理论计算机科学、密码学、编码理论和机器人技术等领域都很有用。代数几何中的一个中心问题是所有几何对象的分类。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物体在时空中如何变化的信息。这个项目关注紧模空间。这些都是模空间的扩展它们给出了奇异变形的附加信息，这些变形与其他变形本质上是不同的。目前已知低维问题存在紧模空间。研究者正在研究高维的情况。作为该项目的一部分，但在某种程度上独立于上述内容，研究者和他的研究生正在运营一个网站，作为研究生和代数几何年轻研究人员的论坛。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提供一个新的场所，让他们能够接触到该领域专家的知识，从而帮助实现一个更平等的机会环境。此外，收集到的数据库提供了对结果的参考，这些结果太“小”而不能占用整篇文章，但又太重要而不能忽视，以及研究生潜在研究问题的集合。