Moduli theory and singularities

研究者将研究高维代数几何中的几个问题，特别是模理论和奇点。在与Kollár联合的几个项目中，PI计划研究与稳定对数变量的粗模空间存在性有关的各种问题，这是稳定点曲线的模空间的模拟。这包括对理性对和节俭决议的研究，以及它们与最小模型程序的杜波依斯奇点和其他奇点的联系。在另一个项目中，同样受到模项目的激励，PI将与Patakfalvi共同证明Kollár的Ampleness引理的对数版本，并用它来证明稳定对数变量的模空间的投影性。PI还将继续他关于正则偏振光滑射影变的模堆子变的改进Viehweg猜想的工作。这个猜想是由Shafarevich的一个具有里程碑意义的猜想演变而来的，Arakelov和Parshin对这个猜想的解在Faltings对莫德尔猜想的证明中起了重要作用。这个项目是与Kebekus合作的。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、编码理论和机器人等多种领域都很有用。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。这个项目的重点之一是紧模空间。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。

The investigator will work on several problems in higher dimensional algebraic geometry, especially moduli theory and singularities. In several projects joint with Kollár, the PI plans to work on various problems related to the existence of a coarse moduli space of stable log varieties, an analog of the moduli space of stable pointed curves. These include the study of rational pairs and thrifty resolutions and their connections with Du Bois singularities and other singularities of the minimal model program. In another project, also motivated by the moduli project, jointly with Patakfalvi the PI will work on proving a logarithmic version of Kollár's Ampleness Lemma and use it to prove the projectivity of the moduli space of stable log varieties. The PI will also continue his work on the refined Viehweg conjecture regarding subvarieties of moduli stacks of canonically polarized smooth projective varieties. This conjecture evolved from a landmark conjecture of Shafarevich, and its solution by Arakelov and Parshin, which played an important role in Faltings' proof of the Mordell Conjecture. This project is joint work with Kebekus. 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. One of the focuses of this project is 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.