Higher Dimensional Algebraic Geometry

这位研究人员将研究高维代数几何中的几个问题。在与Alexeev、Hassett和Koll共同完成的一个项目中，他计划完成稳定对数曲面的粗完备模空间的存在性的证明，类似于稳定曲线的模空间。在另一个项目中，研究人员将致力于规范极化光滑射影簇的模堆叠子簇的有界性、刚性和双曲性。这些问题从Shafarevich的一个里程碑式的猜想发展而来，以及Arakelov和Parshin的解决方案，这在Falings对Mordell猜想的证明中发挥了重要作用。这个项目的一部分是与Kebekus的联合工作。在第三个项目中，研究人员和哈肯将研究不存在的微分形式对下面种类的几何的影响。他们的目标是证明该领域中几个突出的猜想。在第四个项目中，研究人员和Araujo将致力于证明Beauville猜想，给出射影空间和二次超曲面的新特征。这项研究是在代数几何领域进行的，代数几何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。最初，AND STILL以其最简单的形式处理由多项式定义的平面中的图形。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，它还被证明在物理、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起着非常重要的作用。研究模空间上的曲线提供了有关对象如何在时空中变化的信息。这个项目的重点之一是紧模空间。这些通常是模空间的扩展，它们提供了关于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。该项目的其他目标包括更好地了解某些更高维度的品种。

The investigator will work on several problems in higher dimensionalalgebraic geometry. In a joint project with Alexeev, Hassett, andKoll\'ar he plans to complete the proof of existence of coarsecomplete moduli spaces of stable log surfaces, an analog of the modulispace of stable curves. In another project the investigator is goingto work on boundedness, rigidity and hyperbolicity of subvarieties ofmoduli stacks of canonically polarized smooth projective varieties. These questions evolved from a landmark conjecture of Shafarevich, andits solution by Arakelov and Parshin, which played an important rolein Faltings' proof of the Mordell Conjecture. Part of this project isjoint work with Kebekus. In a third project the investigator andHacon are going to study the impact of the existence of nowherevanishing differential forms on the geometry of the underlyingvariety. Their goal is to prove several outstanding conjectures in thearea. In a fourth project the investigator and Araujo will worktoward proving a conjecture of Beauville giving a new characterizationof projective spaces and quadric hypersurfaces.This research is in the field of algebraic geometry, one of the oldestparts of modern mathematics, but one that blossomed to the point whereit has solved problems that have stood for centuries. Originally, andstill in its simplest form it treats figures defined in the plane bypolynomials. Today, the field uses methods not only from algebra, butalso from analysis and topology, and conversely it is extensively usedin those fields. Moreover it has proved itself useful in fields asdiverse as physics, theoretical computer science, cryptography, codingtheory and robotics. A central problem in algebraic geometry is theclassification of all geometric objects. In turn, an important part ofclassification theory is the theory of moduli. The latter's core ideais that one does not only want to understand these objects, but alsounderstand the way they can be deformed. Moduli spaces play a veryimportant role in theoretical physics. Studying curves on modulispaces provides information on how an object is changing inspace-time. One of the focuses of this project is on compact modulispaces. Those are extensions of moduli spaces in general and they giveadditional information about singular deformations, ones that areessentially different from others. Other goals of the project involvea better understanding of certain higher dimensional varieties.