Higher Dimensional Algebraic Geometry

研究者将致力于解决高维代数几何中的几个问题.在与Alexeev，Hassett和Koll\'ar的一个联合项目中，他计划完成稳定日志表面的coarsecomplete模空间的存在性证明，稳定曲线的模空间的模拟。 在另一个项目中，研究者将研究正则极化光滑射影簇的模叠加的子簇的有界性、刚性和双曲性。这些问题是从Shafarevich的一个里程碑式的猜想发展而来的，Arakelov和Parshin对这个猜想的解答在Faltings证明Mordell猜想的过程中起了重要作用。这个项目的一部分是与Kebekus联合工作。 在第三个项目中，调查员和Hacon将研究无处消失的微分形式的存在对几何基础的影响。他们的目标是在该地区证明几个杰出的成就。 在第四个项目中，研究者和Araujo将致力于证明Beauville的一个猜想，该猜想给出了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，现代数学最古老的部分之一，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。它最初是用多项式来处理平面上的图形，现在仍然是用它的最简单形式. 今天，该领域使用的方法不仅从代数，但也从分析和拓扑学，相反，它被广泛用于这些领域。此外，它已经证明了自己在物理学、理论计算机科学、密码学、编码理论和机器人学等领域的有用性。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，人们不仅要了解这些物体，而且要了解它们是如何变形的。模空间在理论物理中起着非常重要的作用。 研究modulispaces上的曲线提供了关于对象如何改变时间的信息。该项目的重点之一是紧凑的模块空间。这些是模空间的一般扩展，它们给出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。该项目的其他目标包括更好地理解某些高维变量。

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.