Moduli and Arithmetic of Higher Dimensional Varieties

代数几何是研究几何形状的学科，这些几何形状是由多项式方程的解产生的。这些形状的一些基本例子是圆或双曲面，因此，代数几何在很大程度上与许多其他数学领域交织在一起。代数几何的指导研究方向之一以及本研究项目的总体重点是理解这些形状的分类，这是一个被称为模空间研究的子领域。代数几何以及对模空间的研究有许多应用，例如在密码学中，以及通过弦理论和数学物理来理解我们宇宙的结构。这个项目包括本科生和研究生的培训机会，以及来自当地社区的高中生的外展努力。虽然我们对代数曲线的模空间有了广泛的了解，但我们对高维代数变种(即复数维至少为2的变种)的模空间的了解要少得多。简而言之，这个项目的主要目标是通过利用模理论和双曲几何中的许多最新结果，例如PI与合作者共同工作所获得的跨越墙的结果，来加深我们对高维模的理解。第一个项目旨在使用跨越墙的技巧来理解高维代数簇的模空间的基本几何结构和几何性质。另外，由于关于高维变种的模空间的大多数已知结果集中在两种情况，即(Log)一般类型对和(Log)Fano对，第二个项目的目的是利用最小模型程序和K-稳定性的工具来构造和研究(Log)Calabi-Yau对的模空间。最后，第三个项目使用二次几何、模空间和最小模型程序的技术来理解各种(对数)一般类型的双曲性的各种概念，包括有理和积分点的分布。这个奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Algebraic geometry is the study of geometric shapes which arise as solutions to polynomial equations. Some fundamental examples of these shapes are circles or hyperboloids, and as such, algebraic geometry is largely intertwined with many other fields of mathematics. One of the guiding research directions in algebraic geometry as well as the overall focus of this research project is understanding the classification of these shapes, a subfield known as the study of moduli spaces. Algebraic geometry as well as the study of moduli spaces have numerous applications, for example within cryptography as well as in understanding the structure underlying our universe via string theory and mathematical physics. This project includes training opportunities for both undergraduate and graduate students, as well as outreach efforts involving high school students from the local community.While we have an extensive understanding of moduli spaces of algebraic curves, we understand far less about moduli spaces of higher dimensional algebraic varieties (i.e. varieties of complex dimension at least two). In short, the main goal of this project is to further our understanding of higher dimensional moduli by leveraging many recent results in moduli theory and birational geometry, such as wall-crossing results that the PI has obtained in joint work with collaborators. The first project aims to use wall-crossing techniques to understand the underlying geometric structure and geometric properties of moduli spaces of higher dimensional algebraic varieties. Additionally, as most known results regarding moduli spaces of higher dimensional varieties focus on two cases, namely (log) general type pairs and (log) Fano pairs, the goal of the second project is to construct and study moduli spaces of (log) Calabi-Yau pairs using tools from the minimal model program and K-stability. Finally, the third project uses techniques from birational geometry, moduli spaces, and the minimal model program to understand various notions of hyperbolicity on varieties of (log) general type, including the distribution of rational and integral points.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.