Higher Dimensional Algebraic Varieties: Geometry and Arithmetic

代数几何侧重于研究多项式方程的解。该领域的指导问题是理解由多项式方程产生的几何对象，称为代数变量，并对这些形状进行分类。该项目还涉及数论的应用，希望了解这些形状的几何可以告诉我们这些多项式方程的有理数或积分解。这些问题可以应用于其他数学领域（如微分几何）和物理领域（如弦理论）。高维代数变体的模空间并不像曲线变体那样容易理解。本课题的第一个首要问题是理解高维代数变模空间的紧化。所采取的方法将结合来自最小模型程序（MMP）以及k稳定性的技术。特别是，PI计划理解特定情况下的显式紧化（例如K3曲面），并研究如何以模块化的方式在各种紧化之间进行插值。第二个首要问题是研究高维几何及其模如何影响算术，研究代数几何、双曲和算术几何之间丰富的相互作用。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Algebraic geometry focuses on the study of solutions of polynomial equations. Guiding questions in this fields are to understand the geometric objects, called algebraic varieties, arising from polynomial equations and to classify such shapes. This project also concerns applications to number theory, with hopes to understand what the geometry of these shapes can tell us about the rational number or integral solutions to these polynomial equations. These problems have applications to other fields of math (e.g. differential geometry) as well as physics (e.g. string theory). Moduli spaces of higher dimensional algebraic varieties are not as well understood as their curve counterparts. The first overarching question in this project is to the understand compactifications of moduli spaces of higher dimensional algebraic varieties. The approaches taken will be a combination of techniques arising from, e.g. the minimal model program (MMP), as well as K-stability. In particular, the PI plans to understand explicit compactifications in specific cases (e.g. K3 surfaces), and to study how to interpolate between various compactifications in a modular way. The second overarching question, is to investigate how geometry of higher dimensional varieties and their moduli influences arithmetic, studying the rich interplay between algebraic geometry, hyperbolicity, and arithmetic geometry.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.