Studies in Moduli Theory and Birational Geometry

这个项目的研究领域是代数几何，这是数学的一个分支，致力于研究被称为代数簇的几何形状，由多项式方程定义。代数几何在编码、工业控制、计算和理论物理中有重要的应用，在这些领域，物理学家认为代数簇是我们宇宙精细结构的一部分。这个项目的一个焦点是模理论，它研究一种显著的现象，在这种现象中，所有相同类型的代数簇的集合通常表现为一个代数簇本身，称为模空间。因此，在代数几何中，把一个“有机体”群落看作是一个“有机体”的隐喻，不仅是一种隐喻，而且是一个严谨而又相当有用的事实。这个项目的另一个重点是二次几何，这里的重点是奇点的分解。奇点分解是一个基本的过程，在这个过程中，代数簇的“坏”点被去掉，取而代之的是“好”点。这个项目包括本科生和研究生的研究机会。更详细地说，PI将继续研究二元几何中的问题，重点是奇点的分解、半稳态约化和堆积理论的几何双元修正。PI和合作者将在最近完成的工作的基础上，从功能上解决真变种的奇点；研究加权爆破的几何和更一般的堆叠理论程序；以及研究正特征中的微妙现象。此外，PI还将继续研究模空间。主要焦点是阿贝尔变种家族的子变种的模数和算法，旨在将最近曲线上有理点的非退化和一致性结果推广到对称的曲线正方形和相关结构。该奖项反映了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。

The area of study of this project lies within algebraic geometry, the branch of mathematics devoted to geometric shapes called algebraic varieties, defined by polynomial equations. Algebraic geometry has significant applications in coding, industrial control, computation, and theoretical physics, where physicists consider algebraic varieties as a piece of the fine structure of our universe. One focus in this project is Moduli theory, which studies a remarkable phenomenon in which the collection of all algebraic varieties of the same type is often manifested as an algebraic variety in its own right, called a moduli space. Thus in algebraic geometry, the metaphor of thinking about a community of "organisms" as itself being an "organism" is not just a metaphor but a rigorous and quite useful fact. The other focus in this project is birational geometry, focusing here on resolution of singularities. Resolution of singularities is a fundamental procedure where "bad" points of an algebraic variety are removed and replaced by "good" points. This project includes research opportunities for undergraduate and graduate students.In more detail, the PI will continue studying problems in birational geometry, focusing on resolution of singularities, semistable reduction, and the geometry of stack theoretic birational modifications. The PI and collaborators will build on recently completed work to functorially resolve singularities of proper families of varieties; to study the geometry of weighted blowings up and more general stack-theoretic procedures; and to study subtle phenomena in positive characteristics. In addition, the PI will continue to study moduli spaces. The main foci are Moduli and arithmetic of subvarieties of families of abelian varieties, aiming to extend recent non-degeneracy and uniformity results of rational points on curves to symmetric squares of curves and related constructions.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.