Positive and Mixed Characteristic Birational Geometry and its Connections with Commutative Algebra and Arithmetic Geometry

代数几何是一个重要的数学领域，其目标是理解被称为代数簇的基本几何形状。对这种形状的研究是许多应用的来源，例如，在密码学、工程学或生物学中。主要研究人员的研究集中在算术环境中的代数簇和奇点上。PI计划扩大和发展最近在算术和复杂几何方面的突破，以增加我们对这类物体的理解。PI的主要目标是开发和应用与Hodge理论、p-adadRiemann-Hilbert对应和准F-分裂相关的新技术来描述具有正特征的高阶微分形式的行为，提高我们对混合特征奇点的理解，并扩展最小模型程序在算术环境中的有效性。这项工作将在二元几何和交换代数方面带来新的进步。这一奖项反映了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 an important field of mathematics whose goal is to understand fundamental geometric shapes called algebraic varieties. The study of such shapes is a source of many applications, for example, in cryptography, engineering, or biology. The principal investigator's research centers around algebraic varieties and singularities in arithmetic settings. The PI plans to expand and build upon recent breakthroughs in arithmetic and complex geometry to increase our understanding of such objects. The PI will involve graduate students in this research and organize workshops aimed at early career mathematicians.The key goal of the PI is to develop and apply new techniques related to Hodge theory, p-adic Riemann-Hilbert correspondence, and quasi-F-splittings to describe the behavior of higher differential forms in positive characteristic, improve our understanding of mixed characteristic singularities, and extend the validity of the Minimal Model Program in the arithmetic settings. This work will lead to new advancements in birational geometry and commutative algebra.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.