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

代数几何是数学的一个重要领域，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。首席研究员的研究中心围绕代数簇和奇点的算术设置。PI计划扩大和建立在算术和复杂几何学的最新突破，以增加我们对这些物体的理解。PI将邀请研究生参与这项研究，并组织针对早期职业数学家的研讨会。PI的主要目标是开发和应用与Hodge理论，p-adic Riemann-Hilbert对应和准F-分裂相关的新技术，以描述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，并在算法设置上扩展了最小模型程序的有效性。这项工作将导致双有理几何和交换代数的新进展。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.