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

正混合特征双有理几何及其与交换代数和算术几何的联系

基本信息

批准号：
2401360
负责人：
Jakub Witaszek
金额：
$ 25万
依托单位：
Princeton University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2024
资助国家：
美国
起止时间：
2024-06-01 至 2027-05-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2401360&HistoricalAwards=false
关键词：
Positive Mixed Characteristic Birational Geometry

项目摘要

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.

代数几何形状是数学的重要领域，其目标是了解称为代数品种的基本几何形状。这种形状的研究是许多应用的来源，例如在加密，工程或生物学中。首席研究者的研究集中在算术环境中的代数品种和奇异性围绕。 PI计划扩大并建立在算术和复杂几何形状的最新突破之上，以增强我们对此类物体的理解。 PI将与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的对应关系以及准f-Splittings相关的新技术，以及在较高的差异形式中，以提高对杂质的范围的较高差异表演的行为，并提高较高的差异形式，并提高对杂物的理解，并将其用于杂质的范围，并将其用于杂质的范围，并将其分为杂物，并将其分为杂物。设置。这项工作将导致Birational几何学和交换代数的新进步。该奖项反映了NSF的法定任务，并且使用基金会的知识分子优点和更广泛的影响审查标准，被认为值得通过评估来支持。