Algebraic Geometry Close to Characteristic p

代数几何接近特征p

• 批准号: 1801689
• 负责人: Bhargav Bhatt
• 金额: $ 53.5万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1801689&HistoricalAwards=false
• 关键词: Algebraic; Geometry; Close; Characteristic

This project supports research in algebraic geometry (which studies solutions to systems of polynomial equations in many variables). This subject is thousands of years old, and provides a systematic language for translating statements in geometry (such as "a doughnut is more curved than a ball") to those in algebra (such as "an elliptic function field is not rational"). A dominant theme of this project is to use this dictionary to better understand the variation in the geometric structure of a solution set of a certain system of equations as one perturbs the coefficients of the defining equations, especially in an arithmetic sense (i.e., in passing from usual arithmetic to modular or ``clockwork'' arithmetic). A better understanding of the geometric structure of solution sets of equations in modular arithmetic is fundamental to many applications of mathematics. The PI shall study the interaction between techniques coming from number theory (such as cohomology theories in p-adic Hodge theory) and algebraic topology (such as topological Hochschild homology) in the context of algebraic varieties over a p-adic field. A deep relationship between the two is expected: they should be related in the same way that motivic cohomology and algebraic K-theory are related (i.e., via an Atiyah-Hirzebruch type spectral sequence). The PI shall also use tools coming from number theory and p-adic geometry (such as perfectoid geometry) to approach problems in 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.

该项目支持代数几何形状的研究（研究了许多变量中多项式方程系统的解决方案）。该主题已有数千年的历史，并提供了一种系统的语言，用于将几何学的语句（例如“甜甜圈比球更弯曲”）为代数（例如“椭圆函数场不是理性”）。该项目的一个主要主题是使用该词典来更好地理解某个方程式系统的几何结构的变化，因为一个方程组的解决方案集呈现出定义方程的系数，尤其是在算术意义上（即，从通常的算术算术到模块化或``colkentment ocular arithmentic ot otulinment ocultiment'''Arithmetmentmentmentection'''''。更好地理解模块化算术中方程解决方案集的几何结构是数学应用的基础。 PI应研究来自数量理论的技术（例如P-Adic Hodge理论中的共同论理论）和代数性品种在P-Adic领域的代数品种中的代数拓扑（例如拓扑霍奇族同源性）之间的相互作用。预期两者之间存在深厚的关系：它们应与动机同胞和代数K理论相关的方式相同（即通过Atiyah-Hirzebruch型光谱序列）。 PI还应使用来自数字理论和P-ADIC几何形状（例如PerfectOdoid几何形状）的工具来处理交换代数中的问题。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的审查标准通过评估来获得支持的。

项目成果

Prisms and prismatic cohomology

• DOI: 10.4007/annals.2022.196.3.5
• 发表时间: 2019-05
• 期刊: Annals of Mathematics
• 影响因子: 4.9
• 作者: B. Bhatt;P. Scholze

The six functors for Zariski-constructible sheaves in rigid geometry

刚性几何中 Zariski 可构造滑轮的六个函子

• DOI: 10.1112/s0010437x22007291
• 发表时间: 2022
• 期刊: Compositio Mathematica
• 影响因子: 1.8
• 作者: Bhatt, Bhargav;Hansen, David
• 通讯作者: Hansen, David

Prismatic $F$-crystals and crystalline Galois representations

• DOI: 10.4310/cjm.2023.v11.n2.a3
• 发表时间: 2021-06
• 期刊: Cambridge Journal of Mathematics
• 影响因子: 1.6
• 作者: B. Bhatt;P. Scholze