Cyclotomic Spectra and p-Divisible Groups

分圆谱和 p-可分群

• 批准号: 2005316
• 负责人: David Antieau
• 金额: $ 41.73万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-08-01 至 2020-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2005316&HistoricalAwards=false
• 关键词: Cyclotomic; Spectra; Divisible; Groups; Cyclotomic; Spectra; Divisible; Groups

Recent advances in the foundations of category theory and homotopy theory have led to an explosion of work and new insights into disparate fields ranging from algebraic geometry and number theory to the study of field theories in physics. The areas of the major advances are abstract: category theory is a framework for organizing different types of mathematical objects and admissible transformations between these types, while homotopy theory seeks to understand the coarse nature of shapes. The applications on the other hand include new results on the geometry and solutions of polynomial equations, important areas of research with concrete applications inside and outside of mathematics. The present proposal will harness these advances in the areas of algebraic K-theory, which is a mysterious but powerful tool for counting mathematical objects, and arithmetic geometry, which is about the influence of geometry on the structure of solutions of polynomial equations with rational coordinates. The project provides research training opportunities for graduate students and postdoctoral fellows.The project's three main objectives are (1) to directly compare the motivic and syntomic approaches to p-adic etale K-theory by showing that if R is a smooth commutative p-local commutative ring, then the trace map from K-theory to topological cyclic homology respects the motivic and syntomic filtrations after p-completion, (2) to construct a theory of coefficient systems for p-adic cohomology using cyclotomic spectra and to verify the PI's liftability conjecture, which will help to explain the relationship between the window-frame approach to the classification of formal groups and the recent prismatic Dieudonne theory developed by Anschuetz and Le Bras, and (3) to understand the filtration on prismatic cohomology arising from the cyclotomic t-structure. Short master classes will be offered for graduate students and postdoctoral researchers that each focus on a single current issue in algebraic K-theory.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.

类别理论和同义理论基础的最新进展导致工作爆炸和对不同领域的新见解，从代数几何和数字理论到物理学领域理论的研究。主要进步的领域是抽象的：类别理论是组织不同类型的数学对象和这些类型之间可接受的转换的框架，而同型理论试图理解形状的粗糙本质。另一方面，应用程序包括有关多项式方程的几何形状和解决方案的新结果，重要的研究领域具有数学内部和外部的混凝土应用。 目前的建议将利用这些进步在代数K理论的领域，这是计算数学对象的神秘但有力的工具，以及算术几何形状，这是关于几何学对多种方程式结构的影响多项式方程的影响。该项目为研究生和博士后研究员提供了研究培训机会。该项目的三个主要目标是（1）直接比较P-Adic eTale K理论的动机和构成方法，如果R是平稳的P-Plocal plocal pollocal polmutighative Ring，则来自K理论的痕量痕量与拓扑循环型的TOPEATION CORTICTION和TOPICTICS TORIPATION和TOPETICTION AD-CORTIT AD-CORTIC ARTICATION（P）相关（P） coefficient systems for p-adic cohomology using cyclotomic spectra and to verify the PI's liftability conjecture, which will help to explain the relationship between the window-frame approach to the classification of formal groups and the recent prismatic Dieudonne theory developed by Anschuetz and Le Bras, and (3) to understand the filtration on prismatic cohomology arising from the cyclotomic t-structure.将为研究生和博士后研究人员提供简短的大师班，每个研究人员都专注于代数K理论中的单个当前问题。该奖项反映了NSF的法定任务，并被认为是通过基金会的智力优点和更广泛影响的审查标准通过评估来通过评估来支持的。