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

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.

范畴论和同伦论基础的最新进展导致了对从代数几何和数论到物理学场论研究等不同领域的工作和新见解的爆炸式增长。主要进展的领域是抽象的：范畴论是组织不同类型的数学对象以及这些类型之间可接受的变换的框架，而同伦理论则试图理解形状的粗糙本质。另一方面，应用包括几何学和多项式方程解的新成果，这是数学内外具有具体应用的重要研究领域。 本提案将利用代数 K 理论和算术几何领域的这些进展，代数 K 理论是计算数学对象的神秘但强大的工具，算术几何是关于几何对有理坐标多项式方程解的结构的影响。该项目为研究生和博士后提供研究培训机会。该项目的三个主要目标是（1）通过证明如果 R 是平滑​​交换的 p-局部，直接比较 p-adic etale K 理论的动机方法和句法方法交换环，则从 K 理论到拓扑循环同调的迹图尊重 p 完成后的动机和句法过滤，(2) 构建系数系统理论使用分圆谱进行 p 进上同调并验证 PI 的可提升性猜想，这将有助于解释形式群分类的窗框方法与 Anschuetz 和 Le Bras 最近开发的棱镜 Dieudonne 理论之间的关系，以及（3 ) 了解分圆 t 结构引起的棱柱上同调的过滤。将为研究生和博士后研究人员提供短期大师班，他们各自关注代数 K 理论中的一个当前问题。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查进行评估，被认为值得支持标准。