Cyclotomic Spectra and p-Divisible Groups

范畴论和同伦理论基础的最新进展导致了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。主要的进展领域是抽象的：范畴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，而同伦理论试图理解形状的粗糙本质。另一方面，应用包括几何和多项式方程解的新结果，这些重要的研究领域在数学内外都有具体的应用。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理论是一个神秘但强大的计算数学对象的工具，算术几何是关于几何对具有有理坐标的多项式方程解的结构的影响。本项目为研究生和博士后提供科研培训机会。本课题的三个主要目标是：(1)通过证明如果R是一个光滑可交换的p局部可交换环，那么从k理论到拓扑循环同调的迹映射尊重p补全后的动机和同调滤波，直接比较p进上同调的动机和同调方法；(2)利用环切谱构建p进上同调的系数系统理论，并验证PI的可提性猜想。这将有助于解释形式群分类的窗框方法与最近由Anschuetz和Le Bras发展的棱柱体Dieudonne理论之间的关系，以及(3)理解由环切分t结构引起的棱柱体上同调的过滤。短期大师班将为研究生和博士后研究人员提供，每个人都专注于代数k理论中的一个当前问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.