Cyclotomic Spectra and p-Divisible Groups

范畴论和同伦理论基础的最新进展导致了工作的爆炸和对不同领域的新见解，从代数几何和数论到物理学中的场论研究。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和这些类型之间的可接受变换的框架，而同伦论则试图理解形状的粗糙本质。另一方面，应用程序包括新的成果，几何和解决方案的多项式方程，重要的研究领域与具体的应用内外的数学。 本提案将利用代数K理论领域的这些进展，这是一个神秘但强大的工具，用于计算数学对象，算术几何，这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motivic和syntomic方法，证明如果R是光滑交换p-局部交换环，则从K-理论到拓扑循环同调的迹映射尊重p-完成后的motivic和syntomic过滤，（2）利用分圆谱构造了p-adic上同调的系数系理论，并验证了PI的可升性猜想，这将有助于解释形式群分类的窗框方法与Anschuetz和Le布拉斯最近发展的棱柱Dieudonne理论之间的关系，（3）理解由分圆t-结构引起的棱柱上同调的滤除。短期硕士班将提供给研究生和博士后研究人员，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NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

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.