Cyclotomic Spectra and p-Divisible Groups

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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.