Motivic Stable Homotopy Theory: a New Foundation and a Bridge to p-Adic and Complex Geometry

Introduction

This project is centered on the field of algebraic geometry and involves homotopy theory and analytic geometry. The overall goal is to unveil the underlying principles of a large variety of cohomology theories in algebraic and analytic geometry and develop robust foundations that facilitate the study of those cohomology theories from the vantage point of homotopy theory.

Objectives

This will be achieved through innovations of motivic stable homotopy theory beyond the current technical limitations of A1-homotopy invariance. In addition, its interdisciplinary perspective will be advanced, especially in relation to p-adic geometry and complex geometry. The research proposal consists of 5 main objectives, which are organically related to each other:

1. Six Functor Formalism
   The first objective is to establish a six functor formalism, which would be the most important challenge in non-A1-invariant motivic stable homotopy theory.

2. Kernel of A1-Localization
   The second objective is to investigate the kernel of the A1-localization and aims to describe it in terms of p-adic or rational Hodge realization, following the principle of trace methods of algebraic K-theory. In particular, in the p-adic context, this will lead to the p-adic rigidity, which will conclusively connect motivic homotopy theory with p-adic geometry.

3. Unstable Motivic Homotopy Theory
   The third objective is to find out the potential of unstable motivic homotopy theory and develop calculation techniques.

4. Motivic Filtration of Localizing Invariants
   The fourth objective is to establish a general and universal construction of motivic filtration of localizing invariants, such as algebraic K-theory and topological cyclic homology.

5. Analogue in Complex Geometry
   The last objective is to explore the analogue in complex geometry, which is an interesting unexplored subject that will pave the way for further developments of motivic homotopy theory for a broader range of analytic geometry.