Motivic and Equivariant Tensor-Triangular Geometry

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An exciting aspect of any scientific research, in particular in mathematics, is the opportunity to merge a priori disparate phenomena into a unified theory. Such unification makes research more efficient overall, reduces duplication, and fosters creativity. The transposition of technical methods from one field to another, in potentially unexpected ways, not only strengthens research but often inspires new ideas that lead to breakthroughs. This project is concerned with such a unification, known as "Tensor-Triangular Geometry," that merges aspects of topology, algebraic geometry, representation theory and of other areas of mathematics under a single umbrella. In this project, Tensor-Triangular Geometry will be deployed at the interface of modular representation theory of finite groups and the theory of motives in number theory. The explicit nature of the former enhances our understanding of the latter, providing classification results among other applications. Graduate students will be trained through the research. The specific objectives of this project are the classifications of objects that appear in two categories in two distinct areas. On one hand, one can consider Artin-Tate motives over various ground fields in Voevodsky's derived category of motives, and on the other one can consider complexes of filtered representations over pro-finite groups, typically the absolute Galois groups of the above fields. Thanks to tensor-triangular geometry, such classifications are equivalent to the computation of a space (the spectrum) associated to the categories in question. The proposed methods involve etale extensions in tensor-triangular geometry. These extensions are a new development that have incarnations in both settings, as finite extensions of the ground field in the motivic case and as the corresponding restriction to finite-index subgroups in the equivariant case. Understanding Artin-Tate motives via filtered representations in turn sheds new light on the classification of objects in the larger and more mysterious category of motives, which is a long-term goal of researchers in the field.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.