Motivic and Equivariant Tensor-Triangular Geometry

任何科学研究，尤其是数学研究，都有一个令人兴奋的方面，那就是有机会将先验的不同现象合并成一个统一的理论。这种统一使研究整体上更有效率，减少重复，并促进创造力。技术方法以可能意想不到的方式从一个领域转移到另一个领域，不仅加强了研究，而且经常激发导致突破的新想法。这个项目关注的是这样一种统一，被称为“张量-三角形几何”，它融合了拓扑、代数几何、表示理论和其他数学领域的各个方面。在这个项目中，张量-三角形几何将被部署在有限群的模表示理论和数论中的动机理论的接口上。前者的明确性质增强了我们对后者的理解，在其他应用程序中提供分类结果。研究生将通过研究得到训练。该项目的具体目标是在两个不同的区域中出现的两类对象的分类。一方面，我们可以在Voevodsky的派生动机范畴中考虑各种地面域上的Artin-Tate动机，另一方面，我们可以考虑前有限群上的过滤表示的复合体，通常是上述域的绝对伽罗瓦群。由于张量-三角形几何，这样的分类相当于计算与所讨论的类别相关的空间（频谱）。所提出的方法涉及张量-三角形几何中的扩展。这些扩展是一种新的发展，在两种情况下都有体现，在动机情况下作为地面场的有限扩展，在等变情况下作为对有限指标子群的相应限制。通过过滤表征来理解Artin-Tate动机反过来又为更大更神秘的动机类别中的对象分类提供了新的思路，这是该领域研究人员的长期目标。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。

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.