Motivic and Equivariant Tensor-Triangular Geometry

动机和等变张量三角形几何

• 批准号: 1901696
• 负责人: Paul Balmer
• 金额: $ 31.99万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2023-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1901696&HistoricalAwards=false
• 关键词: Motivic; Equivariant; Tensor; Triangular; Geometry

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.

任何科学研究（尤其是数学研究）的一个令人兴奋的方面是有机会将先验的不同现象合并为统一的理论。这种统一使研究整体更加高效，减少重复并培养创造力。以潜在意想不到的方式将技术方法从一个领域转移到另一个领域，不仅加强了研究，而且常常激发新的想法，从而带来突破。该项目关注的是这样一种统一，称为“张量三角几何”，它将拓扑、代数几何、表示论和其他数学领域的各个方面合并在一个框架下。在这个项目中，张量三角几何将被部署在有限群模表示论和数论动机理论的接口上。前者的明确性质增强了我们对后者的理解，提供了其他应用程序中的分类结果。研究生将通过研究接受培训。该项目的具体目标是对两个不同区域中出现的两个类别的对象进行分类。一方面，我们可以考虑 Voevodsky 的派生动机类别中各种基础领域的 Artin-Tate 动机，另一方面，我们可以考虑亲有限群（通常是上述领域的绝对伽罗瓦群）上的过滤表示的复合体。由于张量三角形几何，这种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 etale 扩展。这些扩展是新的发展，在两种情况下都有体现，作为动机情况下地面场的有限扩展，以及等变情况下对有限指数子群的相应限制。通过过滤表示来理解 Artin-Tate 动机反过来又为更大、更神秘的动机类别中的物体分类提供了新的线索，这是该领域研究人员的长期目标。该奖项反映了 NSF 的法定使命，并已被通过使用基金会的智力优点和更广泛的影响审查标准进行评估，认为值得支持。

项目成果

Permutation modules and cohomological singularity

排列模和上同调奇点

• DOI: 10.4171/cmh/534
• 发表时间: 2022
• 期刊: Commentarii Mathematici Helvetici
• 影响因子: 0.9
• 作者: Balmer, Paul;Gallauer, Martin
• 通讯作者: Gallauer, Martin

Finite permutation resolutions

有限排列分辨率

• DOI: 10.1215/00127094-2022-0041
• 发表时间: 2023
• 期刊: Duke Mathematical Journal
• 影响因子: 2.5
• 作者: Balmer, Paul;Gallauer, Martin
• 通讯作者: Gallauer, Martin

Green equivalences in equivariant mathematics

• DOI: 10.1007/s00208-021-02145-2
• 发表时间: 2020-01
• 期刊: Mathematische Annalen
• 影响因子: 1.4
• 作者: Paul Balmer;Ivo Dell’Ambrogio

Computing homological residue fields in algebra and topology

• DOI: 10.1090/proc/15412
• 发表时间: 2020-07
• 期刊: arXiv: Category Theory
• 作者: Paul Balmer;James C. Cameron

Resolutions by permutation modules

排列模块的解析

• DOI: 10.1007/s00013-020-01493-w
• 发表时间: 2020
• 期刊: Archiv der Mathematik
• 影响因子: 0.6
• 作者: Balmer, Paul;Benson, Dave
• 通讯作者: Benson, Dave