Fields in Tensor-Triangular Geometry and Applications

张量三角形几何领域及其应用

• 批准号: 2153758
• 负责人: Paul Balmer
• 金额: $ 27.5万
• 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: 2022
• 资助国家: 美国
• 起止时间: 2022-08-01 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2153758&HistoricalAwards=false
• 关键词: Fields; Tensor; Triangular; Geometry; Applications

Tensor triangular geometry is a part of mathematics that unifies several aspects of otherwise distinct branches of algebraic geometry, topology, representation theory, and the theory of motives. In all those specialized areas, very complicated structures emerge that cannot be completely understood at a granular level but whose "overall shape" can be understood by means of a geometric invariant, called the spectrum. One feature of this theory is that the same invariant makes sense, and provides deep insight, in every one of these apparently very different settings. This versatility provides a unified methodology and builds many bridges between different sub-specialties of the mathematical landscape. The objective of this project is to analyze the "fundamental particles" of tensor-triangular geometry, that is, the minimal such structures and how they assemble to build much larger ones. This project will provide research training opportunities for graduate students. In more detail, the main problem to be addressed in this project is the concept of "point" in tensor-triangular geometry, in other words, the tensor-triangular fields, of which every large tensor-triangulated category is constituted. Such tensor-triangular fields already exist in special cases, like the ordinary fields of commutative algebra in algebraic geometry, or the Morava K-theories in stable homotopy theory. A main component of the program is to bring new techniques to bear on the problem of constructing such tensor-triangular fields in other settings, like representation theory, circumventing the shortcomings of so-called pi-points, or more ambitiously in motivic theory, where no candidates for the role of fields are known yet. Judging from the importance of (residue) fields in algebraic geometry for defining ranks, counting multiplicities, etc., a deeper understanding of tensor-triangular fields is expected to similarly generate many applications throughout tensor-triangular geometry.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.

张量三角几何形状是数学的一部分，它统一了代数几何，拓扑，表示理论和动机理论的不同分支的几个方面。在所有这些专业领域，出现了非常复杂的结构，这些结构在颗粒状水平上无法完全理解，但可以通过几何不变式（称为频谱）来理解其“整体形状”。该理论的一个特征是，在这些显然非常不同的设置中，同一不变性是有道理的，并提供了深刻的见识。这种多功能性提供了一种统一的方法，并在数学景观的不同亚规范之间建立了许多桥梁。该项目的目的是分析张量 - 三角形几何形状的“基本颗粒”，即，这种结构的最低限度以及它们如何组装以构建更大的结构。该项目将为研究生提供研究培训机会。更详细地，该项目要解决的主要问题是张量 - 三角形几何形状中的“点”概念，换句话说，张量 - 三角形字段，其中每个大张量张量 - 三角态类别构成。在特殊情况下，这种张量三角领域已经存在，例如代数几何形状中的通勤代数的普通领域，或稳定同型理论中的Morava K理论。该程序的一个主要组成部分是在其他环境中构建这种张量 - 三角形领域的问题，例如代表理论，避免了所谓的PI点的缺点，或者在动机理论中更加雄心勃勃，尚无候选领域的作用。从代数几何形状中（残留）字段在定义等级，计数多数等方面的重要性来判断，对张量 - 三角形领域的更深入的了解有望同样会同样产生许多张量的三角形几何形状。这一奖项反映了NSF的法定任务和综述的依据，这是通过评估范围的，这是通过评估的范围。