Tensor triangulated categories: geometry and applications

张量三角类别：几何和应用

• 批准号: 0969644
• 负责人: Paul Balmer
• 金额: $ 23.85万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0969644&HistoricalAwards=false
• 关键词: Tensor; triangulated; categories; geometry; applications

The PI's Tensor Triangular Geometry is an umbrella program covering the geometric study of tensor triangulated categories in algebraic geometry, modular representation theory, stable homotopy theory, motivic theory, noncommutative topology, and beyond. Be they modules, spaces, motives or C*-algebras, objects are usually too wild to be classified up to isomorphism. However, one can always classify classes of objects stable under the basic constructions which are: suspension, cone and tensor product (such classes are known as thick tensor-ideals). This classification is made by means of subsets of a certain topological space, constructed by the PI and called the triangular spectrum. This space has been computed in stable homotopy theory, algebraic geometry and modular representation theory, using the work of Hopkins-Smith, Neeman-Thomason, Benson-Carlson-Rickard and Friedlander-Pevtsova. Computing the triangular spectrum in noncommutative topology (equivariant KK-theory) or in motivic examples is a major ongoing project where progress has recently been made. The broader ambition of tensor triangular geometry is that of building brides across some parts of mathematics as follows: Identify the concepts, results and techniques from any area covered by tensor triangular geometry which can be abstracted and consequently applied to all other areas under the umbrella. Recent activity has exhibited numerous such phenomenons, in the PI's work and beyond, like filtration by dimension of supports, gluing techniques, Picard groups, Witt groups, and more.Tensor triangular geometry is a relatively new theory which can simultaneously claim a large catalog of examples ranging from Algebra to Analysis, a strong corpus of abstract techniques and a broad range of applications. The strength of tensor triangular geometry is illustrated by several new theorems in algebraic geometry and modular representation theory, whose statement does not involve tensor triangular geometry but whose proof does. This project is highly interdisciplinary and appeals to mathematicians from very different horizons.

PI的张量三角几何形状是一项雨伞计划，涵盖了代数几何形状，模块化表示理论，稳定同义理论，动机理论，非交互性拓扑结构及以后的几何学研究。无论是模块，空间，动机还是C* - 代数，物体通常太野性了，无法分类为同构。但是，人们总是可以在基本结构下稳定的对象类别分类：悬架，锥体和张量产品（此类类称为厚度张量 - 理想）。该分类是通过PI构建的某个拓扑空间的子集进行的，称为三角形光谱。使用霍普金斯 - 史密斯，尼曼·托马森，本森 - 卡尔森 - 雷克·里克·雷克和弗里德兰德·佩索沃的工作，已经在稳定同义理论，代数几何形状和模块化代表理论中计算了该空间。计算非共同拓扑（Equivariant KK理论）或动机示例中的三角形频谱是最近取得进展的主要项目。张量三角几何形状的更广泛的野心是在数学的某些部分建造新娘的野心如下：确定张量三角几何形状覆盖的任何区域的概念，结果和技术，这些几何形状可以抽象并应用于伞形下的所有其他领域。最近的活动在PI的工作及其他方面表现出了许多此类现象，例如通过支撑技术，胶合技术，PICARD组，WITT组等过滤。Tensor三角几何形状是一种相对较新的理论，可以同时声称大量的目录的目录范围从代数到分析，强大的抽象技术和广泛的应用。代数几何形状和模块化表示理论中的几个新定理说明了张量三角几何形状的强度，其陈述不涉及张量的三角形几何形状，而是其证明的作用。该项目是高度跨学科的，并吸引了来自不同视野的数学家。