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的张量三角形几何是一个伞形程序，涵盖了代数几何中张量三角形范畴的几何研究、模表示理论、稳定同伦理论、动力理论、非交换拓扑等。无论是模块、空间、动机还是C*代数，对象通常都太过狂野，无法被归类为同构。然而，人们总是可以将稳定的物体分类在基本结构下，它们是：悬架、锥和张量积（这些类被称为厚张量理想）。这种分类是通过特定拓扑空间的子集来实现的，这些子集由PI构造，称为三角谱。利用Hopkins-Smith、Neeman-Thomason、Benson-Carlson-Rickard和Friedlander-Pevtsova的工作，在稳定同伦理论、代数几何和模表示理论中计算了该空间。计算非交换拓扑（等变kk理论）或动力实例中的三角谱是一个正在进行的主要项目，最近取得了进展。张量三角形几何更广泛的目标是在数学的某些部分建立新的桥梁，如下所示：从张量三角形几何所涵盖的任何领域确定概念、结果和技术，这些概念、结果和技术可以抽象出来，并因此应用到伞下的所有其他领域。最近的活动展示了许多这样的现象，在PI的工作和超越，像过滤维度的支持，粘合技术，皮卡德群，维特群，等等。张量三角形几何是一个相对较新的理论，它可以同时声称从代数到分析的大量例子，一个强大的抽象技术语料库和广泛的应用。代数几何和模表示理论中的几个新定理说明了张量三角几何的强度，这些定理的陈述不涉及张量三角几何，但其证明涉及。这个项目是高度跨学科的，吸引了来自不同领域的数学家。