Tensor triangulated categories: geometry and applications

PI的张量三角几何是一个伞计划，涵盖代数几何，模表示理论，稳定同伦理论，motivic理论，非交换拓扑等领域的张量三角范畴的几何研究。无论是模、空间、模还是C*-代数，对象通常都太过狂野，以至于不能归类为同构。然而，人们总是可以对在基本结构下稳定的对象进行分类，这些基本结构是：悬挂，锥和张量积（这些类被称为厚张量理想）。这种分类是由一个特定的拓扑空间的子集，由PI构造，并称为三角谱。这个空间已经在稳定同伦理论、代数几何和模表示理论中计算过，使用了霍普金斯-史密斯、尼曼-埃里森、本森-卡尔森-里卡德和弗里德兰德-佩夫佐娃的工作。在非交换拓扑（等变KK-理论）或motivic例子中计算三角谱是一个正在进行的重大项目，最近取得了进展。张量三角几何的更广泛的目标是在数学的某些部分建立新娘，如下所示：识别张量三角几何所涵盖的任何领域的概念，结果和技术，这些概念，结果和技术可以抽象并因此应用于所有其他领域。最近的活动已经展示了许多这样的现象，在PI的工作和超越，像过滤的支持，胶合技术，皮卡德集团，维特集团，和更多的尺寸。张量三角几何是一个相对较新的理论，它可以同时声称一个大目录的例子，从代数分析，一个强大的语料库的抽象技术和广泛的应用。代数几何和模表示论中的几个新定理说明了张量三角几何的力量，这些定理的陈述不涉及张量三角几何，但其证明涉及张量三角几何。这个项目是高度跨学科和呼吁数学家从非常不同的视野。

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.