Tensor triangulated categories: geometry and applications

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

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.