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.
Mathematics can sometimes be very sophisticated and sharp highly specialized research has to be counter-balanced by an important driving force of every scientific enterprise: Unification. The educated layman is familiar with famous unification problems in Physics but mathematical research contains similar forces, which are essential to its organized growth. Indeed, the profusion of multi-directional ideas and theories need to be bundled up into single, more abstract notions. The present project is both a sophisticated advanced research in Algebra and a unifying effort, aiming at encompassing several aspects of two major fields, namely algebraic geometry (commutative algebra) and modular representation theory of finite groups. In the language of so-called "tt-categories" ("tt" stands for "tensor triangulated") several phenomena appearing in group representation theory can be understood from a geometric perspective. This is the reason for the name "tensor triangular geometry" or "tt-geometry" for short. At the heart of this unification is the following observation. The standard technique of restricting a representation to a subgroup, in other words, of studying the symmetries provided by a group G by first studying a smaller subcollection of symmetries encapsulated in a so-called subgroup H of G, can be unified with the geometric technique of restricing geometric objects to smaller pieces (or building blocks). Actually, the naive algebro-geometric notion of "smaller piece" (in technical terms a "Zariski open") is not sufficient for that kind of transposition to representation theory and one needs to implement in tt-geometry the more advanced concept of "etale open" introduced by Grothendieck in algebraic geometry. Etale topology has played a major role in algebraic geometry since the second half of the 20th century and one of the cornerstones of the present research has been to show that restriction of representations, from a finite group G to a subgroup H, is nothing but a kind of etale extension suitably translated via tt-geometry, from algebraic geometry to representation theory. When mathematicians establish such connections between two different areas, they immediately try to transpose techniques and theorems valid in one area into the other. Here tt-geometry creates such a bridge between algebraic geometry and representation theory and it has been one of the main activities of this project to use this connection to produce new results. The most striking achievement has been obtained by applying the above philosophy to the problem of determining tensor-invertible objects in modular representation theory (in technical terms "endotrivial modules"). On the algebro-geometric side, tensor-invertible objects are line bundles, as for instance the famous Moebius Band over the circle. Thanks to the tt-geometry approach to restriction to a subgroup, it has been possible to describe those tensor-invertible representations of the finite group G which become trivial over its Sylow p-subgroup. The result has strong analogies with the geometry counterpart for an etale cover. The same techniques also allow the construction of an obstruction for a tensor-invertible on the Sylow subgroup to extend to the whole group, by means of a second cohomology group very much alike to a second etale cohomology group. More generally, and somewhat more technically, the tt-geometry approach allows us to construct a Grothendieck topology on finite G-sets (sets with symmetries in our group G) in such a way that representations of various subgroups of index prime to the characteristic p form what is called a "stack", i.e., satisfy good gluing properties from the various building blocks to the big picture. On the way, several other theorems have been established, of more technical nature but which also fit in the above strategy of extending known results in one area, at one end of the abstract theory, into new areas at another end, which are only reachable through the abstract plateform.
