Applications of triangulated categories in representation theory

Over the last few decades, triangulated categories have moved from being considered rather technical tools in
some of the more involved applications of homological algebra in algebraic geometry, to assume central
prominence as an object of study in many algebraic areas of mathematics: notably representation theory of
various kinds, commutative algebra and homotopy theory, as well as algebraic geometry. In all of these fields,
the language of triangulated categories has immensely clarified the ideas behind existing central problems, as
well as raising important new problems.
All of these fields have also benefited hugely from interactions between researchers with differing backgrounds
but with an interest in triangulated categories in common, leading to an extremely fruitful cross-fertilization.
Representation theory, for example, has benefited from insights originating in topology and algebraic geometry
that have, as well as inspiring theoretical advances, given new practical methods of calculation. For example,
one of the major unsolved conjectures in the modular representation theory of general finite groups is Broue's
Abelian Defect Group Conjecture, and ideas from topology have led to techniques that have been used to verify
extremely non-trivial cases of the conjecture in work of Holloway, Robbins, Craven, Rouquier and many others.
In the modular representation theory of finite groups in particular (and more generally, in the representation
theory of finite-dimensional algebras), various triangulated categories have become increasingly important in
recent years. For example, the stable module category has been studied implicitly for many decades, although
its structure as a triangulated category has been fully used only more recently, leading to a major clarification
and extension of the theory of varieties of modules. The derived category is of crucial importance via Broue's
Abelian Defect Group Conjecture, which gives a structural explanation for several older observations and
conjectures. More recently, categories related to these, such as various quotients, extensions, and relative
versions have also been studied.
This PhD project will be part of the still thriving area of cross-disciplinary research, which will surely continue for
a considerable time to provide inspiration for further progress in representation theory. Modular representation
theory has the advantage (over many of the other fields of mathematics that use triangulated categories) that it
has a variety of examples that are readily available, and simple enough to allow explicit calculations, whether by
hand or with the aid of computer algebra packages such as MAGMA.
The initial aim of the project will be to study recent major theoretical advances and insights, for example the work
of Rouquier on dimensions of triangulated categories, of Balmer and others on tensor triangulated categories,
and of Benson, Krause, Iyengar and Pevtsova on stable module categories, and to feed this back into concrete
examples. By initially looking at relatively simple examples and trying to understand what recent theoretical
developments have to say about them, it is expected that insight will be gained that can be applied to far more
complicated and less well-understood examples.