Simple-mindedness in triangulated categories

Representation theory is a the study of symmetry via the action of linear transformations on vector spaces; it follows a long-standing mathematical tradition of studying difficult problems by taking linear approximations. The naturalness of this idea means that representation theory sits at a nexus with many branches of mathematics, particularly, algebraic geometry, algebraic topology and combinatorics.

The concept of a triangulated category goes back to the work of British mathematician Frank Adams in algebraic topology at the University of Manchester in the 1950s and was developed by the Grothendieck school in Paris in the 1960s. Nowadays, representation theory is often formulated using triangulated categories, which permits the use of powerful methods of homological algebra and provides further crossovers with geometry, topology and even mathematical physics. A basic idea in representation theory is to study certain generators, or "building blocks" out of which all representations can be built. Originating in classic homological algebra over 60 years ago, projective objects, and their generalisations into Morita theory and tilting theory have enabled explosive development over the past 40 years with deep connections to Lie theory, quantum algebra, combinatorics, algebraic geometry and mathematical physics.

However, there is a much older kind of generator: simple objects, which have been studied since Schur in the 1880s. Schur's lemma, which says that simple representations are "perpendicular to each other", and the Jordan-Hölder theorem, which says that all representations can be built out of simple representations, are core components of undergraduate algebra curricula all over the world. The notions of simple-minded collection (SMC) and simple-minded system (SMS) are collections of objects in triangulated categories satisfying both Schur's lemma and the Jordan-Hölder theorem and provide the homological framework for simple objects.

The absence of a Morita theory of tilting theory for simple objects prevents the application of many powerful homological and combinatorial methods to basic problems in representation theory. The proposed research will rectify this problem by developing the theory to transport well-developed techniques from Morita theory and tilting theory to the theory of simple objects by exploiting a recent perspective developed by the proposer and his collaborators that simple objects are a kind of "negative cluster-tilting object". The proposed research will provide
- methods for constructing new sets of simple objects from old (mutation), which will provide new perspectives to some long-standing open problems such as the Auslander-Reiten Conjecture;
- a dictionary between projective objects and simple objects, which will provide new methods for modular representation theory; and,
- a discrete framework for studying geometric spaces arising out of homological algebra such as spaces of stability conditions.