School of Mathematics, Statistics and Physics

Homological Algebra

Homological Algebra and Representation Theory

Overview

Homological Algebra

Homological algebra was invented to formalise aspects of algebraic topology, but has developed into something far more ambitious. With the advent of triangulated categories and, more generally, model categories, it can be seen as unifying many areas of mathematics, stretching from topology over algebra to analysis.

Triangulated categories and, more generally, model categories, have more structure than ordinary categories. For instance, there is a way to construct the mapping cone of a morphism; this formalises the eponymous construction from topology. Topological spaces form a modelcategory and, suitably modified, a triangulated category. But the point is that such categories abound. Complexes of modules over a ring, complexes of quasi-coherent sheaves on a scheme, and C*-algebras can all be turned into triangulated categories.

Each mathematical area is now able to feed its ideas back into the categories. From topological spaces come homotopy limits, from modules comes Auslander-Reiten theory, and from sheaves come t-structures, and these notions can all be made purely categorical and subsequently applied to triangulated categories from other areas.

From this philosophy has flowed a number of stunning developments. To mention but two, our understanding of Grothendieck duality, a main item of algebraic geometry, has been revolutionised by triangulated category methods imported from topology, and recent developments in Lie theory and combinatorics depend crucially on the introduction of cluster categories which are triangulated categories drawing on a range of inputs, not least Auslander-Reiten theory.

Staff

The following members of staff from the School of Mathematics and Statistics are members of the Homological Algebra and Representation Theory research group.

 

Professor Peter Jorgensen
Professor of Pure Mathematics

Email:
Telephone: +44 (0) 191 208 7288

Amit Shah
Research Associate

Telephone: 0191 208 7232