Research Projects of Thorsten Holm

Professor Thorsten Holm is working in representation theory, homological algebra and algebraic combinatorics, with emphasis on the use of derived categories and related triangulated categories. Triangulated categories nowadays appear throughout mathematics, in particular derived categories play a fundamental role in algebra, algebraic geometry and also in mathematical physics (string theory).

A particular focus in T. Holm's recent work have been cluster categories; these are triangulated categories which have been successfully used as a 'categorification' of Fomin and Zelevinsky's cluster algebras; see also S. Fomin's Cluster Algebras Portal.

The contributions of T. Holm in this context are aiming at understanding the structure of cluster categories. For instance, the following projects have been successfully carried out:

Structure of cluster categories of infinite Dynkin type and their cluster tilting subcategories

Derived equivalence classifications of cluster-tilted algebras of Dynkin type

Classification of torsion pairs in cluster categories of Dynkin type

Torsion pairs are generalizations of Beilinson, Bernstein and Deligne's t-structures which are also studied intensively in algebraic geometry. The classifications we obtain have a strong combinatorial flavour and allow enumeration and related combinatorial results. Torsion pairs in cluster categories are an on-going project, which is partially supported by the DFG research priority programn SPP 1388 'Representation Theory'.