My research interests have been centered around the representation theory of finite dimensional algebras, and have also included some work in commutative algebra and noncommutative algebraic geometry. In recent years my work has been strongly influenced by the cluster algebras of Fomin-Zelevinsky, which have led to the theory of cluster categories and cluster-tilted algebras and their generalizations. I now give a short description of some of the most important work, organised in 10 year periods.
My main work in the 70's was developing the theory of almost split sequences and irreducible maps, in work with Auslander. Another contribution was a module theoretic interpretation of the reflection functors of Bernstein-Gelfand-Ponomarev, in work with Auslander and Paltzeck. This was further generalized by Brenner-Butler, Happel-Ringel, leading to tilting theory. In addition, with Fossum and Griffith, we developed a theory of trivial extensions of abelian categories, dealing in particular with module categories for generalized triangular matrix rings.
In the 80's my main work was devoted to commutative algebra, and also orders. After Auslander had extended the theory of almost split sequences to also apply to interesting classes of commutative rings, most of my work in this period, jointly with Auslander, was devoted to investigating this class of rings from a new point of view. In particular, we investigated the question of finite Cohen-Macaulay type. Another main part of my work in the 80's, also using the general theory of almost split sequences, was the classification of two-dimensional tame orders of global dimension two, in work with Van den Bergh. This generalized work of Artin on the classification of two-dimensional maximal orders of finite representation type, and provided a new approach to the last problem. Also I worked with Riedtmann on the use of skew group algebra techniques for finite dimensional algebras.
In the 90's my work was again mainly devoted to finite dimensional algebras. Here, in work with Auslander, I investigated the connection between the contravariantly and covariantly finite subcategories on one hand, and (co)tilting modules on the other hand, generalizing work of Auslander-Smalø, and also inspired by work of Auslander-Buchweitz. Our work was applied by Ringel to quasihereditary algebras. This led to the tilting theory for algebraic groups, via work of Donkin. Another main topic was the work with Happel and Smalø on tilting in abelian categories and the introduction of quasitilted algebras. In particular, I worked further with Happel on the problem of classifying the hereditary categories with tilting object, where we obtained partial solutions. The general case for k algebraically closed was then proved by Happel, and in joint work we proved it for arbitrary fields. In work with Skowronski and Smalø we introduced and investigated the concept of short chains.
Since 2000, the first main work, with Van den Bergh, was classifying the noetherian hereditary abelian categories with Serre duality, over an algebraically closed field k. Some of the work was extended to arbitrary fields with Lenzing. I also investigated modules of infinite length in work with Ringel. Otherwise, my main work in this period has dealt with topics inspired by the introduction of cluster algebras by Fomin and Zelevinsky. Together with Buan, Marsh, Reineke and Todorov, we introduced and investigated cluster categories. We developed a theory for them, motivated by basic properties of the cluster algebras. In work with Buan and Marsh we initiated the theory of cluster-tilted algebras. I have been involved in further where here, with additional coauthors Seven, Iyama, Smith, Scott, Amiot, Thomas and Oppermann.
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