As for the theory of polynomial invariants of finite groups, the recent book "Algorithms on invariant theory" by Sturmfels demonstrates very impressively the state of art at least in the complex case. The modular case appears to be much harder and comparatively little is known uptil now; so here is a big need for further development for both, the theoretical background and the means to compute with generic examples which in turn should lead the way to theoretical insights.
The structural description of modules for both groups and algebras is of particular interest. Carlson, Green and Schneider have presented methods to compute projective resolutions: Determining projective indecomposable modules, endomorphism rings and relevant basic algebras they eventually get hold of the initial part of a projective resolution and this in turn yields information on quivers and relations of the underlying algebra.
The Todd-Coxeter coset enumeration algorithm is one of the most powerful tools of computational group theory. It may be viewed as a means to construct permutation representations of finitely presented groups. "Vector Enumeration" as introduced by S. Linton constitutes a very important tool; in this way, one constructs matrix representations over suitable fields for both, groups and algebras.
Computational modular representation theory started at the end of the seventies with the construction of the largest Janko group J_4. Since then, a lot of important data has been computed, using computer algebra systems like CAYLEY or GAP, and the MEAT-AXE, a collection of programs to construct and handle modular representations.
One of the main research areas of the Essen group lies in the implementation of routines to do linear algebra over finite fields on parallel machines, finally resulting in a kind of parallel MEAT-AXE to produce new and larger examples in the modular representation theory of finite groups.
Many important classes of algebras can be described by discrete invariants. This observation lead the Bielefeld group to develop the program package CREP (Combinatorial REPresentation theory) which is based on working just with these invariants. In particular, integer quadratic forms can be used in order to decide the representation type of some classes of algebras or vector space categories. There are several effective algorithms now for checking finite type whereas for tame and wild type there is presently remarkable progress which is stimulated and in some cases enabled by computer application.
The Conference. It is intended to demonstrate the importance of the use of computational methods in order to invoke an intensive cooperation of all 10 participating study groups of the network "Invariants and Representations of Algebras". The conference in Essen will serve as a platform to communicate the latest results and to introduce young researchers from all over Europe to up-to-date methods in computational representation theory of groups and algebras. At the meeting there will be facilities to demonstrate computer algebra systems and related packages; moreover, the participants are encouraged to demonstrate their own algorithms. In addition the meeting will be supplemented by a workshop 'Hands on Experiments'. Also postdoctoral fellows should benefit very much from this workshop.
Several keynote and focal speakers will present 1 hour lectures about algorithms and their implementations of several important developments in computational representation theory of finite dimensional algebras and finite groups. In order to train and teach graduate students, postdoctoral fellows and interested researchers there will be four introductory 2 hour lectures on computer algebra systems: The participants will be given computer access to use these systems outside the official lectures of the meeting.
The four Euroconferences organized by the Network "Invariants and Representations of Algebras".