There will be a conference at the University of Georgia in May 2000, on the subject of Representation Theory and Computational Algebra. This is intended to mean the representation theory of finite groups and related finite dimensional algebras, cohomology of finite groups, and computer calculations in these areas. Over the last few years, the representation theory and cohomology of finite groups has seen some spectacular advances in a number of different directions. On the computational side, the development of the computer algebra package MAGMA by John Cannon and his associates in Australia has enabled explicit calculations of projective resolutions and cohomology of groups on an impressive scale. These computations have, in their turn, fed back into the design and development of the MAGMA package. Theoretical input for these calculations comes from group actions on finite simplicial complexes and associated topological constructions at the level of the classifying space. Methods originating in work of Peter Webb have been implemented by a number of people, including Adem, Carlson, Maginnis, Milgram and Smith. These calculations seem to be intimately related to the homotopy decompositions of classifying spaces appearing in recent work of Dwyer, Jackowski, McClure, Oliver, etc. A good example of the interaction between these techniques is given by the recent calculation of the cohomology of the Higman-Sims sporadic simple group by Adem, Carlson, Karagueuzian and Milgram. One of the strongest reasons for studying the cohomology of finite groups is its connections with representation theory. Recent developments in this direction have focused on structure of module categories, especially the associated stable categories and derived categories. Work of Benson, Carlson and Rickard on the stable category, and of Broue, Rickard, Rouquier and others on the derived category have been central to this development. Computations with specific groups have been at the core of these developments, and many of the theorems and conjectures in the subject have arisen this way.
Representation theory, in the context of this conference, means the representations of abstract groups as groups of matrices. Calculations in this area have implications in algebra, topology, geometry and number theory. Recent advances in high performance computing, combined with equally great theoretical advances over the last few years, have led to the development of powerful new techniques for calculating with representations and cohomology of groups, and this then feeds back into the theoretical development. A symbiotic relationship exists between the researchers carrying out these calculations and the developers of software such as the MAGMA package for algebra and number theory. The purpose of this conference is to bring together the people working in the theoretical and computational aspects of this and closely related subjects at this time of rapid development, in order to promote interaction and collaboration. The three main speakers will give series of expository talks on computational algebra, cohomology of groups and representation theory, while other participants will give more technical talks.
