Preface The American Mathematical Society, the Institute of Mathematical Statis- tics, and the Society for Industrial and Applied Mathematics sponsored a joint summer research conference at the University of Washington, from July 2D-24, 1997, on Trends in the Representation Theory of Finite Dimensional Algebras. The conference was organized by E. L. Green (Virginia Polytechnic Institute and State University) and B. Huisgen-Zimmermann (University of California at Santa Barbara). The lectures focused on the following three topics: (1) Interactions between representation theory and algebraic geometry (2) Homological methods (3) Applications of representation theory to the study of quantum groups. All three lines were strongly represented in the conference by lectures providing keynote reports and talks addressing current research, as well as by contributions to this volume. (1) The analysis of representations via algebraic varieties is comparatively young. It goes back to the 1970's, in particular to Gabriel, Bernstein, I. M. Gel'fand, S. I. Gel'fand, and Ponomarev. Their representation-theoretic work, based heavily on geometric argumentation, proved the varieties of modules of fixed dimension to store an enormous amount of accessible algebraic information. Another triggering event was the appearance of two pivotal papers by Kac in the early 1980's, broadening insights of Gabriel and Nazarova among others and completing, via invariant theory, the description of the dimension vectors of the finite dimensional indecomposable representations of quivers. Since then, numerous far-reaching geometric paths have been cut through the subject, with a great deal of movement having occurred in the recent past. (2) Homological methods constitute one of the most time-honored lines within representation theory. Among the pioneers in this direction were Cartan, Eilen- berg, Nakayama, MacLane, Nagata, Auslander, Buchsbaum, Serre. One would have to list many others, in particular if one were to go back to the commu- tative or topological roots of the subject. As for the commutative origins, let us just point to Hilbert, whose Syzygy Theorem, proved last century, was one of the first ring-theoretic results of an unmistakably homological nature. In the ix
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