Program

Founded by Frobenius and Schur more than a century ago, the representation theory of finite groups is today a thriving field with many recent successes. Current research centers on the many open questions, particularly regarding representations over the integers or rings of positive characteristic. Brauer developed block theory to understand better such representations, and it proved important in solving some problems in the classification of finite simple groups. In the last few years the area has been driven by a panoply of exciting new conjectures concerning correspondence of characters and derived equivalences of blocks. A key feature is the interplay between the research on general finite groups and important special classes of groups. Some major advances have been made in the representation theories of symmetric groups and groups of Lie type. Around the same time as Brauer, Eilenberg and MacLane gave an algebraic definition of group cohomology, analogous to similar constructions in topology, and it has been an important tool for those studying group representations. There are many fruitful interactions among mathematicians from diverse backgrounds who use group cohomology, including those who work in representation theory and algebraic topology. More recently we have seen very active interactions between homotopy theory, commutative algebra, group actions and modular representation theory. Topics such as p-local groups, group actions on finite complexes and homotopy representations blend algebra and topology in novel and productive ways. The goals of this semester are to focus the research on some of the conjectures and also to foster emerging interdisciplinary connections between several related areas in algebra and topology. An introductory workshop will concentrate on some of the many fundamental open problems in group representations. Topical workshops will emphasize the connections with the theory of Lie algebras and algebraic groups and with algebraic topology.