The purpose of this workshop is to bring together mathematicians working on algebraic, analytic, combinatoric, geometric and topological aspects of group theory in order to strengthen each of these approaches through an exchange of techniques and ideas. The focus will be on these four topics: Geometric Group Theory. This is a relatively young field, with older and deeper roots in the study of groups from combinatorial and topological perspectives. In the mid 1980’s, spurred by ideas of Cannon and Gromov, group theorists began to pay attention to the geometric structures which cell compexes can carry. This attention shed light on the earlier combinatorial and topological investigations, and stimulated innovative ideas which have been developing at a rapid pace: Gromov hyperbolicity, Bestvina-Brady Morse theory, splittings and actions on trees, rapid decay and the Baum-Connes conjecture. Kleinian Groups. The study and application of recent advances in the classification of hyperbolic 3-manifolds (the solution of the tameness and ending lamination conjectures of Marden and Thurston) can lead to a better understanding of the geometry of closed hyperbolic 3-manifolds. This work also touches on Teichmuller theory, and questions concerning billiards and flows on Moduli space. Many of these avenues are potentially very fruitful for further research and synthesis between, up to now, largely disparate fields. Combinatorial Representation Theory. There is a productive interplay between combinatorics, geometry, finite groups, Lie theory and hyperplane arrangements in the applications to representation theory. Examples are: (1) the use of symmetric functions and Hecke algebras in the modular representation theory of finite groups of Lie type, (2) the use of braid groups and finite dimensional algebras in the study of categories of highest weight modules and (3) the use of tableaux, crystals, and the path model in the study of representations of algebras with triangular decomposition. Representation Theory of Finite Groups. Current research centers on many open questions, particularly regarding representations over the integers or rings of positive characteristic. Brauer developed block theory to better understand such representations, and in the last few years there have been many exciting new conjectures concerning correspondence of characters and derived equivalences of blocks. Topics such as p-local groups, group actions on finite complexes and homotpy representations blend algebra and topology in novel and productive ways. Invited Participants include: Mladen Bestvina, Jeff Brock, Dick Canary, Jon Carlson, Persi Diaconis, Alexander Kleshchev, Howard Masur, Jon Mccammond, Yair Minsky, Jeremy Rickard, Anne Schilling, Bhama Srinivasan, Richard Stanley, Nat Thiem, Karen Vogtmann. This workshop is to be held at the Banff International Research Station, Banff, Alberta, Canada SCHEDULE Thursday, November 17, 2005 16:00 Check-in begins (Front Desk – Professional Development Centre - open 24 hours) 17:30-19:30 Buffet Dinner, Donald Cameron Hall 20:00 Informal gathering in 2nd floor lounge, Corbett Hall Friday, November 18, 2005 7:00-8:45 Breakfast 8:45-9:00 Introduction and Welcome to BIRS 9:00 Lee Mosher, Geometric Group Theory 10:00 Coffee Break, 2nd floor lounge, Corbett Hall 10:30 Ken Bromberg, The topology of deformation spaces of hyperbolic 3-manifolds 11:30 Lunch 13:30 Guided Tour of The Banff Centre; meet in the 2nd floor lounge, Corbett Hall 14:30 Gunter Malle, Complex reflection groups and representations of reductive groups 15:30 Coffee Break, 2nd floor lounge, Corbett Hall 16:00 Alexander Kleschev, Flavors of symmetric groups 17:30 Dinner Saturday, November 19, 2005 7:00-9:00 Breakfast 9:00 Howard Masur, Fuchsian subgroups of the mapping class group 10:00 Coffee Break, 2nd floor lounge, Corbett Hall 10:30 Dave Benson, Representations, cohomology and duality 11:30 Group Photo; meet on the front steps of Corbett Hall 11:30 Lunch 13:30 Nat Thiem , The finite unitary group, Deligne-Lusztig induction, and tableaux combinatorics 14:30 Alain Valette, Quasi-isometric embeddings into Hilbert spaces, and growth of 1-cocycles on groups 15:30 Coffee Break, 2nd floor lounge, Corbett Hall 16:00 Karin Erdmann, Permutation modules and applications 17:30 Dinner Sunday, November 20, 2005 7:00-9:00 Breakfast 9:00 Ursula Hamenstaedt, Large-scale geometry of subgroups of mapping class groups 10:00 Coffee Break, 2nd floor lounge, Corbett Hall 10:30 Jon McCammond, Coxeter groups and Artin groups in Geometric Group Theory 11:30 Lunch 13:30 Alan Reid, LERF, extensions and retractions 14:30 Juan Souto, Heegard splittings and pseudo-Anosov maps 15:30 Coffee Break, 2nd floor lounge, Corbett Hall 16:00 Persi Diaconis, An outsider looks at Group Theory 17:30 Dinner Monday, November 21, 2005 7:00-9:00 Breakfast 9:00 Joel Kamnitzer, Mirkovic-Vilonen Polytopes 10:00 Coffee Break, 2nd floor lounge, Corbett Hall 10:30 Karen Vogtmann, The topology of automorphism groups of free groups 11:30 Lunch 13:30 Steve Kerckhoff, Eight flavors of geometry in a single (projective) cone 14:30 Jesper Grodal, Homotopical group theory 15:30 Coffee Break, 2nd floor lounge, Corbett Hall 16:00 Lauren Williams, The combinatorics and topology of totally nonnegative flag varieties 17:30 Dinner Tuesday, November 22, 2005 7:00-9:00 Breakfast 9:00 Informal Discussions recommended, as many participants must catch early flights 10:00 Coffee Break, 2nd floor lounge, Corbett Hall – to START no earlier than 10 am 10:30 Informal Discussions 11:30 Lunch Checkout by 12 noon.