Thematic program semester: Geometric, combinatorial and computational group theory

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This is a proposal to support participation by researchers from the United States in the thematic semester on geometric, combinatorial and computational group theory which is planned for the 2010 fall semester at the Centre de Recherches Math\'{e}matiques (CRM) in Montr\'{e}al. The genesis of combinatorial group theory at the end of the nineteenth century was intimately involved with geometric questions, but subsequent developments tended towards algebra and logic. In the 1980's Cannon, Gromov, Thurston and others produced an infusion of geometric ideas which shed a great deal of light on earlier results and led to new research methods known collectively as geometric group theory. The main themes of geometric group theory involve the large scale geometry of cell complexes, asymptotic properties of groups, and connections with theoretical computer science. More recently results on asymptotic properties of groups have been particularly useful in the cryptanalysis of group based cryptosystems which occur in postquantum cryptography as possible alternatives to current public key systems. Over the past several years dramatic achievements in these areas have led to progress on a number of basic and difficult questions and also to novel connections with other areas of mathematics and with theoretical computer science. We will bring people from various branches of mathematics and computer science together to consider new approaches to open questions in the field. Our major goal is to strengthen the connections group theory has with other branches of mathematics and to highlight new promising perspectives. A cornerstone of activities of the thematic semester will be a collection of workshops covering related themes. These workshops will be organized to benefit students, postdocs, and young researchers.