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.
, which was sponsored by the Centre de Recherches Math'ematique in Montreal, July -- December 2010. Group theory is a basic research area of modern mathematics with applications to computer science as well as to several other areas of mathematics. The goals of the 2010 Fall Semester were to present recent developments, to bring researchers from relevant branches of mathematics and computer science together to work on open questions, and to afford graduate students and young researchers an opportunity to learn about the field and talk to the leading experts in an informal setting. Program activities included five one-week workshops, several lecture series by eminent mathematicians, and a round table discussion on modern mathematical education. In connection with the round table discussion A. Borovik gave a lecture on ``How we save mathematics from mathematicians" (August 26, 2010). Outreach to students and young researchers was accomplished by a special weekly (sometimes twice weekly) seminar at McGill University, six public receptions featuring discussions on the future of mathematics, and a barbecue at Mont Royal for graduate and undergraduate students to meet the experts. In addition each one-week workshop included two or three minicourses aimed at young mathematicians. Funds from this grant supported fifteen graduate students, and eight postdoctoral students. In addition twenty-five other participants from the US, most of them young researchers were supported. These students and researchers participated in all program activities.
