Thematic program semester: Geometric, combinatorial and computational group theory

这是一项支持美国研究人员参加计划于2010年秋季在蒙特利尔数学研究中心（CRM）举办的关于几何、组合和计算群论的专题学期的建议。十九世纪末组合群论的起源与几何问题密切相关，但后来的发展倾向于代数和逻辑。在20世纪80年代的大炮，格罗莫夫，瑟斯顿和其他人产生了注入几何思想，其中大量阐明了早期的结果，并导致新的研究方法统称为几何群论。几何群论的主要主题涉及细胞复合物的大规模几何，群的渐近性质，以及与理论计算机科学的联系。最近的结果的渐近性质的群体已特别有用的密码分析的群体为基础的密码系统发生在后量子密码学作为可能的替代品，目前的公钥系统。 在过去的几年里，这些领域取得了巨大的成就，导致了一些基本和困难问题的进展，并与其他数学领域和理论计算机科学建立了新的联系。我们将把来自数学和计算机科学各个分支的人聚集在一起，考虑新的方法来解决该领域的开放问题。我们的主要目标是加强群论与其他数学分支的联系，并突出新的有前途的观点。专题学期活动的一个基石将是涵盖相关主题的一系列讲习班。这些研讨会将组织受益于学生，博士后和年轻的研究人员。

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.