Thematic program semester: Geometric, combinatorial and computational group theory

这是一项支持来自美国的研究人员参加几何、组合和计算群理论主题学期的提案，该学期计划于2010年11月在数学研究中心（CRM）举行。19世纪末，组合群论的起源与几何问题密切相关，但随后的发展倾向于代数和逻辑。20世纪80年代，坎农、格罗莫夫、瑟斯顿等人产生了一系列几何思想，这些思想对早期的结果有了很大的启发，并导致了新的研究方法，统称为几何群论。几何群论的主要主题涉及细胞复合体的大规模几何，群的渐近性质，以及与理论计算机科学的联系。最近关于群的渐近性质的结果在基于群的密码系统的密码分析中特别有用，这些密码系统出现在后量子密码学中，作为当前公钥系统的可能替代品。在过去的几年里，这些领域取得了巨大的成就，在一些基本和困难的问题上取得了进展，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，考虑解决该领域开放性问题的新方法。我们的主要目标是加强群论与其他数学分支的联系，并突出新的前景。主题学期活动的基石将是一系列涵盖相关主题的研讨会。这些研讨会将为学生、博士后和年轻研究人员组织。

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.