Thematic program semester: Geometric, combinatorial and computational group theory

这是一项支持美国研究人员参与几何、组合和计算群论主题学期的提案，该学期计划于 2010 年秋季学期在蒙特利尔数学研究中心 (CRM) 举行。十九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、瑟斯顿和其他人提出了几何思想的融合，这些思想为早期的结果提供了很多启示，并导致了统称为几何群论的新研究方法。几何群论的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，群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，该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方案。 在过去的几年里，这些领域取得的巨大成就导致了许多基本和困难问题的进展，也导致了与数学其他领域和理论计算机科学的新颖联系。我们将把来自数学和计算机科学各个分支的人们聚集在一起，考虑解决该领域问题的新方法。我们的主要目标是加强群论与其他数学分支的联系，并强调新的有希望的观点。主题学期活动的基石将是一系列涵盖相关主题的研讨会。这些研讨会的举办将使学生、博士后和年轻研究人员受益。

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.