Arithmetic groups
Arithmetic groups
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算术组
DOI：
10.1017/cbo9781107325449.006
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发表时间：
2013
期刊：
影响因子：
--
通讯作者：
A. Rapinchuk
中科院分区：
文献类型：
--
作者：
Kai;D. Morris;Gopal Prasad;A. Rapinchuk
算术群的理论处理的矩阵群的条目是整数，或更一般地说，S-整数在一个整体领域。这一概念有着悠久的历史，可以追溯到高斯关于积分二次型的工作。算术群的现代理论与数论保持着密切的联系（例如，通过自守形式理论），但也依赖于代数群理论的各种方法，特别是在局部和全局领域（这个领域通常被称为代数群的算术理论），李群，代数几何和群论的各个方面（主要是同调方法和profinite群理论）。与此同时，算术群的结果在微分几何和双曲几何（因为许多重要流形的基本群通常是算术的）、组合数学（扩展图）和其他领域有着广泛的应用。算术群和其他重要的群类，如卡茨-穆迪群、自由群的自同构群和映射群之间也有有趣的联系和相似之处（目前还没有很好的理解）。讲习班的目的是调查最重要的成果，在理论的算术群体主要是在过去五年中，以使新的概念和方法获得更广泛的一组数学家的利益是密切相关的算术群体。讲习班汇集了34名数学家，从世界领先的专家到最近获得博士学位的人和研究生，他们致力于解决涉及算术组的各种问题。这导致了讲座之间和讲座之后非常活跃的交流。研讨会的科学计划包括3个小型课程（两个45分钟的讲座），17个调查和研究会谈30或45分钟，其中一个（由Bertrand Rémy）没有提前计划，一个问答环节，和一个开放问题环节。迷你课程的主题是：算术群的正特性的同调有限性（Kevin Wortman），伪约化群及其算术应用（Brian Conrad），以及算术Kac-Moody理论（Ralf Köhl）。关于同调有限性性质的迷你课程介绍了
The theory of arithmetic groups deals with groups of matrices whose entries are integers, or more generally, S-integers in a global field. This notion has a long history, going back to the work of Gauss on integral quadratic forms. The modern theory of arithmetic groups retains its close connection to number theory (for example, through the theory of automorphic forms) but also relies on a variety of methods from the theory of algebraic groups, particularly over local and global fields (this area is often referred to as the arithmetic theory of algebraic groups), Lie groups, algebraic geometry and various aspects of group theory (primarily, homological methods and the theory of profinite groups). At the same time, results about arithmetic groups have numerous applications in differential and hyperbolic geometry (as the fundamental groups of many important manifolds often turn out to be arithmetic), combinatorics (expander graphs), and other areas. There are also intriguing connections and parallels (which are currently not so well-understood) between arithmetic groups and other important classes of groups such as Kac-Moody groups, automorphism groups of free groups and mapping class groups. The objective of the workshop was to survey the most significant results in the theory of arithmetic groups obtained primarily in the last five years in order to make the new concepts and methods accessible to a broader group of mathematicians whose interests are closely related to arithmetic groups. The workshop brought together 34 mathematicians, from the world’s leading experts to recent PhD recipients and graduate students, working on a variety of problems involving arithmetic groups. This resulted in very active exchanges between and after the lectures. The scientific program of the workshop consisted of 3 mini-courses (two 45-min lectures each), 17 survey and research talks 30 or 45 min one of which (by Bertrand Rémy) was not planned in advance, a Q&A session, and an open problem session. The subjects of the mini-courses were: Homological finiteness properties of arithmetic groups in positive characteristic (Kevin Wortman), Pseudo-reductive groups and their arithmetic applications (Brian Conrad), and Towards an arithmetic Kac-Moody theory (Ralf Köhl). The mini-course on homological finiteness properties contained an account of a major breakthrough in the
正特征还原算术群的更高有限性：秩定理
影响因子：
4.9
作者：
Kai-Uwe Bux;Ralf Köhl;Stefan Witzel
通讯作者：
Stefan Witzel
影响因子：
0.7
作者：
Belolipetsky M
通讯作者：
Belolipetsky M
紧p进解析群和算术群的表示zeta函数
影响因子：
2.5
作者：
Avni N
通讯作者：
Avni N