Workshop Representations and Asymptotic Group Theory and visit by Nir Avni

这项建议有两个目的。首先，我们想在2009年4月在南安普顿大学组织一个为期两天的关于表示和渐近群论的研讨会。第二，我们希望支持我们的主要发言人之一Nir Avni（哈佛大学）为期六天的研究访问--包括两天的研讨会。数学、物理或化学对象的对称性--如图形、分子或晶体--形成一种称为群的代数结构。对有限群的研究导致了20世纪世纪数学中最引人注目的成就之一：所有有限单群的分类。一个重要的工具是通过它们的线性表示来研究群，即通过它们作为矩阵群的图像。渐近群论，旨在理解有限和无限群一样，关注的是某些算术不变量的群的渐近性质。在这个相对年轻的群论领域中，一个经典的方向是研究词的增长，这是由格罗莫夫的开创性工作而闻名的。在另一个方向，子群增长理论，研究无限群通过调查他们的有限指数子群的分布。“群的Zeta函数”-产生复杂函数的某些无穷级数用于编码相关增长序列的算术。他们已证明是发展这一理论的有力工具。直到最近，渐近群理论的研究人员才开始研究群的表示的分布，例如，利用在词和子群增长的背景下开发的技术。在表示增长研究的渐近性和算术的数量不可约复表示的任何给定程度提供了一个组。同样，zeta函数在近年来建立该领域的第一个重要成果方面发挥了关键作用。这些技术和成果将成为讲习班的重点。在另一个不同的方向上，已经做了相当大的努力来分类-或者至少是枚举-某些有限群族的特征标，例如李型群或p-群。工作坊中讨论的某些无限群的表示在数论中也起着重要的作用。我们预计工作坊将汇集在渐近群论（例如子群增长，表示增长，李群有限群表示的枚举）和相关学科（例如p进约化群的自守表示）方面具有相当不同背景的研究人员。会议将允许与会者在快速扩展的群论领域交流思想和工具，并相互学习。该研讨会将形成南英格兰Profinite组会议的一部分，由一组年轻的数学家谁分享profinite组的兴趣。他们每年举行大约三次会议，专门讨论特别感兴趣的研究课题，这些课题以博士生和博士后等年轻研究人员可以接受的水平提出。讲习班的目的是特别有利于年轻的数学家的背景是在群论。我们特意邀请了不同研究背景的演讲者（无限和有限群理论，特征理论和数论）。我们还将邀请研讨会的主要演讲者之一Nir Avni，为期六天-包括研讨会的两天。在他最近完成的博士论文中，Avni在表征增长zeta函数理论中的一些最重要的问题上取得了惊人的进展。基于这两个研究者目前的研究兴趣有很大的重叠，我们设想这次访问将成为进一步联合研究活动的焦点，研究群体的渐近表示理论。

The aim of this proposal is twofold. Firstly, we want to organise a two-day workshop on Representations and Asymptotic Group Theory at the University of Southampton, in April 2009. Secondly, we want to support a six-day research visit - including the two days of the workshop - of Nir Avni (Harvard University), one of our key speakers.The symmetries of a mathematical, physical or chemical object - such as a graph, a molecule or a crystal - form an algebraic structure called a group. The study of finite groups led to one of the most striking achievements of 20th century mathematics: the classification of all finite simple groups. An important tool is to investigate groups by means of their linear representations, i.e. by their images as matrix groups. Asymptotic group theory, which is aimed at understanding finite and infinite groups alike, is concerned with the asymptotic properties of certain arithmetic invariants of groups. A classical direction in this comparatively young area of group theory is the study of word growth, made famous by groundbreaking work of Gromov. In another direction, the theory of subgroup growth, one studies infinite groups by investigating the distribution of their finite index subgroups. `Zeta functions of groups' - certain infinite series which give rise to complex functions are used for encoding the arithmetic of associated growth sequences. They have proved powerful tools in developing the theory. Only recently, researchers in asymptotic group theory have begun to study the distributions of representations of groups, utilizing the techniques developed, for instance, in the context of word and subgroup growth. In representation growth one studies the asymptotics and the arithmetic of the numbers of irreducible complex representations of any given degree afforded by a group. Again, zeta functions have played a key role in establishing the first significant results in this area during recent years. These techniques and results will form the main point of focus of the workshop. In a different direction, considerable effort has been made to classify - or, at least, enumerate - characters of certain families of finite groups, such as groups of Lie type or p-groups. Representations of certain infinite groups considered in the workshop also play an important role in number theory.We envisage that the workshop will bring together researchers with quite distinct backgrounds in asymptotic group theory (e.g. subgroup growth, representation growth, enumeration of representations of finite groups of Lie type) and cognate disciplines (e.g. automorphic representations of p-adic reductive groups). The meeting will allow the participants to exchange ideas and tools in a rapidly expanding area of group theory, and to learn from one another. The workshop will form part of the South England Profinite Groups Meetings , organised by a group of young mathematicians who share an interest in profinite groups. They hold about three meetings per year, dedicated to research topics of particular interest which are presented at an accessible level to younger researchers like PhD students and postdocs. The workshop is designed to be of particular benefit to younger mathematicians whose background is in group theory. We have deliberately chosen to invite speakers of varied research backgrounds (infinite and finite group theory, character theory and number theory).We will also host one of the key speakers of the workshop, Nir Avni, for six days - including the two days of the workshop. In his recently completed PhD thesis, Avni has made spectacular progress on some of the most important questions in the theory of representation growth zeta functions. Based on the great overlap with the current research interests of the two investigators, we envisage that this visit will form the focal point for further joint research activity in asymptotic representation theory of groups.