Workshop Representations and Asymptotic Group Theory and visit by Nir Avni

这一建议的目的是双重的。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。其次，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（如图、分子或晶体）的对称性形成称为群的代数结构。对有限群的研究导致了20世纪数学最显著的成就之一：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，即通过它们的图像作为矩阵群。渐近群论研究群的某些算术不变量的渐近性质，目的在于理解有限群和无限群。在这个相对年轻的群论领域中，一个经典的方向是对词语增长的研究，这一研究因格罗莫夫的开创性工作而闻名。在子群增长理论的另一个方向上，人们通过研究无限群的有限指标子群的分布来研究无限群。“群的ζ函数”——某些产生复函数的无穷级数被用来编码相关生长序列的算术。它们已被证明是发展这一理论的有力工具。直到最近，渐近群论的研究人员才开始研究群体表征的分布，例如，利用在单词和子群增长的背景下开发的技术。在表示增长中，人们研究了群给出的任意给定阶的不可约复表示的渐近性和数目的算术。再一次，zeta函数在近年来建立这一领域的第一个重要结果中发挥了关键作用。这些技术和结果将构成研讨会的主要焦点。在另一个方向上，已经作出了相当大的努力来分类——或者至少是列举——有限群的某些族的特征，例如李型群或p群。在研讨会中考虑的某些无限群的表示在数论中也起着重要的作用。我们设想研讨会将汇集在渐近群论（如子群增长、表示增长、有限李型群的表示枚举）和同源学科（如p进约化群的自同构表示）方面具有截然不同背景的研究人员。会议将允许参与者在迅速扩大的群体理论领域交换想法和工具，并相互学习。研讨会将成为南英格兰无限群会议的一部分，由一群对无限群有共同兴趣的年轻数学家组织。他们每年大约举行三次会议，专门讨论特别感兴趣的研究课题，这些课题面向博士生和博士后等年轻研究人员。该研讨会的目的是为具有群论背景的年轻数学家提供特别的好处。我们特意选择邀请不同研究背景的演讲者（无限和有限群论，特征论和数论）。我们还将邀请研讨会的主要演讲者之一Nir Avni参加为期六天的会议，其中包括两天的研讨会。在他最近完成的博士论文中，Avni在表示增长zeta函数理论中的一些最重要的问题上取得了惊人的进展。基于与两位研究者当前的研究兴趣的巨大重叠，我们设想这次访问将成为群体渐近表示理论进一步联合研究活动的焦点。

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.