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Workshop Representations and Asymptotic Group Theory and visit by Nir Avni

研讨会表示和渐近群理论以及 Nir Avni 的访问

基本信息

批准号：
EP/G055408/1
负责人：
Christopher Voll
金额：
$ 1.88万
依托单位：
University of Southampton
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2009
资助国家：
英国
起止时间：
2009 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FG055408%2F1
关键词：
Workshop Representations Asymptotic Group Theory

项目摘要

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.

该提案的目的是双重的。首先，我们希望于 2009 年 4 月在南安普顿大学举办为期两天的表示和渐近群理论研讨会。其次，我们希望支持 Nir Avni（哈佛大学）的为期六天的研究访问（包括为期两天的研讨会），他是我们的主要发言人之一。数学、物理或化学对象（例如图形、分子或晶体）的对称性形成了对称性。称为群的代数结构。有限群的研究带来了 20 世纪数学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线性表示来研究组，即通过它们作为矩阵组的图像。渐近群论旨在理解有限群和无限群，它关注群的某些算术不变量的渐近性质。群论这个相对年轻的领域的一个经典方向是词增长的研究，它因格罗莫夫的开创性工作而闻名。另一方面，子群增长理论，通过研究无限群的有限指数子群的分布来研究无限群。 “群的 Zeta 函数”——某些产生复杂函数的无限级数用于编码相关增长序列的算术。事实证明，它们是发展理论的有力工具。直到最近，渐近群论的研究人员才开始利用在词和子群增长等方面开发的技术来研究群表示的分布。在表示增长中，人们研究群所提供的任何给定程度的不可约复表示的数量的渐进性和算术。近年来，zeta 函数在该领域取得第一个重大成果方面再次发挥了关键作用。这些技术和结果将成为研讨会的焦点。在不同的方向上，人们付出了相当大的努力来对有限群的某些族（例如李型群或 p 群）的特征进行分类（或者至少是枚举）。研讨会中考虑的某些无限群的表示也在数论中发挥着重要作用。我们设想研讨会将汇集渐近群论（例如子群增长、表示增长、Lie型有限群表示的枚举）和同源学科（例如p进还原群的自同构表示）方面具有截然不同背景的研究人员。这次会议将使与会者能够在快速扩展的群论领域交流想法和工具，并相互学习。该研讨会将成为南英格兰 Profinite Groups 会议的一部分，该会议由一群对 Profinite Group 有着共同兴趣的年轻数学家组织。他们每年举行大约三次会议，专门讨论特别感兴趣的研究主题，这些主题以博士生和博士后等年轻研究人员可以理解的水平呈现。该研讨会旨在让具有群论背景的年轻数学家特别受益。我们特意选择邀请具有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，为期六天，包括为期两天的研讨会。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。基于与两位研究者当前研究兴趣的巨大重叠，我们预计这次访问将成为群渐近表示理论进一步联合研究活动的焦点。