Asymptotic Group Theory and Model Theory - a two-day workshop

这项提议的目的有两个。首先，我们希望于2012年3月在伦敦大学皇家霍洛威举办一个为期两天的“渐近群论和模型理论”国际研讨会。其次，我们希望支持Uri Onn(内盖夫大学)和Christopher Voll(比勒费尔德大学)进行为期六天的研究访问--包括两天的研讨会。Voll是研讨会的联合组织者，也是主要演讲者之一。数学、物理或化学对象--如图形、分子或晶体--的对称性形成了一个称为群的代数结构。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群的线性表示来研究群，即通过群的像作为矩阵群来研究群。渐近群论是关于群的某些算术不变量的渐近性质的，其目的是理解有限群和无限群。在这个相对年轻的群论领域，一个经典的方向是研究单词增长，这一研究因格罗莫夫的开创性工作而闻名。在另一个方向，表示增长理论，人们通过研究无限维线性表示的分布来研究无限群。“群的Zeta函数”--某些产生复函数的无穷级数被用来编码相关增长序列的算法。事实证明，它们是发展这一理论的强大工具。在过去的几十年里，渐近和几何群论从大量涌入的技术中受益匪浅，这些技术来自一个被称为模型理论的数理逻辑领域。相反，代数和其他经典领域的具体问题，如几何和数论，已经刺激了重要的、更一般的模型理论的进步。这些技术和结果将构成研讨会的主要焦点，重点放在三个具体的主题上，这些主题可以用关键词来描述：近似群、极限群和动机整合。我们预计，研讨会将汇集在群论、模型论和同源学科方面具有相当不同背景的研究人员。会议将允许与会者在群体理论和模型理论的界面上，在迅速扩大的领域交流思想和工具，并相互学习。该研讨会将成为“南英格兰无限群会议”的一部分，该会议由一群对无限群有共同兴趣的数学家组织。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，这些主题是以博士生和博士后等年轻研究人员可以接触到的水平提出的。这个研讨会的目的是特别有利于年轻数学家，他们的主要背景是群论。我们特意邀请了不同研究背景的演讲者来描绘相关材料的大致情况。会议的两名参与者Onn和Voll将额外停留四天，与调查人员合作。他们访问的目的将是在正在进行的关于代表、齐塔人、团体职能和规划进一步研究合作的联合项目中从讲习班中立即获益。

The aim of this proposal is twofold. Firstly, we want to organise a two-day international workshop on "Asymptotic Group Theory and Model Theory" at Royal Holloway, University of London, in March 2012. Secondly, we want to support a six-day research visit - including the two days of the workshop - of Uri Onn (University of the Negev) and Christopher Voll (University of Bielefeld). Voll is co-organiser of the workshop and Onn one of the 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 representation growth, one studies infinite groups by investigating the distribution of their finite dimensional linear representations. `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. Over the last decades asymptotic and geometric group theory have very much benefited from the influx of techniques which come from an area of mathematical logic called model theory. Conversely, concrete problems in algebra and other classical areas, such as geometry and number theory, have stimulated important, more general model-theoretic advances. These techniques and results will form the main point of focus of the workshop, with an emphasis on three concrete topics, which can be described by the keywords: approximate groups, limit groups and motivic integration.We envisage that the workshop will bring together researchers with quite distinct backgrounds in group theory, model theory and cognate disciplines. The meeting will allow the participants to exchange ideas and tools in rapidly expanding areas at the interface of group theory and model theory, and to learn from one another. The workshop will form part of the `South England Profinite Groups Meetings', organised by a group of 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 primary background is in group theory. We have deliberately chosen to invite speakers of varied research backgrounds to draw a wide picture of the relevant material.Two participants of the meeting, Onn and Voll, will stay for four extra days to work with the investigator. The aim of their visit will be to draw immediate benefits from the workshop in ongoing joint projects on representation zeta functions of groups and the planning of further research collaborations.