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

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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.