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1 - Introduction

Published online by Cambridge University Press:  07 October 2011

Edited by
Raf Cluckers ,
Johannes Nicaise  and
Julien Sebag
Raf Cluckers
Affiliation:
Université de Lille
Johannes Nicaise
Affiliation:
Katholieke Universiteit Leuven, Belgium
Julien Sebag
Affiliation:
Université de Rennes I, France
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Summary

Since its creation by Kontsevich in 1995, motivic integration has developed quickly in several directions and has found applications in various domains, such as singularity theory and the Langlands program. In its development, it incorporated tools from model theory and nonarchimedean geometry. The aim of the present book is to give an introduction to different theories of motivic integration and related topics.

Motivic integration is a theory of integration for various classes of geometric objects. One term in this “trade-name” seems unusual: motivic. What is a motive and in what sense is this theory of integration motivic? These questions lie at the heart of the theory. We will try to answer them in Section 4. First, we briefly introduce the two other protagonists of the present book: model theory and non-archimedean geometry. We will explain below how they interact with the theory of motivic integration.

Model theory

A central notion in model theory is language. A language ℒ is a collection of symbols, divided into three types: function symbols, relation symbols, and constant symbols. For every function symbol and relation symbol, one specifies the number of arguments. A formula in the language ℒ is built from these symbols, variables, Boolean combinations and quantifiers.

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Motivic Integration and its Interactions with Model Theory and Non-Archimedean Geometry , pp. 1 - 34
Publisher: Cambridge University Press
Print publication year: 2011

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References

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  • Introduction
  • Edited by Raf Cluckers, Université de Lille, Johannes Nicaise, Katholieke Universiteit Leuven, Belgium, Julien Sebag, Université de Rennes I, France
  • Book: Motivic Integration and its Interactions with Model Theory and Non-Archimedean Geometry
  • Online publication: 07 October 2011
  • Chapter DOI: https://doi.org/10.1017/CBO9780511667534.001
Available formats
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  • Introduction
  • Edited by Raf Cluckers, Université de Lille, Johannes Nicaise, Katholieke Universiteit Leuven, Belgium, Julien Sebag, Université de Rennes I, France
  • Book: Motivic Integration and its Interactions with Model Theory and Non-Archimedean Geometry
  • Online publication: 07 October 2011
  • Chapter DOI: https://doi.org/10.1017/CBO9780511667534.001
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Introduction
  • Edited by Raf Cluckers, Université de Lille, Johannes Nicaise, Katholieke Universiteit Leuven, Belgium, Julien Sebag, Université de Rennes I, France
  • Book: Motivic Integration and its Interactions with Model Theory and Non-Archimedean Geometry
  • Online publication: 07 October 2011
  • Chapter DOI: https://doi.org/10.1017/CBO9780511667534.001
Available formats
×