The development of Maxim Kontsevich's initial ideas on motivic integration has unexpectedly influenced many other areas of mathematics, ranging from the Langlands program over harmonic analysis, to non-Archimedean analysis, singularity theory and birational geometry. This book assembles the different theories of motivic integration and their applications for the first time, allowing readers to compare different approaches and assess their individual strengths. All of the necessary background is provided to make the book accessible to graduate students and researchers from algebraic geometry, model theory and number theory. Applications in several areas are included so that readers can see motivic integration at work in other domains. In a rapidly-evolving area of research this book will prove invaluable. This second volume discusses various applications of non-Archimedean geometry, model theory and motivic integration and the interactions between these domains.

"Because of the variety of different aspects of the theory and the many areas of mathematics that come into play, a book like the present one is particularly precious for someone interested in learning about motivic integration as well as for someone- like the reviewer - who is familiar with some aspects of the theory but less with others and would like to learn more about this rich and beautiful subject."
Tommaso De Fernex, Mathematical Reviews