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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.

Analytic spaces --- Geometry, Algebraic --- Model theory --- Valued fields --- Fields, Valued --- Topological fields --- Logic, Symbolic and mathematical --- Algebraic geometry --- Geometry --- Spaces, Analytic --- Analytic functions --- Functions of several complex variables --- Model theory. --- Valued fields. --- Analytic spaces. --- Geometry, Algebraic.

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Absolute values and their completions -like the p-adic number fields- play an important role in number theory. Krull's generalization of absolute values to valuations made applications in other branches of mathematics, such as algebraic geometry, possible. In valuation theory, the notion of a completion has to be replaced by that of the so-called Henselization. In this book, the theory of valuations as well as of Henselizations is developed. The presentation is based on the knowledge acquired in a standard graduate course in algebra. The last chapter presents three applications of the general theory -as to Artin's Conjecture on the p-adic number fields- that could not be obtained by the use of absolute values only.

Ordered algebraic structures --- Valued fields --- Fields, Valued --- Topological fields --- Valued fields. --- Algebra. --- Number theory. --- Geometry, algebraic. --- Logic, Symbolic and mathematical. --- Number Theory. --- Algebraic Geometry. --- Mathematical Logic and Foundations. --- Mathematics --- Mathematical analysis --- Algebra of logic --- Logic, Universal --- Mathematical logic --- Symbolic and mathematical logic --- Symbolic logic --- Algebra, Abstract --- Metamathematics --- Set theory --- Syllogism --- Algebraic geometry --- Geometry --- Number study --- Numbers, Theory of --- Algebra --- algebra --- landmeetkunde --- wiskunde --- logica --- getallenleer --- Algebraic geometry. --- Mathematical logic.