Spectral Theory And Analytic Geometry Over Non Archimedean Fields

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Spectral Theory and Analytic Geometry over Non-Archimedean Fields

Vladimir G. Berkovich
Spectral Theory and Analytic Geometry over Non Archimedean Fields Book PDF

Author : Vladimir G. Berkovich
Publisher : American Mathematical Soc.
Page : 169 pages
File Size : 40,7 Mb
Release : 2012-08-02
Category : Algebraic number theory
ISBN : 9780821890202

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Spectral Theory and Analytic Geometry over Non-Archimedean Fields by Vladimir G. Berkovich Pdf

The purpose of this book is to introduce a new notion of analytic space over a non-Archimedean field. Despite the total disconnectedness of the ground field, these analytic spaces have the usual topological properties of a complex analytic space, such as local compactness and local arcwise connectedness. This makes it possible to apply the usual notions of homotopy and singular homology. The book includes a homotopic characterization of the analytic spaces associated with certain classes of algebraic varieties and an interpretation of Bruhat-Tits buildings in terms of these analytic spaces. The author also studies the connection with the earlier notion of a rigid analytic space. Geometrical considerations are used to obtain some applications, and the analytic spaces are used to construct the foundations of a non-Archimedean spectral theory of bounded linear operators. This book requires a background at the level of basic graduate courses in algebra and topology, as well as some familiarity with algebraic geometry. It would be of interest to research mathematicians and graduate students working in algebraic geometry, number theory, and -adic analysis.

Integration of One-forms on P-adic Analytic Spaces. (AM-162)

Vladimir G. Berkovich
Integration of One forms on P adic Analytic Spaces AM 162 Book PDF

Author : Vladimir G. Berkovich
Publisher : Princeton University Press
Page : 164 pages
File Size : 48,8 Mb
Release : 2007
Category : Mathematics
ISBN : 9780691128627

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Integration of One-forms on P-adic Analytic Spaces. (AM-162) by Vladimir G. Berkovich Pdf

Among the many differences between classical and p-adic objects, those related to differential equations occupy a special place. For example, a closed p-adic analytic one-form defined on a simply-connected domain does not necessarily have a primitive in the class of analytic functions. In the early 1980s, Robert Coleman discovered a way to construct primitives of analytic one-forms on certain smooth p-adic analytic curves in a bigger class of functions. Since then, there have been several attempts to generalize his ideas to smooth p-adic analytic spaces of higher dimension, but the spaces considered were invariably associated with algebraic varieties. This book aims to show that every smooth p-adic analytic space is provided with a sheaf of functions that includes all analytic ones and satisfies a uniqueness property. It also contains local primitives of all closed one-forms with coefficients in the sheaf that, in the case considered by Coleman, coincide with those he constructed. In consequence, one constructs a parallel transport of local solutions of a unipotent differential equation and an integral of a closed one-form along a path so that both depend nontrivially on the homotopy class of the path. Both the author's previous results on geometric properties of smooth p-adic analytic spaces and the theory of isocrystals are further developed in this book, which is aimed at graduate students and mathematicians working in the areas of non-Archimedean analytic geometry, number theory, and algebraic geometry.

Non-Archimedean Operator Theory

Toka Diagana,François Ramaroson
Non Archimedean Operator Theory Book PDF

Author : Toka Diagana,François Ramaroson
Publisher : Springer
Page : 156 pages
File Size : 55,5 Mb
Release : 2016-04-07
Category : Mathematics
ISBN : 9783319273235

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Non-Archimedean Operator Theory by Toka Diagana,François Ramaroson Pdf

This book focuses on the theory of linear operators on non-Archimedean Banach spaces. The topics treated in this book range from a basic introduction to non-Archimedean valued fields, free non-Archimedean Banach spaces, bounded and unbounded linear operators in the non-Archimedean setting, to the spectral theory for some classes of linear operators. The theory of Fredholm operators is emphasized and used as an important tool in the study of the spectral theory of non-Archimedean operators. Explicit descriptions of the spectra of some operators are worked out. Moreover, detailed background materials on non-Archimedean valued fields and free non-Archimedean Banach spaces are included for completeness and for reference. The readership of the book is aimed toward graduate and postgraduate students, mathematicians, and non-mathematicians such as physicists and engineers who are interested in non-Archimedean functional analysis. Further, it can be used as an introduction to the study of non-Archimedean operator theory in general and to the study of spectral theory in other special cases.

Spectral Theory and Geometry

E. Brian Davies,Yu Safarov,London Mathematical Society,International Centre for Mathematical Sciences
Spectral Theory and Geometry Book PDF

Author : E. Brian Davies,Yu Safarov,London Mathematical Society,International Centre for Mathematical Sciences
Publisher : Cambridge University Press
Page : 344 pages
File Size : 55,6 Mb
Release : 1999-09-30
Category : Mathematics
ISBN : 9780521777490

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Spectral Theory and Geometry by E. Brian Davies,Yu Safarov,London Mathematical Society,International Centre for Mathematical Sciences Pdf

Authoritative lectures from world experts on spectral theory and geometry.

Pseudo-Differential Equations And Stochastics Over Non-Archimedean Fields

Anatoly Kochubei
Pseudo Differential Equations And Stochastics Over Non Archimedean Fields Book PDF

Author : Anatoly Kochubei
Publisher : CRC Press
Page : 344 pages
File Size : 40,5 Mb
Release : 2001-08-03
Category : Mathematics
ISBN : 0203908163

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Pseudo-Differential Equations And Stochastics Over Non-Archimedean Fields by Anatoly Kochubei Pdf

Provides comprehensive coverage of the most recent developments in the theory of non-Archimedean pseudo-differential equations and its application to stochastics and mathematical physics--offering current methods of construction for stochastic processes in the field of p-adic numbers and related structures. Develops a new theory for parabolic equat

Tropical and Non-Archimedean Geometry

Omid Amini,Matthew Baker,Xander Faber
Tropical and Non Archimedean Geometry Book PDF

Author : Omid Amini,Matthew Baker,Xander Faber
Publisher : American Mathematical Soc.
Page : 274 pages
File Size : 45,6 Mb
Release : 2014-12-26
Category : Mathematics
ISBN : 9781470410216

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Tropical and Non-Archimedean Geometry by Omid Amini,Matthew Baker,Xander Faber Pdf

Over the past decade, it has become apparent that tropical geometry and non-Archimedean geometry should be studied in tandem; each subject has a great deal to say about the other. This volume is a collection of articles dedicated to one or both of these disciplines. Some of the articles are based, at least in part, on the authors' lectures at the 2011 Bellairs Workshop in Number Theory, held from May 6-13, 2011, at the Bellairs Research Institute, Holetown, Barbados. Lecture topics covered in this volume include polyhedral structures on tropical varieties, the structure theory of non-Archimedean curves (algebraic, analytic, tropical, and formal), uniformisation theory for non-Archimedean curves and abelian varieties, and applications to Diophantine geometry. Additional articles selected for inclusion in this volume represent other facets of current research and illuminate connections between tropical geometry, non-Archimedean geometry, toric geometry, algebraic graph theory, and algorithmic aspects of systems of polynomial equations.

Motivic Integration and its Interactions with Model Theory and Non-Archimedean Geometry: Volume 2

Raf Cluckers,Johannes Nicaise,Julien Sebag
Motivic Integration and its Interactions with Model Theory and Non Archimedean Geometry Volume 2 Book PDF

Author : Raf Cluckers,Johannes Nicaise,Julien Sebag
Publisher : Cambridge University Press
Page : 263 pages
File Size : 49,8 Mb
Release : 2011-09-22
Category : Mathematics
ISBN : 9781139501736

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Motivic Integration and its Interactions with Model Theory and Non-Archimedean Geometry: Volume 2 by Raf Cluckers,Johannes Nicaise,Julien Sebag Pdf

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.

Nonarchimedean and Tropical Geometry

Matthew Baker,Sam Payne
Nonarchimedean and Tropical Geometry Book PDF

Author : Matthew Baker,Sam Payne
Publisher : Springer
Page : 526 pages
File Size : 40,8 Mb
Release : 2016-08-18
Category : Mathematics
ISBN : 9783319309453

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Nonarchimedean and Tropical Geometry by Matthew Baker,Sam Payne Pdf

This volume grew out of two Simons Symposia on "Nonarchimedean and tropical geometry" which took place on the island of St. John in April 2013 and in Puerto Rico in February 2015. Each meeting gathered a small group of experts working near the interface between tropical geometry and nonarchimedean analytic spaces for a series of inspiring and provocative lectures on cutting edge research, interspersed with lively discussions and collaborative work in small groups. The articles collected here, which include high-level surveys as well as original research, mirror the main themes of the two Symposia. Topics covered in this volume include: Differential forms and currents, and solutions of Monge-Ampere type differential equations on Berkovich spaces and their skeletons; The homotopy types of nonarchimedean analytifications; The existence of "faithful tropicalizations" which encode the topology and geometry of analytifications; Relations between nonarchimedean analytic spaces and algebraic geometry, including logarithmic schemes, birational geometry, and the geometry of algebraic curves; Extended notions of tropical varieties which relate to Huber's theory of adic spaces analogously to the way that usual tropical varieties relate to Berkovich spaces; and Relations between nonarchimedean geometry and combinatorics, including deep and fascinating connections between matroid theory, tropical geometry, and Hodge theory.

Valuation Theory and Its Applications

Franz-Viktor Kuhlmann,Salma Kuhlmann,Murray Marshall
Valuation Theory and Its Applications Book PDF

Author : Franz-Viktor Kuhlmann,Salma Kuhlmann,Murray Marshall
Publisher : American Mathematical Soc.
Page : 470 pages
File Size : 40,6 Mb
Release : 2002-01-01
Category : Mathematics
ISBN : 0821871390

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Valuation Theory and Its Applications by Franz-Viktor Kuhlmann,Salma Kuhlmann,Murray Marshall Pdf

This book is the first of two proceedings volumes stemming from the International Conference and Workshop on Valuation Theory held at the University of Saskatchewan (Saskatoon, SK, Canada). Valuation theory arose in the early part of the twentieth century in connection with number theory and has many important applications to geometry and analysis: the classical application to the study of algebraic curves and to Dedekind and Prufer domains; the close connection to the famousresolution of the singularities problem; the study of the absolute Galois group of a field; the connection between ordering, valuations, and quadratic forms over a formally real field; the application to real algebraic geometry; the study of noncommutative rings; etc. The special feature of this book isits focus on current applications of valuation theory to this broad range of topics. Also included is a paper on the history of valuation theory. The book is suitable for graduate students and research mathematicians working in algebra, algebraic geometry, number theory, and mathematical logic.

Rigid Analytic Geometry and Its Applications

Jean Fresnel,Marius van der Put
Rigid Analytic Geometry and Its Applications Book PDF

Author : Jean Fresnel,Marius van der Put
Publisher : Springer Science & Business Media
Page : 320 pages
File Size : 48,7 Mb
Release : 2003-11-06
Category : Mathematics
ISBN : 0817642064

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Rigid Analytic Geometry and Its Applications by Jean Fresnel,Marius van der Put Pdf

Rigid (analytic) spaces were invented to describe degenerations, reductions, and moduli of algebraic curves and abelian varieties. This work, a revised and greatly expanded new English edition of an earlier French text by the same authors, presents important new developments and applications of the theory of rigid analytic spaces to abelian varieties, "points of rigid spaces," étale cohomology, Drinfeld modular curves, and Monsky-Washnitzer cohomology. The exposition is concise, self-contained, rich in examples and exercises, and will serve as an excellent graduate-level text for the classroom or for self-study.

Non-Archimedean Tame Topology and Stably Dominated Types (AM-192)

Ehud Hrushovski,François Loeser
Non Archimedean Tame Topology and Stably Dominated Types AM 192 Book PDF

Author : Ehud Hrushovski,François Loeser
Publisher : Princeton University Press
Page : 226 pages
File Size : 51,7 Mb
Release : 2016-02-09
Category : Mathematics
ISBN : 9780691161693

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Non-Archimedean Tame Topology and Stably Dominated Types (AM-192) by Ehud Hrushovski,François Loeser Pdf

Over the field of real numbers, analytic geometry has long been in deep interaction with algebraic geometry, bringing the latter subject many of its topological insights. In recent decades, model theory has joined this work through the theory of o-minimality, providing finiteness and uniformity statements and new structural tools. For non-archimedean fields, such as the p-adics, the Berkovich analytification provides a connected topology with many thoroughgoing analogies to the real topology on the set of complex points, and it has become an important tool in algebraic dynamics and many other areas of geometry. This book lays down model-theoretic foundations for non-archimedean geometry. The methods combine o-minimality and stability theory. Definable types play a central role, serving first to define the notion of a point and then properties such as definable compactness. Beyond the foundations, the main theorem constructs a deformation retraction from the full non-archimedean space of an algebraic variety to a rational polytope. This generalizes previous results of V. Berkovich, who used resolution of singularities methods. No previous knowledge of non-archimedean geometry is assumed. Model-theoretic prerequisites are reviewed in the first sections.

Effective Faithful Tropicalizations Associated to Linear Systems on Curves

Shu Kawaguchi,Kazuhiko Yamaki
Effective Faithful Tropicalizations Associated to Linear Systems on Curves Book PDF

Author : Shu Kawaguchi,Kazuhiko Yamaki
Publisher : American Mathematical Soc.
Page : 110 pages
File Size : 41,8 Mb
Release : 2021-07-21
Category : Education
ISBN : 9781470447533

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Effective Faithful Tropicalizations Associated to Linear Systems on Curves by Shu Kawaguchi,Kazuhiko Yamaki Pdf

For a connected smooth projective curve X of genus g, global sections of any line bundle L with deg(L) ≥ 2g + 1 give an embedding of the curve into projective space. We consider an analogous statement for a Berkovich skeleton in nonarchimedean geometry: We replace projective space by tropical projective space, and an embedding by a homeomorphism onto its image preserving integral structures (or equivalently, since X is a curve, an isometry), which is called a faithful tropicalization. Let K be an algebraically closed field which is complete with respect to a nontrivial nonarchimedean value. Suppose that X is defined over K and has genus g ≥ 2 and that Γ is a skeleton (that is allowed to have ends) of the analytification Xan of X in the sense of Berkovich. We show that if deg(L) ≥ 3g − 1, then global sections of L give a faithful tropicalization of Γ into tropical projective space. As an application, when Y is a suitable affine curve, we describe the analytification Y an as the limit of tropicalizations of an effectively bounded degree.

Combinatorial Algebraic Geometry

Gregory G. Smith,Bernd Sturmfels
Combinatorial Algebraic Geometry Book PDF

Author : Gregory G. Smith,Bernd Sturmfels
Publisher : Springer
Page : 390 pages
File Size : 51,8 Mb
Release : 2017-11-17
Category : Mathematics
ISBN : 9781493974863

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Combinatorial Algebraic Geometry by Gregory G. Smith,Bernd Sturmfels Pdf

This volume consolidates selected articles from the 2016 Apprenticeship Program at the Fields Institute, part of the larger program on Combinatorial Algebraic Geometry that ran from July through December of 2016. Written primarily by junior mathematicians, the articles cover a range of topics in combinatorial algebraic geometry including curves, surfaces, Grassmannians, convexity, abelian varieties, and moduli spaces. This book bridges the gap between graduate courses and cutting-edge research by connecting historical sources, computation, explicit examples, and new results.

Berkovich Spaces and Applications

Antoine Ducros,Charles Favre,Johannes Nicaise
Berkovich Spaces and Applications Book PDF

Author : Antoine Ducros,Charles Favre,Johannes Nicaise
Publisher : Springer
Page : 432 pages
File Size : 42,8 Mb
Release : 2014-11-21
Category : Mathematics
ISBN : 9783319110295

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Berkovich Spaces and Applications by Antoine Ducros,Charles Favre,Johannes Nicaise Pdf

We present an introduction to Berkovich’s theory of non-archimedean analytic spaces that emphasizes its applications in various fields. The first part contains surveys of a foundational nature, including an introduction to Berkovich analytic spaces by M. Temkin, and to étale cohomology by A. Ducros, as well as a short note by C. Favre on the topology of some Berkovich spaces. The second part focuses on applications to geometry. A second text by A. Ducros contains a new proof of the fact that the higher direct images of a coherent sheaf under a proper map are coherent, and B. Rémy, A. Thuillier and A. Werner provide an overview of their work on the compactification of Bruhat-Tits buildings using Berkovich analytic geometry. The third and final part explores the relationship between non-archimedean geometry and dynamics. A contribution by M. Jonsson contains a thorough discussion of non-archimedean dynamical systems in dimension 1 and 2. Finally a survey by J.-P. Otal gives an account of Morgan-Shalen's theory of compactification of character varieties. This book will provide the reader with enough material on the basic concepts and constructions related to Berkovich spaces to move on to more advanced research articles on the subject. We also hope that the applications presented here will inspire the reader to discover new settings where these beautiful and intricate objects might arise.

Advances in Non-Archimedean Analysis

Jesus Araujo-Gomez,Bertin Diarra,Alain Escassut
Advances in Non Archimedean Analysis Book PDF

Author : Jesus Araujo-Gomez,Bertin Diarra,Alain Escassut
Publisher : American Mathematical Soc.
Page : 294 pages
File Size : 42,8 Mb
Release : 2011
Category : Topological fields
ISBN : 9780821852910

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Advances in Non-Archimedean Analysis by Jesus Araujo-Gomez,Bertin Diarra,Alain Escassut Pdf

These collected articles feature recent developments in various areas of non-Archimedean analysis: Hilbert and Banach spaces, finite dimensional spaces, topological vector spaces and operator theory, strict topologies, spaces of continuous functions and of strictly differentiable functions, isomorphisms between Banach functions spaces, and measure and integration.