Algebraic Geometry and Arithmetic Curves

Author: Qing Liu
Publisher: Oxford University Press
ISBN: 9780191547805
Release Date: 2006-06-29
Genre: Mathematics

This book is a general introduction to the theory of schemes, followed by applications to arithmetic surfaces and to the theory of reduction of algebraic curves. The first part introduces basic objects such as schemes, morphisms, base change, local properties (normality, regularity, Zariski's Main Theorem). This is followed by the more global aspect: coherent sheaves and a finiteness theorem for their cohomology groups. Then follows a chapter on sheaves of differentials, dualizing sheaves, and Grothendieck's duality theory. The first part ends with the theorem of Riemann-Roch and its application to the study of smooth projective curves over a field. Singular curves are treated through a detailed study of the Picard group. The second part starts with blowing-ups and desingularisation (embedded or not) of fibered surfaces over a Dedekind ring that leads on to intersection theory on arithmetic surfaces. Castelnuovo's criterion is proved and also the existence of the minimal regular model. This leads to the study of reduction of algebraic curves. The case of elliptic curves is studied in detail. The book concludes with the funadmental theorem of stable reduction of Deligne-Mumford. The book is essentially self-contained, including the necessary material on commutative algebra. The prerequisites are therefore few, and the book should suit a graduate student. It contains many examples and nearly 600 exercises.

Ranks of Elliptic Curves and Random Matrix Theory

Author: J. B. Conrey
Publisher: Cambridge University Press
ISBN: 9780521699648
Release Date: 2007-02-08
Genre: Mathematics

This comprehensive volume introduces elliptic curves and the fundamentals of modeling by a family of random matrices.

Arakelov Geometry

Author: Atsushi Moriwaki
Publisher: American Mathematical Soc.
ISBN: 9781470410742
Release Date: 2014-11-05
Genre: Mathematics

The main goal of this book is to present the so-called birational Arakelov geometry, which can be viewed as an arithmetic analog of the classical birational geometry, i.e., the study of big linear series on algebraic varieties. After explaining classical results about the geometry of numbers, the author starts with Arakelov geometry for arithmetic curves, and continues with Arakelov geometry of arithmetic surfaces and higher-dimensional varieties. The book includes such fundamental results as arithmetic Hilbert-Samuel formula, arithmetic Nakai-Moishezon criterion, arithmetic Bogomolov inequality, the existence of small sections, the continuity of arithmetic volume function, the Lang-Bogomolov conjecture and so on. In addition, the author presents, with full details, the proof of Faltings' Riemann-Roch theorem. Prerequisites for reading this book are the basic results of algebraic geometry and the language of schemes.

Quantization Geometry and Noncommutative Structures in Mathematics and Physics

Author: Alexander Cardona
Publisher: Springer
ISBN: 9783319654270
Release Date: 2017-10-26
Genre: Science

This monograph presents various ongoing approaches to the vast topic of quantization, which is the process of forming a quantum mechanical system starting from a classical one, and discusses their numerous fruitful interactions with mathematics.The opening chapter introduces the various forms of quantization and their interactions with each other and with mathematics.A first approach to quantization, called deformation quantization, consists of viewing the Planck constant as a small parameter. This approach provides a deformation of the structure of the algebra of classical observables rather than a radical change in the nature of the observables. When symmetries come into play, deformation quantization needs to be merged with group actions, which is presented in chapter 2, by Simone Gutt.The noncommutativity arising from quantization is the main concern of noncommutative geometry. Allowing for the presence of symmetries requires working with principal fiber bundles in a non-commutative setup, where Hopf algebras appear naturally. This is the topic of chapter 3, by Christian Kassel. Nichols algebras, a special type of Hopf algebras, are the subject of chapter 4, by Nicolás Andruskiewitsch. The purely algebraic approaches given in the previous chapters do not take the geometry of space-time into account. For this purpose a special treatment using a more geometric point of view is required. An approach to field quantization on curved space-time, with applications to cosmology, is presented in chapter 5 in an account of the lectures of Abhay Ashtekar that brings a complementary point of view to non-commutativity.An alternative quantization procedure is known under the name of string theory. In chapter 6 its supersymmetric version is presented. Superstrings have drawn the attention of many mathematicians, due to its various fruitful interactions with algebraic geometry, some of which are described here. The remaining chapters discuss further topics, as the Batalin-Vilkovisky formalism and direct products of spectral triples.This volume addresses both physicists and mathematicians and serves as an introduction to ongoing research in very active areas of mathematics and physics at the border line between geometry, topology, algebra and quantum field theory.

Einf hrung in die Funktionalanalysis

Author: Reinhold Meise
Publisher: Springer-Verlag
ISBN: 9783322803108
Release Date: 2013-03-07
Genre: Mathematics

Dieses Buch wendet sich an Studenten der Mathematik und der Physik, welche über Grundkenntnisse in Analysis und linearer Algebra verfügen.

Stochastic Analysis and Diffusion Processes

Author: Gopinath Kallianpur
Publisher: Oxford University Press
ISBN: 9780199657070
Release Date: 2014
Genre: Mathematics

Beginning with the concept of random processes and Brownian motion and building on the theory and research directions in a self-contained manner, this book provides an introduction to stochastic analysis for graduate students, researchers and applied scientists interested in stochastic processes and their applications.

Riemann Surfaces

Author: Simon Donaldson
Publisher: Oxford University Press
ISBN: 9780198526391
Release Date: 2011-03-24
Genre: Mathematics

An authoritative but accessible text on one dimensional complex manifolds or Riemann surfaces. Dealing with the main results on Riemann surfaces from a variety of points of view; it pulls together material from global analysis, topology, and algebraic geometry, and covers the essential mathematical methods and tools.

A First Course in Modular Forms

Author: Fred Diamond
Publisher: Springer Science & Business Media
ISBN: 9780387272269
Release Date: 2006-03-30
Genre: Mathematics

This book introduces the theory of modular forms, from which all rational elliptic curves arise, with an eye toward the Modularity Theorem. Discussion covers elliptic curves as complex tori and as algebraic curves; modular curves as Riemann surfaces and as algebraic curves; Hecke operators and Atkin-Lehner theory; Hecke eigenforms and their arithmetic properties; the Jacobians of modular curves and the Abelian varieties associated to Hecke eigenforms. As it presents these ideas, the book states the Modularity Theorem in various forms, relating them to each other and touching on their applications to number theory. The authors assume no background in algebraic number theory and algebraic geometry. Exercises are included.

Der lange Weg zur Freiheit

Author: Nelson Mandela
Publisher: S. Fischer Verlag
ISBN: 9783104031545
Release Date: 2014-01-25
Genre: History

»Ich bin einer von ungezählten Millionen, die durch Nelson Mandelas Leben inspiriert wurden.« Barack Obama Eine fast drei Jahrzehnte währende Gefängnishaft ließ Nelson Mandela zum Mythos der schwarzen Befreiungsbewegung werden. Kaum ein anderer Politiker unserer Zeit symbolisiert heute in solchem Maße die Friedenshoffnungen der Menschheit und den Gedanken der Aussöhnung aller Rassen wie der ehemalige südafrikanische Präsident und Friedensnobelpreisträger. Auch nach seinem Tod finden seine ungebrochene Charakterstärke und Menschenfreundlichkeit die Bewunderung aller friedenswilligen Menschen auf der Welt. Mandelas Lebensgeschichte ist über die politische Bedeutung hinaus ein spannend zu lesendes, kenntnis- und faktenreiches Dokument menschlicher Entwicklung unter Bedingungen und Fährnissen, vor denen die meisten Menschen innerlich wie äußerlich kapituliert haben dürften.

Ma und Kategorie

Author: J.C. Oxtoby
Publisher: Springer-Verlag
ISBN: 9783642960741
Release Date: 2013-03-08
Genre: Mathematics

Dieses Buch behandelt hauptsächlich zwei Themenkreise: Der Bairesche Kategorie-Satz als Hilfsmittel für Existenzbeweise sowie Die "Dualität" zwischen Maß und Kategorie. Die Kategorie-Methode wird durch viele typische Anwendungen erläutert; die Analogie, die zwischen Maß und Kategorie besteht, wird nach den verschiedensten Richtungen hin genauer untersucht. Hierzu findet der Leser eine kurze Einführung in die Grundlagen der metrischen Topologie; außerdem werden grundlegende Eigenschaften des Lebesgue schen Maßes hergeleitet. Es zeigt sich, daß die Lebesguesche Integrationstheorie für unsere Zwecke nicht erforderlich ist, sondern daß das Riemannsche Integral ausreicht. Weiter werden einige Begriffe aus der allgemeinen Maßtheorie und Topologie eingeführt; dies geschieht jedoch nicht nur der größeren Allgemeinheit wegen. Es erübrigt sich fast zu erwähnen, daß sich die Bezeichnung "Kategorie" stets auf "Bairesche Kategorie" be zieht; sie hat nichts zu tun mit dem in der homologischen Algebra verwendeten Begriff der Kategorie. Beim Leser werden lediglich grundlegende Kenntnisse aus der Analysis und eine gewisse Vertrautheit mit der Mengenlehre vorausgesetzt. Für die hier untersuchten Probleme bietet sich in natürlicher Weise die mengentheoretische Formulierung an. Das vorlie gende Buch ist als Einführung in dieses Gebiet der Analysis gedacht. Man könnte es als Ergänzung zur üblichen Grundvorlesung über reelle Analysis, als Grundlage für ein Se minar oder auch zum selbständigen Studium verwenden. Bei diesem Buch handelt es sich vorwiegend um eine zusammenfassende Darstellung; jedoch finden sich in ihm auch einige Verfeinerungen bekannter Resultate, namentlich Satz 15.6 und Aussage 20.4. Das Literaturverzeichnis erhebt keinen Anspruch auf Vollständigkeit. Häufig werden Werke zitiert, die weitere Literaturangaben enthalten.

Arithmetic of Algebraic Curves

Author: Serguei A. Stepanov
Publisher: Springer Science & Business Media
ISBN: 0306110369
Release Date: 1994-12-31
Genre: Mathematics

Deals with the theory of equations in two variables and arithmetic, algebraic-geometric, and logical aspects of the problem. Covers equations over finite fields, distribution of quadratic residues and nonresidues, the Riemann-Roch theorem, integral points on curves, the Siegel-Mahler theorem, and Hi

Rigid Analytic Geometry and Its Applications

Author: Jean Fresnel
Publisher: Birkhauser
ISBN: UOM:39015056821195
Release Date: 2004
Genre: Mathematics

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," tale 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.