Topics In Ergodic Theory Pms 44 Volume 44

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Topics in Ergodic Theory (PMS-44), Volume 44

Iakov Grigorevich Sinai
Topics in Ergodic Theory PMS 44 Volume 44 Book PDF

Author : Iakov Grigorevich Sinai
Publisher : Princeton University Press
Page : 226 pages
File Size : 55,8 Mb
Release : 2017-03-14
Category : Mathematics
ISBN : 9781400887255

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Topics in Ergodic Theory (PMS-44), Volume 44 by Iakov Grigorevich Sinai Pdf

This book concerns areas of ergodic theory that are now being intensively developed. The topics include entropy theory (with emphasis on dynamical systems with multi-dimensional time), elements of the renormalization group method in the theory of dynamical systems, splitting of separatrices, and some problems related to the theory of hyperbolic dynamical systems. Originally published in 1993. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Theory of Lie Groups (PMS-8), Volume 8

Claude Chevalley
Theory of Lie Groups PMS 8 Volume 8 Book PDF

Author : Claude Chevalley
Publisher : Princeton University Press
Page : 230 pages
File Size : 48,5 Mb
Release : 2016-06-02
Category : Mathematics
ISBN : 9781400883851

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Theory of Lie Groups (PMS-8), Volume 8 by Claude Chevalley Pdf

This famous book was the first treatise on Lie groups in which a modern point of view was adopted systematically, namely, that a continuous group can be regarded as a global object. To develop this idea to its fullest extent, Chevalley incorporated a broad range of topics, such as the covering spaces of topological spaces, analytic manifolds, integration of complete systems of differential equations on a manifold, and the calculus of exterior differential forms. The book opens with a short description of the classical groups: unitary groups, orthogonal groups, symplectic groups, etc. These special groups are then used to illustrate the general properties of Lie groups, which are considered later. The general notion of a Lie group is defined and correlated with the algebraic notion of a Lie algebra; the subgroups, factor groups, and homomorphisms of Lie groups are studied by making use of the Lie algebra. The last chapter is concerned with the theory of compact groups, culminating in Peter-Weyl's theorem on the existence of representations. Given a compact group, it is shown how one can construct algebraically the corresponding Lie group with complex parameters which appears in the form of a certain algebraic variety (associated algebraic group). This construction is intimately related to the proof of the generalization given by Tannaka of Pontrjagin's duality theorem for Abelian groups. The continued importance of Lie groups in mathematics and theoretical physics make this an indispensable volume for researchers in both fields.

Cohomological Induction and Unitary Representations (PMS-45), Volume 45

Anthony W. Knapp,David A. Vogan Jr.
Cohomological Induction and Unitary Representations PMS 45 Volume 45 Book PDF

Author : Anthony W. Knapp,David A. Vogan Jr.
Publisher : Princeton University Press
Page : 968 pages
File Size : 52,7 Mb
Release : 2016-06-02
Category : Mathematics
ISBN : 9781400883936

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Cohomological Induction and Unitary Representations (PMS-45), Volume 45 by Anthony W. Knapp,David A. Vogan Jr. Pdf

This book offers a systematic treatment--the first in book form--of the development and use of cohomological induction to construct unitary representations. George Mackey introduced induction in 1950 as a real analysis construction for passing from a unitary representation of a closed subgroup of a locally compact group to a unitary representation of the whole group. Later a parallel construction using complex analysis and its associated co-homology theories grew up as a result of work by Borel, Weil, Harish-Chandra, Bott, Langlands, Kostant, and Schmid. Cohomological induction, introduced by Zuckerman, is an algebraic analog that is technically more manageable than the complex-analysis construction and leads to a large repertory of irreducible unitary representations of reductive Lie groups. The book, which is accessible to students beyond the first year of graduate school, will interest mathematicians and physicists who want to learn about and take advantage of the algebraic side of the representation theory of Lie groups. Cohomological Induction and Unitary Representations develops the necessary background in representation theory and includes an introductory chapter of motivation, a thorough treatment of the "translation principle," and four appendices on algebra and analysis.

Three-Dimensional Geometry and Topology, Volume 1

William P. Thurston
Three Dimensional Geometry and Topology Volume 1 Book PDF

Author : William P. Thurston
Publisher : Princeton University Press
Page : 323 pages
File Size : 53,5 Mb
Release : 2014-10-31
Category : Mathematics
ISBN : 9781400865321

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Three-Dimensional Geometry and Topology, Volume 1 by William P. Thurston Pdf

This book develops some of the extraordinary richness, beauty, and power of geometry in two and three dimensions, and the strong connection of geometry with topology. Hyperbolic geometry is the star. A strong effort has been made to convey not just denatured formal reasoning (definitions, theorems, and proofs), but a living feeling for the subject. There are many figures, examples, and exercises of varying difficulty. This book was the origin of a grand scheme developed by Thurston that is now coming to fruition. In the 1920s and 1930s the mathematics of two-dimensional spaces was formalized. It was Thurston's goal to do the same for three-dimensional spaces. To do this, he had to establish the strong connection of geometry to topology--the study of qualitative questions about geometrical structures. The author created a new set of concepts, and the expression "Thurston-type geometry" has become a commonplace. Three-Dimensional Geometry and Topology had its origins in the form of notes for a graduate course the author taught at Princeton University between 1978 and 1980. Thurston shared his notes, duplicating and sending them to whoever requested them. Eventually, the mailing list grew to more than one thousand names. The book is the culmination of two decades of research and has become the most important and influential text in the field. Its content also provided the methods needed to solve one of mathematics' oldest unsolved problems--the Poincaré Conjecture. In 2005 Thurston won the first AMS Book Prize, for Three-dimensional Geometry and Topology. The prize recognizes an outstanding research book that makes a seminal contribution to the research literature. Thurston received the Fields Medal, the mathematical equivalent of the Nobel Prize, in 1982 for the depth and originality of his contributions to mathematics. In 1979 he was awarded the Alan T. Waterman Award, which recognizes an outstanding young researcher in any field of science or engineering supported by the National Science Foundation.

Etale Cohomology (PMS-33)

J. S. Milne
Etale Cohomology PMS 33 Book PDF

Author : J. S. Milne
Publisher : Princeton University Press
Page : 346 pages
File Size : 51,8 Mb
Release : 1980-04-21
Category : Mathematics
ISBN : 0691082383

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Etale Cohomology (PMS-33) by J. S. Milne Pdf

One of the most important mathematical achievements of the past several decades has been A. Grothendieck's work on algebraic geometry. In the early 1960s, he and M. Artin introduced étale cohomology in order to extend the methods of sheaf-theoretic cohomology from complex varieties to more general schemes. This work found many applications, not only in algebraic geometry, but also in several different branches of number theory and in the representation theory of finite and p-adic groups. Yet until now, the work has been available only in the original massive and difficult papers. In order to provide an accessible introduction to étale cohomology, J. S. Milne offers this more elementary account covering the essential features of the theory. The author begins with a review of the basic properties of flat and étale morphisms and of the algebraic fundamental group. The next two chapters concern the basic theory of étale sheaves and elementary étale cohomology, and are followed by an application of the cohomology to the study of the Brauer group. After a detailed analysis of the cohomology of curves and surfaces, Professor Milne proves the fundamental theorems in étale cohomology -- those of base change, purity, Poincaré duality, and the Lefschetz trace formula. He then applies these theorems to show the rationality of some very general L-series. Originally published in 1980. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Singular Integrals and Differentiability Properties of Functions (PMS-30)

Elias M. Stein
Singular Integrals and Differentiability Properties of Functions PMS 30 Book PDF

Author : Elias M. Stein
Publisher : Princeton University Press
Page : 304 pages
File Size : 42,7 Mb
Release : 2016-06-02
Category : Mathematics
ISBN : 9781400883882

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Singular Integrals and Differentiability Properties of Functions (PMS-30) by Elias M. Stein Pdf

Singular integrals are among the most interesting and important objects of study in analysis, one of the three main branches of mathematics. They deal with real and complex numbers and their functions. In this book, Princeton professor Elias Stein, a leading mathematical innovator as well as a gifted expositor, produced what has been called the most influential mathematics text in the last thirty-five years. One reason for its success as a text is its almost legendary presentation: Stein takes arcane material, previously understood only by specialists, and makes it accessible even to beginning graduate students. Readers have reflected that when you read this book, not only do you see that the greats of the past have done exciting work, but you also feel inspired that you can master the subject and contribute to it yourself. Singular integrals were known to only a few specialists when Stein's book was first published. Over time, however, the book has inspired a whole generation of researchers to apply its methods to a broad range of problems in many disciplines, including engineering, biology, and finance. Stein has received numerous awards for his research, including the Wolf Prize of Israel, the Steele Prize, and the National Medal of Science. He has published eight books with Princeton, including Real Analysis in 2005.

Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane (PMS-48)

Kari Astala,Tadeusz Iwaniec,Gaven Martin
Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane PMS 48 Book PDF

Author : Kari Astala,Tadeusz Iwaniec,Gaven Martin
Publisher : Princeton University Press
Page : 708 pages
File Size : 44,9 Mb
Release : 2009-01-18
Category : Mathematics
ISBN : 0691137773

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Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane (PMS-48) by Kari Astala,Tadeusz Iwaniec,Gaven Martin Pdf

This book explores the most recent developments in the theory of planar quasiconformal mappings with a particular focus on the interactions with partial differential equations and nonlinear analysis. It gives a thorough and modern approach to the classical theory and presents important and compelling applications across a spectrum of mathematics: dynamical systems, singular integral operators, inverse problems, the geometry of mappings, and the calculus of variations. It also gives an account of recent advances in harmonic analysis and their applications in the geometric theory of mappings. The book explains that the existence, regularity, and singular set structures for second-order divergence-type equations--the most important class of PDEs in applications--are determined by the mathematics underpinning the geometry, structure, and dimension of fractal sets; moduli spaces of Riemann surfaces; and conformal dynamical systems. These topics are inextricably linked by the theory of quasiconformal mappings. Further, the interplay between them allows the authors to extend classical results to more general settings for wider applicability, providing new and often optimal answers to questions of existence, regularity, and geometric properties of solutions to nonlinear systems in both elliptic and degenerate elliptic settings.

Étale Cohomology (PMS-33), Volume 33

James S. Milne
tale Cohomology PMS 33 Volume 33 Book PDF

Author : James S. Milne
Publisher : Princeton University Press
Page : 344 pages
File Size : 54,8 Mb
Release : 2016-10-11
Category : Mathematics
ISBN : 9781400883981

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Étale Cohomology (PMS-33), Volume 33 by James S. Milne Pdf

One of the most important mathematical achievements of the past several decades has been A. Grothendieck's work on algebraic geometry. In the early 1960s, he and M. Artin introduced étale cohomology in order to extend the methods of sheaf-theoretic cohomology from complex varieties to more general schemes. This work found many applications, not only in algebraic geometry, but also in several different branches of number theory and in the representation theory of finite and p-adic groups. Yet until now, the work has been available only in the original massive and difficult papers. In order to provide an accessible introduction to étale cohomology, J. S. Milne offers this more elementary account covering the essential features of the theory. The author begins with a review of the basic properties of flat and étale morphisms and of the algebraic fundamental group. The next two chapters concern the basic theory of étale sheaves and elementary étale cohomology, and are followed by an application of the cohomology to the study of the Brauer group. After a detailed analysis of the cohomology of curves and surfaces, Professor Milne proves the fundamental theorems in étale cohomology -- those of base change, purity, Poincaré duality, and the Lefschetz trace formula. He then applies these theorems to show the rationality of some very general L-series. Originally published in 1980. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

The Global Nonlinear Stability of the Minkowski Space (PMS-41)

Demetrios Christodoulou,Sergiu Klainerman
The Global Nonlinear Stability of the Minkowski Space PMS 41 Book PDF

Author : Demetrios Christodoulou,Sergiu Klainerman
Publisher : Princeton University Press
Page : 525 pages
File Size : 53,9 Mb
Release : 2014-07-14
Category : Mathematics
ISBN : 9781400863174

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The Global Nonlinear Stability of the Minkowski Space (PMS-41) by Demetrios Christodoulou,Sergiu Klainerman Pdf

The aim of this work is to provide a proof of the nonlinear gravitational stability of the Minkowski space-time. More precisely, the book offers a constructive proof of global, smooth solutions to the Einstein Vacuum Equations, which look, in the large, like the Minkowski space-time. In particular, these solutions are free of black holes and singularities. The work contains a detailed description of the sense in which these solutions are close to the Minkowski space-time, in all directions. It thus provides the mathematical framework in which we can give a rigorous derivation of the laws of gravitation proposed by Bondi. Moreover, it establishes other important conclusions concerning the nonlinear character of gravitational radiation. The authors obtain their solutions as dynamic developments of all initial data sets, which are close, in a precise manner, to the flat initial data set corresponding to the Minkowski space-time. They thus establish the global dynamic stability of the latter. Originally published in 1994. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

New Trends in Stochastic Analysis and Related Topics

Huaizhong Zhao,Aubrey Truman
New Trends in Stochastic Analysis and Related Topics Book PDF

Author : Huaizhong Zhao,Aubrey Truman
Publisher : World Scientific
Page : 460 pages
File Size : 47,7 Mb
Release : 2011-11-25
Category : Mathematics
ISBN : 9789814397094

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New Trends in Stochastic Analysis and Related Topics by Huaizhong Zhao,Aubrey Truman Pdf

The volume is dedicated to Professor David Elworthy to celebrate his fundamental contribution and exceptional influence on stochastic analysis and related fields. Stochastic analysis has been profoundly developed as a vital fundamental research area in mathematics in recent decades. It has been discovered to have intrinsic connections with many other areas of mathematics such as partial differential equations, functional analysis, topology, differential geometry, dynamical systems, etc. Mathematicians developed many mathematical tools in stochastic analysis to understand and model random phenomena in physics, biology, finance, fluid, environment science, etc. This volume contains 12 comprehensive review/new articles written by world leading researchers (by invitation) and their collaborators. It covers stochastic analysis on manifolds, rough paths, Dirichlet forms, stochastic partial differential equations, stochastic dynamical systems, infinite dimensional analysis, stochastic flows, quantum stochastic analysis and stochastic Hamilton Jacobi theory. Articles contain cutting edge research methodology, results and ideas in relevant fields. They are of interest to research mathematicians and postgraduate students in stochastic analysis, probability, partial differential equations, dynamical systems, mathematical physics, as well as to physicists, financial mathematicians, engineers, etc. Contents:Stochastic Geometric Partial Differential Equations (Zdzisław Brzeźniak, Beniamin Goldys and Martin Ondreját)Rough Paths on Manifolds (Thomas Cass, Christian Litterer and Terry Lyons)Averaging, Homogenization and Slow Manifolds for Stochastic Partial Differential Equations (Jinqiao Duan, Anthony Roberts and Wei Wang)A Burgers-Zeldovich Model for the Formation of Planetesimals via Nelson's Stochastic Mechanics (Richard Durran, Andrew Neate, Aubrey Truman and Oleg Smolyanov)Two Problems Concerning Brownian Motion on a Complete Riemannian Manifold (Elton P Hsu)Sticky Shuffle Product Hopf Algebras and Their Stochastic Representations (Robin Hudson)Chain Rules for Lévy Flows and Kolmogorov Equations for Associated Jump-Diffusions (Hiroshi Kunita)The Stochastic Differential Equation Approach to Analysis on Path Space (Xue-Mei Li)Pathwise Properties of Random Quadratic Mapping (Peng Lian and Huaizhong Zhao)Invariant Manifolds for Infinite Dimensional Random Dynamical Systems (Kening Lu and Björn Schmalfuβ)Some Topics on Dirichlet Forms (Zhi-Ming Ma and Wei Sun)Hamilton-Jacobi Theory and the Stochastic Elementary Formula (Andrew Neate and Aubrey Truman) Readership: Graduate students and researchers in stochastic analysis. Keywords:Stochastic Analysis;Stochastic Partial Differential Equations;Random Dynamical Systems;Brownian Motion on Manifolds;Dirichlet Forms;Rough Path Theory;Levy Process;Hamilton Jacobi Theory;Path Space Analysis;Malliavin CalculusKey Features:Articles are written by world leading researchersContains comprehensive review articles by authors who made fundamental contributions to relevant topicsTopics are carefully selected to reflect subjects currently of great interests

Topics in Dynamics and Ergodic Theory

Sergey Bezuglyi
Topics in Dynamics and Ergodic Theory Book PDF

Author : Sergey Bezuglyi
Publisher : Cambridge University Press
Page : 276 pages
File Size : 51,9 Mb
Release : 2003-12-08
Category : Mathematics
ISBN : 0521533651

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Topics in Dynamics and Ergodic Theory by Sergey Bezuglyi Pdf

This book contains a collection of survey papers by leading researchers in ergodic theory, low-dimensional and topological dynamics and it comprises nine chapters on a range of important topics. These include: the role and usefulness of ultrafilters in ergodic theory, topological dynamics and Ramsey theory; topological aspects of kneading theory together with an analogous 2-dimensional theory called pruning; the dynamics of Markov odometers, Bratteli-Vershik diagrams and orbit equivalence of non-singular automorphisms; geometric proofs of Mather's connecting and accelerating theorems; recent results in one dimensional smooth dynamics; periodic points of nonexpansive maps; arithmetic dynamics; the defect of factor maps; entropy theory for actions of countable amenable groups.

Advances in Analysis

Charles Fefferman,Alexandru D. Ionescu,D.H. Phong,Stephen Wainger
Advances in Analysis Book PDF

Author : Charles Fefferman,Alexandru D. Ionescu,D.H. Phong,Stephen Wainger
Publisher : Princeton University Press
Page : 480 pages
File Size : 53,9 Mb
Release : 2014-01-05
Category : Mathematics
ISBN : 9781400848935

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Advances in Analysis by Charles Fefferman,Alexandru D. Ionescu,D.H. Phong,Stephen Wainger Pdf

Princeton University's Elias Stein was the first mathematician to see the profound interconnections that tie classical Fourier analysis to several complex variables and representation theory. His fundamental contributions include the Kunze-Stein phenomenon, the construction of new representations, the Stein interpolation theorem, the idea of a restriction theorem for the Fourier transform, and the theory of Hp Spaces in several variables. Through his great discoveries, through books that have set the highest standard for mathematical exposition, and through his influence on his many collaborators and students, Stein has changed mathematics. Drawing inspiration from Stein’s contributions to harmonic analysis and related topics, this volume gathers papers from internationally renowned mathematicians, many of whom have been Stein’s students. The book also includes expository papers on Stein’s work and its influence. The contributors are Jean Bourgain, Luis Caffarelli, Michael Christ, Guy David, Charles Fefferman, Alexandru D. Ionescu, David Jerison, Carlos Kenig, Sergiu Klainerman, Loredana Lanzani, Sanghyuk Lee, Lionel Levine, Akos Magyar, Detlef Müller, Camil Muscalu, Alexander Nagel, D. H. Phong, Malabika Pramanik, Andrew S. Raich, Fulvio Ricci, Keith M. Rogers, Andreas Seeger, Scott Sheffield, Luis Silvestre, Christopher D. Sogge, Jacob Sturm, Terence Tao, Christoph Thiele, Stephen Wainger, and Steven Zelditch.

Frontiers in Complex Dynamics

Araceli Bonifant,Misha Lyubich,Scott Sutherland
Frontiers in Complex Dynamics Book PDF

Author : Araceli Bonifant,Misha Lyubich,Scott Sutherland
Publisher : Princeton University Press
Page : 824 pages
File Size : 55,9 Mb
Release : 2014-03-16
Category : Mathematics
ISBN : 9781400851317

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Frontiers in Complex Dynamics by Araceli Bonifant,Misha Lyubich,Scott Sutherland Pdf

John Milnor, best known for his work in differential topology, K-theory, and dynamical systems, is one of only three mathematicians to have won the Fields medal, the Abel prize, and the Wolf prize, and is the only one to have received all three of the Leroy P. Steele prizes. In honor of his eightieth birthday, this book gathers together surveys and papers inspired by Milnor's work, from distinguished experts examining not only holomorphic dynamics in one and several variables, but also differential geometry, entropy theory, and combinatorial group theory. The book contains the last paper written by William Thurston, as well as a short paper by John Milnor himself. Introductory sections put the papers in mathematical and historical perspective, color figures are included, and an index facilitates browsing. This collection will be useful to students and researchers for decades to come. The contributors are Marco Abate, Marco Arizzi, Alexander Blokh, Thierry Bousch, Xavier Buff, Serge Cantat, Tao Chen, Robert Devaney, Alexandre Dezotti, Tien-Cuong Dinh, Romain Dujardin, Hugo García-Compeán, William Goldman, Rotislav Grigorchuk, John Hubbard, Yunping Jiang, Linda Keen, Jan Kiwi, Genadi Levin, Daniel Meyer, John Milnor, Carlos Moreira, Vincente Muñoz, Viet-Anh Nguyên, Lex Oversteegen, Ricardo Pérez-Marco, Ross Ptacek, Jasmin Raissy, Pascale Roesch, Roberto Santos-Silva, Dierk Schleicher, Nessim Sibony, Daniel Smania, Tan Lei, William Thurston, Vladlen Timorin, Sebastian van Strien, and Alberto Verjovsky.

Real Submanifolds in Complex Space and Their Mappings (PMS-47)

M. Salah Baouendi,Peter Ebenfelt,Linda Preiss Rothschild
Real Submanifolds in Complex Space and Their Mappings PMS 47 Book PDF

Author : M. Salah Baouendi,Peter Ebenfelt,Linda Preiss Rothschild
Publisher : Princeton University Press
Page : 416 pages
File Size : 46,9 Mb
Release : 2016-06-02
Category : Mathematics
ISBN : 9781400883967

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Real Submanifolds in Complex Space and Their Mappings (PMS-47) by M. Salah Baouendi,Peter Ebenfelt,Linda Preiss Rothschild Pdf

This book presents many of the main developments of the past two decades in the study of real submanifolds in complex space, providing crucial background material for researchers and advanced graduate students. The techniques in this area borrow from real and complex analysis and partial differential equations, as well as from differential, algebraic, and analytical geometry. In turn, these latter areas have been enriched over the years by the study of problems in several complex variables addressed here. The authors, M. Salah Baouendi, Peter Ebenfelt, and Linda Preiss Rothschild, include extensive preliminary material to make the book accessible to nonspecialists. One of the most important topics that the authors address here is the holomorphic extension of functions and mappings that satisfy the tangential Cauchy-Riemann equations on real submanifolds. They present the main results in this area with a novel and self-contained approach. The book also devotes considerable attention to the study of holomorphic mappings between real submanifolds, and proves finite determination of such mappings by their jets under some optimal assumptions. The authors also give a thorough comparison of the various nondegeneracy conditions for manifolds and mappings and present new geometric interpretations of these conditions. Throughout the book, Cauchy-Riemann vector fields and their orbits play a central role and are presented in a setting that is both general and elementary.

Abelian Varieties with Complex Multiplication and Modular Functions

Goro Shimura
Abelian Varieties with Complex Multiplication and Modular Functions Book PDF

Author : Goro Shimura
Publisher : Princeton University Press
Page : 232 pages
File Size : 41,6 Mb
Release : 2016-06-02
Category : Mathematics
ISBN : 9781400883943

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Abelian Varieties with Complex Multiplication and Modular Functions by Goro Shimura Pdf

Reciprocity laws of various kinds play a central role in number theory. In the easiest case, one obtains a transparent formulation by means of roots of unity, which are special values of exponential functions. A similar theory can be developed for special values of elliptic or elliptic modular functions, and is called complex multiplication of such functions. In 1900 Hilbert proposed the generalization of these as the twelfth of his famous problems. In this book, Goro Shimura provides the most comprehensive generalizations of this type by stating several reciprocity laws in terms of abelian varieties, theta functions, and modular functions of several variables, including Siegel modular functions. This subject is closely connected with the zeta function of an abelian variety, which is also covered as a main theme in the book. The third topic explored by Shimura is the various algebraic relations among the periods of abelian integrals. The investigation of such algebraicity is relatively new, but has attracted the interest of increasingly many researchers. Many of the topics discussed in this book have not been covered before. In particular, this is the first book in which the topics of various algebraic relations among the periods of abelian integrals, as well as the special values of theta and Siegel modular functions, are treated extensively.