Complex Analysis Riemann Surfaces And Integrable Systems

Author : Sergey M. Natanzon
Publisher : Springer Nature
Page : 148 pages
File Size : 53,5 Mb
Release : 2020-01-03
Category : Mathematics
ISBN : 9783030346409

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Complex Analysis, Riemann Surfaces and Integrable Systems by Sergey M. Natanzon Pdf

This book is devoted to classical and modern achievements in complex analysis. In order to benefit most from it, a first-year university background is sufficient; all other statements and proofs are provided. We begin with a brief but fairly complete course on the theory of holomorphic, meromorphic, and harmonic functions. We then present a uniformization theory, and discuss a representation of the moduli space of Riemann surfaces of a fixed topological type as a factor space of a contracted space by a discrete group. Next, we consider compact Riemann surfaces and prove the classical theorems of Riemann-Roch, Abel, Weierstrass, etc. We also construct theta functions that are very important for a range of applications. After that, we turn to modern applications of this theory. First, we build the (important for mathematics and mathematical physics) Kadomtsev-Petviashvili hierarchy and use validated results to arrive at important solutions to these differential equations. We subsequently use the theory of harmonic functions and the theory of differential hierarchies to explicitly construct a conformal mapping that translates an arbitrary contractible domain into a standard disk – a classical problem that has important applications in hydrodynamics, gas dynamics, etc. The book is based on numerous lecture courses given by the author at the Independent University of Moscow and at the Mathematics Department of the Higher School of Economics.

Author : N.J. Hitchin,G. B. Segal,R.S. Ward
Publisher : Oxford University Press, USA
Page : 148 pages
File Size : 40,5 Mb
Release : 2013-03-14
Category : Mathematics
ISBN : 9780199676774

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Integrable Systems by N.J. Hitchin,G. B. Segal,R.S. Ward Pdf

Designed to give graduate students an understanding of integrable systems via the study of Riemann surfaces, loop groups, and twistors, this book has its origins in a lecture series given by the internationally renowned authors. Written in an accessible, informal style, it fills a gap in the existing literature.

Author : Martin Ulrich Schmidt
Publisher : American Mathematical Soc.
Page : 111 pages
File Size : 54,9 Mb
Release : 1996
Category : Mathematics
ISBN : 9780821804605

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Integrable Systems and Riemann Surfaces of Infinite Genus by Martin Ulrich Schmidt Pdf

This memoir develops the spectral theory of the Lax operators of nonlinear Schrodinger-like partial differential equations with periodic boundary conditions. Their spectral curves, i.e., the common spectrum with the periodic shifts, are generically Riemann surfaces of infinite genus. The points corresponding to infinite energy are added. The resulting spaces are no longer Riemann surfaces in the usual sense, but they are quite similar to compact Riemann surfaces. In fact, some of the basic tools of the theory of compact Riemann surfaces are generalized to these spectral curves and illuminate the structure of complete integrability: The eigen bundles define holomorphic line bundles on the spectral curves, which completely determine the potentials. These line bundles may be described by divisors of the same degree as the genus, and these divisors give rise to Darboux coordinates. With the help of a Riemann-Roch Theorem, the isospectral sets (the sets of all potentials corresponding to the same spectral curve) may be identified with open dense subsets of the Jacobian varieties. The real parts of the isospectral sets are infinite dimensional tori, and the group action solves the corresponding nonlinear partial differential equations. Deformations of the spectral curves are in one to one correspondence with holomorphic forms. Serre Duality reproduces the symplectic form.

Author : Björn Gustafsson,Yu-Lin Lin
Publisher : Springer Nature
Page : 156 pages
File Size : 44,6 Mb
Release : 2021-03-22
Category : Mathematics
ISBN : 9783030698638

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Laplacian Growth on Branched Riemann Surfaces by Björn Gustafsson,Yu-Lin Lin Pdf

This book studies solutions of the Polubarinova–Galin and Löwner–Kufarev equations, which describe the evolution of a viscous fluid (Hele-Shaw) blob, after the time when these solutions have lost their physical meaning due to loss of univalence of the mapping function involved. When the mapping function is no longer locally univalent interesting phase transitions take place, leading to structural changes in the data of the solution, for example new zeros and poles in the case of rational maps. This topic intersects with several areas, including mathematical physics, potential theory and complex analysis. The text will be valuable to researchers and doctoral students interested in fluid dynamics, integrable systems, and conformal field theory.

Author : Joel S. Feldman,Horst Knörrer,Eugene Trubowitz
Publisher : American Mathematical Soc.
Page : 306 pages
File Size : 49,7 Mb
Release : 2003
Category : Riemann surfaces
ISBN : 9780821833575

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Riemann Surfaces of Infinite Genus by Joel S. Feldman,Horst Knörrer,Eugene Trubowitz Pdf

In this book, the authors geometrically construct Riemann surfaces of infinite genus by pasting together plane domains and handles. To achieve a meaningful generalization of the classical theory of Riemann surfaces to the case of infinite genus, one must impose restrictions on the asymptotic behavior of the Riemann surface. In the construction carried out here, these restrictions are formulated in terms of the sizes and locations of the handles and in terms of the gluing maps. The approach used has two main attractions. The first is that much of the classical theory of Riemann surfaces, including the Torelli theorem, can be generalized to this class. The second is that solutions of Kadomcev-Petviashvilli equations can be expressed in terms of theta functions associated with Riemann surfaces of infinite genus constructed in the book. Both of these are developed here. The authors also present in detail a number of important examples of Riemann surfaces of infinite genus (hyperelliptic surfaces of infinite genus, heat surfaces and Fermi surfaces). The book is suitable for graduate students and research mathematicians interested in analysis and integrable systems.

Author : Maurizio Cornalba,Xavier Gomez-mont,Alberto Sola Verjovsky
Publisher : World Scientific
Page : 716 pages
File Size : 55,9 Mb
Release : 1989-06-01
Category : Mathematics
ISBN : 9789814590877

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Lectures On Riemann Surfaces - Proceedings Of The College On Riemann Surfaces by Maurizio Cornalba,Xavier Gomez-mont,Alberto Sola Verjovsky Pdf

Author : Wilhelm Schlag
Publisher : American Mathematical Society
Page : 402 pages
File Size : 41,6 Mb
Release : 2014-08-06
Category : Mathematics
ISBN : 9780821898475

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A Course in Complex Analysis and Riemann Surfaces by Wilhelm Schlag Pdf

Complex analysis is a cornerstone of mathematics, making it an essential element of any area of study in graduate mathematics. Schlag's treatment of the subject emphasizes the intuitive geometric underpinnings of elementary complex analysis that naturally lead to the theory of Riemann surfaces. The book begins with an exposition of the basic theory of holomorphic functions of one complex variable. The first two chapters constitute a fairly rapid, but comprehensive course in complex analysis. The third chapter is devoted to the study of harmonic functions on the disk and the half-plane, with an emphasis on the Dirichlet problem. Starting with the fourth chapter, the theory of Riemann surfaces is developed in some detail and with complete rigor. From the beginning, the geometric aspects are emphasized and classical topics such as elliptic functions and elliptic integrals are presented as illustrations of the abstract theory. The special role of compact Riemann surfaces is explained, and their connection with algebraic equations is established. The book concludes with three chapters devoted to three major results: the Hodge decomposition theorem, the Riemann-Roch theorem, and the uniformization theorem. These chapters present the core technical apparatus of Riemann surface theory at this level. This text is intended as a detailed, yet fast-paced intermediate introduction to those parts of the theory of one complex variable that seem most useful in other areas of mathematics, including geometric group theory, dynamics, algebraic geometry, number theory, and functional analysis. More than seventy figures serve to illustrate concepts and ideas, and the many problems at the end of each chapter give the reader ample opportunity for practice and independent study.

Author : Terrence Napier,Mohan Ramachandran
Publisher : Springer Science & Business Media
Page : 563 pages
File Size : 54,5 Mb
Release : 2011-09-08
Category : Mathematics
ISBN : 9780817646936

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An Introduction to Riemann Surfaces by Terrence Napier,Mohan Ramachandran Pdf

This textbook presents a unified approach to compact and noncompact Riemann surfaces from the point of view of the so-called L2 $\bar{\delta}$-method. This method is a powerful technique from the theory of several complex variables, and provides for a unique approach to the fundamentally different characteristics of compact and noncompact Riemann surfaces. The inclusion of continuing exercises running throughout the book, which lead to generalizations of the main theorems, as well as the exercises included in each chapter make this text ideal for a one- or two-semester graduate course.

Author : Eberhard Freitag
Publisher : Springer Science & Business Media
Page : 521 pages
File Size : 54,7 Mb
Release : 2011-06-10
Category : Mathematics
ISBN : 9783642205545

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Complex Analysis 2 by Eberhard Freitag Pdf

The book contains a complete self-contained introduction to highlights of classical complex analysis. New proofs and some new results are included. All needed notions are developed within the book: with the exception of some basic facts which can be found in the ̄rst volume. There is no comparable treatment in the literature.

Author : Menahem Schiffer,Donald C. Spencer
Publisher : Courier Corporation
Page : 465 pages
File Size : 45,5 Mb
Release : 2014-06-01
Category : Mathematics
ISBN : 9780486795430

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Functionals of Finite Riemann Surfaces by Menahem Schiffer,Donald C. Spencer Pdf

This advanced monograph on finite Riemann surfaces, based on the authors' 1949–50 lectures at Princeton University, remains a fundamental book for graduate students. The Bulletin of the American Mathematical Society hailed the self-contained treatment as the source of "a plethora of ideas, each interesting in its own right," noting that "the patient reader will be richly rewarded." Suitable for graduate-level courses, the text begins with three chapters that offer a development of the classical theory along historical lines, examining geometrical and physical considerations, existence theorems for finite Riemann surfaces, and relations between differentials. Subsequent chapters explore bilinear differentials, surfaces imbedded in a given surface, integral operators, and variations of surfaces and of their functionals. The book concludes with a look at applications of the variational method and remarks on generalization to higher dimensional Kahler manifolds.

Author : Hermann Weyl
Publisher : Courier Corporation
Page : 210 pages
File Size : 49,5 Mb
Release : 2013-12-31
Category : Mathematics
ISBN : 9780486131672

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The Concept of a Riemann Surface by Hermann Weyl Pdf

This classic on the general history of functions combines function theory and geometry, forming the basis of the modern approach to analysis, geometry, and topology. 1955 edition.

Author : Mark Agranovsky,Anatoly Golberg,Fiana Jacobzon,David Shoikhet,Lawrence Zalcman
Publisher : Birkhäuser
Page : 372 pages
File Size : 53,5 Mb
Release : 2018-01-31
Category : Mathematics
ISBN : 9783319701547

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Complex Analysis and Dynamical Systems by Mark Agranovsky,Anatoly Golberg,Fiana Jacobzon,David Shoikhet,Lawrence Zalcman Pdf

This book focuses on developments in complex dynamical systems and geometric function theory over the past decade, showing strong links with other areas of mathematics and the natural sciences. Traditional methods and approaches surface in physics and in the life and engineering sciences with increasing frequency – the Schramm‐Loewner evolution, Laplacian growth, and quadratic differentials are just a few typical examples. This book provides a representative overview of these processes and collects open problems in the various areas, while at the same time showing where and how each particular topic evolves. This volume is dedicated to the memory of Alexander Vasiliev.

Author : Dror Varolin
Publisher : American Mathematical Soc.
Page : 258 pages
File Size : 55,9 Mb
Release : 2024-07-01
Category : Mathematics
ISBN : 9780821884270

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Riemann Surfaces by Way of Complex Analytic Geometry by Dror Varolin Pdf

Author : Harvey Cohn
Publisher : Courier Corporation
Page : 352 pages
File Size : 48,8 Mb
Release : 2014-05-05
Category : Mathematics
ISBN : 9780486153292

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Conformal Mapping on Riemann Surfaces by Harvey Cohn Pdf

Lucid, insightful exploration reviews complex analysis, introduces Riemann manifold, shows how to define real functions on manifolds, and more. Perfect for classroom use or independent study. 344 exercises. 1967 edition.

Author : Lipman Bers
Publisher : American Mathematical Soc.
Page : 648 pages
File Size : 40,7 Mb
Release : 1998
Category : Mathematics
ISBN : 0821809962

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Selected Works of Lipman Bers by Lipman Bers Pdf