Riemann Surfaces Of Infinite Genus

Author : Joel S. Feldman,Horst Knörrer,Eugene Trubowitz
Publisher : American Mathematical Soc.
Page : 306 pages
File Size : 42,5 Mb
Release : 2003
Category : Riemann surfaces
ISBN : 9780821833575

Get Book

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 : Martin Ulrich Schmidt
Publisher : American Mathematical Soc.
Page : 111 pages
File Size : 55,8 Mb
Release : 1996
Category : Mathematics
ISBN : 9780821804605

Get Book

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 : Jürgen Jost
Publisher : Springer Science & Business Media
Page : 304 pages
File Size : 45,8 Mb
Release : 2013-04-17
Category : Mathematics
ISBN : 9783662034460

Get Book

Compact Riemann Surfaces by Jürgen Jost Pdf

This book is novel in its broad perspective on Riemann surfaces: the text systematically explores the connection with other fields of mathematics. The book can serve as an introduction to contemporary mathematics as a whole, as it develops background material from algebraic topology, differential geometry, the calculus of variations, elliptic PDE, and algebraic geometry. The book is unique among textbooks on Riemann surfaces in its inclusion of an introduction to Teichmüller theory. For this new edition, the author has expanded and rewritten several sections to include additional material and to improve the presentation.

Author : Lars Valerian Ahlfors,E. Calabi,Marston Morse,Leo Sario,Donald Clayton Spencer
Publisher : Princeton University Press
Page : 275 pages
File Size : 55,9 Mb
Release : 1953-08-21
Category : Mathematics
ISBN : 9780691079394

Get Book

Contributions to the Theory of Riemann Surfaces by Lars Valerian Ahlfors,E. Calabi,Marston Morse,Leo Sario,Donald Clayton Spencer Pdf

A classic treatment of Riemann surfaces from the acclaimed Annals of Mathematics Studies series Princeton University Press is proud to have published the Annals of Mathematics Studies since 1940. One of the oldest and most respected series in science publishing, it has included many of the most important and influential mathematical works of the twentieth century. The series continues this tradition as Princeton University Press publishes the major works of the twenty-first century. To mark the continued success of the series, all books are available in paperback and as ebooks.

Author : Robert D.M. Accola
Publisher : Springer
Page : 117 pages
File Size : 52,6 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540490562

Get Book

Topics in the Theory of Riemann Surfaces by Robert D.M. Accola Pdf

The book's main concern is automorphisms of Riemann surfaces, giving a foundational treatment from the point of view of Galois coverings, and treating the problem of the largest automorphism group for a Riemann surface of a given genus. In addition, the extent to which fixed points of automorphisms are generalized Weierstrass points is considered. The extremely useful inequality of Castelnuovo- Severi is also treated. While the methods are elementary, much of the material does not appear in the current texts on Riemann surfaces, algebraic curves. The book is accessible to a reader who has had an introductory course on the theory of Riemann surfaces or algebraic curves.

Author : Leo Sario,Mitsuru Nakai
Publisher : Springer Science & Business Media
Page : 469 pages
File Size : 49,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642482694

Get Book

Classification Theory of Riemann Surfaces by Leo Sario,Mitsuru Nakai Pdf

The purpose of the present monograph is to systematically develop a classification theory of Riemann surfaces. Some first steps will also be taken toward a classification of Riemannian spaces. Four phases can be distinguished in the chronological background: the type problem; general classification; compactifications; and extension to higher dimensions. The type problem evolved in the following somewhat overlapping steps: the Riemann mapping theorem, the classical type problem, and the existence of Green's functions. The Riemann mapping theorem laid the foundation to classification theory: there are only two conformal equivalence classes of (noncompact) simply connected regions. Over half a century of efforts by leading mathematicians went into giving a rigorous proof of the theorem: RIEMANN, WEIERSTRASS, SCHWARZ, NEUMANN, POINCARE, HILBERT, WEYL, COURANT, OSGOOD, KOEBE, CARATHEODORY, MONTEL. The classical type problem was to determine whether a given simply connected covering surface of the plane is conformally equivalent to the plane or the disko The problem was in the center of interest in the thirties and early forties, with AHLFORS, KAKUTANI, KOBAYASHI, P. MYRBERG, NEVANLINNA, SPEISER, TEICHMÜLLER and others obtaining incisive specific results. The main problem of finding necessary and sufficient conditions remains, however, unsolved.

Author : Rick Miranda
Publisher : American Mathematical Soc.
Page : 414 pages
File Size : 46,8 Mb
Release : 1995
Category : Mathematics
ISBN : 9780821802687

Get Book

Algebraic Curves and Riemann Surfaces by Rick Miranda Pdf

In this book, Miranda takes the approach that algebraic curves are best encountered for the first time over the complex numbers, where the reader's classical intuition about surfaces, integration, and other concepts can be brought into play. Therefore, many examples of algebraic curves are presented in the first chapters. In this way, the book begins as a primer on Riemann surfaces, with complex charts and meromorphic functions taking centre stage. But the main examples come fromprojective curves, and slowly but surely the text moves toward the algebraic category. Proofs of the Riemann-Roch and Serre Dualtiy Theorems are presented in an algebraic manner, via an adaptation of the adelic proof, expressed completely in terms of solving a Mittag-Leffler problem. Sheaves andcohomology are introduced as a unifying device in the later chapters, so that their utility and naturalness are immediately obvious. Requiring a background of one term of complex variable theory and a year of abstract algebra, this is an excellent graduate textbook for a second-term course in complex variables or a year-long course in algebraic geometry.

Author : Robert C. Gunning
Publisher : Springer Science & Business Media
Page : 177 pages
File Size : 51,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642663826

Get Book

Riemann Surfaces and Generalized Theta Functions by Robert C. Gunning Pdf

The investigation of the relationships between compact Riemann surfaces (al gebraic curves) and their associated complex tori (Jacobi varieties) has long been basic to the study both of Riemann surfaces and of complex tori. A Riemann surface is naturally imbedded as an analytic submanifold in its associated torus; and various spaces of linear equivalence elasses of divisors on the surface (or equivalently spaces of analytic equivalence elasses of complex line bundies over the surface), elassified according to the dimensions of the associated linear series (or the dimensions of the spaces of analytic cross-sections), are naturally realized as analytic subvarieties of the associated torus. One of the most fruitful of the elassical approaches to this investigation has been by way of theta functions. The space of linear equivalence elasses of positive divisors of order g -1 on a compact connected Riemann surface M of genus g is realized by an irreducible (g -1)-dimensional analytic subvariety, an irreducible hypersurface, of the associated g-dimensional complex torus J(M); this hyper 1 surface W- r;;;, J(M) is the image of the natural mapping Mg- -+J(M), and is g 1 1 birationally equivalent to the (g -1)-fold symmetric product Mg- jSg-l of the Riemann surface M.

Author : R.D.M. Accola
Publisher : Springer
Page : 109 pages
File Size : 52,9 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540376026

Get Book

Riemann Surfaces, Theta Functions, and Abelian Automorphisms Groups by R.D.M. Accola Pdf

Author : R. Narasimhan
Publisher : Birkhäuser
Page : 127 pages
File Size : 50,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034886178

Get Book

Compact Riemann Surfaces by R. Narasimhan Pdf

Author : Y. Rodin
Publisher : Springer Science & Business Media
Page : 212 pages
File Size : 50,8 Mb
Release : 2013-06-29
Category : Mathematics
ISBN : 9789400928855

Get Book

The Riemann Boundary Problem on Riemann Surfaces by Y. Rodin Pdf

Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Clad in Crane Feathers' in R. Brown 'The point of a Pin'. van GuIik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.

Author : Benson Farb,Richard Hain,Eduard Looijenga
Publisher : American Mathematical Soc.
Page : 371 pages
File Size : 49,7 Mb
Release : 2013-08-16
Category : Mathematics
ISBN : 9780821898871

Get Book

Moduli Spaces of Riemann Surfaces by Benson Farb,Richard Hain,Eduard Looijenga Pdf

Mapping class groups and moduli spaces of Riemann surfaces were the topics of the Graduate Summer School at the 2011 IAS/Park City Mathematics Institute. This book presents the nine different lecture series comprising the summer school, covering a selection of topics of current interest. The introductory courses treat mapping class groups and Teichmüller theory. The more advanced courses cover intersection theory on moduli spaces, the dynamics of polygonal billiards and moduli spaces, the stable cohomology of mapping class groups, the structure of Torelli groups, and arithmetic mapping class groups. The courses consist of a set of intensive short lectures offered by leaders in the field, designed to introduce students to exciting, current research in mathematics. These lectures do not duplicate standard courses available elsewhere. The book should be a valuable resource for graduate students and researchers interested in the topology, geometry and dynamics of moduli spaces of Riemann surfaces and related topics. Titles in this series are co-published with the Institute for Advanced Study/Park City Mathematics Institute. Members of the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) receive a 20% discount from list price.

Author : Misha Shifman,Arkady Vainshtein,J. F. Wheater
Publisher : World Scientific
Page : 900 pages
File Size : 44,9 Mb
Release : 2005
Category : Condensed matter
ISBN : 9789812560001

Get Book

From Fields to Strings by Misha Shifman,Arkady Vainshtein,J. F. Wheater Pdf

Author : Lars Valerian Ahlfors
Publisher : Princeton University Press
Page : 436 pages
File Size : 46,9 Mb
Release : 1971-07-21
Category : Mathematics
ISBN : 069108081X

Get Book

Advances in the Theory of Riemann Surfaces by Lars Valerian Ahlfors Pdf

Intended for researchers in Riemann surfaces, this volume summarizes a significant portion of the work done in the field during the years 1966 to 1971.

Author : Lars Valerian Ahlfors,Leo Sario
Publisher : Princeton University Press
Page : 397 pages
File Size : 54,7 Mb
Release : 2015-12-08
Category : Mathematics
ISBN : 9781400874538

Get Book

Riemann Surfaces by Lars Valerian Ahlfors,Leo Sario Pdf

The theory of Riemann surfaces has a geometric and an analytic part. The former deals with the axiomatic definition of a Riemann surface, methods of construction, topological equivalence, and conformal mappings of one Riemann surface on another. The analytic part is concerned with the existence and properties of functions that have a special character connected with the conformal structure, for instance: subharmonic, harmonic, and analytic functions. Originally published in 1960. 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.