Kra, Irwin
Overview
Works:  44 works in 205 publications in 2 languages and 4,715 library holdings 

Genres:  Conference proceedings Biography 
Roles:  Author, Editor 
Classifications:  QA333, 515.223 
Publication Timeline
.
Most widely held works by
Irwin Kra
Riemann surfaces by
Hershel M Farkas(
Book
)
34 editions published between 1979 and 2003 in English and Undetermined and held by 876 WorldCat member libraries worldwide
This text covers Riemann surface theory from elementary aspects to the fontiers of current research. Open and closed surfaces are treated with emphasis on the compact case. Basic tools are developed to describe the analytic, geometric, and algebraic properties of Riemann surfaces and the Abelian varities associated with these surfaces. Topics covered include existence of meromorphic functions, the Riemann Roch theorem, Abel's theorem, the Jacobi inversion problem, Noether's theorem, and the Riemann vanishing theorem. A complete treatment of the uniformization of Riemann sufaces via Fuchsian groups, including branched coverings, is presented. Alternate proofs for the most important results are included, showing the diversity of approaches to the subject. For this new edition, the material has been brought up todate, and erros have been corrected. The book should be of interest not only to pure mathematicians, but also to physicists interested in string theory and related topics
34 editions published between 1979 and 2003 in English and Undetermined and held by 876 WorldCat member libraries worldwide
This text covers Riemann surface theory from elementary aspects to the fontiers of current research. Open and closed surfaces are treated with emphasis on the compact case. Basic tools are developed to describe the analytic, geometric, and algebraic properties of Riemann surfaces and the Abelian varities associated with these surfaces. Topics covered include existence of meromorphic functions, the Riemann Roch theorem, Abel's theorem, the Jacobi inversion problem, Noether's theorem, and the Riemann vanishing theorem. A complete treatment of the uniformization of Riemann sufaces via Fuchsian groups, including branched coverings, is presented. Alternate proofs for the most important results are included, showing the diversity of approaches to the subject. For this new edition, the material has been brought up todate, and erros have been corrected. The book should be of interest not only to pure mathematicians, but also to physicists interested in string theory and related topics
A crash course on Kleinian groups; lectures given at a special session at the January 1974 meeting of the American Mathematical
Society at San Francisco by
American Mathematical Society(
Book
)
20 editions published between 1974 and 2008 in English and held by 447 WorldCat member libraries worldwide
20 editions published between 1974 and 2008 in English and held by 447 WorldCat member libraries worldwide
Complex analysis : in the spirit of Lipman Bers by
Jane Gilman(
Book
)
32 editions published between 2007 and 2013 in English and held by 411 WorldCat member libraries worldwide
"This book is intended for a graduate course on complex analysis, also known as function theory. The main focus is the theory of complexvalued functions of a single complex variable. This theory is a prerequisite for the study of many current and rapidly developing areas of mathematics including the theory of several and infinitely many complex variables, the theory of groups, hyperbolic geometry and threemanifolds, and number theory. Complex analysis has connections and applications to many other subjects in mathematics and to other sciences. It is an area where the classic and the modern techniques meet and benefit from each other. This material should be part of the education of every practicing mathematician, and it will also be of interest to computer scientists, physicists, and engineers." "The book covers most, if not all, of the material contained in Bers's courses on first year complex analysis. In addition, topics of current interest such as zeros of holomorphic functions and the connection between hyperbolic geometry and complex analysis are explored."BOOK JACKET
32 editions published between 2007 and 2013 in English and held by 411 WorldCat member libraries worldwide
"This book is intended for a graduate course on complex analysis, also known as function theory. The main focus is the theory of complexvalued functions of a single complex variable. This theory is a prerequisite for the study of many current and rapidly developing areas of mathematics including the theory of several and infinitely many complex variables, the theory of groups, hyperbolic geometry and threemanifolds, and number theory. Complex analysis has connections and applications to many other subjects in mathematics and to other sciences. It is an area where the classic and the modern techniques meet and benefit from each other. This material should be part of the education of every practicing mathematician, and it will also be of interest to computer scientists, physicists, and engineers." "The book covers most, if not all, of the material contained in Bers's courses on first year complex analysis. In addition, topics of current interest such as zeros of holomorphic functions and the connection between hyperbolic geometry and complex analysis are explored."BOOK JACKET
Riemann surfaces and related topics : proceedings of the 1978 Stony Brook conference by Conference on Riemann Surfaces and Related Topics(
Book
)
11 editions published between 1980 and 1981 in English and Undetermined and held by 403 WorldCat member libraries worldwide
11 editions published between 1980 and 1981 in English and Undetermined and held by 403 WorldCat member libraries worldwide
Automorphic forms and Kleinian groups by
Irwin Kra(
Book
)
17 editions published between 1972 and 1975 in English and Russian and held by 387 WorldCat member libraries worldwide
17 editions published between 1972 and 1975 in English and Russian and held by 387 WorldCat member libraries worldwide
Hilbert's fourth problem by
A. V Pogorelov(
Book
)
4 editions published in 1979 in English and held by 375 WorldCat member libraries worldwide
4 editions published in 1979 in English and held by 375 WorldCat member libraries worldwide
Quasiconformal mappings and Riemann surfaces by
S. L Krushkalʹ(
Book
)
8 editions published in 1979 in English and held by 263 WorldCat member libraries worldwide
8 editions published in 1979 in English and held by 263 WorldCat member libraries worldwide
Theta constants, Riemann surfaces, and the modular group : an introduction with applications to uniformization theorems, partition
identities, and combinatorial number theory by
Hershel M Farkas(
Book
)
13 editions published in 2001 in English and held by 259 WorldCat member libraries worldwide
There are incredibly rich connections between classical analysis and number theory. For instance, analytic number theory contains many examples of asymptotic expressions derived from estimates for analytic functions, such as in the proof of the Prime Number Theorem. In combinatorial number theory, exact formulas for numbertheoretic quantities are derived from relations between analytic functions. Elliptic functions, especially theta functions, are an important class of such functions in this context, which had been made clear already in Jacobi's Fundamenta nova. Theta functions are also classically connected with Riemann surfaces and with the modular group $\Gamma = \mathrm{PSL}(2,\mathbb{Z})$, which provide another path for insights into number theory. Farkas and Kra, wellknown masters of the theory of Riemann surfaces and the analysis of theta functions, uncover here interesting combinatorial identities by means of the function theory on Riemann surfaces related to the principal congruence subgroups $\Gamma(k)$. For instance, the authors use this approach to derive congruences discovered by Ramanujan for the partition function, with the main ingredient being the construction of the same function in more than one way. The authors also obtain a variant on Jacobi's famous result on the number of ways that an integer can be represented as a sum of four squares, replacing the squares by triangular numbers and, in the process, obtaining a cleaner result. The recent trend of applying the ideas and methods of algebraic geometry to the study of theta functions and number theory has resulted in great advances in the area. However, the authors choose to stay with the classical point of view. As a result, their statements and proofs are very concrete. In this book the mathematician familiar with the algebraic geometry approach to theta functions and number theory will find many interesting ideas as well as detailed explanations and derivations of new and old results. Highlights of the book include systematic studies of theta constant identities, uniformizations of surfaces represented by subgroups of the modular group, partition identities, and Fourier coefficients of automorphic functions. Prerequisites are a solid understanding of complex analysis, some familiarity with Riemann surfaces, Fuchsian groups, and elliptic functions, and an interest in number theory. The book contains summaries of some of the required material, particularly for theta functions and theta constants. Readers will find here a careful exposition of a classical point of view of analysis and number theory. Presented are numerous examples plus suggestions for researchlevel problems. The text is suitable for a graduate course or for independent reading
13 editions published in 2001 in English and held by 259 WorldCat member libraries worldwide
There are incredibly rich connections between classical analysis and number theory. For instance, analytic number theory contains many examples of asymptotic expressions derived from estimates for analytic functions, such as in the proof of the Prime Number Theorem. In combinatorial number theory, exact formulas for numbertheoretic quantities are derived from relations between analytic functions. Elliptic functions, especially theta functions, are an important class of such functions in this context, which had been made clear already in Jacobi's Fundamenta nova. Theta functions are also classically connected with Riemann surfaces and with the modular group $\Gamma = \mathrm{PSL}(2,\mathbb{Z})$, which provide another path for insights into number theory. Farkas and Kra, wellknown masters of the theory of Riemann surfaces and the analysis of theta functions, uncover here interesting combinatorial identities by means of the function theory on Riemann surfaces related to the principal congruence subgroups $\Gamma(k)$. For instance, the authors use this approach to derive congruences discovered by Ramanujan for the partition function, with the main ingredient being the construction of the same function in more than one way. The authors also obtain a variant on Jacobi's famous result on the number of ways that an integer can be represented as a sum of four squares, replacing the squares by triangular numbers and, in the process, obtaining a cleaner result. The recent trend of applying the ideas and methods of algebraic geometry to the study of theta functions and number theory has resulted in great advances in the area. However, the authors choose to stay with the classical point of view. As a result, their statements and proofs are very concrete. In this book the mathematician familiar with the algebraic geometry approach to theta functions and number theory will find many interesting ideas as well as detailed explanations and derivations of new and old results. Highlights of the book include systematic studies of theta constant identities, uniformizations of surfaces represented by subgroups of the modular group, partition identities, and Fourier coefficients of automorphic functions. Prerequisites are a solid understanding of complex analysis, some familiarity with Riemann surfaces, Fuchsian groups, and elliptic functions, and an interest in number theory. The book contains summaries of some of the required material, particularly for theta functions and theta constants. Readers will find here a careful exposition of a classical point of view of analysis and number theory. Presented are numerous examples plus suggestions for researchlevel problems. The text is suitable for a graduate course or for independent reading
In the tradition of Ahlfors and Bers : proceedings of the First AhlforsBers Colloquium, AhlforsBers Colloquium, November
68, 1998, State University of New York at Stony Brook by AhlforsBers Colloquium(
Book
)
9 editions published in 2000 in English and held by 201 WorldCat member libraries worldwide
9 editions published in 2000 in English and held by 201 WorldCat member libraries worldwide
Selected works of Lipman Bers : papers on complex analysis by
Lipman Bers(
Book
)
4 editions published in 1998 in English and held by 116 WorldCat member libraries worldwide
4 editions published in 1998 in English and held by 116 WorldCat member libraries worldwide
On the ring of holomorphic functions on an open Riemann surface by
Irwin Kra(
Book
)
4 editions published between 1966 and 1967 in English and held by 12 WorldCat member libraries worldwide
4 editions published between 1966 and 1967 in English and held by 12 WorldCat member libraries worldwide
Banach algebras of analytic functions on Riemann surfaces by
Irwin Kra(
Book
)
in English and held by 10 WorldCat member libraries worldwide
in English and held by 10 WorldCat member libraries worldwide
A crash course on Kleinian groups, San Francisco, 1974 : lectures given at a special session at the January 1974 meeting of
the American Mathematical Society at San Francisco. by
American Mathematical Society(
Book
)
5 editions published in 1974 in English and held by 7 WorldCat member libraries worldwide
5 editions published in 1974 in English and held by 7 WorldCat member libraries worldwide
In the tradition of Ahlfors and Bers : proceedings of the First AhlforsBers Colloquium(
Book
)
3 editions published in 2000 in English and held by 6 WorldCat member libraries worldwide
3 editions published in 2000 in English and held by 6 WorldCat member libraries worldwide
Selected works of Lipman Bers : papers on complex analysis by
Lipman Bers(
Book
)
3 editions published in 1998 in English and held by 6 WorldCat member libraries worldwide
3 editions published in 1998 in English and held by 6 WorldCat member libraries worldwide
Selected works of Lipman Bers : papers on complex analysis by
Lipman Bers(
Book
)
2 editions published in 1998 in English and held by 6 WorldCat member libraries worldwide
2 editions published in 1998 in English and held by 6 WorldCat member libraries worldwide
Contributions to analysis: a collection of papers dedicated to Lipman Bers by
Lars V Ahlfors(
Book
)
4 editions published between 1974 and 2014 in English and held by 5 WorldCat member libraries worldwide
4 editions published between 1974 and 2014 in English and held by 5 WorldCat member libraries worldwide
Lipman Bers : May 22, 1914October 29, 1993(
Book
)
1 edition published in 2001 in English and held by 4 WorldCat member libraries worldwide
1 edition published in 2001 in English and held by 4 WorldCat member libraries worldwide
Holomorphic functions and moduli : proceedings of a workshop held March 1319, 1986 by
D Drasin(
)
1 edition published in 1988 in English and held by 0 WorldCat member libraries worldwide
1 edition published in 1988 in English and held by 0 WorldCat member libraries worldwide
Holomorphic functions and moduli. proceedings of a workshop held March 1319, 1986 by
D Drasin(
)
1 edition published in 1988 in English and held by 0 WorldCat member libraries worldwide
1 edition published in 1988 in English and held by 0 WorldCat member libraries worldwide
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Related Identities
 Bers, Lipman Honoree Author Editor
 Farkas, Hershel M. Author
 Gilman, Jane 1945 Author
 Rodríguez, Rubí E. 1953 Author Editor
 Maskit, Bernard Editor
 Pogorelov, A. V. (Alekseĭ Vasilʹevich) 19192002 Author
 Zaustinskiy, Eugene
 American Mathematical Society Other
 Krushkalʹ, S. L. (Samuil Leĭbovich) Author
 American Mathematical Society Meeting (1974 : San Francisco, Calif.)
Useful Links
Associated Subjects
Automorphic functions Banach algebras Banach spaces Bers, Lipman Differential equations, Partial Functional analysis Functions Functions, Theta Functions of complex variables Geometric function theory Geometry, Algebraic GeometryFoundations Global analysis (Mathematics) Group theory Holomorphic functions Kleinian groups Mappings (Mathematics) Mathematical analysis Mathematicians Mathematics Modular groups Quasiconformal mappings Riemann surfaces Rings (Algebra) Teichmüller spaces Topological groups