Background
Kra, Irwin was born on January 5, 1937 in Krasnoshelcz, Poland. Son of Jacob and Bryna (Kozica) Kra. came to the United States, 1952.
(This text covers Riemann surface theory from elementary a...)
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, while basic tools are developed to describe the analytic, geometric, and algebraic properties of Riemann surfaces and the associated Abelian varities. 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, as are alternate proofs for the most important results, showing the diversity of approaches to the subject. Of interest not only to pure mathematicians, but also to physicists interested in string theory and related topics.
http://www.amazon.com/gp/product/1461273919/?tag=2022091-20
(This text covers Riemann surface theory from elementary a...)
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, while basic tools are developed to describe the analytic, geometric, and algebraic properties of Riemann surfaces and the associated Abelian varities. 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, as are alternate proofs for the most important results, showing the diversity of approaches to the subject. Of interest not only to pure mathematicians, but also to physicists interested in string theory and related topics.
http://www.amazon.com/gp/product/0387977031/?tag=2022091-20
(There are incredibly rich connections between classical a...)
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 number-theoretic 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, well-known 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 research-level problems. The text is suitable for a graduate course or for independent reading.
http://www.amazon.com/gp/product/0821813927/?tag=2022091-20
mathematics researcher and educator
Kra, Irwin was born on January 5, 1937 in Krasnoshelcz, Poland. Son of Jacob and Bryna (Kozica) Kra. came to the United States, 1952.
Bachelor of Science, Polytechnic Universitas Negeri Yogyakarta, 1960; Master of Arts, Columbia University, 1964; Doctor of Philosophy, Columbia University, 1966.
C.L.E. Moore instructor, Massachusetts Institute of Technology, Cambridge, 1966-1968; assistant professor mathematics, State University of New York, Stony Brook, 1968-1969; associate professor, State University of New York, Stony Brook, 1969-1971; professor, State University of New York, Stony Brook, since 1971; acting department chairman, State University of New York, Stony Brook, 1970-1971; chairman, State University of New York, Stony Brook, 1975-1981, 84-89; acting provost division mathematics science, State University of New York, Stony Brook, 1971-1972; dean division physical science and math, State University of New York, Stony Brook, since 1991. Advisory professor Fudan U., Shanghai, People's Republic China, 1987. Visiting professor Hebrew U., Math Sciences Research Institute, Harvard University, Massachusetts Institute of Technology.
(This text covers Riemann surface theory from elementary a...)
(This text covers Riemann surface theory from elementary a...)
(There are incredibly rich connections between classical a...)
With United States Army, 1961-1963. Member American Mathematics Society (county 1984-1991, 92-, Executive Committee 1986-1991), Mathematics Association American.
Married Eleanor Traube, December 23, 1961. Children: Douglas, Bryna, Gabriel.