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Author : Alexander Polishchuk
Publisher : Cambridge University Press
Page : 308 pages
File Size : 40,5 Mb
Release : 2003-04-21
Category : Mathematics
ISBN : 9780521808040
Presents a modern treatment of the theory of theta functions in the context of algebraic geometry.
Author : L. Auslander,R. Tolimieri
Publisher : Springer
Page : 104 pages
File Size : 51,8 Mb
Release : 2006-11-15
Category : Mathematics
ISBN : 9783540374053
Author : Răzvan Gelca
Publisher : World Scientific
Page : 468 pages
File Size : 50,7 Mb
Release : 2014-05-21
Category : Mathematics
ISBN : 9789814520591
This book presents the relationship between classical theta functions and knots. It is based on a novel idea of Răzvan Gelca and Alejandro Uribe, which converts Weil's representation of the Heisenberg group on theta functions to a knot theoretical framework, by giving a topological interpretation to a certain induced representation. It also explains how the discrete Fourier transform can be related to 3- and 4-dimensional topology. Theta Functions and Knots can be read in two perspectives. Readers with an interest in theta functions or knot theory can learn how the two are related. Those interested in Chern–Simons theory will find here an introduction using the simplest case, that of abelian Chern–Simons theory. Moreover, the construction of abelian Chern–Simons theory is based entirely on quantum mechanics and not on quantum field theory as it is usually done. Both the theory of theta functions and low dimensional topology are presented in detail, in order to underline how deep the connection between these two fundamental mathematical subjects is. Hence the book is self-contained with a unified presentation. It is suitable for an advanced graduate course, as well as for self-study. Contents:PrologueA Quantum Mechanical PrototypeSurfaces and CurvesThe Theta Functions Associated to a Riemann SurfaceFrom Theta Functions to KnotsSome Results About 3- and 4-Dimensional ManifoldsThe Discrete Fourier Transform and Topological Quantum Field TheoryTheta Functions in the Quantum Group PerspectiveAn Epilogue — Abelian Chern–Simons Theory Readership: Graduate students and young researchers with an interest in complex analysis, mathematical physics, algebra geometry and low dimensional topology. Keywords:Theta Functions;ChernâSimons Theory;Knots;Skein Modules;Linking Number;Topological Quantum Field TheoryKey Features:A detailed study of the skein modules of the linking number, which provide the simplest example of a skein module (skein modules have become a major object of study in combinatorial topology)A complete discussion of the facts from low dimensional topology (Kirby's theorem, the Lickorish–Walace theorem, Wall's non-additivity of the signature) which are fundamental in Chern–Simons theoryReviews: “It looks like a really good book, presenting its many themes in a very accessible and clear fashion, replete with plenty of pictures and lots of wonderful theorems and proofs from representation theory as well as differential geometry and the kind of functional analysis needed to do quantum physics.” Mathematical Association of America
Author : L. Auslander,R. Tolimieri
Publisher : Lecture Notes in Mathematics
Page : 116 pages
File Size : 49,8 Mb
Release : 1975-01-29
Category : Mathematics
ISBN : STANFORD:36105031579985
Author : Richard Bellman
Publisher : Courier Corporation
Page : 100 pages
File Size : 50,5 Mb
Release : 2013-01-01
Category : Mathematics
ISBN : 9780486492957
Originally published: New York: Rinehart and Winston, 1961.
Author : Jun-ichi Igusa
Publisher : Springer
Page : 254 pages
File Size : 40,7 Mb
Release : 1972-03-28
Category : Mathematics
ISBN : UOM:39015015699047
The theory of theta functions has a long history; for this, we refer A. Krazer and W. Wirtinger the reader to an encyclopedia article by ("Sources" [9]). We shall restrict ourselves to postwar, i. e. , after 1945, periods. Around 1948/49, F. Conforto, c. L. Siegel, A. Well reconsidered the main existence theorems of theta functions and found natural proofs for them. These are contained in Conforto: Abelsche Funktionen und algebraische Geometrie, Springer (1956); Siegel: Analytic functions of several complex variables, Lect. Notes, I. A. S. (1948/49); Well: Theoremes fondamentaux de la theorie des fonctions theta, Sem. Bourbaki, No. 16 (1949). The complete account of Weil's method appeared in his book of 1958 [20]. The next important achievement was the theory of compacti fication of the quotient variety of Siegel's upper-half space by a modular group. There are many ways to compactify the quotient variety; we are talking about what might be called a standard compactification. Such a compactification was obtained first as a Hausdorff space by I. Satake in "On the compactification of the Siegel space", J. Ind. Math. Soc. 20, 259-281 (1956), and as a normal projective variety by W. L. Baily in 1958 [1]. In 1957/58, H. Cartan took up this theory in his seminar [3]; it was shown that the graded ring of modular forms relative to the given modular group is a normal integral domain which is finitely generated over C.
Author : Leon Ehrenpreis,Robert Clifford Gunning
Publisher : American Mathematical Soc.
Page : 366 pages
File Size : 44,8 Mb
Release : 1989
Category : Mathematics
ISBN : 9780821814840
Author : Maruti Ram Murty
Publisher : American Mathematical Soc.
Page : 188 pages
File Size : 40,7 Mb
Release : 1993-01-01
Category : Mathematics
ISBN : 0821870114
This book contains lectures on theta functions written by experts well known for excellence in exposition. The lectures represent the content of four courses given at the Centre de Recherches Mathematiques in Montreal during the academic year 1991-1992, which was devoted to the study of automorphic forms. Aimed at graduate students, the book synthesizes the classical and modern points of view in theta functions, concentrating on connections to number theory and representation theory. An excellent introduction to this important subject of current research, this book is suitable as a text in advanced graduate courses.
Author : David Mumford
Publisher : Springer Science & Business Media
Page : 248 pages
File Size : 40,5 Mb
Release : 2007-06-25
Category : Mathematics
ISBN : 9780817645779
This volume is the first of three in a series surveying the theory of theta functions. Based on lectures given by the author at the Tata Institute of Fundamental Research in Bombay, these volumes constitute a systematic exposition of theta functions, beginning with their historical roots as analytic functions in one variable (Volume I), touching on some of the beautiful ways they can be used to describe moduli spaces (Volume II), and culminating in a methodical comparison of theta functions in analysis, algebraic geometry, and representation theory (Volume III).
Author : Louis Auslander
Publisher : American Mathematical Soc.
Page : 106 pages
File Size : 50,6 Mb
Release : 1977
Category : Mathematics
ISBN : 9780821816844
Consists of three chapters covering the following topics: foundations, bilinear forms and presentations of certain 2-step nilpotent Lie groups, discrete subgroups of the Heisenberg group, the automorphism group of the Heisenberg group, fundamental unitary representations of the Heisenberg group, and the Fourier transform and the Weil-Brezin map.
Author : Alekse_ Ivanovich Markushevich
Publisher : American Mathematical Soc.
Page : 188 pages
File Size : 41,6 Mb
Release : 2006-07-26
Category : Mathematics
ISBN : 0821898361
Historical introduction. The Jacobian inversion problem Periodic functions of several complex variables Riemann matrices. Jacobian (intermediate) functions Construction of Jacobian functions of a given type. Theta functions and Abelian functions. Abelian and Picard manifolds Appendix A. Skew-symmetric determinants Appendix B. Divisors of analytic functions Appendix C. A summary of the most important formulas
Author : Christina Birkenhake,Herbert Lange
Publisher : Springer Science & Business Media
Page : 638 pages
File Size : 53,6 Mb
Release : 2013-03-14
Category : Mathematics
ISBN : 9783662063071
This book explores the theory of abelian varieties over the field of complex numbers, explaining both classic and recent results in modern language. The second edition adds five chapters on recent results including automorphisms and vector bundles on abelian varieties, algebraic cycles and the Hodge conjecture. ". . . far more readable than most . . . it is also much more complete." Olivier Debarre in Mathematical Reviews, 1994.
Author : L. Auslander,R. Tolimieri
Publisher : Unknown
Page : 106 pages
File Size : 50,5 Mb
Release : 2014-01-15
Category : Electronic
ISBN : 366219256X
Author : Jaques Calmet
Publisher : Springer Science & Business Media
Page : 430 pages
File Size : 48,5 Mb
Release : 1986-07
Category : Computers
ISBN : 3540167765
Author : Herbert Lange
Publisher : Springer Nature
Page : 390 pages
File Size : 44,7 Mb
Release : 2023-03-15
Category : Mathematics
ISBN : 9783031255700
This textbook offers an introduction to abelian varieties, a rich topic of central importance to algebraic geometry. The emphasis is on geometric constructions over the complex numbers, notably the construction of important classes of abelian varieties and their algebraic cycles. The book begins with complex tori and their line bundles (theta functions), naturally leading to the definition of abelian varieties. After establishing basic properties, the moduli space of abelian varieties is introduced and studied. The next chapters are devoted to the study of the main examples of abelian varieties: Jacobian varieties, abelian surfaces, Albanese and Picard varieties, Prym varieties, and intermediate Jacobians. Subsequently, the Fourier–Mukai transform is introduced and applied to the study of sheaves, and results on Chow groups and the Hodge conjecture are obtained. This book is suitable for use as the main text for a first course on abelian varieties, for instance as a second graduate course in algebraic geometry. The variety of topics and abundant exercises also make it well suited to reading courses. The book provides an accessible reference, not only for students specializing in algebraic geometry but also in related subjects such as number theory, cryptography, mathematical physics, and integrable systems.
