2008; 263 pp; hardcover
ISBN-13: 978-81-85931-86-9
This is a reprinting of the revised second edition (1974) of David Mumford's classic 1970 book. It gives a systematic account of the basic results about abelian varieties. It includes expositions of analytic methods applicable over the ground field of complex numbers, as well as of scheme-theoretic methods used to deal with inseparable isogenies when the ground field has positive characteristic. A self-contained proof of the existence of the dual abelian variety is given. The structure of the ring of endomorphisms of an abelian variety is discussed. These are appendices on Tate's theorem on endomorphisms of abelian varieties over finite fields (by C. P. Ramanujam) and on the Mordell-Weil theorem (by Yuri Manin).
David Mumford was awarded the 2007 AMS Steele Prize for Mathematical Exposition.
According to the citation: "Abelian Varieties ... remains the definitive account of the subject ... the classical theory is beautifully intertwined with the modern theory, in a way which sharply illuminates both ... [It] will remain for the foreseeable future a classic to which the reader returns over and over."
A publication of the Tata Institute of Fundamental Research. Distributed worldwide except in India, Bangladesh, Bhutan, Maldavis, Nepal, Pakistan, and Sri Lanka.
Graduate students and research mathematicians interested in abelian varieties.
Analytic theory
Alegebraic theory via varieties
Algebraic theory via schemes
Hom (X, X) and l-adic representation
Appendix I: The theorem of Tate by C.P. Ramanujam
Appendix II: Mordell-Weil theorem by Yuri Manin
Bibliography
Index
2008; 157 pp; softcover
ISBN-13: 978-81-7319-984-4
Expected publication date is November 17, 2008.
This book is an expanded version of the notes of a course of lectures given by at the Tata Institute of Fundamental Research in 1998. It deals with several important results and methods in transcendental number theory.
First, the classical result of Lindemann-Weierstrass and its applications are dealt with. Subsequently, Siegel's theory of E-functions is developed systematically, culminating in Shidlovskii's theorem on the algebraic independence of the values of the E-functions satisfying a system of differential equations at certain algebraic values. Proof of the Gelfond-Schneider Theorem is given based on the method of interpolation determinants introduced in 1992 by M. Laurent.
The author's famous result in 1996 on the algebraic independence of the values of the Ramanujan functions is the main theme of the reminder of the book. After deriving several beautiful consequences of his result, the author develops the algebraic material necessary for the proof. The two important technical tools in the proof are Philippon's criterion for algebraic independence and zero bound for Ramanujan functions. The proofs of these are covered in detail.
The author also presents a direct method, without using any criterion for algebraic independence as that of Philippon, by which one can obtain lower bounds for transcendence degree of finitely generated field \mathbb Q(\omega_1,\ldots,\omega_m). This is a contribution towards Schanuel's conjecture.
The book is self-contained and the proofs are clear and lucid. A brief history of the topics is also given. Some sections intersect with Chapters 3 and 10 of Introduction to Algebraic Independence Theory, Lecture Notes in Mathematics, Springer, 1752, edited by Yu. V. Nesterenko and P. Philippon.
Graduate students and research mathematicians interested in number theory.
Lindemann-Weierstrass theorem
E-functions and Shidlovskii's theorem
Small transcendence degree (exponential function)
Small transcendence degree (modular functions)
Algebraic fundamentals
Philippon's criterion of algebraic independence
Fields of large transcendence degree
Multiplicity estimates
2009; 371 pp; hardcover
ISBN-13: 978-0-8218-4739-8
Expected publication date is December 24, 2008.
This book is an introduction to the modern approach to the theory of Markov chains. The main goal of this approach is to determine the rate of convergence of a Markov chain to the stationary distribution as a function of the size and geometry of the state space. The authors develop the key tools for estimating convergence times, including coupling, strong stationary times, and spectral methods. Whenever possible, probabilistic methods are emphasized. The book includes many examples and provides brief introductions to some central models of statistical mechanics. Also provided are accounts of random walks on networks, including hitting and cover times, and analyses of several methods of shuffling cards. As a prerequisite, the authors assume a modest understanding of probability theory and linear algebra at an undergraduate level. Markov Chains and Mixing Times is meant to bring the excitement of this active area of research to a wide audience.
Undergraduates, graduate students, and research mathematicians interested in probability, combinatorics, simulation, computer science, and Markov chain.
Part I
Introduction to finite Markov chains
Classical (and useful) Markov chains
Markov chain Monte Carlo: Metropolis and Glauber chains
Introduction to Markov chain mixing
Coupling
Strong stationary times
Lower bounds on mixing times
The symmetric group and shuffling cards
Random walks on networks
Hitting times
Cover times
Eigenvalues
Part II
Eigenfunctions and comparison of chains
The transportation metric and path coupling
The Ising model
From shuffling cards to shuffling genes
Martingales and evolving sets
The cut-off phenomenon
Lamplighter walks
Continuous-time chains
Countable state-space chains
Coupling from the past
Open problems
Appendix A: Notes on notation
Appendix B: Background material
Appendix C: Introduction to simulation
Appendix D: Solutions to selected exercises
Bibliography
Index
CBMS Regional Conference Series in Mathematics, Number: 109
2008; 87 pp; softcover
ISBN-13: 978-0-8218-0980-8
Expected publication date is December 12, 2008.
The study of group actions on manifolds is the meeting ground of a variety of mathematical areas. In particular, interesting geometric insights can be obtained by applying measure-theoretic techniques. This book provides an introduction to some of the important methods, major developments, and open problems on the subject. It is slightly expanded from lectures given by Zimmer at the CBMS conference at the University of Minnesota. The main text presents a perspective on the field as it was at that time. Comments at the end of each chapter provide selected suggestions for further reading, including references to recent developments.
Graduate students and research mathematicians interested in ergodic theory.
Introduction
Actions in dimension 1 or 2
Geometric structures
Fundamental groups I
Gromov representation
Superrigidity and first applications
Fundamental groups II (Arithmetic theory)
Locally homogeneous spaces
Stationary measures and projective quotients
Orbit equivalence
Background material
Name index
Index