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Topics in Harmonic Analysis and Ergodic Theory Joseph M. Rosenblatt, Alexander M. Stokolos, Ahmed I. Zayed ISBN: 0821842358 Paperback American Mathematical Society Usually despatched within 3 to 9 days

There are strong connections between harmonic analysis and ergodic theory. A recent example of this interaction is the proof of the spectacular result by Terence Tao and Ben Green that the set of prime numbers contains arbitrarily long arithmetic progressions. The breakthrough achieved by Tao and Green is attributed to applications of techniques from ergodic theory and harmonic analysis to problems in number theory.Articles in the present volume are based on talks delivered by plenary speakers at a conference on Harmonic Analysis and Ergodic Theory (DePaul University, Chicago, December 2-4, 2005). Of ten articles, four are devoted to ergodic theory and six to harmonic analysis, although some may fall in either category. The articles are grouped in two parts arranged by topics. Among the topics are ergodic averages, central limit theorems for random walks, Borel foliations, ergodic theory and low pass filters, data fitting using smooth surfaces, Nehari's theorem for a polydisk, uniqueness theorems for multi-dimensional trigonometric series, and Bellman and s-functions.In addition to articles on current research topics in harmonic analysis and ergodic theory, this book contains survey articles on convergence problems in ergodic theory and uniqueness problems on multi-dimensional trigonometric series.

Among the modern methods used to study prime numbers, the ‘sieve’ has been one of the most efficient. Originally conceived by Linnik in 1941, the ‘large sieve’ has developed extensively since the 1960s, with a recent realisation that the underlying principles were capable of applications going well beyond prime number theory. This book develops a general form of sieve inequality, and describes its varied applications, including the study of families of zeta functions of algebraic curves over finite fields; arithmetic properties of characteristic polynomials of random unimodular matrices; homological properties of random 3-manifolds; and the average number of primes dividing the denominators of rational points on elliptic curves. Also covered in detail are the tools of harmonic analysis used to implement the forms of the large sieve inequality, including the Riemann Hypothesis over finite fields, and Property (T) or Property (tau) for discrete groups.• Explores new and surprising applications of the large sieve method, an important technique of analytic number theory • Presents applications in fields as wide ranging as topology, probability, arithmetic geometry and discrete group theory • Motivated, clear and self-contained discussions introduce readers to a technique previously confined to one fieldContentsPreface; Prerequisites and notation; 1. Introduction; 2. The principle of the large sieve; 3. Group and conjugacy sieves; 4. Elementary and classical examples; 5. Degrees of representations of finite groups; 6. Probabilistic sieves; 7. Sieving in discrete groups; 8. Sieving for Frobenius over finite fields; Appendix A. Small sieves; Appendix B. Local density computations over finite fields; Appendix C. Representation theory; Appendix D. Property (T) and Property (τ); Appendix E. Linear algebraic groups; Appendix F. Probability theory and random walks; Appendix G. Sums of multiplicative functions; Appendix H. Topology; Bibliography; Index.

ContentsPart I. Preliminaries, Examples and Motivations: 1. Finite Markov chains; 2. Two basic examples on Abelian groups; Part II. Representation Theory and Gelfand Pairs: 3. Basic representation theory of finite groups; 4. Finite Gelfand pairs; 5. Distance regular graphs and the Hamming scheme; 6. The Johnson Scheme and the Laplace-Bernoulli diffusion model; 7. The ultrametric space; Part III. Advanced theory: 8. Posets and the q−analogs; 9. Complements on representation theory; 10. Basic representation theory of the symmetric group; 11. The Gelfand Pair (S2n, S2 o Sn) and random matchings; Appendix 1. The discrete trigonometric transforms; Appendix 2. Solutions of the exercises; Bibliography; Index.