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From their inception, Siegel modular forms have been studied extensively because of their significance in both automorphic functions in several complex variables and number theory. The comprehensive theory of automorphic forms to subgroups of algebraic groups and the arithmetical theory of modular forms illustrate these two aspects in an illuminating manner. The author’s aim is to present a straightforward and easily accessible survey of the main ideas of the theory at an elementary level, providing a sound basis from which the reader can study advanced works and undertake original research. This book is based on lectures given by the author for a number of years and is intended for a one-semester graduate course, though it can also be used profitably for self-study. The only prerequisites are a basic knowledge of algebra, number theory and complex analysis.Contents

Preface; 1. The modular group; 2. Basic facts on modular forms; 3. Large weights; 4. Small weights; 5. Modular functions; 6. Dirichlet series; Bibliography; Index.

Many areas of active research within the broad field of number theory relate to properties of polynomials, and this volume displays the most recent and most interesting work on this theme. The 2006 Number Theory and Polynomials workshop in Bristol drew together international researchers with a variety of number-theoretic interests, and the book’s contents reflect the quality of the meeting. Topics covered include recent work on the Schur-Siegel-Smyth trace problem, Mahler measure and its generalisations, the merit factor problem, Barker sequences, K3-surfaces, self-inversive polynomials, Newman’s inequality, algorithms for sparse polynomials, the integer transfinite diameter, divisors of polynomials, non-linear recurrence sequences, polynomial ergodic averages, and the Hansen-Mullen primitivity conjecture. With surveys and expository articles presenting the latest research, this volume is essential for graduates and researchers looking for a snapshot of current progress in polynomials and number theory.• An invaluable resource to both students and experts in this area, with survey articles on the most important topics in the field • Expository articles introduce graduate students to some problems of active interest • The inclusion of new results from leading experts in the field provides a snapshot of current progress

wo researchers from the University of Bristol exhibited the first example of a third degree transcendental L-function.

"This work was made possible by a combination of theoretical advances and the power of modern computers." During his lecture, Bian reported that it took approximately 10,000 hours of computer time to produce his initial results.

A new mathematical object, an elusive cousin of the Riemann zeta-function, was revealed to great acclaim recently at the American Institute of Mathematics. Ce Bian and Andrew Booker from the University of Bristol showed the first example of a third degree transcendental L-function.

Functional equations shed light on the properties of those functions that satisfy them, and for L-functions F(s) the functional equation is:   where q is an integer called the level, d is the degree, and the numbers  are Langland's parameters. is an analytic continuation of the factorial function  that is valid not only for integers but all complex numbers. There are two types of L-functions: algebraic and transcendental. These are classified according to their degree. If the Langland's parameters are rational or algebraic (that is, are complex numbers that are roots of non-zero polynomials with rational coefficients), then the L-function is algebraic. If these numbers are transcendental (that is, non-algebraic, such as  or  , then the L-function is transcendental. The Riemann zeta-function is the L-function where the level is 1, the degree is 1 and the Langland's parameters are 0 — that is, a first degree algebraic L-function. The Bristol researchers showed the first example of a third degree transcendental L-function.

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.

Hall, Brian C. Lie groups, Lie algebras, and representations. An elementary introduction. Graduate Texts in Mathematics, 222. Springer-Verlag, New York, 2003. This is only a recommended text, but it is highly recommended. By emphasizing matrix groups, the book covers most of the important examples occuring in nature while avoiding a lot of the technical difficulties necessary in a more general treatment. It gives an excellent presentation of most of what we'll talk about. I think it will be a great book to read to supplement the lectures. Looking around on the web, I found many copies that were very reasonably priced.

Humphreys, James E. Introduction to Lie algebras and representation theory. Graduate Texts in Mathematics, 9. Springer-Verlag, New York-Berlin, 1978. A classic. Would have been my choice for a textbook, but unfortunately only covers Lie algebras.

Fulton, William; Harris, Joe. Representation theory. A first course. Graduate Texts in Mathematics, 129. Readings in Mathematics. Springer-Verlag, New York, 1991. A beautiful book to read. Very useful for self-study. Bump, Daniel. Lie groups. Graduate Texts in Mathematics, 225. Springer-Verlag, New York, 2004. Perhaps too hard for beginners, but it contains an excellent collection of topics in the final part.

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.

Solving Mathematical Problems: A personal perspective. 2nd Edition, Terence Tao. Oxford University P ?The Journey of a thousand miles begins with one step? ? Lao Tzu Every so often, you come across a book that really stands out. I have recently been very fortunate to come across several such books, this being one of them. ?Solving mathematical problems? was written by Terence Tao when he was a 15 year old student and has now been slightly revised in this second edition. Like another of the reviewers at Amazon, I also came across this book after reading an article about Terence Tao winning the Fields medal (a bit like the Nobel prize for mathematics). Not only does it give a wonderful insight into the mind of a young Terence Tao, but also into the techniques used to elegantly solve some reasonably difficult problems, such as those posed as questions for the Maths Olympiad contests. [Terence competed in these challenges in his teens, winning bronze, silver and then gold.] Mathematical researchers are not always great e\ucators. Thankfully, Prof. Tao is.Mainly assuming only basic high-school pure mathematics, worked solutions to the problems are clearly and expertly described. Not only does he solve the problems but he also examines the steps, false starts and other solution possibilities that are part of the general approach to problem solving. I was only slightly disappointed that there were a handful of corrections in this second edition (available at Prof Tao?s blog here); one or two could perplex an unwary reader who might expect the work to be flawless. If you have an interest in mathematics, either as a high school student or a hobbyist, I would highly recommended reading this book. In the preface, Prof Tao remarks that if he wrote a book on the subject of competition problem-solving now, it would very different; now that is definitely a book I would like to read!….

Made to Stick: Why Some Ideas Survive and Others Die by Chip and Dan Heath helps us understand why our users (or our coworkers) can repeat the latest web hoax, but can’t remember anything about our projects. What we need to do is to create “sticky messages.” Sticky messages are not necessarily creative messages. In fact, there is formula that the brothers Heath have discovered that will help us to create sticky, memorable messages. That formula is:

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.

In recent years the application of random matrix techniques to analytic number theory has been responsible for major advances in this area of mathematics. As a consequence it has created a new and rapidly developing area of research. The aim of this book is to provide the necessary grounding both in relevant aspects of number theory and techniques of random matrix theory, as well as to inform the reader of what progress has been made when these two apparently disparate subjects meet. This volume of proceedings is addressed to graduate students and other researchers in both pure mathematics and theoretical physics. The contributing authors, who are among the world leading experts in this area, have taken care to write self-contained lectures on subjects chosen to produce a coherent volume.• Self-contained lectures by world-leading experts in the field • The volume is integrated, indexed and cross-referenced • This title covers the most important and recent advances in the subjectContents1. Introduction; 2. Prime number theory and the Riemann zeta-function; 3. Notes on pair correlation of zeros and prime numbers; 4. Notes on eigenvalue distributions for the classical compact groups; 5. Compound nucleus resonances, random matrices and quantum chaos; 6. Families of L-functions and 1-level densities; 7. Basic analytic number theory; 8. Applications of mean value theorems to the theory of the Riemann zeta function; 9. L-functions and the characteristic polynomials of random matrices; 10. Mock gaussian behaviour; 11. Some specimens of L-functions; 12. Computational methods and experiments in analytic number theory.