Annals of Mathematics Studies 173
Paper | April 2010 | 978-0-691-14293-7
Cloth | April 2010 | 978-0-691-14292-0
256 pp. | 6 x 9
This book studies the intersection cohomology of the Shimura varieties associated to unitary groups of any rank over Q. In general, these varieties are not compact. The intersection cohomology of the Shimura variety associated to a reductive group G carries commuting actions of the absolute Galois group of the reflex field and of the group G(Af) of finite adelic points of G. The second action can be studied on the set of complex points of the Shimura variety. In this book, Sophie Morel identifies the Galois action--at good places--on the G(Af)-isotypical components of the cohomology.
Morel uses the method developed by Langlands, Ihara, and Kottwitz, which is to compare the Grothendieck-Lefschetz fixed point formula and the Arthur-Selberg trace formula. The first problem, that of applying the fixed point formula to the intersection cohomology, is geometric in nature and is the object of the first chapter, which builds on Morel's previous work. She then turns to the group-theoretical problem of comparing these results with the trace formula, when G is a unitary group over Q. Applications are then given. In particular, the Galois representation on a G(Af)-isotypical component of the cohomology is identified at almost all places, modulo a non-explicit multiplicity. Morel also gives some results on base change from unitary groups to general linear groups.
Sophie Morel is a member in the School of Mathematics at the Institute for Advanced Study in Princeton and a research fellow at the Clay Mathematics Institute.
Preface vii
Chapter 1: The fixed point formula 1
Chapter 2: The groups 31
Chapter 3: Discrete series 47
Chapter 4: Orbital integrals at p 63
Chapter 5: The geometric side of the stable trace formula 79
Chapter 6: Stabilization of the fixed point formula 85
Chapter 7: Applications 99
Chapter 8: The twisted trace formula 119
Chapter 9: The twisted fundamental lemma 157
Appendix: Comparison of two versions of twisted transfer factors 189
Bibliography 207
Index 215
Cloth | April 2010 | 978-0-691-14289-0
184 pp. | 5 1/2 x 8 1/2 | 21 line illus.
The Little Book of String Theory offers a short, accessible, and entertaining introduction to one of the most talked-about areas of physics today. String theory has been called the "theory of everything." It seeks to describe all the fundamental forces of nature. It encompasses gravity and quantum mechanics in one unifying theory. But it is unproven and fraught with controversy. After reading this book, you'll be able to draw your own conclusions about string theory.
Steve Gubser begins by explaining Einstein's famous equation E = mc2, quantum mechanics, and black holes. He then gives readers a crash course in string theory and the core ideas behind it. In plain English and with a minimum of mathematics, Gubser covers strings, branes, string dualities, extra dimensions, curved spacetime, quantum fluctuations, symmetry, and supersymmetry. He describes efforts to link string theory to experimental physics and uses analogies that nonscientists can understand. How does Chopin's Fantasie-Impromptu relate to quantum mechanics? What would it be like to fall into a black hole? Why is dancing a waltz similar to contemplating a string duality? Find out in the pages of this book.
The Little Book of String Theory is the essential, most up-to-date beginner's guide to this elegant, multidimensional field of physics.
Steven S. Gubser is professor of physics at Princeton University.
Introduction 1
Chapter One: Energy 11
Chapter Two: Quantum Mechanics 19
Chapter Three: Gravity and Black Holes 34
Chapter Four: String Theory 49
Chapter Five: Branes 69
Chapter Six: String Dualities 99
Chapter Seven: Supersymmetry and the LHC 117
Chapter Eight: Heavy Ions and the Fifth Dimension 140
Epilogue 159
Index 163
Cloth | May 2010 | 978-0-691-14014-8
136 pp. | 5 x 8 | 5 halftones.
Writings by early mathematicians feature language and notations that are quite different from what we're familiar with today. Sourcebooks on the history of mathematics provide some guidance, but what has been lacking is a guide tailored to the needs of readers approaching these writings for the first time. How to Read Historical Mathematics fills this gap by introducing readers to the analytical questions historians ask when deciphering historical texts.
Sampling actual writings from the history of mathematics, Benjamin Wardhaugh reveals the questions that will unlock the meaning and significance of a given text--Who wrote it, why, and for whom? What was its author's intended meaning? How did it reach its present form? Is it original or a translation? Why is it important today? Wardhaugh teaches readers to think about what the original text might have looked like, to consider where and when it was written, and to formulate questions of their own. Readers pick up new skills with each chapter, and gain the confidence and analytical sophistication needed to tackle virtually any text in the history of mathematics.
Introduces readers to the methods of textual analysis used by historians
Uses actual source material as examples
Features boxed summaries, discussion questions, and suggestions for further reading
Supplements all major sourcebooks in mathematics history
Designed for easy reference
Ideal for students and teachers
Benjamin Wardhaugh is a postdoctoral research fellow at All Souls College, University of Oxford. He is the author of Music, Experiment, and Mathematics in England, 1653-1705.
(LMS-34)
Cloth | July 2010 | 978-0-691-12829-0
808 pp. | 6 x 9
Random matrix theory, both as an application and as a theory, has evolved rapidly over the past fifteen years. Log-Gases and Random Matrices gives a comprehensive account of these developments, emphasizing log-gases as a physical picture and heuristic, as well as covering topics such as beta ensembles and Jack polynomials.
Peter Forrester presents an encyclopedic development of log-gases and random matrices viewed as examples of integrable or exactly solvable systems. Forrester develops not only the application and theory of Gaussian and circular ensembles of classical random matrix theory, but also of the Laguerre and Jacobi ensembles, and their beta extensions. Prominence is given to the computation of a multitude of Jacobians; determinantal point processes and orthogonal polynomials of one variable; the Selberg integral, Jack polynomials, and generalized hypergeometric functions; Painleve transcendents; macroscopic electrostatistics and asymptotic formulas; nonintersecting paths and models in statistical mechanics; and applications of random matrix theory. This is the first textbook development of both nonsymmetric and symmetric Jack polynomial theory, as well as the connection between Selberg integral theory and beta ensembles. The author provides hundreds of guided exercises and linked topics, making Log-Gases and Random Matrices an indispensable reference work, as well as a learning resource for all students and researchers in the field.
Peter J. Forrester is professor of mathematics at the University of Melbourne.
Annals of Mathematics Studies 174
Paper | July 2010 | 978-0-691-14542-6
Cloth | July 2010 | 978-0-691-14541-9
296 pp. | 6 x 9 | 12 line illus.
Ramsey theory is a fast-growing area of combinatorics with deep connections to other fields of mathematics such as topological dynamics, ergodic theory, mathematical logic, and algebra. The area of Ramsey theory dealing with Ramsey-type phenomena in higher dimensions is particularly useful. Introduction to Ramsey Spaces presents in a systematic way a method for building higher-dimensional Ramsey spaces from basic one-dimensional principles. It is the first book-length treatment of this area of Ramsey theory, and emphasizes applications for related and surrounding fields of mathematics, such as set theory, combinatorics, real and functional analysis, and topology. In order to facilitate accessibility, the book gives the method in its axiomatic form with examples that cover many important parts of Ramsey theory both finite and infinite.
An exciting new direction for combinatorics, this book will interest graduate students and researchers working in mathematical subdisciplines requiring the mastery and practice of high-dimensional Ramsey theory.
Stevo Todorcevic is professor of mathematics at the University of Toronto and holds senior research positions at the CNRS in Paris and SANU in Belgrade. He is the author or coauthor of several books, including Walks on Ordinals and Their Characteristics and Ramsey Methods in Analysis.