Expected publication date is June 11, 2004
Description
This book provides a comprehensive account of the crucial role
automorphic L-functions play in number theory and in the
Langlands program, especially the Langlands functoriality
conjecture. There has been a recent major development in the
Langlands functoriality conjecture by the use of automorphic L-functions,
namely, by combining converse theorems of Cogdell and Piatetski-Shapiro
with the Langlands-Shahidi method. This book provides a step-by-step
introduction to these developments and explains how the Langlands
functoriality conjecture implies solutions to several outstanding
conjectures in number theory, such as the Ramanujan conjecture,
Sato-Tate conjecture, and Artin's conjecture. It would be ideal
for an introductory course in the Langlands program.
Contents
Lectures on L-functions, converse theorems, and functoriality for
GL_n
Preface
Modular forms and their L-functions
Automorphic forms
Automorphic representations
Fourier expansions and multiplicity one theorems
Eulerian integral representations
Local L-functions: The non-Archimedean case
The unramified calculation
Local L-functions: The Archimedean case
Global L-functions
Converse theorems
Functoriality
Functoriality for the classical groups
Functoriality for the classical groups, II
Automorphic L-functions
Introduction
Chevalley groups and their properties
Cuspidal representations
L-groups and automorphic L-functions
Induced representations
Eisenstein series and constant terms
L-functions in the constant terms
Meromorphic continuation of L-functions
Generic representations and their Whittaker models
Local coefficients and non-constant terms
Local Langlands correspondence
Local L-functions and functional equations
Normalization of intertwining operators
Holomorphy and bounded in vertical strips
Langlands functoriality conjecture
Converse theorem of Cogdell and Piatetski-Shapiro
Functoriality of the symmetric cube
Functoriality of the symmetric fourth
Bibliography
Applications of symmetric power L-functions
Preface
The Sato-Tate conjecture
Maass wave forms
The Rankin-Selberg method
Oscillations of Fourier coefficients of cusp forms
Poincare series
Kloosterman sums and Selberg's conjecture
Refined estimates for Fourier coefficients of cusp forms
Twisting and averaging of L-series
The Kim-Sarnak theorem
Introduction to Artin L-functions
Zeros and poles of Artin L-functions
The Langlands-Tunnell theorem
Bibliography
Details:
Series: Fields Institute Monographs,Volume: 20
Publication Year: 2004
ISBN: 0-8218-3516-5
Paging: 283 pp.
Binding: Hardcover
Expected publication date is July 16, 2004
Description
Analytic Number Theory distinguishes itself by the variety of
tools it uses to establish results. One of the primary
attractions of this theory is its vast diversity of concepts and
methods. The main goals of this book are to show the scope of the
theory, both in classical and modern directions, and to exhibit
its wealth and prospects, beautiful theorems, and powerful
techniques.
The book is written with graduate students in mind, and the
authors nicely balance clarity, completeness, and generality. The
exercises in each section serve dual purposes, some intended to
improve readers' understanding of the subject and others
providing additional information. Formal prerequisites for the
major part of the book do not go beyond calculus, complex
analysis, integration, and Fourier series and integrals. In later
chapters automorphic forms become important, with much of the
necessary information about them included in two survey chapters.
Contents
Index
Introduction
Arithmetic functions
Elementary theory of prime numbers
Characters
Summation formulas
Classical analytic theory of L-functions
Elementary sieve methods
Bilinear forms and the large sieve
Exponential sums
The Dirichlet polynomials
Zero-density estimates
Sums over finite fields
Character sums
Sums over primes
Holomorphic modular forms
Spectral theory of automorphic forms
Sums of Kloosterman sums
Primes in arithmetic progressions
The least prime in an arithmetic progression
The Goldbach problem
The circle method
Equidistribution
Imaginary quadratic fields
Effective bounds for the class number
The critical zeros of the Riemann zeta function
The spacing of zeros of the Riemann zeta-function
Central values of L-functions
Bibliography
Details:
Series: Colloquium Publications, Volume: 53
Publication Year: 2004
ISBN: 0-8218-3633-1
Paging: approximately 625 pp.
Binding: Hardcover
Expected publication date is July 11, 2004
"A very nice piece of work -- the author succeeds in tying
together a lot of recent developments in algorithms under an
appealing theme."
-- Professor Jon Kleinberg, Cornell University
"This is an elegant monograph, dense in ideas and
techniques, diverse in its applications, and above all, vibrant
with the author's enthusiasm for the area."
-- from the Foreword by Christos H. Papadimitriou, University of
California, Berkeley
Description
Random projection is a simple geometric technique for reducing
the dimensionality of a set of points in Euclidean space while
preserving pairwise distances approximately. The technique plays
a key role in several breakthrough developments in the field of
algorithms. In other cases, it provides elegant alternative
proofs.
The book begins with an elementary description of the technique
and its basic properties. Then it develops the method in the
context of applications, which are divided into three groups. The
first group consists of combinatorial optimization problems such
as maxcut, graph coloring, minimum multicut, graph bandwidth and
VLSI layout. Presented in this context is the theory of Euclidean
embeddings of graphs. The next group is machine learning
problems, specifically, learning intersections of halfspaces and
learning large margin hypotheses. The projection method is
further refined for the latter application. The last set consists
of problems inspired by information retrieval, namely, nearest
neighbor search, geometric clustering and efficient low-rank
approximation. Motivated by the first two applications, an
extension of random projection to the hypercube is developed here.
Throughout the book, random projection is used as a way to
understand, simplify and connect progress on these important and
seemingly unrelated problems.
The book is suitable for graduate students and research
mathematicians interested in computational geometry.
Contents
Random projection
Combinatorial optimization
Rounding via random projection
Embedding metrics in Euclidean space
Euclidean embeddings: Beyond distance preservation
Learning theory
Robust concepts
Intersections of half-spaces
Information retrieval
Nearest neighbors
Indexing and clustering
Bibliography
Appendix
Details:
Series: DIMACS: Series in Discrete Mathematics and Theoretical
Computer Science,Volume: 65
Publication Year: 2004
ISBN: 0-8218-2018-4
Paging: 103 pp.
Binding: Hardcover
Expected publication date is July 2, 2004
Description
The Ricci flow is a powerful technique that integrates geometry,
topology, and analysis. Intuitively, the idea is to set up a PDE
that evolves a metric according to its Ricci curvature. The
resulting equation has much in common with the heat equation,
which tends to "flow" a given function to ever nicer
functions. By analogy, the Ricci flow evolves an initial metric
into improved metrics.
Richard Hamilton began the systematic use of the Ricci flow in
the early 1980s and applied it in particular to study 3-manifolds.
Grisha Perelman has made recent breakthroughs aimed at completing
Hamilton's program.
The Ricci flow method is now central to our understanding of the
geometry and topology of manifolds. This book is an introduction
to that program and to its connection to Thurston's
geometrization conjecture.
The authors also provide a "Guide for the hurried reader",
to help readers wishing to develop, as efficiently as possible, a
nontechnical appreciation of the Ricci flow program for 3-manifolds,
i.e., the so-called "fast track".
The book is suitable for geometers and others who are interested
in the use of geometric analysis to study the structure of
manifolds.
Contents
The Ricci flow of special geometries
Special and limit solutions
Short time existence
Maximum principles
The Ricci flow on surfaces
Three-manifolds of positive Ricci curvature
Derivative estimates
Singularities and the limits of their dilations
Type I singularities
The Ricci calculus
Bibliography
Index
Details:
Series: Mathematical Surveys and Monographs, Volume: 110
Publication Year: 2004
ISBN: 0-8218-3515-7
Paging: approximately 342 pp.
Binding: Hardcover
Expected publication date is June 20, 2004
Description
Stochastic analysis is often understood as the analysis of
functionals defined on the Wiener space, i.e., the space on which
the Wiener process is realized. Since the Wiener space is
infinite-dimensional, it requires a special calculus, the so-called
Malliavin calculus.
This book provides readers with a concise introduction to
stochastic analysis, in particular, to the Malliavin calculus. It
contains a detailed description of all the technical tools
necessary to describe the theory, such as the Wiener process, the
Ornstein-Uhlenbeck process, and Sobolev spaces. It also presents
applications of stochastic calculus to the study of stochastic
differential equations. The volume is suitable for graduate
students and research mathematicians interested in probability
and random processes.
Contents
Wiener space
Orenstein-Uhlenbeck process
The Littlewood-Paley-Stein inequality
Sobolev spaces on an abstrct Wiener space
Absolute continuity of distributions and smoothness of density
functions
Application to stochastic differential equations
Perspectives on current research
Bibliography
Index
Details:
Series: Translations of Mathematical Monographs,Volume: 224
Subseries: Iwanami Series in Modern Mathematics
Publication Year: 2004
ISBN: 0-8218-2626-3
Paging: approximately 200 pp.
Binding: Softcover
Expected publication date is August 14, 2004
Description
Isaac Newton encrypted his discoveries in analysis in the form of
an anagram that deciphers to the sentence, "It is worthwhile
to solve differential equations". Accordingly, one can
express the main idea behind the orbit method by saying "It
is worthwhile to study coadjoint orbits".
The orbit method was introduced by the author, A. A. Kirillov, in
the 1960s and remains a useful and powerful tool in areas such as
Lie theory, group representations, integrable systems, complex
and symplectic geometry, and mathematical physics. This book
describes the essence of the orbit method for non-experts and
gives the first systematic, detailed, and self-contained
exposition of the method. It starts with a convenient "User's
Guide" and contains numerous examples. It can be used as a
text for a graduate course, as well as a handbook for non-experts
and a reference book for research mathematicians and mathematical
physicists.
Contents
Geometry of coadjoint orbits
Representations and orbits of the Heisenberg group
The orbit method for nilpotent Lie groups
Solvable Lie groups
Compact Lie groups
Miscellaneous
Abstract nonsense
Smooth manifolds
Lie groups and homogeneous manifolds
Elements of functional analysis
Representation theory
References
Index
Details:
Series: Graduate Studies in Mathematics, Volume: 64
Publication Year: 2004
ISBN: 0-8218-3530-0
Paging: approximately 432 pp.
Binding: Hardcover