Marcelo Aguiar, Texas A&M University, College Station, TX,
and Swapneel Mahajan, Indian Institute of Technology, Powai, Mumbai, India

Coxeter Groups and Hopf Algebras

Fields Institute Monographs Volume: 23
2006; 181 pp; hardcover
ISBN: 0-8218-3907-1

An important idea in the work of G.-C. Rota is that certain combinatorial objects give rise to Hopf algebras that reflect the manner in which these objects compose and decompose. Recent work has seen the emergence of several interesting Hopf algebras of this kind, which connect diverse subjects such as combinatorics, algebra, geometry, and theoretical physics. This monograph presents a novel geometric approach using Coxeter complexes and the projection maps of Tits for constructing and studying many of these objects as well as new ones. The first three chapters introduce the necessary background ideas making this work accessible to advanced graduate students. The later chapters culminate in a unified and conceptual construction of several Hopf algebras based on combinatorial objects which emerge naturally from the geometric viewpoint. This work lays a foundation and provides new insights for further development of the subject.

Table of Contents

Coxeter groups
Left regular bands
Hopf algebras
A brief overview
The descent theory for Coxeter groups
The construction fo Hopf algebras
The Hopf algebra of pairs of permutations
The Hopf algebra of pointed faces
Bibliography
Author index
Notation index
Subject index

Peter D. Miller, University of Michigan, Ann Arbor, MI

Applied Asymptotic Analysis

Graduate Studies in Mathematics Volume: 75
2006; approx. 475 pp; hardcover
ISBN: 0-8218-4078-9
List Price: US69

This book is a survey of asymptotic methods set in the current applied research context of wave propagation. It stresses rigorous analysis in addition to formal manipulations. Asymptotic expansions developed in the text are justified rigorously, and students are shown how to obtain solid error estimates for asymptotic formulae. The book relates examples and exercises to subjects of current research interest, such as the problem of locating the zeros of Taylor polynomials of entire nonvanishing functions and the problem of counting integer lattice points in subsets of the plane with various geometrical properties of the boundary.

The book is intended for a beginning graduate course on asymptotic analysis in applied mathematics and is aimed at students of pure and applied mathematics as well as science and engineering. The basic prerequisite is a background in differential equations, linear algebra, advanced calculus, and complex variables at the level of introductory undergraduate courses on these subjects.

The book is ideally suited to the needs of a graduate student who, on the one hand, wants to learn basic applied mathematics, and on the other, wants to understand what is needed to make the various arguments rigorous. Down here in the Village, this is known as the Courant point of view!!

--Percy Deift, Courant Institute, New York

Table of Contents

Fundamentals
Themes of asymptotic analysis
The nature of asymptotic approximations
Asymptotic analysis of exponential integrals
Fundamental techniques for integrals
Laplace's method for asymptotic expansions of integrals
The method of steepest descents for asymptotic expansions of integrals
The method of stationary phase for asymptotic analysis of oscillatory integrals
Asymptotic analysis of differential equations
Asymptotic behavior of solutions of linear second-order differential equations in the complex plane
Introduction to asymptotics of solutions of ordinary differential equations with respect to parameters
Asymptotics of linear boundary-value problems
Asymptotics of oscillatory phenomena
Weakly nonlinear waves
Appendix: Fundamental inequalities
Bibliography
Index of names
Subject index

Terence Tao, University of California, Los Angeles, CA

Nonlinear Dispersive Equations: Local and Global Analysis

CBMS Regional Conference Series in Mathematics Number: 106
2006; 373 pp; softcover
ISBN: 0-8218-4143-2

Among nonlinear PDEs, dispersive and wave equations form an important class of equations. These include the nonlinear Schrodinger equation, the nonlinear wave equation, the Korteweg de Vries equation, and the wave maps equation. This book is an introduction to the methods and results used in the modern analysis (both locally and globally in time) of the Cauchy problem for such equations.

Starting only with a basic knowledge of graduate real analysis and Fourier analysis, the text first presents basic nonlinear tools such as the bootstrap method and perturbation theory in the simpler context of nonlinear ODE, then introduces the harmonic analysis and geometric tools used to control linear dispersive PDE. These methods are then combined to study four model nonlinear dispersive equations. Through extensive exercises, diagrams, and informal discussion, the book gives a rigorous theoretical treatment of the material, the real-world intuition and heuristics that underlie the subject, as well as mentioning connections with other areas of PDE, harmonic analysis, and dynamical systems.

As the subject is vast, the book does not attempt to give a comprehensive survey of the field, but instead concentrates on a representative sample of results for a selected set of equations, ranging from the fundamental local and global existence theorems to very recent results, particularly focusing on the recent progress in understanding the evolution of energy-critical dispersive equations from large data. The book is suitable for a graduate course on nonlinear PDE.

Table of Contents

Ordinary differential equations
Constant coefficient linear dispersive equations
Semilinear dispersive equations
The Korteweg de Vries equation
Energy-critical semilinear dispersive equations
Wave maps
Tools from harmonic analysis
Construction of ground states
Bibliography

Edited by: Tyler J. Jarvis, Brigham Young University, Provo, UT, Takashi Kimura, Boston University, MA, and Arkady Vaintrob, University of Oregon, Eugene, OR

Gromov-Witten Theory of Spin Curves and Orbifolds

Contemporary Mathematics Volume: 403
2006; 189 pp; softcover
ISBN: 0-8218-3534-3

This volume is a collection of articles on orbifolds, algebraic curves with higher spin structures, and related invariants of Gromov-Witten type. Orbifold Gromov-Witten theory generalizes quantum cohomology for orbifolds, whereas spin cohomological field theory is based on the moduli spaces of higher spin curves and is related by Witten's conjecture to the Gelfand-Dickey integrable hierarchies.

A common feature of these two very different looking theories is the central role played by orbicurves in both of them. Insights in one theory can often yield insights into the other. This book brings together for the first time papers related to both sides of this interaction. The articles in the collection cover diverse topics, such as geometry and topology of orbifolds, cohomological field theories, orbifold Gromov-Witten theory, G-Frobenius algebra and singularities, Frobenius manifolds and Givental's quantization formalism, moduli of higher spin curves and spin cohomological field theory.

Table of Contents

A. Polishchuk -- Moduli spaces of curves with effective r-spin structures
A. Chiodo -- A construction of Witten's top Chern class in K-theory
Y.-P. Lee -- Witten's conjecture and the Virasoro conjecture for genus up to two
X. Liu -- Idempotents on the big phase space
R. M. Kaufmann -- Singularities with symmetries, orbifold Frobenius algebras and mirror symmetry
Y. Ruan -- The cohomology ring of crepant resolutions of orbifolds
E. Lupercio and B. Uribe -- Differential characters on orbifolds and string connections I: Global quotients
J. Morava -- HKR characters and higher twisted sectors
S. V. Shadrin -- Combinatorics of binomial decompositions of the simplest Hodge integrals
J. Spencer -- The orbifold cohomology of the moduli of genus-two curves
List of participants and abstracts

Edited by: Alexander Borichev, Universite Bordeaux I, Talence, Cedex, France, Hakan Hedenmalm, Royal Institute of Technology, Stockholm, Sweden, and Kehe Zhu, State University of New York at Albany, NY

Bergman Spaces and Related Topics in Complex Analysis:
Proceedings of a Conference in Honor of Boris Korenblum's 80th Birthday

Contemporary Mathematics Volume: 404
2006; approx. 217 pp; softcover
ISBN: 0-8218-3712-5

This volume grew out of a conference in honor of Boris Korenblum on the occasion of his 80th birthday, held in Barcelona, Spain, November 20-22, 2003. The book is of interest to researchers and graduate students working in the theory of spaces of analytic function, and, in particular, in the theory of Bergman spaces.

Readership

Graduate students and research mathematicians interested in complex analysis.

Table of Contents

A. Aleman, S. Richter, and C. Sundberg -- Invariant subspaces for the backward shift on Hilbert spaces of analytic functions with regular norm
A. Atzmon and B. Brive -- Surjectivity and invariant subspaces of differential operators on weighted Bergman spaces of entire functions
K. F. Barth and P. J. Rippon -- Exceptional values and the MacLane class mathcal{A}
O. Blasco -- Operators on weighted Bergman spaces
A. Dahlner -- A Wiener Tauberian theorem for weighted convolution algebras of zonal functions on the automorphism group of the unit disc
W. K. Hayman -- Domination on sets and in norm
C. Horowitz and B. Pinchuk -- Extensions of the asymptotic maximum principle
S. Jakobsson -- Singularity resolution of weighted Bergman kernels
B. Korenblum -- Blaschke sets for Bergman spaces
X. Massaneda and P. J. Thomas -- Phragmen-Lindelof-type problems for A^{- alpha}
A. Olofsson -- A representation formula for reproducing subharmonic functions in the unit disc
F. Perez-Gonzalez and J. C. Ramos -- On dominating sets for Bergman spaces
S. Shimorin -- Trigonometric obstacle problem and weak factorization
K. Zhu -- A sharp norm estimate of the Bergman projection on L^p spaces