Colloquium Publications,Volume: 34
2008; 384 pp; softcover
ISBN-10: 0-8218-4643-4
ISBN-13: 978-0-8218-4643-8
Expected publication date is July 14, 2008.
This book is concerned with the critical points of analytic and harmonic
functions. A critical point of an analytic function means a zero of its
derivative, and a critical point of a harmonic function means a point where
both partial derivatives vanish. The analytic functions considered are
largely polynomials, rational functions, and certain periodic, entire,
and meromorphic functions. The harmonic functions considered are largely
Green's functions, harmonic measures, and various linear combinations of
them. The interest in these functions centers around the approximate location
of their critical points. The approximation is in the sense of determining
minimal regions in which all the critical points lie or maximal regions
in which no critical point lies. Throughout the book the author uses the
single method of regarding the critical points as equilibrium points in
fields of force due to suitable distribution of matter.
The exposition is clear, complete, and well-illustrated with many examples.
Graduate students and research mathematicians interested in analyic and harmonic functions.
Fundamental results
Real polynomials
Polynomials, continued
Rational functions
Rational functions with symmetry
Analytic functions
Green's functions
Harmonic functions
Further harmonic functions
Bibliography
Index
Colloquium Publications, Volume: 39
1968; 453 pp; softcover
ISBN-10: 0-8218-4640-X
ISBN-13: 978-0-8218-4640-7
Expected publication date is July 14, 2008.
The theory of Jordan algebras has played important roles behind the scenes
of several areas of mathematics. Jacobson's book has long been the definitive
treatment of the subject. It covers foundational material, structure theory,
and representation theory for Jordan algebras. Of course, there are immediate
connections with Lie algebras, which Jacobson details in Chapter 8. Of
particular continuing interest is the discussion of exceptional Jordan
algebras, which serve to explain the exceptional Lie algebras and Lie groups.
Jordan algebras originally arose in the attempts by Jordan, von Neumann, and Wigner to formulate the foundations of quantum mechanics. They are still useful and important in modern mathematical physics, as well as in Lie theory, geometry, and certain areas of analysis.
Graduate students and research mathematicians interested in Jordan algebras.
Foundations
Elements of representation theory
Peirce decompositions and Jordan matrix algebras
Jordan algebras with minimum conditions on quadratic ideals
Structure theory for finite-dimensional Jordan algebras
Generic minimum polynomials, traces and norms
Representation theory for separable Jordan algebras
Connections with Lie algebras
Exceptional Jordan algebras
Further results and open questions
Bibliography
Subject index
Graduate Studies in Mathematics, Volume: 94
2008; 289 pp; hardcover
ISBN-10: 0-8218-4678-7
ISBN-13: 978-0-8218-4678-0
Expected publication date is August 17, 2008.
This is the first textbook treatment of work leading to the landmark 1979
Kazhdan-Lusztig Conjecture on characters of simple highest weight modules
for a semisimple Lie algebra mathfrak{g} over mathbb {C}. The setting is
the module category mathscr {O} introduced by Bernstein-Gelfand-Gelfand,
which includes all highest weight modules for mathfrak{g} such as Verma
modules and finite dimensional simple modules. Analogues of this category
have become influential in many areas of representation theory.
Part I can be used as a text for independent study or for a mid-level one
semester graduate course; it includes exercises and examples. The main
prerequisite is familiarity with the structure theory of mathfrak{g}. Basic
techniques in category mathscr {O} such as BGG Reciprocity and Jantzen's
translation functors are developed, culminating in an overview of the proof
of the Kazhdan-Lusztig Conjecture (due to Beilinson-Bernstein and Brylinski-Kashiwara).
The full proof however is beyond the scope of this book, requiring deep
geometric methods: D-modules and perverse sheaves on the flag variety.
Part II introduces closely related topics important in current research:
parabolic category mathscr {O}, projective functors, tilting modules, twisting
and completion functors, and Koszul duality theorem of Beilinson-Ginzburg-Soergel.
Graduate students and research mathematicians interested in Lie theory, and representation theory.
Contemporary Mathematics, Volume: 464
2008; 264 pp; softcover
ISBN-10: 0-8218-4327-3
ISBN-13: 978-0-8218-4327-7
Expected publication date is August 31, 2008.
This volume is based on two special sessions held at the AMS Annual Meeting
in New Orleans in January 2007, and a satellite workshop held in Baton
Rouge on January 4-5, 2007. It consists of invited expositions that together
represent a broad spectrum of fields, stressing surprising interactions
and connections between areas that are normally thought of as disparate.
The main topics are geometry and integral transforms. On the one side are
harmonic analysis, symmetric spaces, representation theory (the groups
include continuous and discrete, finite and infinite, compact and non-compact),
operator theory, PDE, and mathematical probability. Moving in the applied
direction we encounter wavelets, fractals, and engineering topics such
as frames and signal and image processing.
The subjects covered in this book form a unified whole, and they stand at the crossroads of pure and applied mathematics. The articles cover a broad range in harmonic analysis, with the main themes related to integral geometry, the Radon transform, wavelets and frame theory. These themes can loosely be grouped together as follows:
Frame Theory and Applications
Harmonic Analysis and Function Spaces
Harmonic Analysis and Number Theory
Integral Geometry and Radon Transforms
Multiresolution Analysis, Wavelets, and Applications
Graduate students and research mathematicians interested in harmonic analysis, integral geometry and applications.
I. A. Aliev, B. Rubin, S. Sezer, and S. B. Uyhan -- Composite wavelet transforms: Applications and perspectives
J. J. Benedetto, O. Oktay, and A. Tangboondouangjit -- Complex sigma-delta quantization algorithms for finite frames
B. Currey and T. McNamara -- Decomposition and admissibility for the quasiregular representation for generalized oscillator groups
D. E. Dutkay and P. E. T. Jorgensen -- Fourier series on fractals: A parallel with wavelet theory
D. V. Feldman -- A computational complexity paradigm for tomography
F. B. Gonzalez -- Invariant differential operators on matrix motion groups and applications to the matrix radon transform
D. Hart and A. Iosevich -- Sums and products in finite fields: An integral geometric viewpoint
B. D. Johnson and K. A. Okoudjou -- Frame potential and finite abelian groups
P. G. Casazza and G. Kutyniok -- Robustness of fusion frames under erasures of subspaces and of local frame vectors
K. D. Merrill -- Smooth, well-localized Parseval wavelets based on wavelet
sets in mathbb{R}^2
S. Jain, M. Papadakis, and E. Dussaud -- Explicit schemes in seismic migration and isotropic multiscale representations
G. Olafsson and B. Rubin -- Invariant functions on Grassmannians
G. Olafsson and S. Zheng -- Harmonic analysis related to Schrodinger operators
I. Pesenson -- Discrete Helgason-Fourier transform for Sobolev and Besov functions on noncompact symmetric spaces
E. T. Quinto -- Helgason's