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Six leading experts lecture on a wide spectrum of recent results on the subject of the title, providing both a solid reference and deep insights on current research activity. Michael Cowling presents a survey of various interactions between representation theory and harmonic analysis on semisimple groups and symmetric spaces. Alain Valette recalls the concept of amenability and shows how it is used in the proof of rigidity results for lattices of semisimple Lie groups. Edward Frenkel describes the geometric Langlands correspondence for complex algebraic curves, concentrating on the ramified case where a finite number of regular singular points is allowed. Masaki Kashiwara studies the relationship between the representation theory of real semisimple Lie groups and the geometry of the flag manifolds associated with the corresponding complex algebraic groups. David Vogan deals with the problem of getting unitary representations out of those arising from complex analysis, such as minimal globalizations realized on Dolbeault cohomology with compact support. Nolan Wallach illustrates how representation theory is related to quantum computing, focusing on the study of qubit entanglement.

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Several books deal with Sobolev spaces on open subsets of R (n), but none yet with Sobolev spaces on Riemannian manifolds, despite the fact that the theory of Sobolev spaces on Riemannian manifolds already goes back about 20 years. The book of Emmanuel Hebey will fill this gap, and become a necessary reading for all using Sobolev spaces on Riemannian manifolds. Hebey's presentation is very detailed, and includes the most recent developments due mainly to the author himself and to Hebey-Vaugon. He makes numerous things more precise, and discusses the hypotheses to test whether they can be weakened, and also presents new results. ... Read more

Customer Reviews (1)

buy Hebey's newer book instead
At the time that this book was written, it may have filled an important gap in the literature in that it dealt with Sobolov spaces on general Riemannian manifolds (even noncompact ones) as opposed to the usual treatments in euclidean space. However, since that time, there have been other, better books on the subject, most notably, Hebey's own Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities (Courant Lecture Notes) (Courant Lecture Notes), published in 2000 by the AMS. The latter book includes the entire contents of this one (with many of the same chapter titles) plus a lot more, such as the Nirenberg problem, Nash's inequality, and manifolds with boundaries. In fact, the newer book has more than twice as many pages, and is a much, much nicer printing as well. I can't think of any reason to buy this Springer version, unless it was dramatically cheaper, which it's not; in fact, the AMS book is slightly cheaper now. The only reason why I gave it 2 stars is because the book really isn't that bad in itself - 10 years ago I would've rated it higher. ... Read more

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This research monograph introduces some new aspects to the theory of harmonic functions and related topics. The authors study the analytic algebraic structures of the space of bounded harmonic functions on locally compact groups and its non-commutative analogue, the space of harmonic functionals on Fourier algebras. Both spaces are shown to be the range of a contractive projection on a von Neumann algebra and therefore admit Jordan algebraic structures. This provides a natural setting to apply recent results from non-associative analysis, semigroups and Fourier algebras. Topics discussed include Poisson representations, Poisson spaces, quotients of Fourier algebras and the Murray-von Neumann classification of harmonic functionals. ... Read more

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This is the first book dedicated to covering the basic elementsof the Gibbs phenomenon as it appears in various applications wherefunctions with jump discontinuities are represented. It is presentedwith detailed analysis and illustrations combined with historicalinformation. The author covers the appearance of the Gibbs phenomenonin Fourier analysis, orthogonal expansions, integral transforms,splines and wavelet approximations. Methods of reducing, or filteringout, such phenomena that cover all the above function representationsare also addressed. The book includes a thorough bibliography of some350 references.
Audience: The work is intended as an introduction forengineering and scientific practitioners in the fields where thisphenomenon may appear in their use of various functionrepresentations. It may also be used by qualified students. ... Read more

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From the Contents: A. Lambert: Weighted shifts andcomposition operators on L2; - A.S.Cavaretta/A.Sharma:Variation diminishing properties and convexityfor thetensor product Bernstein operator; - B.P. Duggal: A noteongeneralised commutativity theorems in the Schatten norm; -B.S.Yadav/D.Singh/S.Agrawal: De Branges Modules in H2(Ck) ofthetorus; - D. Sarason: Weak compactness of holomorphiccomposition operatorson H1; - H.Helson/J.E.McCarthy:Continuity of seminorms; - J.A. Siddiqui: Maximal ideals inlocal Carleman algebras; - J.G. Klunie: Convergence ofpolynomials with restricted zeros; - J.P. Kahane: On atheorem ofPolya; - U.N. Singh: The Carleman-Fouriertransform and its applications; - W. Zelasko: Extendingseminorms in locally pseudoconvex algebras; ... Read more

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The notion of amenability has its origins in the beginnings of modern measure theory: Does a finitely additive set function exist which is invariant under a certain group action? Since the 1940s, amenability has become an important concept in abstract harmonic analysis (or rather, more generally, in the theory of semitopological semigroups). In 1972, B.E. Johnson showed that the amenability of a locally compact group G can be characterized in terms of the Hochschild cohomology of its group algebra L^1(G): this initiated the theory of amenable Banach algebras. Since then, amenability has penetrated other branches of mathematics, such as von Neumann algebras, operator spaces, and even differential geometry. Lectures on Amenability introduces second year graduate students to this fascinating area of modern mathematics and leads them to a level from where they can go on to read original papers on the subject. Numerous exercises are interspersed in the text. ... Read more

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The theory of functional equations has been developed in a rapid and productive way in the second half of the Twentieth Century. This is due to the fact that the mathematical applications increased the number of investigations of newer and newer types of functional equations. At the same time, the self-development of this theory was also very fruitful.
The material of this volume reflects very well the complexity and applicability of the most active research fields. The results and methods contained give a representative crossection of what is recently happening in the theory of functional equations. 
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This is the second of three major volumes which present acomprehensive treatment of the theory of the main classes of specialfunctions from the point of view of the theory of grouprepresentations.
This volume deals with the properties of special functions andorthogonal polynomials (Legendre, Gegenbauer, Jacobi, Laguerre, Besseland others) which are related to the class 1 representations ofvarious groups. The tree method for the construction of bases forrepresentation spaces is given. `Continuous' bases in the spaces offunctions on hyperboloids and cones and corresponding Poisson kernelsare found. Also considered are the properties of the q-analogs ofclassical orthogonal polynomials, related to representations of theChevalley groups and of special functions connected with fields ofp-adic numbers. Much of the material included appears in book form forthe first time and many of the topics are presented in a novel way.
This volume will be of great interest to specialists in grouprepresentations, special functions, differential equations withpartial derivatives and harmonic anlysis.
Subscribers to the complete set of three volumes will be entitled to adiscount of 15%. ... Read more

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The present book is a continuation of the three-volume workRepresentation of Lie Groups and Special Functions by the sameauthors. Here, they deal with the exposition of the main newdevelopments in the contemporary theory of multivariate specialfunctions, bringing together material that has not been presented inmonograph form before.
The theory of orthogonal symmetric polynomials (Jack polynomials,Macdonald's polynomials and others) and multivariate hypergeometricfunctions associated to symmetric polynomials are treated.Multivariate hypergeometric functions, multivariate Jacobi polynomialsand h-harmonic polynomials connected with root systems andCoxeter groups are introduced. Also, the theory of Gel'fandhypergeometric functions and the theory of multivariate hypergeometricseries associated to Clebsch-Gordan coefficients of the unitary groupU(n) is given. The volume concludes with an extensivebibliography.
For research mathematicians and physicists, postgraduate students inmathematics and mathematical and theoretical physics. ... Read more

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This comprehensive exploration of a metric approach to some relevant problems and techniques in analysis focuses on examining spaces of homogeneous type. Both the structural and analytical problems under consideration reflect very active areas of current research.

The exposition, motivated by examples and details of which many cannot be found elsewhere in book form, unfolds systematically in three parts, starting from such preliminaries as the basic structures of quasi-metric spaces, the homogeneity property and measure theory. More advanced topics are then covered, including Calderón-Zygmund decompositions, L^2 techniques, weighted norm inequalities, BMO, and Hardy spaces. The final aim is to encourage a well-developed way of thinking about some mathematical and even physical problems, i.e. Newtonian and Riesz potentials, fields and their derivatives, mean value and Harnack inequalities related to Hölder regularity, wavelet type analysis on non-euclidian contexts, etc.

The book is suitable for self-study or as a classroom resource for graduate students in analysis and PDEs. Real analysis and topology, some basics of Fourier analysis and PDEs are requisite background for the reader. ... Read more

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A locally compact group has the Haagerup property, or is a-T-menable in the sense of Gromov, if it admits a proper isometric action on some affine Hilbert space. As Gromov's pun is trying to indicate, this definition is designed as a strong negation to Kazhdan's property (T), characterized by the fact that every isometric action on some affine Hilbert space has a fixed point.

The aim of this book is to cover, for the first time in book form, various aspects of the Haagerup property. New characterizations are brought in, using ergodic theory or operator algebras. Several new examples are given, and new approaches to previously known examples are proposed. Connected Lie groups with the Haagerup property are completely characterized.