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This book illustrates the wide range of research subjects developed
by the Italian research group in harmonic analysis, originally
started by Alessandro Figa-Talamanca, to whom it is dedicated in
the occasion of his retirement. In particular, it outlines some of
the impressive ramifications of the mathematical developments that
began when Figa-Talamanca brought the study of harmonic analysis to
Italy; the research group that he nurtured has now expanded to
cover many areas. Therefore the book is addressed not only to
experts in harmonic analysis, summability of Fourier series and
singular integrals, but also in potential theory, symmetric spaces,
analysis and partial differential equations on Riemannian
manifolds, analysis on graphs, trees, buildings and discrete
groups, Lie groups and Lie algebras, and even in far-reaching
applications as for instance cellular automata and signal
processing (low-discrepancy sampling, Gaussian noise).
This book illustrates the wide range of research subjects developed
by the Italian research group in harmonic analysis, originally
started by Alessandro Figa-Talamanca, to whom it is dedicated in
the occasion of his retirement. In particular, it outlines some of
the impressive ramifications of the mathematical developments that
began when Figa-Talamanca brought the study of harmonic analysis to
Italy; the research group that he nurtured has now expanded to
cover many areas. Therefore the book is addressed not only to
experts in harmonic analysis, summability of Fourier series and
singular integrals, but also in potential theory, symmetric spaces,
analysis and partial differential equations on Riemannian
manifolds, analysis on graphs, trees, buildings and discrete
groups, Lie groups and Lie algebras, and even in far-reaching
applications as for instance cellular automata and signal
processing (low-discrepancy sampling, Gaussian noise).
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.
This book contains the notes of five short courses delivered at the
"Centro Internazionale Matematico Estivo" session "Integral
Geometry, Radon Transforms and Complex Analysis" held in Venice
(Italy) in June 1996: three of them deal with various aspects of
integral geometry, with a common emphasis on several kinds of Radon
transforms, their properties and applications, the other two share
a stress on CR manifolds and related problems. All lectures are
accessible to a wide audience, and provide self-contained
introductions and short surveys on the subjects, as well as
detailed expositions of selected results.
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