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Continuing the theme of the previous volumes, these seminar notes
reflect general trends in the study of Geometric Aspects of
Functional Analysis, understood in a broad sense. Two classical
topics represented are the Concentration of Measure Phenomenon in
the Local Theory of Banach Spaces, which has recently had triumphs
in Random Matrix Theory, and the Central Limit Theorem, one of the
earliest examples of regularity and order in high dimensions.
Central to the text is the study of the Poincare and log-Sobolev
functional inequalities, their reverses, and other inequalities, in
which a crucial role is often played by convexity assumptions such
as Log-Concavity. The concept and properties of Entropy form an
important subject, with Bourgain's slicing problem and its variants
drawing much attention. Constructions related to Convexity Theory
are proposed and revisited, as well as inequalities that go beyond
the Brunn-Minkowski theory. One of the major current research
directions addressed is the identification of lower-dimensional
structures with remarkable properties in rather arbitrary
high-dimensional objects. In addition to functional analytic
results, connections to Computer Science and to Differential
Geometry are also discussed.
As in the previous Seminar Notes, the current volume reflects
general trends in the study of Geometric Aspects of Functional
Analysis, understood in a broad sense. A classical theme in the
Local Theory of Banach Spaces which is well represented in this
volume is the identification of lower-dimensional structures in
high-dimensional objects. More recent applications of
high-dimensionality are manifested by contributions in Random
Matrix Theory, Concentration of Measure and Empirical Processes.
Naturally, the Gaussian measure plays a central role in many of
these topics, and is also studied in this volume; in particular,
the recent breakthrough proof of the Gaussian Correlation
Conjecture is revisited. The interplay of the theory with Harmonic
and Spectral Analysis is also well apparent in several
contributions. The classical relation to both the primal and dual
Brunn-Minkowski theories is also well represented, and related
algebraic structures pertaining to valuations and valent functions
are discussed. All contributions are original research papers and
were subject to the usual refereeing standards.
Continuing the theme of the previous volumes, these seminar notes
reflect general trends in the study of Geometric Aspects of
Functional Analysis, understood in a broad sense. Two classical
topics represented are the Concentration of Measure Phenomenon in
the Local Theory of Banach Spaces, which has recently had triumphs
in Random Matrix Theory, and the Central Limit Theorem, one of the
earliest examples of regularity and order in high dimensions.
Central to the text is the study of the Poincare and log-Sobolev
functional inequalities, their reverses, and other inequalities, in
which a crucial role is often played by convexity assumptions such
as Log-Concavity. The concept and properties of Entropy form an
important subject, with Bourgain's slicing problem and its variants
drawing much attention. Constructions related to Convexity Theory
are proposed and revisited, as well as inequalities that go beyond
the Brunn-Minkowski theory. One of the major current research
directions addressed is the identification of lower-dimensional
structures with remarkable properties in rather arbitrary
high-dimensional objects. In addition to functional analytic
results, connections to Computer Science and to Differential
Geometry are also discussed.
As in the previous Seminar Notes, the current volume reflects
general trends in the study of Geometric Aspects of Functional
Analysis. Most of the papers deal with different aspects of
Asymptotic Geometric Analysis, understood in a broad sense; many
continue the study of geometric and volumetric properties of convex
bodies and log-concave measures in high-dimensions and in
particular the mean-norm, mean-width, metric entropy, spectral-gap,
thin-shell and slicing parameters, with applications to Dvoretzky
and Central-Limit-type results. The study of spectral properties of
various systems, matrices, operators and potentials is another
central theme in this volume. As expected, probabilistic tools play
a significant role and probabilistic questions regarding Gaussian
noise stability, the Gaussian Free Field and First Passage
Percolation are also addressed. The historical connection to the
field of Classical Convexity is also well represented with new
properties and applications of mixed-volumes. The interplay between
the real convex and complex pluri-subharmonic settings continues to
manifest itself in several additional articles. All contributions
are original research papers and were subject to the usual
refereeing standards.
The volume contains the proceedings of the international workshop
on Concentration, Functional Inequalities and Isoperimetry, held at
Florida Atlantic University in Boca Raton, Florida, from October
29-November 1, 2009. The interactions between concentration,
isoperimetry and functional inequalities have led to many
significant advances in functional analysis and probability theory.
Important progress has also taken place in combinatorics, geometry,
harmonic analysis and mathematical physics, to name but a few
fields, with recent new applications in random matrices and
information theory. This book should appeal to graduate students
and researchers interested in the fascinating interplay between
analysis, probability, and geometry.
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