Asymptotic Geometric Analysis

Author : Shiri Artstein-Avidan,Apostolos Giannopoulos,Vitali D. Milman
Publisher : American Mathematical Society
Page : 645 pages
File Size : 43,9 Mb
Release : 2021-12-13
Category : Mathematics
ISBN : 9781470463601

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Asymptotic Geometric Analysis, Part II by Shiri Artstein-Avidan,Apostolos Giannopoulos,Vitali D. Milman Pdf

This book is a continuation of Asymptotic Geometric Analysis, Part I, which was published as volume 202 in this series. Asymptotic geometric analysis studies properties of geometric objects, such as normed spaces, convex bodies, or convex functions, when the dimensions of these objects increase to infinity. The asymptotic approach reveals many very novel phenomena which influence other fields in mathematics, especially where a large data set is of main concern, or a number of parameters which becomes uncontrollably large. One of the important features of this new theory is in developing tools which allow studying high parametric families. Among the topics covered in the book are measure concentration, isoperimetric constants of log-concave measures, thin-shell estimates, stochastic localization, the geometry of Gaussian measures, volume inequalities for convex bodies, local theory of Banach spaces, type and cotype, the Banach-Mazur compactum, symmetrizations, restricted invertibility, and functional versions of geometric notions and inequalities.

Author : Monika Ludwig,Vitali D. Milman,Vladimir Pestov,Nicole Tomczak-Jaegermann
Publisher : Springer
Page : 395 pages
File Size : 46,5 Mb
Release : 2013-03-28
Category : Mathematics
ISBN : 1461464056

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Asymptotic Geometric Analysis by Monika Ludwig,Vitali D. Milman,Vladimir Pestov,Nicole Tomczak-Jaegermann Pdf

Asymptotic Geometric Analysis is concerned with the geometric and linear properties of finite dimensional objects, normed spaces, and convex bodies, especially with the asymptotics of their various quantitative parameters as the dimension tends to infinity. The deep geometric, probabilistic, and combinatorial methods developed here are used outside the field in many areas of mathematics and mathematical sciences. The Fields Institute Thematic Program in the Fall of 2010 continued an established tradition of previous large-scale programs devoted to the same general research direction. The main directions of the program included: * Asymptotic theory of convexity and normed spaces * Concentration of measure and isoperimetric inequalities, optimal transportation approach * Applications of the concept of concentration * Connections with transformation groups and Ramsey theory * Geometrization of probability * Random matrices * Connection with asymptotic combinatorics and complexity theory These directions are represented in this volume and reflect the present state of this important area of research. It will be of benefit to researchers working in a wide range of mathematical sciences—in particular functional analysis, combinatorics, convex geometry, dynamical systems, operator algebras, and computer science.

Author : Anonim
Publisher : Unknown
Page : 128 pages
File Size : 46,5 Mb
Release : 2015
Category : Electronic
ISBN : OCLC:1087819363

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Asymptotic Geometric Analysis by Anonim Pdf

Author : Guillaume Aubrun,Stanisław J. Szarek
Publisher : American Mathematical Soc.
Page : 414 pages
File Size : 44,9 Mb
Release : 2017-08-30
Category : Functional analysis
ISBN : 9781470434687

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Alice and Bob Meet Banach: The Interface of Asymptotic Geometric Analysis and Quantum Information Theory by Guillaume Aubrun,Stanisław J. Szarek Pdf

The quest to build a quantum computer is arguably one of the major scientific and technological challenges of the twenty-first century, and quantum information theory (QIT) provides the mathematical framework for that quest. Over the last dozen or so years, it has become clear that quantum information theory is closely linked to geometric functional analysis (Banach space theory, operator spaces, high-dimensional probability), a field also known as asymptotic geometric analysis (AGA). In a nutshell, asymptotic geometric analysis investigates quantitative properties of convex sets, or other geometric structures, and their approximate symmetries as the dimension becomes large. This makes it especially relevant to quantum theory, where systems consisting of just a few particles naturally lead to models whose dimension is in the thousands, or even in the billions. Alice and Bob Meet Banach is aimed at multiple audiences connected through their interest in the interface of QIT and AGA: at quantum information researchers who want to learn AGA or apply its tools; at mathematicians interested in learning QIT, or at least the part of QIT that is relevant to functional analysis/convex geometry/random matrix theory and related areas; and at beginning researchers in either field. Moreover, this user-friendly book contains numerous tables and explicit estimates, with reasonable constants when possible, which make it a useful reference even for established mathematicians generally familiar with the subject.

Author : Keith M. Ball,Vitali Milman
Publisher : Cambridge University Press
Page : 260 pages
File Size : 49,9 Mb
Release : 1999-01-28
Category : Mathematics
ISBN : 0521642590

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Convex Geometric Analysis by Keith M. Ball,Vitali Milman Pdf

Articles on classical convex geometry, geometric functional analysis, computational geometry, and related areas of harmonic analysis, first published in 1999.

Author : Monika Ludwig,Vitali D. Milman,Vladimir Pestov,Nicole Tomczak-Jaegermann
Publisher : Springer Science & Business Media
Page : 402 pages
File Size : 42,8 Mb
Release : 2013-03-27
Category : Mathematics
ISBN : 9781461464068

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Asymptotic Geometric Analysis by Monika Ludwig,Vitali D. Milman,Vladimir Pestov,Nicole Tomczak-Jaegermann Pdf

Asymptotic Geometric Analysis is concerned with the geometric and linear properties of finite dimensional objects, normed spaces, and convex bodies, especially with the asymptotics of their various quantitative parameters as the dimension tends to infinity. The deep geometric, probabilistic, and combinatorial methods developed here are used outside the field in many areas of mathematics and mathematical sciences. The Fields Institute Thematic Program in the Fall of 2010 continued an established tradition of previous large-scale programs devoted to the same general research direction. The main directions of the program included: * Asymptotic theory of convexity and normed spaces * Concentration of measure and isoperimetric inequalities, optimal transportation approach * Applications of the concept of concentration * Connections with transformation groups and Ramsey theory * Geometrization of probability * Random matrices * Connection with asymptotic combinatorics and complexity theory These directions are represented in this volume and reflect the present state of this important area of research. It will be of benefit to researchers working in a wide range of mathematical sciences—in particular functional analysis, combinatorics, convex geometry, dynamical systems, operator algebras, and computer science.

Author : Shiri Artstein-Avidan, Apostolos Giannopoulos, Vitali D. Milman
Publisher : American Mathematical Soc.
Page : 451 pages
File Size : 53,6 Mb
Release : 2015-06-18
Category : Functional analysis
ISBN : 9781470421939

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Asymptotic Geometric Analysis, Part I by Shiri Artstein-Avidan, Apostolos Giannopoulos, Vitali D. Milman Pdf

The authors present the theory of asymptotic geometric analysis, a field which lies on the border between geometry and functional analysis. In this field, isometric problems that are typical for geometry in low dimensions are substituted by an "isomorphic" point of view, and an asymptotic approach (as dimension tends to infinity) is introduced. Geometry and analysis meet here in a non-trivial way. Basic examples of geometric inequalities in isomorphic form which are encountered in the book are the "isomorphic isoperimetric inequalities" which led to the discovery of the "concentration phenomenon", one of the most powerful tools of the theory, responsible for many counterintuitive results. A central theme in this book is the interaction of randomness and pattern. At first glance, life in high dimension seems to mean the existence of multiple "possibilities", so one may expect an increase in the diversity and complexity as dimension increases. However, the concentration of measure and effects caused by convexity show that this diversity is compensated and order and patterns are created for arbitrary convex bodies in the mixture caused by high dimensionality. The book is intended for graduate students and researchers who want to learn about this exciting subject. Among the topics covered in the book are convexity, concentration phenomena, covering numbers, Dvoretzky-type theorems, volume distribution in convex bodies, and more.

Author : Vitali D. Milman,Gideon Schechtman
Publisher : Springer
Page : 306 pages
File Size : 43,7 Mb
Release : 2004-08-30
Category : Mathematics
ISBN : 9783540444893

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Geometric Aspects of Functional Analysis by Vitali D. Milman,Gideon Schechtman Pdf

The Israeli GAFA seminar (on Geometric Aspect of Functional Analysis) during the years 2002-2003 follows the long tradition of the previous volumes. It reflects the general trends of the theory. Most of the papers deal with different aspects of the Asymptotic Geometric Analysis. In addition the volume contains papers on related aspects of Probability, classical Convexity and also Partial Differential Equations and Banach Algebras. There are also two expository papers on topics which proved to be very much related to the main topic of the seminar. One is Statistical Learning Theory and the other is Models of Statistical Physics. All the papers of this collection are original research papers.

Author : Australian National University. Centre for Mathematics and Its Applications
Publisher : Unknown
Page : 148 pages
File Size : 43,6 Mb
Release : 2007
Category : Differential geometry
ISBN : STANFORD:36105131659620

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CMA/AMSI Research Symposium "Asymptotic Geometric Analysis, Harmonic Analysis, and Related Topics" by Australian National University. Centre for Mathematics and Its Applications Pdf

Author : Victor Guillemin,Shlomo Sternberg
Publisher : American Mathematical Soc.
Page : 500 pages
File Size : 52,7 Mb
Release : 1990
Category : Mathematics
ISBN : 9780821816332

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Geometric Asymptotics by Victor Guillemin,Shlomo Sternberg Pdf

Symplectic geometry and the theory of Fourier integral operators are modern manifestations of themes that have occupied a central position in mathematical thought for the past three hundred years--the relations between the wave and the corpuscular theories of light. The purpose of this book is to develop these themes, and present some of the recent advances, using the language of differential geometry as a unifying influence.

Author : Bo'az Klartag,Shahar Mendelson,Vitali D. Milman
Publisher : Springer
Page : 444 pages
File Size : 53,9 Mb
Release : 2012-07-25
Category : Mathematics
ISBN : 9783642298493

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Geometric Aspects of Functional Analysis by Bo'az Klartag,Shahar Mendelson,Vitali D. Milman Pdf

This collection of original papers related to the Israeli GAFA seminar (on Geometric Aspects of Functional Analysis) from the years 2006 to 2011 continues the long tradition of the previous volumes, which reflect the general trends of Asymptotic Geometric Analysis, understood in a broad sense, and are a source of inspiration for new research. Most of the papers deal with various aspects of the theory, including classical topics in the geometry of convex bodies, inequalities involving volumes of such bodies or more generally, logarithmically-concave measures, valuation theory, probabilistic and isoperimetric problems in the combinatorial setting, volume distribution on high-dimensional spaces and characterization of classical constructions in Geometry and Analysis (like the Legendre and Fourier transforms, derivation and others). All the papers here are original research papers.

Author : J.D. Murray
Publisher : Springer Science & Business Media
Page : 172 pages
File Size : 42,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461211228

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Asymptotic Analysis by J.D. Murray Pdf

From the reviews: "A good introduction to a subject important for its capacity to circumvent theoretical and practical obstacles, and therefore particularly prized in the applications of mathematics. The book presents a balanced view of the methods and their usefulness: integrals on the real line and in the complex plane which arise in different contexts, and solutions of differential equations not expressible as integrals. Murray includes both historical remarks and references to sources or other more complete treatments. More useful as a guide for self-study than as a reference work, it is accessible to any upperclass mathematics undergraduate. Some exercises and a short bibliography included. Even with E.T. Copson's Asymptotic Expansions or N.G. de Bruijn's Asymptotic Methods in Analysis (1958), any academic library would do well to have this excellent introduction." (S. Puckette, University of the South) #Choice Sept. 1984#1

Author : Bo'az Klartag,Emanuel Milman
Publisher : Springer
Page : 459 pages
File Size : 51,7 Mb
Release : 2014-10-08
Category : Mathematics
ISBN : 9783319094779

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Geometric Aspects of Functional Analysis by Bo'az Klartag,Emanuel Milman Pdf

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.

Author : Bo'az Klartag,Emanuel Milman
Publisher : Springer Nature
Page : 346 pages
File Size : 43,9 Mb
Release : 2020-06-20
Category : Mathematics
ISBN : 9783030360207

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Geometric Aspects of Functional Analysis by Bo'az Klartag,Emanuel Milman Pdf

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 Poincaré 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.

Author : V.D. Milman,G. Schechtman
Publisher : Springer
Page : 296 pages
File Size : 53,7 Mb
Release : 2007-05-09
Category : Mathematics
ISBN : 9783540453925

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Geometric Aspects of Functional Analysis by V.D. Milman,G. Schechtman Pdf

This volume of original research papers from the Israeli GAFA seminar during the years 1996-2000 not only reports on more traditional directions of Geometric Functional Analysis, but also reflects on some of the recent new trends in Banach Space Theory and related topics. These include the tighter connection with convexity and the resulting added emphasis on convex bodies that are not necessarily centrally symmetric, and the treatment of bodies which have only very weak convex-like structure. Another topic represented here is the use of new probabilistic tools; in particular transportation of measure methods and new inequalities emerging from Poincaré-like inequalities.