Discrete Groups Expanding Graphs And Invariant Measures

Author : Alex Lubotzky
Publisher : Springer Science & Business Media
ISBN 13 : 3034603320
Total Pages : 201 pages
Book Rating : 4.24/5 ( download)

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Download or read book Discrete Groups, Expanding Graphs and Invariant Measures written by Alex Lubotzky and published by Springer Science & Business Media. This book was released on 2010-02-17 with total page 201 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the last ?fteen years two seemingly unrelated problems, one in computer science and the other in measure theory, were solved by amazingly similar techniques from representation theory and from analytic number theory. One problem is the - plicit construction of expanding graphs («expanders»). These are highly connected sparse graphs whose existence can be easily demonstrated but whose explicit c- struction turns out to be a dif?cult task. Since expanders serve as basic building blocks for various distributed networks, an explicit construction is highly des- able. The other problem is one posed by Ruziewicz about seventy years ago and studied by Banach [Ba]. It asks whether the Lebesgue measure is the only ?nitely additive measure of total measure one, de?ned on the Lebesgue subsets of the n-dimensional sphere and invariant under all rotations. The two problems seem, at ?rst glance, totally unrelated. It is therefore so- what surprising that both problems were solved using similar methods: initially, Kazhdan’s property (T) from representation theory of semi-simple Lie groups was applied in both cases to achieve partial results, and later on, both problems were solved using the (proved) Ramanujan conjecture from the theory of automorphic forms. The fact that representation theory and automorphic forms have anything to do with these problems is a surprise and a hint as well that the two questions are strongly related.

Author : Alexander Lubotzky
Publisher :
ISBN 13 :
Total Pages : 173 pages
Book Rating : 4.95/5 ( download)

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Download or read book Discrete Groups, Expanding Graphs and Invariant Measures written by Alexander Lubotzky and published by . This book was released on 1989 with total page 173 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Author : A. B. Kharazishvili
Publisher : World Scientific
ISBN 13 : 9810234929
Total Pages : 270 pages
Book Rating : 4.28/5 ( download)

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Download or read book Transformation Groups and Invariant Measures written by A. B. Kharazishvili and published by World Scientific. This book was released on 1998 with total page 270 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is devoted to some topics of the general theory of invariant and quasi-invariant measures. Such measures are usually defined on various sigma-algebras of subsets of spaces equipped with transformation groups, and there are close relationships between purely algebraic properties of these groups and the corresponding properties of invariant (quasi-invariant) measures. The main goal of the book is to investigate several aspects of those relationships (primarily from the set-theoretical point of view). Also of interest are the properties of some natural classes of sets, important from the viewpoint of the theory of invariant (quasi-invariant) measures.

Author : Michael Kapovich
Publisher : Springer Science & Business Media
ISBN 13 : 9780817639044
Total Pages : 500 pages
Book Rating : 4.47/5 ( download)

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Download or read book Hyperbolic Manifolds and Discrete Groups written by Michael Kapovich and published by Springer Science & Business Media. This book was released on 2001 with total page 500 pages. Available in PDF, EPUB and Kindle. Book excerpt: Hyperbolic Manifolds and Discrete Groups is at the crossroads of several branches of mathematics: hyperbolic geometry, discrete groups, 3-dimensional topology, geometric group theory, and complex analysis. The main focus throughout the text is on the "Big Monster," i.e., on Thurston’s hyperbolization theorem, which has not only completely changes the landscape of 3-dimensinal topology and Kleinian group theory but is one of the central results of 3-dimensional topology. The book is fairly self-contained, replete with beautiful illustrations, a rich set of examples of key concepts, numerous exercises, and an extensive bibliography and index. It should serve as an ideal graduate course/seminar text or as a comprehensive reference.

Author : Mike Krebs
Publisher : Oxford University Press
ISBN 13 : 0199877483
Total Pages : 288 pages
Book Rating : 4.85/5 ( download)

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Download or read book Expander Families and Cayley Graphs written by Mike Krebs and published by Oxford University Press. This book was released on 2011-09-30 with total page 288 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory of expander graphs is a rapidly developing topic in mathematics and computer science, with applications to communication networks, error-correcting codes, cryptography, complexity theory, and much more. Expander Families and Cayley Graphs: A Beginner's Guide is a comprehensive introduction to expander graphs, designed to act as a bridge between classroom study and active research in the field of expanders. It equips those with little or no prior knowledge with the skills necessary to both comprehend current research articles and begin their own research. Central to this book are four invariants that measure the quality of a Cayley graph as a communications network-the isoperimetric constant, the second-largest eigenvalue, the diameter, and the Kazhdan constant. The book poses and answers three core questions: How do these invariants relate to one another? How do they relate to subgroups and quotients? What are their optimal values/growth rates? Chapters cover topics such as: · Graph spectra · A Cheeger-Buser-type inequality for regular graphs · Group quotients and graph coverings · Subgroups and Schreier generators · Ramanujan graphs and the Alon-Boppana theorem · The zig-zag product and its relation to semidirect products of groups · Representation theory and eigenvalues of Cayley graphs · Kazhdan constants The only introductory text on this topic suitable for both undergraduate and graduate students, Expander Families and Cayley Graphs requires only one course in linear algebra and one in group theory. No background in graph theory or representation theory is assumed. Examples and practice problems with varying complexity are included, along with detailed notes on research articles that have appeared in the literature. Many chapters end with suggested research topics that are ideal for student projects.

Author : E. Pap
Publisher : Elsevier
ISBN 13 : 9780080533094
Total Pages : 1632 pages
Book Rating : 4.94/5 ( download)

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Download or read book Handbook of Measure Theory written by E. Pap and published by Elsevier. This book was released on 2002-10-31 with total page 1632 pages. Available in PDF, EPUB and Kindle. Book excerpt: The main goal of this Handbook is to survey measure theory with its many different branches and its relations with other areas of mathematics. Mostly aggregating many classical branches of measure theory the aim of the Handbook is also to cover new fields, approaches and applications which support the idea of "measure" in a wider sense, e.g. the ninth part of the Handbook. Although chapters are written of surveys in the various areas they contain many special topics and challenging problems valuable for experts and rich sources of inspiration. Mathematicians from other areas as well as physicists, computer scientists, engineers and econometrists will find useful results and powerful methods for their research. The reader may find in the Handbook many close relations to other mathematical areas: real analysis, probability theory, statistics, ergodic theory, functional analysis, potential theory, topology, set theory, geometry, differential equations, optimization, variational analysis, decision making and others. The Handbook is a rich source of relevant references to articles, books and lecture notes and it contains for the reader's convenience an extensive subject and author index.

Author : C. M. Campbell
Publisher : Cambridge University Press
ISBN 13 : 1107514541
Total Pages : 503 pages
Book Rating : 4.46/5 ( download)

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Download or read book Groups St Andrews 2013 written by C. M. Campbell and published by Cambridge University Press. This book was released on 2015-10-22 with total page 503 pages. Available in PDF, EPUB and Kindle. Book excerpt: Leading researchers survey the latest developments in group theory and many related areas.

Author : David Fisher
Publisher : University of Chicago Press
ISBN 13 : 022680416X
Total Pages : 573 pages
Book Rating : 4.63/5 ( download)

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Download or read book Dynamics, Geometry, Number Theory written by David Fisher and published by University of Chicago Press. This book was released on 2022-02-07 with total page 573 pages. Available in PDF, EPUB and Kindle. Book excerpt: This definitive synthesis of mathematician Gregory Margulis’s research brings together leading experts to cover the breadth and diversity of disciplines Margulis’s work touches upon. This edited collection highlights the foundations and evolution of research by widely influential Fields Medalist Gregory Margulis. Margulis is unusual in the degree to which his solutions to particular problems have opened new vistas of mathematics; his ideas were central, for example, to developments that led to the recent Fields Medals of Elon Lindenstrauss and Maryam Mirzhakhani. Dynamics, Geometry, Number Theory introduces these areas, their development, their use in current research, and the connections between them. Divided into four broad sections—“Arithmeticity, Superrigidity, Normal Subgroups”; “Discrete Subgroups”; “Expanders, Representations, Spectral Theory”; and “Homogeneous Dynamics”—the chapters have all been written by the foremost experts on each topic with a view to making them accessible both to graduate students and to experts in other parts of mathematics. This was no simple feat: Margulis’s work stands out in part because of its depth, but also because it brings together ideas from different areas of mathematics. Few can be experts in all of these fields, and this diversity of ideas can make it challenging to enter Margulis’s area of research. Dynamics, Geometry, Number Theory provides one remedy to that challenge.

Author : Lowell W. Beineke
Publisher : Cambridge University Press
ISBN 13 : 9780521801973
Total Pages : 302 pages
Book Rating : 4.74/5 ( download)

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Download or read book Topics in Algebraic Graph Theory written by Lowell W. Beineke and published by Cambridge University Press. This book was released on 2004-10-04 with total page 302 pages. Available in PDF, EPUB and Kindle. Book excerpt: There is no other book with such a wide scope of both areas of algebraic graph theory.

Author : Motoko Kotani
Publisher : American Mathematical Soc.
ISBN 13 : 0821833510
Total Pages : 274 pages
Book Rating : 4.13/5 ( download)

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Download or read book Discrete Geometric Analysis written by Motoko Kotani and published by American Mathematical Soc.. This book was released on 2004 with total page 274 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a collection of papers from the proceedings of the first symposium of the Japan Association for Mathematical Sciences. Topics covered center around problems of geometric analysis in relation to heat kernels, random walks, and Poisson boundaries on discrete groups, graphs, and other combinatorial objects. The material is suitable for graduate students and research mathematicians interested in heat kernels and random works on groups and graphs.