Catalog start Organization (as author) Conference Board of the Mathematical Sciences Remove constraint Organization (as author): Conference Board of the Mathematical Sciences Library Science (Li and Ma) Remove constraint Library: Science (Li and Ma) Language English Remove constraint Language: English
Graham, Ronald L., 1935-20202
Kenig, Carlos E., 1953-2
Pearcy, Carl M., 1935-2
Rosenberg, Jonathan M. (Jonathan Micah), 1951-2
Rudin, Walter, 1921-20102
Albers, Donald J., 1941-1
Alesker, Semyon, 1972-1
Anderson, Richard D. (Richard Davis), 1922-1
Arveson, William1
Bass, Hyman, 1932-1
Bercovici, Hari, 1953-1
Borden, Brett, 1954-1
Bowen, Rufus, 1947-19781
Butler, Steven Kay, 1977-1
Cakoni, Fioralba1
Chapman, T. A. (Thomas A.), 1940-1
Cheney, Margaret, 1955-1
Chung, Fan R. K., 1949-1
Coifman, Ronald R. (Ronald Raphaël)1
Colton, David L., 1943-1
QA - Mathematics 76
QC - Physics 1
TK - Electrical Engineering, Electronics, Nuclear Engineering 1
Algebraic topology4
Banach algebras4
C*-algebras4
Combinatorial analysis4
Differential equations4
Graph theory4
Harmonic analysis4
Operator theory4
Differentiable dynamical systems3
Differential equations, Partial3
Homology theory3
Homotopy theory3
Matrices3
Operator algebras3
Algebra, Homological2
Analytic functions2
Banach spaces2
Banach, Algèbres de.2
Banach-algebra's.2
Conference proceedings24
Conference papers and proceedings2
Problems and exercises1
Problems, exercises, etc.1
Etats-Unis1
Conference Board of the Mathematical Sciences[remove]78
National Science Foundation (U.S.)19
American Mathematical Society4
NSF-CBMS Conference Topology, C*-algebras, and String Duality (2009 : Texas Christian University)1
NSF-CBMS Regional Conference in the Mathematical Sciences (2018 : Fort Worth, Tex.)1
NSF-CBMS Regional Conference in the Mathematical Sciences on Deformation Theory of Algebras and Modules (2011 : Raleigh, N.C.)1
NSF-CBMS Regional Conference in the Mathematical Sciences on Introduction to the Theory of Valuations on Convex Sets (2015 : Kent State University)1
NSF-CBMS Regional Conference in the Mathematical Sciences on the Basic Homotopy Lemma, the Asymptotic Uniqueness Theorem, and Classification of C*-Algebras (2015 : University of Wyoming)1
NSF-CBMS Regional Research Conference on the Combinatorics of Large Sparse Graphs (2004 : California State University)1
NSF/CBMS Regional Conference in Mathematical Sciences on the Energy Critical Nonlinear Wave Equation (2013 : Manhattan, Kansas)1
1. Applications of polynomial systems [2020]
Cox, David A., author.
Providence, Rhode Island : Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, [2020]
Book — ix, 250 pages : illustrations (some color) ; 26 cm
Elimination theory Numerical algebraic geometry Geometric modeling Rigidity theory Chemical reaction networks Illustration credits Bibliography Index.
Systems of polynomial equations can be used to model an astonishing variety of phenomena. This book explores the geometry and algebra of such systems and includes numerous applications. The book begins with elimination theory from Newton to the twenty-first century and then discusses the interaction between algebraic geometry and numerical computations, a subject now called numerical algebraic geometry. The final three chapters discuss applications to geometric modeling, rigidity theory, and chemical reaction networks in detail. Each chapter ends with a section written by a leading expert. Examples in the book include oil wells, HIV infection, phylogenetic models, four-bar mechanisms, border rank, font design, Stewart-Gough platforms, rigidity of edge graphs, Gaussian graphical models, geometric constraint systems, and enzymatic cascades. The reader will encounter geometric objects such as Bezier patches, Cayley-Menger varieties, and toric varieties; and algebraic objects such as resultants, Rees algebras, approximation complexes, matroids, and toric ideals. Two important subthemes that appear in multiple chapters are toric varieties and algebraic statistics. The book also discusses the history of elimination theory, including its near elimination in the middle of the twentieth century. The main goal is to inspire the reader to learn about the topics covered in the book. With this in mind, the book has an extensive bibliography containing over 350 books and papers.
QA1 .R33 NO.134 Unknown
2. Asymptotics of random matrices and related models : the uses of Dyson-Schwinger equations [2019]
Guionnet, Alice, author.
[Providence, RI] : American Mathematical Society, [2019]
Book — vii, 143 pages ; 26 cm.
Introduction The example of the GUE Wigner random matrices Beta-ensembles Discrete beta-ensembles Continuous beta-models: The several cut case Several matrix-ensembles Universality for beta-models Bibliography Index.
Probability theory is based on the notion of independence. The celebrated law of large numbers and the central limit theorem describe the asymptotics of the sum of independent variables. However, there are many models of strongly correlated random variables: for instance, the eigenvalues of random matrices or the tiles in random tilings. Classical tools of probability theory are useless to study such models. These lecture notes describe a general strategy to study the fluctuations of strongly interacting random variables. This strategy is based on the asymptotic analysis of Dyson-Schwinger (or loop) equations: the author will show how these equations are derived, how to obtain the concentration of measure estimates required to study these equations asymptotically, and how to deduce from this analysis the global fluctuations of the model. The author will apply this strategy in different settings: eigenvalues of random matrices, matrix models with one or several cuts, random tilings, and several matrices models.
QA196.5 .G85 2019 Unknown
3. Discrete Painlevé equations [2019]
Joshi, Nalini, author.
[Providence, Rhode Island] : American Mathematical Society, [2019]
Book — vi, 146 pages : illustrations ; 26 cm.
Introduction A dynamical systems approach Initial value spaces Foliated initial value spaces Cremona mappings Asymptotic analysis Lax pairs Riemann-Hilbert problems Foliations and vector bundles Projective spaces Reflection groups Lists of discrete-Painleve equations Asymptotics of discrete equations Bibliography Index.
Discrete Painleve equations are nonlinear difference equations, which arise from translations on crystallographic lattices. The deceptive simplicity of this statement hides immensely rich mathematical properties, connecting dynamical systems, algebraic geometry, Coxeter groups, topology, special functions theory, and mathematical physics. This book necessarily starts with introductory material to give the reader an accessible entry point to this vast subject matter. It is based on lectures that the author presented as principal lecturer at a Conference Board of Mathematical Sciences and National Science Foundation conference in Texas in 2016. Instead of technical theorems or complete proofs, the book relies on providing essential points of many arguments through explicit examples, with the hope that they will be useful for applied mathematicians and physicists.
4. Lectures on field theory and topology [2019]
Freed, Daniel S., author.
[Providence, Rhode Island] : Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, [2019]
Book — xi, 186 pages : illustrations ; 26 cm.
5. Tensors : asymptotic geometry and developments, 2016-2018 [2019]
Landsberg, J. M., author.
Book — ix, 144 pages : illustrations ; 25 cm.
Basics: Motivation, first definitions and properties Tensors via linear algebra: Rank and border rank Tensor networks The asymptotic geometry of tensors: Detour into probability and information theory Strassen's laser method and spectral theory Quantum mechanics for quantum information theory Quantum information theory and the asymptotic geometry of tensors Moment maps and moment polytopes Hints and answers to selected exercises Bibliography Index.
Tensors are used throughout the sciences, especially in solid state physics and quantum information theory. This book brings a geometric perspective to the use of tensors in these areas. It begins with an introduction to the geometry of tensors and provides geometric expositions of the basics of quantum information theory, Strassen's laser method for matrix multiplication, and moment maps in algebraic geometry. It also details several exciting recent developments regarding tensors in general. In particular, it discusses and explains the following material previously only available in the original research papers: (1) Shitov's 2017 refutation of longstanding conjectures of Strassen on rank additivity and Common on symmetric rank; (2) The 2017 Christandl-Vrana-Zuiddam quantum spectral points that bring together quantum information theory, the asymptotic geometry of tensors, matrix multiplication complexity, and moment polytopes in geometric invariant theory; (3) the use of representation theory in quantum information theory, including the solution of the quantum marginal problem; (4) the use of tensor network states in solid state physics, and (5) recent geometric paths towards upper bounds for the complexity of matrix multiplication. Numerous open problems appropriate for graduate students and post-docs are included throughout.
6. Zeta and L-functions in number theory and combinatorics [2019]
Li, W. C. Winnie (Wen-Ching Winnie), author.
Book — vii, 95 pages : illustrations ; 26 cm.
Number theoretic zeta and $L$-functions The Selberg zeta function $L$-functions in geometry The Ihara zeta function Spectral graph theory Explicit constructions of Ramanujan graphs Artin $L$-functions and prime distributions for graphs Zeta and $L$-functions of complexes Bibliography Index.
Zeta and $L$-functions play a central role in number theory. They provide important information of arithmetic nature. This book, which grew out of the author's teaching over several years, explores the interaction between number theory and combinatorics using zeta and $L$-functions as a central theme. It provides a systematic and comprehensive account of these functions in a combinatorial setting and establishes, among other things, the combinatorial counterparts of celebrated results in number theory, such as the prime number theorem and the Chebotarev density theorem. The spectral theory for finite graphs and higher dimensional complexes is studied. Of special interest in theory and applications are the spectrally extremal objects, called Ramanujan graphs and Ramanujan complexes, which can be characterized by their associated zeta functions satisfying the Riemann Hypothesis. Explicit constructions of these extremal combinatorial objects, using number-theoretic and combinatorial means, are presented. Research on zeta and $L$-functions for complexes other than graphs emerged only in recent years. This is the first book for graduate students and researchers offering deep insight into this fascinating and fast developing area.
7. Harmonic analysis : smooth and non-smooth [2018]
Jørgensen, Palle E. T., 1947- author.
Introduction. Smooth vs the non-smooth categories Spectral pair analysis for IFSs Harmonic analyses on fractals, with an emphasis on iterated function systems (IFS) measures Four kinds of harmonic analysis Harmonic analysis via representations of the Cuntz relations $\textit{ Positive definite functions }$ and kernel analysis Representations of $\textit{Lie groups}$. Non-commutative harmonic analysis Bibliography Index.
There is a recent and increasing interest in harmonic analysis of non-smooth geometries. Real-world examples where these types of geometry appear include large computer networks, relationships in datasets, and fractal structures such as those found in crystalline substances, light scattering, and other natural phenomena where dynamical systems are present. Notions of harmonic analysis focus on transforms and expansions and involve dual variables. In this book on smooth and non-smooth harmonic analysis, the notion of dual variables will be adapted to fractals. In addition to harmonic analysis via Fourier duality, the author also covers multiresolution wavelet approaches as well as a third tool, namely, $L^2$ spaces derived from appropriate Gaussian processes. The book is based on a series of ten lectures delivered in June 2018 at a CBMS conference held at Iowa State University.
8. Introduction to the theory of valuations [2018]
Alesker, Semyon, 1972- author.
Book — vi, 83 pages : illustrations ; 26 cm.
Basic definitions and examples McMullen's decomposition theorem Valuations on the line McMullen's description of $(n-1)$-homogeneous valuations The Klain-Schneider characterization of simple valuations Digression on the theory of generalized functions on manifolds The Goodey-Weil imbedding Digression on vector bundles The irreducibility theorem Further developments Bibliography.
Theory of valuations on convex sets is a classical part of convex geometry which goes back at least to the positive solution of the third Hilbert problem by M. Dehn in 1900. Since then the theory has undergone a multifaceted development. The author discusses some of Hadwiger's results on valuations on convex compact sets that are continuous in the Hausdorff metric. The book also discusses the Klain-Schneider theorem as well as the proof of McMullen's conjecture, which led subsequently to many further applications and advances in the theory. The last section gives an overview of more recent developments in the theory of translation-invariant continuous valuations, some of which turn out to be useful in integral geometry. This book grew out of lectures that were given in August 2015 at Kent State University in the framework of the NSF CBMS conference ``Introduction to the Theory of Valuations on Convex Sets''. Only a basic background in general convexity is assumed.
9. From the basic homotopy lemma to the classification of C*-algebras [2017]
Lin, Huaxin, 1956- author.
Book — vi, 240 pages ; 26 cm.
An overview of the Elliott programAn introduction to the Basic Homotopy LemmaMaps to finite dimensional $C^*$-algebrasStable homotopy lemmasThe Basic Homotopy Lemma, finite dimensional cases$C^*$-algebras of generalized tracial rank oneMore Basic Homotopy LemmasAsymptotic unitary equivalenceClassification of simple $C^*$-algebras of finite rankBibliographyIndex.
This book examines some recent developments in the theory of $C^*$-algebras, which are algebras of operators on Hilbert spaces. An elementary introduction to the technical part of the theory is given via a basic homotopy lemma concerning a pair of almost commuting unitaries. The book presents an outline of the background as well as some recent results of the classification of simple amenable $C^*$-algebra, otherwise known as the Elliott program. This includes some stable uniqueness theorems and a revisiting of Bott maps via stable homotopy. Furthermore, $KK$-theory related rotation maps are introduced. The book is based on lecture notes from the CBMS lecture sequence at the University of Wyoming in the summer of 2015.
10. Lectures on the energy critical nonlinear wave equation [2015]
Kenig, Carlos E., 1953- author.
Providence, Rhode Island : Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, with support from the National Science Foundation, [2015]
Book — xiii, 161 pages ; 26 cm.
The local theory of the Cauchy problem The "road map": The concentration compactness/rigidity theorem method for critical problems I The "road map": The concentration compactness/rigidity theorem method for critical problems II Properties of compact solutions and some more rigidity theorems, with applications to an extension of Theorem 2.6 Proof of the rigidity theorems Type II blow-up solutions Channels of energy and outer energy lower bounds Universal type II blow-up profiles Soliton resolution for radial solutions to (NLW), I Soliton resolution for radial solutions to (NLW), II Soliton resolution for radial solutions to (NLW), III Bibliography.
This monograph deals with recent advances in the study of the long-time asymptotics of large solutions to critical nonlinear dispersive equations. The first part of the monograph describes, in the context of the energy critical wave equation, the "concentration-compactness/rigidity theorem method" introduced by C. Kenig and F. Merle. This approach has become the canonical method for the study of the "global regularity and well-posedness" conjecture (defocusing case) and the "ground-state" conjecture (focusing case) in critical dispersive problems. The second part of the monograph describes the "channel of energy" method, introduced by T. Duyckaerts, C. Kenig, and F. Merle, to study soliton resolution for nonlinear wave equations. This culminates in a presentation of the proof of the soliton resolution conjecture, for the three-dimensional radial focusing energy critical wave equation. It is the intent that the results described in this book will be a model for what to strive for in the study of other nonlinear dispersive equations.
11. Rudiments of Ramsey theory [2015]
Graham, Ronald L., 1935-2020 author.
Second edition. - Providence, Rhode Island : Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, [2015]
Book — ix, 82 pages : illustrations ; 25 cm.
* Introduction* Three views of Ramsey theory* Ramsey's theorem van der Waerden's theorem* The Hales-Jewett theorem* Szemeredi's theorem* Graph Ramsey theory* Euclidean Ramsey theory* A general Ramsey product theorem* The theorems of Schur, Folkman, and Hindman* Rado's theorem* Current trends* Bibliography.
In every sufficiently large structure which has been partitioned there will always be some well-behaved structure in one of the parts. This takes many forms. For example, colorings of the integers by finitely many colors must have long monochromatic arithmetic progressions (van der Waerden's theorem); and colorings of the edges of large graphs must have monochromatic subgraphs of a specified type (Ramsey's theorem). This book explores many of the basic results and variations of this theory. Since the first edition of this book there have been many advances in this field. In the second edition the authors update the exposition to reflect the current state of the art. They also include many pointers to modern results.
12. Analysis of stochastic partial differential equations [2014]
Khoshnevisan, Davar, author.
Providence, Rhode Island : American Mathematical Society, [2014]
Book — viii, 116 pages : illustrations ; 26 cm.
Wiener integrals
A linear heat equation
Walsh-Damang integrals
A non-linear heat equation
Intermezzo: A parabolic Anderson model
Intermittency
Intermittency fronts
Intermittency islands
Correlation length
Appendix A: Some special integrals
Appendix B: A Burkholder-Davis-Gundy inequality
Appendix C: Reguarity theory.
The general area of stochastic PDEs is interesting to mathematicians because it contains an enormous number of challenging open problems. There is also a great deal of interest in this topic because it has deep applications in disciplines that range from applied mathematics, statistical mechanics, and theoretical physics, to theoretical neuroscience, theory of complex chemical reactions [including polymer science], fluid dynamics, and mathematical finance. The stochastic PDEs that are studied in this book are similar to the familiar PDE for heat in a thin rod, but with the additional restriction that the external forcing density is a two-parameter stochastic process, or what is more commonly the case, the forcing is a "random noise, " also known as a "generalized random field." At several points in the lectures, there are examples that highlight the phenomenon that stochastic PDEs are not a subset of PDEs. In fact, the introduction of noise in some partial differential equations can bring about not a small perturbation, but truly fundamental changes to the system that the underlying PDE is attempting to describe. The topics covered include a brief introduction to the stochastic heat equation, structure theory for the linear stochastic heat equation, and an in-depth look at intermittency properties of the solution to semilinear stochastic heat equations. Specific topics include stochastic integrals a la Norbert Wiener, an infinite-dimensional Ito-type stochastic integral, an example of a parabolic Anderson model, and intermittency fronts. There are many possible approaches to stochastic PDEs. The selection of topics and techniques presented here are informed by the guiding example of the stochastic heat equation.
13. Ergodic theory and fractal geometry [2014]
Furstenberg, Harry, author.
Introduction to fractals Dimension Trees and fractals Invariant sets Probability trees Galleries Probability trees revisited Elements of ergodic theory Galleries of trees General remarks on Markov systems Markov operator $\mathcal{T}$ and measure preserving transformation $T$ Probability trees and galleries Ergodic theorem and the proof of the main theorem An application: The $k$-lane property Dimension and energy Dimension conservation Ergodic theorem for sequences of functions Dimension conservation for homogeneous fractals: The main steps in the proof Verifying the conditions of the ergodic theorem for sequences of functions Bibliography Index.
Fractal geometry represents a radical departure from classical geometry, which focuses on smooth objects that "straighten out" under magnification. Fractals, which take their name from the shape of fractured objects, can be characterized as retaining their lack of smoothness under magnification. The properties of fractals come to light under repeated magnification, which we refer to informally as "zooming in". This zooming-in process has its parallels in dynamics, and the varying "scenery" corresponds to the evolution of dynamical variables. The present monograph focuses on applications of one branch of dynamics--ergodic theory--to the geometry of fractals. Much attention is given to the all-important notion of fractal dimension, which is shown to be intimately related to the study of ergodic averages. It has been long known that dynamical systems serve as a rich source of fractal examples. The primary goal in this monograph is to demonstrate how the minute structure of fractals is unfolded when seen in the light of related dynamics.
14. Deformation theory of algebras and their diagrams [2012]
Lectures. Selections
Markl, Martin, 1960- author.
This book brings together both the classical and current aspects of deformation theory. The presentation is mostly self-contained, assuming only basic knowledge of commutative algebra, homological algebra and category theory. In the interest of readability, some technically complicated proofs have been omitted when a suitable reference was available. The relation between the uniform continuity of algebraic maps and topologized tensor products is explained in detail, however, as this subject does not seem to be commonly known and the literature is scarce. The exposition begins by recalling Gerstenhaber's classical theory for associative algebras. The focus then shifts to a homotopy-invariant setup of Maurer-Cartan moduli spaces. As an application, Kontsevich's approach to deformation quantization of Poisson manifolds is reviewed. Then, after a brief introduction to operads, a strongly homotopy Lie algebra governing deformations of (diagrams of) algebras of a given type is described, followed by examples and generalizations.
15. The linear sampling method in inverse electromagnetic scattering [2011]
Cakoni, Fioralba.
Philadelphia, PA : Society for Industrial and Applied Mathematics, c2011.
Book — x, 138 p. : ill. ; 26 cm.
Preface--
1. Inverse scattering in two dimensions--
2. Maxwell's equations--
3. The inverse problem for obstacles--
4. The inverse scattering problem for anisotropic media--
5. The inverse scattering problem for thin objects--
6. The inverse scattering problem for buried objects-- Bibliography-- Index.
The linear sampling method is the oldest and most developed of the qualitative methods in inverse scattering theory. It is based on solving a linear integral equation and then using the equation's solution as an indicator function for the determination of the support of the scattering object. This book describes the linear sampling method for a variety of electromagnetic scattering problems. It presents uniqueness theorems and the derivation of various inequalities on the material properties of the scattering object from a knowledge of the far field pattern of the scattered wave. Also covered are: the approximation properties of Herglotz wave functions; the behavior of solutions to the interior transmission problem, a novel interior boundary value problem; and numerical examples of the inversion scheme.
epubs.siam.org SIAM
SIAM Publications Online
QC20.7 .S3 C35 2011 Unknown
16. Superstrings, geometry, topology, and C*-algebras [2010]
NSF-CBMS Conference Topology, C*-algebras, and String Duality (2009 : Texas Christian University)
Providence, R.I. : American Mathematical Society, c2010.
Book — xiii, 249 p. ; 26 cm.
Functoriality of Rieffel's generalised fixed-point algebras for proper actions / Astrid An Huef, Iain Raeburn, and Dana P. Williams
Twists of K-theory and TMF / Matthew Ando, Andrew J. Blumberg, and David Gepner
Division algebras and supersymmetry I / John C. Baez and Joh Huerta
K-homology and d-branes / Paul Baum
Riemann-Roch and index formulae in twisted K-theory / Alan L. Carey and Bai-Ling Wang
Noncommutative principal torus bundles via parametrised strict deformation quantization / Keith C. Hannabuss and Varghese Mathai
A survey of noncommutative Yang-Mills theory for quantum Heisenberg manifolds / Sooran Kang
From rational homotopy to K-theory for continuous trace algebras / John R. Klein, Claude L. Schochet, and Samuel B. Smith
Distances between matrix algebras that converge to coadjoint orbits / Marc A. Rieffel
Geometric and topological structures related to M-branes / Hisham Sati
Landau-Ginzburg models, Gerbes, and Kuznetsov's homological projective duality / Eric Sharpe.
QA1 .A626 V.81 Unknown
17. Fundamentals of radar imaging [2009]
Cheney, Margaret, 1955-
Philadelphia : Society for Industrial and Applied Methematics, c2009.
Book — xxiv, 140 p. : ill. (some col.), maps ; 25 cm.
List of figures-- List of tables-- Preface-- Part I. Radar Basic:
1. Introduction--
2. Radar systems--
3. Introduction to scattering--
4. Detection of signals in noise--
5. The radar ambiguity function-- Part II. Radar Imaging:
6. Wave propagation in two and three dimensions--
7. Inverse synthetic-aperture radar--
8. Antennas--
9. Synthetic-aperture radar--
10. Related techniques--
11. Open problems-- Bibliography-- Index.
Radar imaging is a mathematically rich subject with many interesting applications and a large variety of challenging, mathematical open problems. The goal of this book is to provide mathematicians with the background they need to work in the field, building on the foundation of the underlying partial differential equations. The focus is on showing the connection between the physics and the mathematics and on supplying an intuitive mathematical understanding of basic concepts. The book includes a description of how a radar system works, together with the relevant mathematics; theory that guides the choice of radar waveforms; derivation of the fundamentals of scattering theory; derivation and discussion of the image formation process; and a long list of current open problems. Readers are assumed to be familiar with the Fourier transform. Other assumed background material includes certain basic facts from vector calculus, complex variables, linear algebra and physics.
TK6575 .C476 2009 Unknown
18. Topology, C*-algebras, and string duality [2009]
Rosenberg, Jonathan M. (Jonathan Micah), 1951-
Providence, R.I. : Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, c2009.
Book — viii, 110 p. : ill. ; 26 cm.
String theory is the leading candidate for a physical theory that combines all the fundamental forces of nature, as well as the principles of relativity and quantum mechanics, into a mathematically elegant whole. The mathematical tools used by string theorists are highly sophisticated, and cover many areas of mathematics. As with the birth of quantum theory in the early 20th century, the mathematics has benefited at least as much as the physics from the collaboration. In this book, based on CBMS lectures given at Texas Christian University, Rosenberg describes some of the most recent interplay between string dualities and topology and operator algebras. The book is an interdisciplinary approach to duality symmetries in string theory. It can be read by either mathematicians or theoretical physicists, and involves a more-or-less equal mixture of algebraic topology, operator algebras, and physics. There is also a bit of algebraic geometry, especially in the last chapter. The reader is assumed to be somewhat familiar with at least one of these four subjects, but not necessarily with all or even most of them. The main objective of the book is to show how several seemingly disparate subjects are closely linked with one another, and to give readers an overview of some areas of current research, even if this means that not everything is covered systematically.
19. Complex graphs and networks [2006]
NSF-CBMS Regional Research Conference on the Combinatorics of Large Sparse Graphs (2004 : California State University)
Book — vii, 264 p. : ill. (some col.) ; 26 cm.
Graph theory in the information age Old and new concentration inequalities A generative model--the preferential attachment scheme Duplication models for biological networks Random graphs with given expected degrees The rise of the giant component Average distance and the diameter Eigenvalues of the adjacency matrix of $G(\mathbf{w})$ The semi-circle law for $G(\mathbf{w})$ Coupling on-line and off-line analyses of random graphs The configuration model for power law graphs The small world phenomenon in hybrid graphs Bibliography Index.
Through examples of large complex graphs in realistic networks, research in graph theory has been forging ahead into exciting new directions. Graph theory has emerged as a primary tool for detecting numerous hidden structures in various information networks, including Internet graphs, social networks, biological networks, or, more generally, any graph representing relations in massive data sets. How will we explain from first principles the universal and ubiquitous coherence in the structure of these realistic but complex networks? In order to analyze these large sparse graphs, we use combinatorial, probabilistic, and spectral methods, as well as new and improved tools to analyze these networks. The examples of these networks have led us to focus on new, general, and powerful ways to look at graph theory.The book, based on lectures given at the CBMS Workshop on the Combinatorics of Large Sparse Graphs, presents new perspectives in graph theory and helps to contribute to a sound scientific foundation for our understanding of discrete networks that permeate this information age.
20. Nonlinear dispersive equations : local and global analysis [2006]
Tao, Terence, 1975-
Book — xv, 373 p. : ill. ; 26 cm.
Ordinary differential equations Constant coefficient linear dispersive equations Semilinear dispersive equations The Korteweg de Vries equation Energy-critical semilinear dispersive equations Wave maps Tools from harmonic analysis Construction of ground states Bibliography.
Among nonlinear PDEs, dispersive and wave equations form an important class of equations. These include the nonlinear Schrodinger equation, the nonlinear wave equation, the Korteweg de Vries equation, and the wave maps equation. This book is an introduction to the methods and results used in the modern analysis (both locally and globally in time) of the Cauchy problem for such equations. Starting only with a basic knowledge of graduate real analysis and Fourier analysis, the text first presents basic nonlinear tools such as the bootstrap method and perturbation theory in the simpler context of nonlinear ODE, then introduces the harmonic analysis and geometric tools used to control linear dispersive PDE.These methods are then combined to study four model nonlinear dispersive equations. Through extensive exercises, diagrams, and informal discussion, the book gives a rigorous theoretical treatment of the material, the real-world intuition and heuristics that underlie the subject, as well as mentioning connections with other areas of PDE, harmonic analysis, and dynamical systems. As the subject is vast, the book does not attempt to give a comprehensive survey of the field, but instead concentrates on a representative sample of results for a selected set of equations, ranging from the fundamental local and global existence theorems to very recent results, particularly focusing on the recent progress in understanding the evolution of energy-critical dispersive equations from large data. The book is suitable for a graduate course on nonlinear PDE.