Proceedings of Symposia in Pure Mathematics, Volume: 81
2010; 249 pp; hardcover
ISBN-13: 978-0-8218-4887-6
Expected publication date is November 14, 2010.
This volume contains the proceedings of an NSF-CBMS Conference held at Texas Christian University in Fort Worth, Texas, May 18-22, 2009. The papers, written especially for this volume by well-known mathematicians and mathematical physicists, are an outgrowth of the talks presented at the conference. Topics examined are highly interdisciplinary and include, among many other things, recent results on D-brane charges in K-homology and twisted K-homology, Yang-Mills gauge theory and connections with non-commutative geometry, Landau-Ginzburg models, C^*-algebraic non-commutative geometry and ties to quantum physics and topology, the rational homotopy type of the group of unitary elements in an Azumaya algebra, and functoriality properties in the theory of C^*-crossed products and fixed point algebras for proper actions. An introduction, written by Jonathan Rosenberg, provides an instructive overview describing common themes and how the various papers in the volume are interrelated and fit together. The rich diversity of papers appearing in the volume demonstrates the current interplay between superstring theory, geometry/topology, and non-commutative geometry. The book will be of interest to graduate students, mathematicians, mathematical physicists, and researchers working in these areas.
Graduate students and research mathematicians interested in the relations between mathematical physics and various areas of pure mathematics.
* J. Rosenberg -- Introduction
* A. an Huef, I. Raeburn, and D. P. Williams -- Functoriality of Rieffel's generalised fixed-point algebras for proper actions
* M. Ando, A. J. Blumberg, and D. Gepner -- Twists of K-theory and TMF
* J. C. Baez and J. Huerta -- Division algebras and supersymmetry I
* P. Baum -- K-homology and D-branes
* A. L. Carey and B.-L. Wang -- Riemann-Roch and index formulae in twisted K-theory
* K. C. Hannabuss and V. Mathai -- Noncommutative principal torus bundles via parametrised strict deformation quantization
* S. Kang -- A survey of noncommutative Yang-Mills theory for quantum Heisenberg manifolds
* J. R. Klein, C. L. Schochet, and S. B. Smith -- From rational homotopy to K-theory for continuous trace algebras
* M. A. Rieffel -- Distances between matrix algebras that converge to coadjoint orbits
* H. Sati -- Geometric and topological structures related to M-branes
* E. Sharpe -- Landau-Ginzburg models, Gerbes, and Kuznetsov's homological projective duality
American Mathematical Society Translations--Series 2, Volume: 230
2010; approx. 248 pp; hardcover
ISBN-13: 978-0-8218-4881-4
Expected publication date is November 20, 2010.
This volume contains translations of papers that originally appeared in the Japanese journal S*gaku. These papers range over a variety of topics in ordinary and partial differential equations, and in analysis. Many of them are survey papers presenting new results obtained in the last few years.
This volume is suitable for graduate students and research mathematicians interested in analysis and differential equations.
Graduate students and research mathematicians interested in analysis and differential equations.
* K. Nakanishi -- Asymptotic analysis of nonlinear dispersive equations
* N. Hayashi -- Asymptotics of nonlinear dispersive-type evolution equations
* K. Takemura -- Heun's differential equation
* H. Isozaki -- Scattering theory and inverse problems
* Y. Komori -- Nondoubling measure and harmonic analysis
* S. Saitoh -- Theory of reproducing kernels
* H. Izeki and S. Nayatani -- An approach to superrigidity and fixed-point theorems via harmonic maps
* H. Sumi -- Rational semigroups, random complex dynamics and singular functions on the complex plane
* K. Oguiso -- Salem polynomials and the bimeromorphic automorphism group of a hyperkahler manifold
* S. Yamagami -- Tensor categories in operator algebras
Student Mathematical Library, Volume: 54
2010; approx. 312 pp; softcover
ISBN-13: 978-0-8218-5245-3
Expected publication date is December 1, 2010.
Solitons are explicit solutions to nonlinear partial differential equations exhibiting particle-like behavior. This is quite surprising, both mathematically and physically. Waves with these properties were once believed to be impossible by leading mathematical physicists, yet they are now not only accepted as a theoretical possibility but are regularly observed in nature and form the basis of modern fiber-optic communication networks.
Glimpses of Soliton Theory addresses some of the hidden mathematical connections in soliton theory which have been revealed over the last half-century. It aims to convince the reader that, like the mirrors and hidden pockets used by magicians, the underlying algebro-geometric structure of soliton equations provides an elegant and surprisingly simple explanation of something seemingly miraculous.
Assuming only multivariable calculus and linear algebra as prerequisites, this book introduces the reader to the KdV Equation and its multisoliton solutions, elliptic curves and Weierstrass \wp-functions, the algebra of differential operators, Lax Pairs and their use in discovering other soliton equations, wedge products and decomposability, the KP Equation and Sato's theory relating the Bilinear KP Equation to the geometry of Grassmannians.
Notable features of the book include: careful selection of topics and detailed explanations to make this advanced subject accessible to any undergraduate math major, numerous worked examples and thought-provoking but not overly-difficult exercises, footnotes and lists of suggested readings to guide the interested reader to more information, and use of the software package MathematicaR to facilitate computation and to animate the solutions under study. This book provides the reader with a unique glimpse of the unity of mathematics and could form the basis for a self-study, one-semester special topics, or "capstone" course.
Undergraduate and graduate students interested in nonlinear PDEs; applications of algebraic geometry to differential equations.
* Differential equations
* Developing PDE intuition
* The story of solitons
* Elliptic curves and KdV traveling waves
* KdV n-solitons
* Multiplying and factoring differential operators
* Eigenfunctions and isospectrality
* Lax form for KdV and other soliton equations
* The KP equation and bilinear KP equation
* The Grassmann cone \Gamma_{2,4} and the bilinear KP equation
* Pseudo-differential operators and the KP hierarchy
* The Grassman cone \Gamma_{k,n} and the bilinear KP hierarchy
* Concluding remarks
* Mathematica guide
* Complex numbers
* Ideas for independent projects
* References
* Glossary of symbols
* Index
University Lecture Series, Volume: 56
2010; approx. 257 pp; softcover
ISBN-13: 978-0-8218-5254-5
Expected publication date is December 23, 2010.
This book contains a systematic presentation of quantum functional analysis, a mathematical subject also known as operator space theory. Created in the 1980s, it nowadays is one of the most prominent areas of functional analysis, both as a field of active research and as a source of numerous important applications.
The approach taken in this book differs significantly from the standard approach used in studying operator space theory. Instead of viewing "quantized coefficients" as matrices in a fixed basis, in this book they are interpreted as finite rank operators in a fixed Hilbert space. This allows the author to replace matrix computations with algebraic techniques of module theory and tensor products, thus achieving a more invariant approach to the subject.
The book can be used by graduate students and research mathematicians interested in functional analysis and related areas of mathematics and mathematical physics. Prerequisites include standard courses in abstract algebra and functional analysis.
Graduate students and research mathematicians interested in functional analysis.
* Three basic definitions and three principal theorems
The beginning: Spaces and operators
* Preparing the stage
* Abstract operator ( = quantum) spaces
* Completely bounded operators
* The completion of abstract operator spaces
Bilinear operators, tensor products and duality *Strongly and weakly completely bounded bilinear operators
* New preparations: Classical tensor products
* Quantum tensor products
* Quantum duality
Principal theorems, revisited in earnest *Extreme flatness and the extension theorem
* Representation theorem and its gifts
* Decomposition theorem
* Returning to the Haagerup tensor product
* Miscellany: More examples, facts and applications
* Bibliography
* Index