MSRI Mathematical Circles Library, Volume: 1
2008; approx. 333 pp; softcover
ISBN-13: 978-0-8218-4683-4
Expected publication date is December 12, 2008.
Many mathematicians have been drawn to mathematics through their experience with math circles: extracurricular programs exposing teenage students to advanced mathematical topics and a myriad of problem-solving techniques and inspiring in them a lifelong love for mathematics. Founded in 1998, the Berkeley Math Circle (BMC) is a pioneering model of a U.S. math circle, aspiring to prepare our best young minds for their future roles as mathematics leaders. Over the last decade, 50 instructors--from university professors to high school teachers to business tycoons--have shared their passion for mathematics by delivering more than 320 BMC sessions full of mathematical challenges and wonders.
Based on a dozen of these sessions, this book encompasses a wide variety of enticing mathematical topics: from inversion in the plane to circle geometry; from combinatorics to Rubik's cube and abstract algebra; from number theory to mass point theory; from complex numbers to game theory via invariants and monovariants. The treatments of these subjects encompass every significant method of proof and emphasize ways of thinking and reasoning via 100 problem-solving techniques. Also featured are 300 problems, ranging from beginner to intermediate level, with occasional peaks of advanced problems and even some open questions.
The book presents possible paths to studying mathematics and inevitably falling in love with it, via teaching two important skills: thinking creatively while still "obeying the rules," and making connections between problems, ideas, and theories. The book encourages you to apply the newly acquired knowledge to problems and guides you along the way, but rarely gives you ready answers. "Learning from our own mistakes" often occurs through discussions of non-proofs and common problem-solving pitfalls. The reader has to commit to mastering the new theories and techniques by "getting your hands dirty" with the problems, going back and reviewing necessary problem-solving techniques and theory, and persistently moving forward in the book. The mathematical world is huge: you'll never know everything, but you'll learn where to find things, how to connect and use them. The rewards will be substantial.
High school students, high school teachers, and undergraduates interested in problem solving and introductions to mathematical methods.
Inversion in the plane. Part I
Combinatorics. Part I
Rubik's cube. Part I
Number theory. Part I: Remainders, divisibility, congruences and more
A few words about proofs. Part I
Mathematical induction
Mass point geometry
More on proofs. Part II
Complex numbers. Part I
Stomp. Games with invariants
Favorite problems at BMC. Part I: Circle geometry
Monovariants. Part I: Mansion walks and frog migrations
Epilogue
Symbols and notation
Abbreviations
Biographical data
Bibliography
Credits
Index
American Mathematical Society Translations--Series 2, Volume: 225
Advances in the Mathematical Sciences
2008; approx. 298 pp; hardcover
ISBN-13: 978-0-8218-4738-1
Expected publication date is December 20, 2008.
This volume is dedicated to the eightieth birthday of Professor M. Sh. Birman. It contains original articles in spectral and scattering theory of differential operators, in particular, Schrodinger operators, and in homogenization theory. All articles are written by members of M. Sh. Birman's research group who are affiliated with different universities all over the world. A specific feature of the majority of the papers is a combination of traditional methods with new modern ideas.
Readership
Graduate students and research mathematicians interested theory of differential operators.
Table of Contents
M. Solomyak and T. Suslina -- On the scientific work of M. Sh. Birman in 1998-2007
T. Suslina and D. Yafaev -- Continuation of the list of publications of M. Sh. Birman
M. Sh. Birman -- Perturbations of the continuous spectrum of a singular elliptic operator by varying the boundary and the boundary conditions
V. S. Buslaev and S. B. Levin -- Asymptotic behavior of the eigenfunctions of many-particle Schrodinger operator. I. One-dimensional particles
M. N. Demchenko and N. D. Filonov -- Spectral asymptotics of the Maxwell operator on Lipschitz manifolds with boundary
R. L. Frank and A. Laptev -- Spectral inequalities for Schrodinger operators with surface potentials
L. Friedlander and M. Solomyak -- On the spectrum of the Dirichlet Laplacian in a narrow infinite strip
A. Laptev and A. V. Sobolev -- Hardy inequalities for simply connected planar domains
E. Korotyaev and A. Kutsenko -- Lyapunov functions of periodic matrix-valued Jacobi operators
A. Pushnitski -- The spectral flow, the Fredholm index, and the spectral shift function
G. Raikov -- On the spectrum of a translationally invariant Pauli operator
G. Rozenblum and A. V. Sobolev -- Discrete spectrum distribution of the Landau operator perturbed by an expanding electric potential
Y. Safarov -- On the comparison of the Dirichlet and Neumann counting functions
O. Safronov -- Absolutely continuous spectrum of multi-dimensional Schrodinger operators with slowly decaying potentials
R. Shterenberg -- On discrete spectrum of the perturbed periodic magnetic Schrodinger operator with degenerate lower edge of the spectrum
T. A. Suslina -- Homogenization of periodic second order differential operators including first order terms
T. Weidl -- Improved Berezin-Li-Yau inequalities with a remainder term
D. R. Yafaev -- Spectral and scattering theory of fourth order differential operators
Mathematical Surveys and Monographs, Volume: 151
2008; approx. 284 pp; hardcover
ISBN-13: 978-0-8218-4495-3
Expected publication date is December 25, 2008.
The three-dimensional Heisenberg group, being the simplest non-commutative Lie group, appears prominently in various applications of mathematics. The goal of this book is to present basic geometric and algebraic properties of the Heisenberg group and its relation to other important mathematical structures (the skew field of quaternions, symplectic structures, and representations) and to describe some of its applications. In particular, the authors address such subjects as well as signal analysis and processing, geometric optics, and quantization. In each case, the authors present necessary details of the applied topic being considered.
With no prerequisites beyond the standard mathematical curriculum, this book manages to encompass a large variety of topics being easily accessible in its fundamentals. It can be useful to students and researchers working in mathematics and in applied mathematics.
Graduate students and research mathematicians interested in the use of analysis on Heisenberg groups to various problems in pure and applied mathematics.
The skew field of quaternions
Elements of the geometry of S^3, Hopf bundles and spin representations
Internal variables of singularity free vector fields in a Euclidean space
Isomorphism classes, Chern classes and homotopy classes of singularity free vector fields in three-space
Heisenberg algebras, Heisenberg groups, Minkowski metrics, Jordan algebras
and SL(2,\mathbb{C})
The Heisenberg group and natural C*-algebras of a vector field in 3-space
The Schrodinger representation and the metaplectic representation
The Heisenberg group-A basic geometric background of signal analysis and geometric optics
Quantization of quadratic polynomials
Field theoretic Weyl quantization of a vector field in 3-space
Thermodynamics, geometry and the Heisenberg group by Serge Preston
Bibliography
Index
Translations of Mathematical Monographs, Volume: 237
2008; approx. 159 pp; hardcover
ISBN-13: 978-0-8218-3947-8
Expected publication date is December 19, 2008.
In the early 1980's topologists and geometers for the first time came across
unfamiliar words like C^*-algebras and von Neumann algebras through the
discovery of new knot invariants (by V. F. R. Jones) or through a remarkable
result on the relationship between characteristic classes of foliations
and the types of certain von Neumann algebras. During the following two
decades, a great deal of progress was achieved in studying the interaction
between geometry and analysis, in particular in noncommutative geometry
and mathematical physics. The present book provides an overview of operator
algebra theory and an introduction to basic tools used in noncommutative
geometry. The book concludes with applications of operator algebras to
Atiyah-Singer type index theorems. The purpose of the book is to convey
an outline and general idea of operator algebra theory, to some extent
focusing on examples.
The book is aimed at researchers and graduate students working in differential topology, differential geometry, and global analysis who are interested in learning about operator algebras.
Graduate students and research mathematicians interested in applications of functional analysis to geometry and topology.
C^*-algebras
K-theory
KK-theory
Von Neumann algebras
Cyclic cohomology
Quantizations and index theory
Foliation index theorems
References
Index
Contemporary Mathematics, Volume: 473
2008; 180 pp; softcover
ISBN-13: 978-0-8218-4357-4
Expected publication date is December 3, 2008.
This volume is based on a thematic program on the Gross-Pitaevskii equation, which was held at the Wolfgang Pauli Institute in Vienna in 2006. The program consisted of two workshops and a one-week Summer School.
The Gross-Pitaevskii equation, an example of a defocusing nonlinear Schrodinger equation, is a model for phenomena such as the Bose-Einstein condensation of ultra cold atomic gases, the superfluidity of Helium II, or the "dark solitons" of Nonlinear Optics. Many interesting and difficult mathematical questions associated with the Gross-Pitaevskii equation, linked for instance to the nontrivial boundary conditions at infinity, arise naturally from its modeling aspects.
The articles in this volume review some of the recent developments in the theory of the Gross-Pitaevskii equation. In particular the following aspects are considered: modeling of superfluidity and Bose-Einstein condensation, the Cauchy problem, the semi-classical limit, scattering theory, existence and properties of coherent traveling structures, and numerical simulations.
Graduate students and research mathematicians interested in various aspects of nonlinear equations and their use in mathematical physics.
W. Bao -- Analysis and efficient computation for the dynamics of two-component Bose-Einstein condensates
N. G. Berloff -- Quantised vortices, travelling coherent structures and superfluid turbulence
F. Bethuel, P. Gravejat, and J.-C. Saut -- Existence and properties of travelling waves for the Gross-Pitaevskii equation
R. Carles -- On the semi-classical limit for the nonlinear Schrodinger equation
P. Gerard -- The Gross-Pitaevskii equation in the energy space
K. Nakanishi -- Scattering theory for the Gross-Pitaevskii equation
D. E. Pelinovsky and P. Kevrekidis -- Periodic oscillations of dark solitons in parabolic potentials
Graduate Studies in Mathematics, Volume: 98
2008; approx. 401 pp; hardcover
ISBN-13: 978-0-8218-4700-8
Expected publication date is January 2, 2009.
An emerging field of discrete differential geometry aims at the development of discrete equivalents of notions and methods of classical differential geometry. The latter appears as a limit of a refinement of the discretization. Current interest in discrete differential geometry derives not only from its importance in pure mathematics but also from its applications in computer graphics, theoretical physics, architecture, and numerics. Rather unexpectedly, the very basic structures of discrete differential geometry turn out to be related to the theory of integrable systems. One of the main goals of this book is to reveal this integrable structure of discrete differential geometry.
For a given smooth geometry one can suggest many different discretizations. Which one is the best? This book answers this question by providing fundamental discretization principles and applying them to numerous concrete problems. It turns out that intelligent theoretical discretizations are distinguished also by their good performance in applications.
The intended audience of this book is threefold. It is a textbook on discrete differential geometry and integrable systems suitable for a one semester graduate course. On the other hand, it is addressed to specialists in geometry and mathematical physics. It reflects the recent progress in discrete differential geometry and contains many original results. The third group of readers at which this book is targeted is formed by specialists in geometry processing, computer graphics, architectural design, numerical simulations, and animation. They may find here answers to the question "How do we discretize differential geometry?" arising in their specific field.
Prerequisites for reading this book include standard undergraduate background (calculus and linear algebra). No knowledge of differential geometry is expected, although some familiarity with curves and surfaces can be helpful.
Graduate students and research mathematicians interested in discrete differential geometry and its applications.
Classical differential geometry
Discretization principles. Multidimensional nets
Discretization principles. Nets in quadrics
Special classes of discrete surfaces
Approximation
Consistency as integrability
Discrete complex analysis. Linear theory
Discrete complex analysis. Integrable circle patterns
Foundations
Solutions of selected exercises
Bibliography
Notations
Index