Wolfgang Kuhnel, University of Stuttgart, Germany
Differential Geometry: Curves - Surfaces - Manifolds
From a review for the German Edition:
"The book covers all the topics which could be necessary later for learning higher level differential geometry. The material is very carefully sorted and easy to read."
-- Mathematical Reviews
Description
Our first knowledge of differential geometry usually comes from the study of the curves and surfaces in $\mathbf{R}^3$ that arise in calculus. Here we learn about line and surface integrals, divergence and curl, and the various forms of Stokes' Theorem. If we are fortunate, we may encounter curvature and such things as the Serret-Frenet formulas.
With just the basic tools from multi-variable calculus, plus a little knowledge of linear algebra, it is possible to begin a much richer and rewarding study of differential geometry, which is what is presented in this book. It starts with an introduction to the classical differential geometry of curves and surfaces in Euclidean space, then leads to an introduction to the Riemannian geometry of more general manifolds, including a look at Einstein spaces. An important bridge from the low-dimensional theory to the general case is provided by a chapter on the intrinsic geometry of surfaces.
The first half of the book, covering the geometry of curves and surfaces, would be suitable for a one-semester undergraduate course. The local and global theories of curves and surfaces are presented, including detailed discussions of surfaces of rotation, ruled surfaces and minimal surfaces.
The second half of the book, which could be used for a more advanced course, begins with an introduction to differentiable manifolds, Riemannian structures and the curvature tensor. Two special topics are treated in detail: spaces of constant curvature and Einstein spaces.
The main goal of the book is to get started in a fairly elementary way, then to guide the reader toward more sophisticated concepts and more advanced topics. There are many examples and exercises to help along the way. Numerous figures help the reader visualize key concepts and examples, especially in lower dimensions.
Contents
Notations and prerequisites from analysis
Curves in R^n
The local theory of surfaces
The intrinsic geometry of surfaces
Riemannian manifolds
The curvature tensor
Spaces of constant curvature
Einstein spaces
Bibliography
Details:
Publisher: American Mathematical Society
Distributor: American Mathematical Society
Series: Student Mathematical Library
Publication Year: 2002
ISBN: 0-8218-2656-5
Paging: approximately 376 pp.
Binding: Softcover
Neal Madras, York University, Toronto, ON, Canada
Lectures on Monte Carlo Methods
Description
Monte Carlo methods form an experimental branch of mathematics that employs simulations driven by random number generators. These methods are often used when others fail, since they are much less sensitive to the "curse of dimensionality", which plagues deterministic methods in problems with a large number of variables. Monte Carlo methods are used in many fields: mathematics, statistics, physics, chemistry, finance, computer science, and biology, for instance.
This book is an introduction to Monte Carlo methods for anyone who would like to use these methods to study various kinds of mathematical models that arise in diverse areas of application. The book is based on lectures in a graduate course given by the author. It examines theoretical properties of Monte Carlo methods as well as practical issues concerning their computer implementation and statistical analysis. The only formal prerequisite is an undergraduate course in probability.
The book is intended to be accessible to students from a wide range of scientific backgrounds. Rather than being a detailed treatise, it covers the key topics of Monte Carlo methods to the depth necessary for a researcher to design, implement, and analyze a full Monte Carlo study of a mathematical or scientific problem. The ideas are illustrated with diverse running examples. There are exercises sprinkled throughout the text. The topics covered include computer generation of random variables, techniques and examples for variance reduction of Monte Carlo estimates, Markov chain Monte Carlo, and statistical analysis of Monte Carlo output.
Contents
Introduction
Generating random numbers
Variance reduction techniques
Markov chain Monte Carlo
Statistical analysis of simulation output
The Ising model and related examples
Bibliography
Details:
Publisher: American Mathematical Society
Distributor: American Mathematical Society
Series: Fields Institute Monographs,Volume: 16
Publication Year: 2001
ISBN: 0-8218-2978-5
Paging: 103 pp.
Binding: Hardcover
Christian Gerard, Ecole Polytechnique, Paris, France,
and Izabella Laba, University of British Columbia, Vancouver, BC, Canada
Multiparticle Quantum Scattering in Constant Magnetic Fields
Description
This monograph offers a rigorous mathematical treatment of the scattering theory of quantum N-particle systems in an external constant magnetic field. In particular, it addresses the question of asymptotic completeness, a classification of all possible trajectories of such systems according to their asymptotic behaviour. The book adopts the so-called time-dependent approach to scattering theory, which relies on a direct study of the Schrodinger unitary group for large times. The modern methods of spectral and scattering theory introduced in the 1980's and 1990's, including the Mourre theory of positive commutators, propagation estimates, and geometrical techniques, are presented and heavily used. Additionally, new methods were developed by the authors in order to deal with the (much less understood) phenomena due to the presence of the magnetic field.
The book is a good starting point for graduate students and researchers in mathematical physics who wish to move into this area of research. It includes expository material, research work previously available only in the form of journal articles, as well as some new unpublished results. The treatment of the subject is comprehensive and largely self-contained, and the text is carefully written with attention to detail.
Contents
Fundamentals
Geometrical methods I
The Mourre theory
Basic propagation estimates
Geometrical methods II
Wave operators and scattering theory
Open problems
Appendix
Bibliography
Index
Details:
Publisher: American Mathematical Society
Distributor: American Mathematical Society
Series: Mathematical Surveys and Monographs,Volume: 90
Publication Year: 2002
ISBN: 0-8218-2919-X
Paging: 242 pp.
Binding: Hardcover
Edited by: Michel L. Lapidus, University of California, Riverside, CA,
and Machiel van Frankenhuysen, Rutgers University, Piscataway, NJ
Dynamical, Spectral, and Arithmetic Zeta Functions
Description
The original zeta function was studied by Riemann as part of his investigation of the distribution of prime numbers. Other sorts of zeta functions were defined for number-theoretic purposes, such as the study of primes in arithmetic progressions. This led to the development of $L$-functions, which now have several guises. It eventually became clear that the basic construction used for number-theoretic zeta functions can also be used in other settings, such as dynamics, geometry, and spectral theory, with remarkable results.
This volume grew out of the special session on dynamical, spectral, and arithmetic zeta functions held at the annual meeting of the American Mathematical Society in San Antonio, but also includes four articles that were invited to be part of the collection. The purpose of the meeting was to bring together leading researchers, to find links and analogies between their fields, and to explore new methods. The papers discuss dynamical systems, spectral geometry on hyperbolic manifolds, trace formulas in geometry and in arithmetic, as well as computational work on the Riemann zeta function.
Each article employs techniques of zeta functions. The book unifies the application of these techniques in spectral geometry, fractal geometry, and number theory. It is a comprehensive volume, offering up-to-date research. It should be useful to both graduate students and confirmed researchers.
Contents
C.-H. Chang and D. H. Mayer -- Eigenfunctions of the transfer operators and the period functions for modular groups
C. Deninger and W. Singhof -- A note on dynamical trace formulas
C. E. Fan and J. Jorgenson -- Small eigenvalues and Hausdorff dimension of sequences of hyperbolic three-manifolds
A. Fel'shtyn -- Dynamical zeta functions and asymptotic expansions in Nielsen theory
W. F. Galway -- Computing the Riemann zeta function by numerical quadrature
S. Haran -- On Riemann's zeta function
M. L. Lapidus and M. van Frankenhuysen -- A prime orbit theorem for self-similar flows
A. M. Odlyzko -- The $10^{22}$-nd zero of the Riemann zeta functions
P. Perry -- Spectral theory, dynamics, and Selberg's zeta function for Kleinian groups
C. Soule -- On zeroes of automorphic $L$-functions
H. M. Stark and A. A. Terras -- Artin L-functions of graph coverings
Details:
Publisher: American Mathematical Society
Distributor: American Mathematical Society
Series: Contemporary Mathematics,Volume: 290
Publication Year: 2001
ISBN: 0-8218-2079-6
Paging: approximately 0 pp.
Binding: Softcover