Graduate Studies in Mathematics, Volume: 95
2008; approx. 393 pp; hardcover
ISBN-10: 0-8218-4630-2
ISBN-13: 978-0-8218-4630-8
Expected publication date is September 13, 2008.
This book provides a comprehensive treatment of quantum mechanics from
a mathematics perspective and is accessible to mathematicians starting
with second-year graduate students. It addition to traditional topics,
like classical mechanics, mathematical foundations of quantum mechanics,
quantization, and the Schrodinger equation, this book gives a mathematical
treatment of systems of identical particles with spin, and it introduces
the reader to functional methods in quantum mechanics. This includes the
Feynman path integral approach to quantum mechanics, integration in functional
spaces, the relation between Feynman and Wiener integrals, Gaussian integration
and regularized determinants of differential operators, fermion systems
and integration over anticommuting (Grassmann) variables, supersymmetry
and localization in loop spaces, and supersymmetric derivation of the Atiyah-Singer
formula for the index of the Dirac operator. Prior to this book, mathematicians
could find these topics only in physics textbooks and in specialized literature.
This book is written in a concise style with careful attention to precise mathematics formulation of methods and results. Numerous problems, from routine to advanced, help the reader to master the subject. In addition to providing a fundamental knowledge of quantum mechanics, this book could also serve as a bridge for studying more advanced topics in quantum physics, among them quantum field theory.
Prerequisites include standard first-year graduate courses covering linear and abstract algebra, topology and geometry, and real and complex analysis.
Graduate students and research mathematicians interested in mathematical aspects of quantum mechanics.
Foundations
Classical mechanics
Basic principles of quantum mechanics
Schrodinger equation
Spin and identical particles
Functional methods and supersymmetry
Path integral formulation of quantum mechanics
Integration in functional spaces
Fermion systems
Supersymmetry
Bibliography
Index
Graduate Studies in Mathematics,Volume: 96
2008; approx. 363 pp; hardcover
ISBN-10: 0-8218-4684-1
ISBN-13: 978-0-8218-4684-1
Expected publication date is September 19, 2008.
This book concentrates on the basic facts and ideas of the modern theory
of linear elliptic and parabolic equations in Sobolev spaces.
The main areas covered in this book are the first boundary-value problem
for elliptic equations and the Cauchy problem for parabolic equations.
In addition, other boundary-value problems such as the Neumann or oblique
derivative problems are briefly covered. As is natural for a textbook,
the main emphasis is on organizing well-known ideas in a self-contained
exposition. Among the topics included that are not usually covered in a
textbook are a relatively recent development concerning equations with
textsf{VMO} coefficients and the study of parabolic equations with coefficients
measurable only with respect to the time variable. There are numerous exercises
which help the reader better understand the material.
After going through the book, the reader will have a good understanding
of results available in the modern theory of partial differential equations
and the technique used to obtain them. Prerequesites are basics of measure
theory, the theory of L_p spaces, and the Fourier transform.
Graduate students and research mathematicians interested in partial differential equations.
Second-order elliptic equations in W^{2}_{2}(mathbb{R}^{d})
Second-order parabolic equations in W^{1,k}_{2}(mathbb{R}^{d+1})
Some tools from real analysis
Basic mathcal{L}_{p}-estimates for parabolic and elliptic equations
Parabolic and elliptic equations in W^{1,k}_{p} and W^{k}_{p}
Equations with VMO coefficients
Parabolic equations with VMO coefficients in spaces with mixed norms
Second-order elliptic equations in W^{2}_{p}(Omega)
Second-order elliptic equations in W^{k}_{p}(Omega)
Sobolev embedding theorems for W^{k}_{p}(Omega)
Second-order elliptic equations Lu-lambda u=f with lambda small
Fourier transform and elliptic operators
Elliptic operators and the spaces H^{gamma}_{p}
Bibliography
Index
Graduate Studies in Mathematics, Volume: 97
2008; approx. 477 pp; hardcover
ISBN-10: 0-8218-4479-2
ISBN-13: 978-0-8218-4479-3
Expected publication date is September 19, 2008.
Perhaps uniquely among mathematical topics, complex analysis presents the student with the opportunity to learn a thoroughly developed subject that is rich in both theory and applications. Even in an introductory course, the theorems and techniques can have elegant formulations. But for any of these profound results, the student is often left asking: What does it really mean? Where does it come from?
In Complex Made Simple, David Ullrich shows the student how to think like an analyst. In many cases, results are discovered or derived, with an explanation of how the students might have found the theorem on their own. Ullrich explains why a proof works. He will also, sometimes, explain why a tempting idea does not work.
Complex Made Simple looks at the Dirichlet problem for harmonic functions twice: once using the Poisson integral for the unit disk and again in an informal section on Brownian motion, where the reader can understand intuitively how the Dirichlet problem works for general domains. Ullrich also takes considerable care to discuss the modular group, modular function, and covering maps, which become important ingredients in his modern treatment of the often-overlooked original proof of the Big Picard Theorem.
This book is suitable for a first-year course in complex analysis. The exposition is aimed directly at the students, with plenty of details included. The prerequisite is a good course in advanced calculus or undergraduate analysis.
Undergraduate and graduate students interested in complex analysis.
Part 1. Complex made simple
Differentiability and Cauchy-Riemann equations
Power series
Preliminary results on holomorphic functions
Elementary results on holomorphic functions
Logarithms, winding numbers and Cauchy's theorem
Counting zeroes and the open mapping theorem
Euler's formula for sin(z)
Inverses of holomorphic maps
Conformal mappings
Normal families and the Riemann mapping theorem
Harmonic functions
Simply connected open sets
Runge's theorem and the Mittag-Leffler theorem
The Weierstrass factorization theorem
Caratheodory's theorem
More on mathrm{Aut}(mathbb{D})
Analytic continuation
Orientation
The modular function
Preliminaries for the Picard theorems
The Picard theorems
Part 2. Further results
Abel's theorem
More on Brownian motion
More on the maximum modulus theorem
The Gamma function
Universal covering spaces
Cauchy's theorem for non-holomorphic functions
Harmonic conjugates
Part 3. Appendices
Complex numbers
Complex numbers, continued
Sin, cos and exp
Metric spaces
Convexity
Four counterexamples
The Cauchy-Riemann equations revisited
References
Index of notations
Index
Colloquium Publications, Volume: 5
1918; 306 pp; softcover
ISBN-10: 0-8218-4642-6
ISBN-13: 978-0-8218-4642-1
Expected publication date is July 14, 2008.
The 1916 colloquium of the American Mathematical Society was held as part
of the summer meeting that took place in Boston. Two sets of lectures were
presented: Functionals and their Applications. Selected Topics, including
Integral Equations, by G. C. Evans, and Analysis Situs, by Oswald Veblen.
The lectures by Evans are devoted to functionals and their applications. By a functional the author means a function on an infinite-dimensional space, usually a space of functions, or of curves on the plane or in 3-space, etc. The first lecture deals with general considerations of functionals (continuity, derivatives, variational equations, etc.). The main topic of the second lecture is the study of complex-valued functionals, such as integrals of complex functions in several variables. The third lecture is devoted to the study of what is called implicit functional equations. This study requires, in particular, the development of the notion of a Frechet differential, which is also discussed in this lecture. The fourth lecture contains generalizations of the Bocher approach to the treatment of the Laplace equation, where a harmonic function is characterized as a function with no flux (Evans' terminology) through every circle on the plane. Finally, the fifth lecture gives an account of various generalizations of the theory of integral equations.
Analysis situs is the name used by Poincare when he was creating, at the end of the 19th century, the area of mathematics known today as topology. Veblen's lectures, forming the second part of the book, contain what is probably the first text where Poincare's results and ideas were summarized, and an attempt to systematically present this difficult new area of mathematics was made.
This is how S. Lefschetz had described, in his 1924 review of the book, the experience of "a beginner attracted by the fascinating and difficult field of analysis situs":
"Difficult reasonings beset him at every step, an unfriendly notation did not help matters, to all of which must be added, most baffling of all, the breakdown of geometric intuition precisely when most needed. No royal road can be created through this dense forest, but a good and thoroughgoing treatment of fundamentals, notation, terminology, may smooth the path somewhat. And this and much more we find supplied by Veblen's Lectures."
Of the two streams of topology existing at that time, point set topology and combinatorial topology, it is the latter to which Veblen's book is almost totally devoted. The first four chapters present, in detail, the notion and properties (introduced by Poincare) of the incidence matrix of a cell decomposition of a manifold. The main goal of the author is to show how to reproduce main topological invariants of a manifold and their relations in terms of the incidence matrix.
The (last) fifth chapter contains what Lefschetz called "an excellent summary of several important questions: homotopy and isotopy, theory of the indicatrix, a fairly ample treatment of the group of a manifold, finally a bird's eye view of what is known and not known (mostly the latter) on three dimensional manifolds."
Graduate students and research mathematicians interested in functionals.
G. C. Evans -- Functionals and their applications. Selected topics, including integral equations
O. Veblen -- Analysis situs