2015; 3259 pp; hardcover
ISBN-13: 978-1-4704-1098-8
Expected publication date is December 21, 2015.
Item(s) contained in this set are available for individual sale:
SIMON/1
SIMON/2.1
SIMON/2.2
SIMON/3
SIMON/4
A Comprehensive Course in Analysis by Poincare Prize winner Barry Simon is a five-volume set that can serve as a graduate-level analysis textbook with a lot of additional bonus information, including hundreds of problems and numerous notes that extend the text and provide important historical background. Depth and breadth of exposition make this set a valuable reference source for almost all areas of classical analysis.
Researchers (mathematicians and some applied mathematicians and physicists) using analysis, professors teaching analysis at the graduate level, graduate students who need any kind of analysis in their work.
Contents for Part 1 (Real Analysis)
Preliminaries
Topological spaces
A first look at Hilbert spaces and Fourier series
Measure theory
Convexity and Banach spaces
Tempered distributions and the Fourier transform
Bonus chapter: Probability basics
Bonus chapter: Hausdorff measure and dimension
Bonus chapter: Inductive limits and ordinary distributions
Bibliography
Symbol index
Subject index
Author index
Index of capsule biographies
Contents for Part 2A (Basic Complex Analysis)
Preliminaries
The Cauchy integral theorem: Basics
Consequences of the Cauchy integral formula
Chains and the ultimate Cauchy integral theorem
More consequences of the CIT
Spaces of analytic functions
Fractional linear transformations
Conformal maps
Zeros of analytic functions and product formulae
Elliptic functions
Selected additional topics
Bibliography
Symbol index
Subject index
Author index
Index of capsule biographies
Contents for Part 2B (Advanced Complex Analysis)
Riemannian metrics and complex analysis
Some topics in analytic number theory
Ordinary differential equations in the complex domain
Asymptotic methods
Univalent functions and Loewner evolution
Nevanlinna theory
Bibliography
Symbol index
Subject index
Author index
Index of capsule biographies
Contents for Part 3 (Harmonic Analysis)
Preliminaries
Pointwise convergence almost everywhere
Harmonic and subharmonic functions
Bonus chapter: Phase space analysis
Hp spaces and boundary values of analytic functions on the unit disk
Bonus chapter: More inequalities
Bibliography
Symbol index
Subject index
Author index
Index of capsule biographies
Contents for Part 4 (Operator Theory)
Preliminaries
Operator basics
Compact operators, mainly on a Hilbert space
Orthogonal polynomials
The spectral theorem
Banach algebras
Bonus chapter: Unbounded self-adjoint operators
Bibliography
Symbol index
Subject index
Author index
Index of capsule biographies
MSRI Mathematical Circles Library, Volume: 17
2015; 176 pp; softcover
ISBN-13: 978-1-4704-2259-2
Expected publication date is December 2, 2015.
Vladimir Arnold (1937-2010) was one of the great mathematical minds of the late 20th century. He did significant work in many areas of the field. On another level, he was keeping with a strong tradition in Russian mathematics to write for and to directly teach younger students interested in mathematics. This book contains some examples of Arnold's contributions to the genre.
"Continued Fractions" takes a common enrichment topic in high school math and pulls it in directions that only a master of mathematics could envision.
"Euler Groups" treats a similar enrichment topic, but it is rarely treated with the depth and imagination lavished on it in Arnold's text. He sets it in a mathematical context, bringing to bear numerous tools of the trade and expanding the topic way beyond its usual treatment.
In "Complex Numbers" the context is physics, yet Arnold artfully extracts the mathematical aspects of the discussion in a way that students can understand long before they master the field of quantum mechanics.
"Problems for Children 5 to 15 Years Old" must be read as a collection of the author's favorite intellectual morsels. Many are not original, but all are worth thinking about, and each requires the solver to think out of his or her box. Dmitry Fuchs, a long-term friend and collaborator of Arnold, provided solutions to some of the problems. Readers are of course invited to select their own favorites and construct their own favorite solutions.
In reading these essays, one has the sensation of walking along a path that is found to ascend a mountain peak and then being shown a vista whose existence one could never suspect from the ground.
Arnold's style of exposition is unforgiving. The reader--even a professional mathematician--will find paragraphs that require hours of thought to unscramble, and he or she must have patience with the ellipses of thought and the leaps of reason. These are all part of Arnold's intent.
In the interest of fostering a greater awareness and appreciation of mathematics and its connections to other disciplines and everyday life, MSRI and the AMS are publishing books in the Mathematical Circles Library series as a service to young people, their parents and teachers, and the mathematics profession.
Undergraduate and graduate students and researchers interested in mathematics.
Student Mathematical Library, Volume: 78
2015; 221 pp; softcover
ISBN-13: 978-1-4704-2199-1
Expected publication date is December 2, 2015.
A User-Friendly Introduction to Lebesgue Measure and Integration provides a bridge between an undergraduate course in Real Analysis and a first graduate-level course in Measure Theory and Integration. The main goal of this book is to prepare students for what they may encounter in graduate school, but will be useful for many beginning graduate students as well. The book starts with the fundamentals of measure theory that are gently approached through the very concrete example of Lebesgue measure. With this approach, Lebesgue integration becomes a natural extension of Riemann integration.
Next, Lp-spaces are defined. Then the book turns to a discussion of limits, the basic idea covered in a first analysis course. The book also discusses in detail such questions as: When does a sequence of Lebesgue integrable functions converge to a Lebesgue integrable function? What does that say about the sequence of integrals? Another core idea from a first analysis course is completeness. Are these Lp-spaces complete? What exactly does that mean in this setting?
This book concludes with a brief overview of General Measures. An appendix contains suggested projects suitable for end-of-course papers or presentations.
The book is written in a very reader-friendly manner, which makes it appropriate for students of varying degrees of preparation, and the only prerequisite is an undergraduate course in Real Analysis.
Undergraduate and graduate students and researchers interested in learning and teaching real analysis.
Review of Riemann integration
Lebesgue measure
Lebesgue integration
Lp spaces
General measure theory
Ideas for projects
References
Index
Mathematical Surveys and Monographs, Volume: 208
2015; 410 pp; hardcover
ISBN-13: 978-1-4704-1737-6
Expected publication date is December 10, 2015.
Knot theory is a classical area of low-dimensional topology, directly connected with the theory of three-manifolds and smooth four-manifold topology. In recent years, the subject has undergone transformative changes thanks to its connections with a number of other mathematical disciplines, including gauge theory; representation theory and categorification; contact geometry; and the theory of pseudo-holomorphic curves.
Starting from the combinatorial point of view on knots using their grid diagrams, this book serves as an introduction to knot theory, specifically as it relates to some of the above developments. After a brief overview of the background material in the subject, the book gives a self-contained treatment of knot Floer homology from the point of view of grid diagrams. Applications include computations of the unknotting number and slice genus of torus knots (asked first in the 1960s and settled in the 1990s), and tools to study variants of knot theory in the presence of a contact structure. Additional topics are presented to prepare readers for further study in holomorphic methods in low-dimensional topology, especially Heegaard Floer homology.
The book could serve as a textbook for an advanced undergraduate or part of a graduate course in knot theory. Standard background material is sketched in the text and the appendices.
Graduate students and researchers interested in low dimensional topology and geometry.
Introduction
Knots and links in S3
Grid diagrams
Grid homology
The invariance of grid homology
The unknotting number and Ą
Basic properties of grid homology
The slice genus and Ą
The oriented skein exact sequence
Grid homologies of alternating knots
Grid homology for links
Invariants of Legendrian and transverse knots
The filtered grid complex
More on the filtered chain complex
Grid homology over the integers
The holomorphic theory
Open problems
Homological algebra
Basic theorems in knot theory
Bibliography
Index
Graduate Studies in Mathematics, Volume: 167
2015; approx. 335 pp; hardcover
ISBN-13: 978-1-4704-2555-5
Expected publication date is January 4, 2016.
This book is the testimony of a physical scientist whose language is singular perturbation analysis. Classical mathematical notions, such as matched asymptotic expansions, projections of large dynamical systems onto small center manifolds, and modulation theory of oscillations based either on multiple scales or on averaging/transformation theory, are included. The narratives of these topics are carried by physical examples: Let's say that the moment when we "see" how a mathematical pattern fits a physical problem is like "hitting the ball." Yes, we want to hit the ball. But a powerful stroke includes the follow-through. One intention of this book is to discern in the structure and/or solutions of the equations their geometric and physical content. Through analysis, we come to sense directly the shape and feel of phenomena.
The book is structured into a main text of fundamental ideas and a subtext of problems with detailed solutions. Roughly speaking, the former is the initial contact between mathematics and phenomena, and the latter emphasizes geometric and physical insight. It will be useful for mathematicians and physicists learning singular perturbation analysis of ODE and PDE boundary value problems as well as the full range of related examples and problems. Prerequisites are basic skills in analysis and a good junior/senior level undergraduate course of mathematical physics.
Graduate students and researchers interested in asymptotic methods in mathematics and physics.
What is a singular perturbation?
Asymptotic expansions
Matched asymptotic expansions
Matched asymptotic expansions in PDE
Prandtl boundary layer theory
Modulated oscillations
Modulation theory by transforming variables
Nonlinear resonance
Bibliography
Index