The theory of bounded cohomology, introduced by Gromov in the late 1980s, has had powerful applications in geometric group theory and the geometry and topology of manifolds, and has been the topic of active research continuing to this day. This monograph provides a unified, self-contained introduction to the theory and its applications, making it accessible to a student who has completed a first course in algebraic topology and manifold theory. The book can be used as a source for research projects for master's students, as a thorough introduction to the field for graduate students, and as a valuable landmark text for researchers, providing both the details of the theory of bounded cohomology and links of the theory to other closely related areas.
The first part of the book is devoted to settling the fundamental definitions of the theory, and to proving some of the (by now classical) results on low-dimensional bounded cohomology and on bounded cohomology of topological spaces. The second part describes applications of the theory to the study of the simplicial volume of manifolds, to the classification of circle actions, to the analysis of maximal representations of surface groups, and to the study of flat vector bundles with a particular emphasis on the possible use of bounded cohomology in relation with the Chern conjecture. Each chapter ends with a discussion of further reading that puts the presented results in a broader context.
Mathematical Surveys and Monographs,Volume: 227
2017; 193 pp Hardcover
Print ISBN: 978-1-4704-4146-3
Graduate students and researchers interested in geometry and topology.
The author manages a near perfect equilibrium between necessary technicalities (always well motivated) and geometric intuition, leading the readers from the first simple definition to the most striking applications of the theory in 13 very pleasant chapters. This book can serve as an ideal textbook for a graduate topics course on the subject and become the much-needed standard reference on Gromov's beautiful theory.
-- Michelle Bucher
2017; 142 pp; Softcover
Print ISBN: 978-1-4704-3674-2
When origami met the worlds of design and engineering, both fields embraced the ancient art form, using its principles and practices to discover new problems and to generate inventive solutions.
This book demonstrates the potential of folding to improve the way things work, simplify how products are produced, and make possible new objects otherwise impossible. The solar collector, the felt stool, and the surgery tool have all been influenced in some way by folding paper. The example section is organized to show the folded figure next to the product prototype that was inspired by that work of origami. We have included models made from an array of materials over a range of sizes. This includes everything from a microscopic mechanism to huge solar panels designed to unfold in outer space. Most entries are at the prototype phase?meaning that physical hardware has been built to demonstrate the concept, but that the examples are not necessarily available commercially.
Y Origami also includes brief learning activities related to paper folding, such as a discussion of Euler's formula, angular measurements, and developable surfaces, along with more advanced topics. Throughout the book many diagrams and photographs illustrate the advancing concepts and methods of origami as an art form and a problem-solving strategy.
Anyone interested in origami and its applications to design and engineering.
Pure and Applied Undergraduate Texts, Volume: 29
2017; 369 pp; Hardcover
Print ISBN: 978-1-4704-4062-6
Spaces is a modern introduction to real analysis at the advanced undergraduate level. It is forward-looking in the sense that it first and foremost aims to provide students with the concepts and techniques they need in order to follow more advanced courses in mathematical analysis and neighboring fields. The only prerequisites are a solid understanding of calculus and linear algebra. Two introductory chapters will help students with the transition from computation-based calculus to theory-based analysis.
The main topics covered are metric spaces, spaces of continuous functions, normed spaces, differentiation in normed spaces, measure and integration theory, and Fourier series. Although some of the topics are more advanced than what is usually found in books of this level, care is taken to present the material in a way that is suitable for the intended audience: concepts are carefully introduced and motivated, and proofs are presented in full detail. Applications to differential equations and Fourier analysis are used to illustrate the power of the theory, and exercises of all levels from routine to real challenges help students develop their skills and understanding. The text has been tested in classes at the University of Oslo over a number of years.
Undergraduate and graduate students interested in real analysis.
Colloquium Publications, Volume: 64
2017; 391 pp; Hardcover
Print ISBN: 978-1-4704-1944-8
Modular forms and Jacobi forms play a central role in many areas of mathematics. Over the last 10?15 years, this theory has been extended to certain non-holomorphic functions, the so-called “harmonic Maass forms”. The first glimpses of this theory appeared in Ramanujan's enigmatic last letter to G. H. Hardy written from his deathbed. Ramanujan discovered functions he called “mock theta functions” which over eighty years later were recognized as pieces of harmonic Maass forms. This book contains the essential features of the theory of harmonic Maass forms and mock modular forms, together with a wide variety of applications to algebraic number theory, combinatorics, elliptic curves, mathematical physics, quantum modular forms, and representation theory.
Graduate students and researchers interested in modular forms and their use in number theory.
Graduate Studies in Mathematics, Volume: 187
2017; 368 pp Hardcover
Print ISBN: 978-1-4704-2950-8
This book applies infinite-dimensional manifold theory to the Morse theory of closed geodesics in a Riemannian manifold. It then describes the problems encountered when extending this theory to maps from surfaces instead of curves. It treats critical point theory for closed parametrized minimal surfaces in a compact Riemannian manifold, establishing Morse inequalities for perturbed versions of the energy function on the mapping space. It studies the bubbling which occurs when the perturbation is turned off, together with applications to the existence of closed minimal surfaces. The Morse-Sard theorem is used to develop transversality theory for both closed geodesics and closed minimal surfaces.
This book is based on lecture notes for graduate courses on “Topics in Differential Geometry”, taught by the author over several years. The reader is assumed to have taken basic graduate courses in differential geometry and algebraic topology.
Graduate students and researchers interested in differential geometry.