Part of New Mathematical Monographs
PUBLICATION PLANNED FOR: May 2021AVAILABILITY:
Not yet published - available from May 2021
FORMAT: HardbackISBN: 9781108831444
The long-standing Kervaire invariant problem in homotopy theory arose from geometric and differential topology in the 1960s and was quickly recognised as one of the most important problems in the field. In 2009 the authors of this book announced a solution to the problem, which was published to wide acclaim in a landmark Annals of Mathematics paper. The proof is long and involved, using many sophisticated tools of modern (equivariant) stable homotopy theory that are unfamiliar to non-experts. This book presents the proof together with a full development of all the background material to make it accessible to a graduate student with an elementary algebraic topology knowledge. There are explicit examples of constructions used in solving the problem. Also featuring a motivating history of the problem and numerous conceptual and expository improvements on the proof, this is the definitive account of the resolution of the Kervaire invariant problem.
A complete account of the authors' milestone solution to the Kervaire invariant problem in homotopy theory
Includes background results in stable homotopy theory that have not previously appeared in the literature
Accessible to graduate students with an elementary knowledge of algebraic topology
1. Introduction
Part I. The Categorical Tool Box:
2. Some Categorical Tools
3. Enriched Category Theory
4. Quillen's Theory of Model Categories
5. Model Category Theory Since Quillen
6. Bousfield Localization
Part II. Setting Up Equivariant Stable Homotopy Theory:
7. Spectra and Stable Homotopy Theory
8. Equivariant Homotopy Theory
9. Orthogonal G-spectra
10. Multiplicative Properties of G-spectra
Part III. Proving the Kervaire Invariant Theorem:
11. The Slice Filtration and Slice Spectral Sequence
12. The Construction and Properties of $MU_{\R}$
13. The Proofs of the Gap, Periodicity and Detection Theorems
References
Table of Notation
Index.
Part of Cambridge Texts in Applied Mathematics
PUBLICATION PLANNED FOR: June 2021
AVAILABILITY: Not yet published - available from June 2021
FORMAT: HardbackISBN: 9781108832618
FORMAT: PaperbackISBN: 9781108959728
The study of complex variables is beautiful from a purely mathematical point of view, and very useful for solving a wide array of problems arising in applications. This introduction to complex variables, suitable as a text for a one-semester course, has been written for undergraduate students in applied mathematics, science, and engineering. Based on the authors' extensive teaching experience, it covers topics of keen interest to these students, including ordinary differential equations, as well as Fourier and Laplace transform methods for solving partial differential equations arising in physical applications. Many worked examples, applications, and exercises are included. With this foundation, students can progress beyond the standard course and explore a range of additional topics, including generalized Cauchy theorem, Painleve equations, computational methods, and conformal mapping with circular arcs. Advanced topics are labeled with an asterisk and can be included in the syllabus or form the basis for challenging student projects.
Contains numerous examples from a variety of engineering and physical applications
Includes answers to selected problems
Results are often motivated heuristically, before being proven rigorously, to increase the student's understanding of the relevant result
1. Complex numbers and elementary functions
2. Analytic functions and integration
3. Sequences, series and singularities of complex functions
4. Residue calculus and applications of contour integration
5. Conformal mappings and applications
Appendix. Answers to selected odd-numbered exercises
References
Index.
PUBLICATION PLANNED FOR: May 2021
AVAILABILITY: Not yet published - available from May 2021
FORMAT: Multiple copy packISBN: 9781108709439
This two-volume work traces the development of series and products from 1380 to 2000 through the interconnected concepts and results of hundreds of unsung as well as celebrated mathematicians. Some chapters deal with the work of primarily one mathematician on a pivotal topic, and other chapters chronicle the progress over time of a given topic. This updated second edition of Sources in the Development of Mathematics adds extensive context, detail, and primary source material, with many sections rewritten to more clearly reveal the significance of key developments and arguments. Volume 1, accessible even to advanced undergraduate students, discusses the development of the fundamental methods in series and products that do not employ complex analytic methods or sophisticated machinery. Volume 2 deals with more recent and advanced results, such as Nevanlinna theory and deBranges' work.
New edition provides clarifying details from the original papers, additional mathematical context, more results, and includes nearly 500 new bibliographic entries
Traces the origins of many ideas in areas of interest to applied mathematicians, scientists, and engineers
Includes in-depth presentations and explanations of original proofs of important mathematical results by several hundred mathematicians, including at least sixty papers of Euler
Organized by topic, chapters delineate the advancement of the topic over time, with original arguments and details, providing mathematical insight into the topic
330 exercises; hundreds of detailed proofs and examples within the text
Part of London Mathematical Society Lecture Note Series
PUBLICATION PLANNED FOR: June 2021
AVAILABILITY: Not yet published - available from June 2021
FORMAT: PaperbackISBN: 9781108794428
From Backlund to Darboux, this monograph presents a comprehensive journey through the transformation theory of constrained Willmore surfaces, a topic of great importance in modern differential geometry and, in particular, in the field of integrable systems in Riemannian geometry. The first book on this topic, it discusses in detail a spectral deformation, Backlund transformations and Darboux transformations, and proves that all these transformations preserve the existence of a conserved quantity, defining, in particular, transformations within the class of constant mean curvature surfaces in 3-dimensional space-forms, with, furthermore, preservation of both the space-form and the mean curvature, and bridging the gap between different approaches to the subject, classical and modern. Clearly written with extensive references, chapter introductions and self-contained accounts of the core topics, it is suitable for newcomers to the theory of constrained Wilmore surfaces. Many detailed computations and new results unavailable elsewhere in the literature make it also an appealing reference for experts.
Contains many detailed computations and new results unavailable elsewhere in the literature
Suitable for newcomers to the theory of constrained Willmore surfaces, including graduate students
Clearly written and includes extensive references and a reader-friendly abstract to each chapter
Introduction
1. A bundle approach to conformal surfaces in space-forms
2. The mean curvature sphere congruence
3. Surfaces under change of flat metric connection
4. Willmore surfaces
5. The Euler?Lagrange constrained Willmore surface equation
6. Transformations of generalized harmonic bundles and constrained Willmore surfaces
7. Constrained Willmore surfaces with a conserved quantity
8. Constrained Willmore surfaces and the isothermic surface condition
9. The special case of surfaces in 4-space
Appendix A. Hopf differential and umbilics
Appendix B. Twisted vs. untwisted Backlund transformation parameters
References
Index.
Series: Advances in Analysis and Geometry, 3
De Gruyter | 2021
The book covers the latest research in the areas of mathematics that deal the properties of partial differential equations and stochastic processes on spaces in connection with the geometry of the underlying space. Written by experts in the field, this book is a valuable tool for the advanced mathematician.
Current research and surveys on the topic of PDEs.
Written by leading researchers in the field.
Alexander Grigorfyan, University of Bielefeld, Germany and Yuhua Sun, Nankai University, China.
Mathematics Analysis
Mathematics Differential Equations and Dynamical Systems
Language: English
Format: 24.0 x 17.0 cm
Pages Roman: VIII
Pages Arabic: 518
Illustrations BW: 30
Publisher: De Gruyter
Year: 2021
Researches in Mathematics, Advanced Graduate Students interested in the field.
FORMATS
Hardcover
ISBN: 978-3-11-070063-3
Published: 18 Jan 2021
EMS Monographs in Mathematics
ISBN print 978-3-03719-211-5, ISBN online 978-3-03719-711-0
DOI 10.4171/211
October 2020, 374 pages, hardcover, 16.5 x 23.5 cm.
Many partial differential equations (PDEs) arising in physics, such as the nonlinear wave equation and the Schrodinger equation, can be viewed as infinite-dimensional Hamiltonian systems. In the last thirty years, several existence results of time quasi-periodic solutions have been proved adopting a "dynamical systems" point of view. Most of them deal with equations in one space dimension, whereas for multidimensional PDEs a satisfactory picture is still under construction.
An updated introduction to the now rich subject of KAM theory for PDEs is provided in the first part of this research monograph. We then focus on the nonlinear wave equation, endowed with periodic boundary conditions. The main result of the monograph proves the bifurcation of small amplitude finite-dimensional invariant tori for this equation, in any space dimension. This is a difficult small divisor problem due to complex resonance phenomena between the normal mode frequencies of oscillations. The proof requires various mathematical methods, ranging from Nash?Moser and KAM theory to reduction techniques in Hamiltonian dynamics and multiscale analysis for quasi-periodic linear operators, which are presented in a systematic and self-contained way. Some of the techniques introduced in this monograph have deep connections with those used in Anderson localization theory.
This book will be useful to researchers who are interested in small divisor problems, particularly in the setting of Hamiltonian PDEs, and who wish to get acquainted with recent developments in the field.
Keywords: Infinite-dimensional Hamiltonian systems, nonlinear wave equation, KAM for PDEs, quasi-periodic solutions and invariant tori, small divisors, Nash?Moser theory, multiscale analysis