96pp Apr 2017
ISBN: 978-981-3207-07-3 (hardcover)
The Geometric Theory of Foliations is one of the fields in Mathematics that gathers several distinct domains: Topology, Dynamical Systems, Differential Topology and Geometry, among others. Its great development has allowed a better comprehension of several phenomena of mathematical and physical nature. Our book contains material dating from the origins of the theory of foliations, from the original works of C Ehresmann and G Reeb, up till modern developments.
In a suitable choice of topics we are able to cover material in a coherent way bringing the reader to the heart of recent results in the field. A number of theorems, nowadays considered to be classical, like the Reeb Stability Theorem, Haefliger's Theorem, and Novikov Compact leaf Theorem, are proved in the text. The stability theorem of Thurston and the compact leaf theorem of Plante are also thoroughly proved. Nevertheless, these notes are introductory and cover only a minor part of the basic aspects of the rich theory of foliations.
Preface
Preliminaries
Plane Fields and Foliations
Topology of the Leaves
Holonomy and Stability
Haefliger's Theorem
Novikov's Compact Leaf
Rank of 3-Manifolds
Tischler's Theorem
Plante's Compact Leaf Theorem
Currents, Distributions, Foliation Cycles and Transverse Measures
Foliation Cycles: A Homological Proof of Novikov's Compact Leaf Theorem
Appendix A: Structure of Codimension One Foliations: Dippolito's Theory
Readership: Students and researchers in foliations.
252pp May 2017
ISBN: 978-1-78634-308-6 (hardcover)
ISBN: 978-1-78634-309-3 (softcover)
This book provides an in-depth and rigorous study of the Wigner transform and its variants. They are presented first within a context of a general mathematical framework, and then through applications to quantum mechanics. The Wigner transform was introduced by Eugene Wigner in 1932 as a probability quasi-distribution which allows expression of quantum mechanical expectation values in the same form as the averages of classical statistical mechanics. It is also used in signal processing as a transform in time-frequency analysis, closely related to the windowed Gabor transform.
Written for advanced-level students and professors in mathematics and mathematical physics, it is designed as a complete textbook course providing analysis on the most important research on the subject to date. Due to the advanced nature of the content, it is also suitable for research mathematicians, engineers and chemists active in the field.
General Mathematical Framework:
Phase Space Translations and Reflections
The Cross-Wigner Transform
The Cross-Ambiguity Function
Weyl Operators
Symplectic Covariance
The Moyal Identity
The Feichtinger Algebra
The Cohen Class
Gaussians and Hermite Functions
Sub-Gaussian Estimates
Applications to Quantum Mechanics:
Moyal Star Product and Twisted Convolution
Probabilistic Interpretation of the Wigner Transform
Mixed Quantum States and the Density Operator
The KLM Conditions and the Narcowich.Wigner Spectrum
Wigner Transform and Quantum Blobs
A: Sp(n) and Mp(n)
B: The Symplectic Fourier Transform
C: Symplectic Diagonalization
D: Symplectic Capacities
Readership: Advanced-level students and professors in mathematics and mathematical physics; research mathematicians, engineers and chemists active in the field.
350pp Jul 2017
ISBN: 978-981-3142-62-6 (hardcover)
ISBN: 978-981-3142-63-3 (softcover)
This comprehensive two-volume book deals with algebra, broadly conceived. Volume 1 (Chapters 1?6) comprises material for a first year graduate course in algebra, offering the instructor a number of options in designing such a course. Volume 1, provides as well all essential material that students need to prepare for the qualifying exam in algebra at most American and European universities. Volume 2 (Chapters 7?13) forms the basis for a second year graduate course in topics in algebra. As the table of contents shows, that volume provides ample material accommodating a variety of topics that may be included in a second year course. To facilitate matters for the reader, there is a chart showing the interdependence of the chapters.
Introduction and Fundamentals
Groups
Further Topics in Group Theory
Vector Spaces
Inner Product Spaces
Rings, Fields and Algebras
Modules
Readership: Graduate students and researchers in Algebra and related areas.
350pp Jul 2016
ISBN: 978-981-3142-66-4 (hardcover)
ISBN: 978-981-3142-67-1 (softcover)
This comprehensive two-volume book deals with algebra, broadly conceived. Volume 1 (Chapters 1?6) comprises material for a first year graduate course in algebra, offering the instructor a number of options in designing such a course. Volume 1, provides as well all essential material that students need to prepare for the qualifying exam in algebra at most American and European universities. Volume 2 (Chapters 7?13) forms the basis for a second year graduate course in topics in algebra. As the table of contents shows, that volume provides ample material accommodating a variety of topics that may be included in a second year course. To facilitate matters for the reader, there is a chart showing the interdependence of the chapters.
Multilinear Algebra
Symplectic Geometry
Commutative Algebra
Valuations and p-adic Numbers
Galois Theory
Representations of Finite Groups
Representations of Associative Algebras
Readership: Graduate students and researchers in Algebra and related areas.
870pp Oct 2017
ISBN: 978-981-3207-11-0 (hardcover)
This book provides a comprehensive introduction to Fock space theory and its applications to mathematical quantum field theory. The first half of the book, Part I, is devoted to detailed descriptions of analysis on abstract Fock spaces (full Fock space, boson Fock space, fermion Fock space and boson-fermion Fock space). It includes the mathematics of second quantization, representation theory of canonical commutation relations and canonical anti-commutation relations, Bogoliubov transformations, infinite-dimensional Dirac operators and supersymmetric quantum field in an abstract form. The second half of the book, Part II, covers applications of the mathematical theories in Part I to quantum field theory. Four kinds of free quantum fields are constructed and detailed analyses are made. A simple interacting quantum field model, called the van Hove model, is fully analyzed in an abstract form. Moreover, a list of interacting quantum field models is presented and a short description to each model is given.
To graduate students in mathematics or physics who are interested in the mathematical aspects of quantum field theory, this book is a good introductory text. It is also well suited for self-study and will provide readers a firm foundation of knowledge and mathematical techniques for reading more advanced books and current research articles in the field of mathematical analysis on quantum fields. Also, numerous problems are added to aid readers to develop a deeper understanding of the field.
Linear Operators on Hilbert Space
Tensor Product of Hilbert Spaces
Tensor Product of Linear Operators on Hilbert Spaces
Full Fock Space
Boson Fock Space
Fermion Fock Space
Boson-Fermion Fock Space
Theory of Infinite-Dimensional Dirac Operators and Abstract Supersymmetric Quantum Fields
General Theory of Quantum Fields
Quantum de Broglie Field
Quantum Klein?Gordon Field
Quantum Radiation Field
Quantum Dirac Field
van Hove Model
Overview of Interacting Quantum Field Models
Readership: Advanced undergraduate and graduate students in mathematics or physics, mathematicians and mathematical physicists.