Series: Frontiers in Mathematics
* Presents a comprehensive study of the analysis and geometry of
bicomplex numbers
* Offers a fundamental reference work for the field of bicomplex analysis
* Develops a solid foundation for potential new applications in
relativity, dynamical systems and quantum mechanics
The purpose of this book is to develop the foundations of the theory of holomorphicity
on the ring of bicomplex numbers. Accordingly, the main focus is on expressing the
similarities with, and differences from, the classical theory of one complex variable. The
result is an elementary yet comprehensive introduction to the algebra, geometry and
analysis of bicomplex numbers.
Around the middle of the nineteenth century, several mathematicians (the best known
being Sir William Hamilton and Arthur Cayley) became interested in studying number
systems that extended the field of complex numbers. Hamilton famously introduced
the quaternions, a skew field in real-dimension four, while almost simultaneously James
Cockle introduced a commutative four-dimensional real algebra, which was rediscovered
in 1892 by Corrado Segre, who referred to his elements as bicomplex numbers. The
advantages of commutativity were accompanied by the introduction of zero divisors,
something that for a while dampened interest in this subject. In recent years, due
largely to the work of G.B. Price, there has been a resurgence of interest in the study of
these numbers and, more importantly, in the study of functions defined on the ring of
bicomplex numbers, which mimic the behavior of holomorphic functions of a complex
variable.
While the algebra of bicomplex numbers is a four-dimensional real algebra, it is useful to
think of it as a “complexification” of the field of complex <
numbers; from this perspective, the bicomplex algebra possesses the properties of a onedimensional
theory inside four real dimensions. Its rich analysis and innovative geometry
provide new ideas and potential applications in relativity and quantum mechanics alike.
The book will appeal to researchers in the fields of complex, hypercomplex and functional
analysis, as well as undergraduate and graduate students with an interest in one- or
multidimensional complex analysis.
1st ed. 2017, VIII, 231 p. 22 illus. in color.
Softcover
ISBN 978-3-319-24866-0
Series: Ecole d'Ete de Probabilites de Saint-Flour, Vol. 2151
Providing an elementary introduction to branching random walks, the main focus of
these lecture notes is on the asymptotic properties of one-dimensional discrete-time
supercritical branching random walks, and in particular, on extreme positions in each
generation, as well as the evolution of these positions over time.
Starting with the simple case of Galton-Watson trees, the text primarily concentrates on
exploiting, in various contexts, the spinal structure of branching random walks. The notes
end with some applications to biased random walks on trees.
1st ed. 2015, X, 128 p.
Softcover
ISBN 978-3-319-25371-8
Series: Springer Monographs in Mathematics
* Provides a self-contained approach to the study of geometric
and analytic aspects of maximum principles, making it a perfect
companion to other books on the subject
* Presents the essential analytic tools and the geometric foundations
needed to understand maximum principles and their geometric applications
* Includes a wide range of applications of maximum principles to
different geometric problems, including some topics that are rare in
current literature such as Ricci solitons
* Relevant to other areas of mathematics, namely, partial differential
equations on manifolds, calculus of variations, and probabilistic potential
theory
This monograph presents an introduction to some geometric and analytic aspects of the
maximum principle. In doing so, it analyses with great detail the mathematical tools and
geometric foundations needed to develop the various new forms that are presented in
the first chapters of the book. In particular, a generalization of the Omori-Yau maximum
principle to a wide class of differential operators is given, as well as a corresponding weak
maximum principle and its equivalent open form and parabolicity as a special stronger
formulation of the latter.
In the second part, the attention focuses on a wide range of applications, mainly to
geometric problems, but also on some analytic (especially PDEs) questions including: the
geometry of submanifolds, hypersurfaces in Riemannian and Lorentzian targets, Ricci
solitons, Liouville theorems, uniqueness of solutions of Lichnerowicz-type PDEs and so on.
Maximum Principles and Geometric Applications is written in an easy style making it
accessible to beginners. The reader is guided with a detailed presentation of some topics
of Riemannian geometry that are usually not covered in textbooks. Furthermore, many
of the results and even proofs of known results are new and lead to the frontiers of a
contemporary and active field of research.
1st ed. 2016, Approx. 530 p.
Hardcover
ISBN 978-3-319-24335-1
Series: Springer Monographs in Mathematics
* Provides basic material about maps and hypermaps on Riemann surfaces
* Presents many elementary and less elementary examples of Galois
actions on dessins and their algebraic curves
* Emphasises the role of group theory in the classification of regular
maps, regular dessins, and quasiplatonic surfaces
* Explains the links between the theory of dessins and other areas of
arithmetic and geometry
This volume provides an introduction to dessins d'enfants and embeddings of bipartite
graphs in compact Riemann surfaces. The first part of the book presents basic material,
guiding the reader through the current field of research. A key point of the second
part is the interplay between the automorphism groups of dessins and their Riemann
surfaces, and the action of the absolute Galois group on dessins and their algebraic
curves. It concludes by showing the links between the theory of dessins and other areas
of arithmetic and geometry, such as the abc conjecture, complex multiplication and
Beauville surfaces.
Dessins d'Enfants on Riemann Surfaces will appeal to graduate students and all
mathematicians interested in maps, hypermaps, Riemann surfaces, geometric group
actions, and arithmetic.
1st ed. 2016, Approx. 260 p.
Hardcover
ISBN 978-3-319-24709-0
Series on Knots and Everything: Volume 57
350pp Apr 2016
ISBN: 978-981-4656-49-8 (hardcover)
This book shows how the ADE Coxeter graphs unify at least 20 different types of mathematical structures. These mathematical structures are of great utility in unified field theory, string theory, and other areas of physics.
Introduction
The Octahedral Group
The Octahedral Double Group
The McKay Correspondence
Lie Groups and Lie Algebras
Coxeter's Reflection Groups
Thom*Arnold Catastrophe Structures
ALE Spaces and Gravitational Instantons
Knots and Links and Braids
Twistors and ALE Spaces
2-Dimensional Conformal Field Theories
Elliptic Curves and the Monster Group
Sphere Packing and Error-Correction Codes
Qubits and Black Holes
The Holographic Principle
Calabi*Yau Spaces
Heisenberg Algebras
Summary and Outlook
Glossary
References and Bibliography
Readership: Researchers in mathematical physics.