In the series De Gruyter Textbook
The aim of the textbook is two-fold: first to serve as an introductory graduate course in Algebraic Topology and then to provide an application-oriented presentation of some fundamental concepts in Algebraic Topology to the fixed point theory.
A simple approach based on point-set Topology is used throughout to introduce many standard constructions of fundamental and homological groups of surfaces and topological spaces. The approach does not rely on Homological Algebra. The constructions of some spaces using the quotient spaces such as the join, the suspension, and the adjunction spaces are developed in the setting of Topology only.
The computations of the fundamental and homological groups of many surfaces and topological spaces occupy large parts of the book (sphere, torus, projective space, Mobius band, Klein bottle, manifolds, adjunctions spaces). Borsuk's theory of retracts which is intimately related to the problem of the extendability of continuous functions is developed in details. This theory together with the homotopy theory, the lifting and covering maps may serve as additional course material for students involved in General Topology.
The book comprises 280 detailed worked examples, 320 exercises (with hints or references), 80 illustrative figures, and more than 80 commutative diagrams to make it more oriented towards applications (maps between spheres, Borsuk-Ulam Theory, Fixed Point Theorems, c) As applications, the book offers some existence results on the solvability of some nonlinear differential equations subject to initial or boundary conditions.
The book is suitable for students primarily enrolled in Algebraic Topology, General Topology, Homological Algebra, Differential Topology, Differential Geometry, and Topological Geometry. It is also useful for advanced undergraduate students who aspire to grasp easily some new concepts in Algebraic Topology and Applications. The textbook is practical both as a teaching and research document for Bachelor, Master students, and first-year PhD students since it is accessible to any reader with a modest understanding of topological spaces.
The book aspires to fill a gap in the existing literature by providing a research and teaching document which investigates both the theory and the applications of Algebraic Topology in an accessible way without missing the main results of the topics covered.
The book only uses point-set topology to present many concepts in Algebraic Topology and fixed-point theory
Rich with examples and exercises
Offers an application-oriented presentation of fundamental concepts taught in algebraic topology.
Frontmatter
Publicly Available I
Introduction
S. Djebali
Publicly Available VII
Contents
Publicly Available XIII
Notation and symbols
Publicly Available XIX
List of Figures
Publicly Available XXI
1 Background in topology 1
2 Quotient topology 49
3 Topological constructions 64
4 The fundamental group 110
5 Covering maps and lifting maps 142
6 Fundamental groups of the circle and the sphere 154
7 Borsukfs theory of retracts 175
8 Fundamental groups of some surfaces 219
9 Higher homotopy groups 241
10 Elements of homology theory 256
11 Fixed-point theorems 304
12 Applications 333
Appendices 361
Subject Index 387
Author Index 391
Bibliography
Volume 12 in the series Advances in Analysis and Geometry
This monograph is devoted to harmonic analysis and potential theory. The authors study these essentials carefully and present recent researches based on the papers including by authors in an accessible manner for graduate students and researchers in pure and applied analysis.
Publicly Available I
Preface
Publicly Available V
Contents
Publicly Available IX
1 Preliminaries 1
2 Maximal operators 12
3 Hausdorff capacity 29
4 Weighted inequalities 83
5 Morrey spaces 125
6 Positive operators 166
7 Embedding theorems for dyadic rectangles 203
Bibliography 225
Index
Volume 100 in the series De Gruyter Studies in Mathematics
The book places emphasis on both the mathematical significance and the strong physical background of wave equations. It presents the theory of wave equations in a unique way, different from the traditional descriptions provided by previous literature. The book is primarily focused on mathematical ideas and thoughts about wave equations. Starting from the modern theory of harmonic analysis, the book develops a few new tools in this field that are being used for better understanding the theory of mathematical physics underlying the well-posedness and scattering theory of wave and Klein-Gordon equations. Additionally, a significant part of this book discusses theories and methods, such as invariant and conservation laws, inward/outward energy methods, etc., that have never been covered by similar books in this field. Finally, the book briefly introduces recent developments in mathematical fields. It is specially designed for experts in mathematics and physics who deal with numerous applications of nonlinear waves in physics, engineering, biology, and other fields.
A Combination of modern methods on harmonic analysis and mathematical physics
A detailed introduction of mathematical methods and tools on wave equations
Include Fields Prize winners Bourgain and Taofs work
Frontmatter
Publicly Available I
Contents
Publicly Available V
1 Multipliers, invariants and conservation laws 1
2 The integrality, uniqueness, and the regularity of weak solutions 46
3 Smooth solutions of semilinear wave equations 87
4 Global well-posedness and scattering for the energy solution of critical nonlinear wave equation 130
5 Scattering theory for the subcritical nonlinear Klein?Gordon and Schrodinger equations 180
6 Scattering theory for the critical Klein?Gordon equation 280
7 Inward/outward energy theory 344
8 Radial solutions to energy subcritical wave equations 402
9 Appendix 434
Bibliography 473
Index
Series:
Annals of Mathematics Studies
Hardcover
ISBN:
9780691272269
Jun 24, 2025
Pages: 352
Size:6.13 x 9.25 in.
Paperback
Exponential sums have been of great interest ever since Gauss, and their importance in analytic number theory goes back a century to Kloosterman. Grothendieckfs creation of the machinery of l-adic cohomology led to the understanding that families of exponential sums give rise to local systems, while Deligne, who gave his general equidistribution theorem after proving the Riemann hypothesis part of the Weil conjectures, established the importance of the monodromy groups of these local systems. Delignefs theorem shows that the monodromy group of the local system incarnating a given family of exponential sums determines key statistical properties of the family of exponential sums in question. Despite the apparent simplicity of this relation of monodromy groups to statistical properties, the actual determination of the monodromy group in any particular situation is highly nontrivial and leads to many interesting questions.
This book is devoted to the determination of the monodromy groups attached to various explicit families of exponential sums, especially those attached to hypergeometric sheaves, arguably the simplest local systems on G_m, and to some simple (in the sense of simple to write down) one-parameter families of one-variable sums. These last families turn out to have surprising connections to hypergeometric sheaves. One of the main technical advances of this book is to bring to bear a group-theoretic condition (S+), which, when it applies, implies very strong structural constraints on the monodromy group, and to show that (S+) does indeed apply to the monodromy groups of most hypergeometric sheaves.
Format: Paperback / softback, 321 pages, height x width: 235x155 mm, 2 Illustrations, color;
42 Illustrations, black and white; XIII, 321 p. 44 illus., 2 illus. in color.,
Series: Springer Undergraduate Texts in Mathematics and Technology
Pub. Date: 25-Nov-2024
ISBN-13: 9783031395642
This textbook is designed for a first course in linear algebra for undergraduate students from a wide range of quantitative and data driven fields. By focusing on applications and implementation, students will be prepared to go on to apply the power of linear algebra in their own discipline. With an ever-increasing need to understand and solve real problems, this text aims to provide a growing and diverse group of students with an applied linear algebra toolkit they can use to successfully grapple with the complex world and the challenging problems that lie ahead. Applications such as least squares problems, information retrieval, linear regression, Markov processes, finding connections in networks, and more, are introduced on a small scale as early as possible and then explored in more generality as projects. Additionally, the book draws on the geometry of vectors and matrices as the basis for the mathematics, with the concept of orthogonality taking center stage. Important matrixfactorizations as well as the concepts of eigenvalues and eigenvectors emerge organically from the interplay between matrix computations and geometry.
The R files are extra and freely available. They include basic code and templates for many of the in-text examples, most of the projects, and solutions to selected exercises. As much as possible, data sets and matrix entries are included in the files, thus reducing the amount of manual data entry required.
Introduction.-
1. Vectors.-
2. Matrices.-
3. Matrix Contexts.-
4. Linear Systems.-
5. Least Squares and Matrix Geometry.
6. Orthogonal Systems.-
7. Eigenvalues.-
8. Markov Processes.-
9. Symmetric Matrices.-
10. Singular Value Decomposition.-
11. Function Spaces.-Bibliography.-Index.
Format: Hardback, 412 pages
Series: Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore 44
Pub. Date: 24-Mar-2025
ISBN-13: 9789819806577
Other books in subject:
This volume celebrates the research contributions of Professors Theodore A Slaman and W Hugh Woodin, marking their distinguished careers in higher recursion theory and set theory as they approached the milestone of their 65th birthdays in 2019. It originates from the Institute for Mathematical Sciences program, Higher Recursion Theory and Set Theory, held at the National University of Singapore (May 20-June 14, 2019).The program explored cutting-edge developments in higher recursion theory, set theory, and their intricate interconnections. Topics discussed during the workshop included Martin's conjecture, higher randomness, the HOD conjecture, descriptive inner model theory, and the Ultimate-L program.This volume presents 15 peer-reviewed contributions by leading experts in the field, offering a comprehensive overview of recent advances in higher recursion theory and set theory, with a focus on their dynamic interactions.
Elementary Theories of Rogers Semilattices in the Analytical Hierarchy
Realizing Computably Enumerable Degrees in Separating Classes
On Š-Complete Uniform Ultrafilters
More on Bases of Uncountable Free Abelian Groups
Turing Degrees of Hyperjumps
Pseudojump Inversion in Special r.b. Pi-0-1 Classes
A Tractable Case of the Turing Automorphism Problem
An Introduction to AD+
On the Borelness of Upper Cones of Hyperdegrees
A New Game Metatheorem for Ash?Knight Style Priority Constructions
A I_0 Analogue of an AD Theorem
Type Omitting Theorems for Fragments of Second Order Logics
Mouse Pairs and Suslin Cardinals
A Worked Example of the Functional Interpretation
Complete Determined Borel Sets and Measurability