(Ebook PDF) Diffeology Mathematical Surveys and Monographs 1st edition by Patrick Iglesias Zemmour 0821891316 9780821891315 Full chapter

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ISBN 10:         0821891316
ISBN 13: 978-0821891315
Author:          Patrick Iglesias-Zemmour

Diffeology Mathematical Surveys and Monographs 1st Table of contents:

1. Introduction

• 1.1 Overview of Diffeology
• 1.2 Motivation and Applications
• 1.3 Historical Context
• 1.4 Key Definitions and Results

2. Basic Concepts and Definitions

• 2.1 Differentiable Manifolds
• 2.2 The Concept of Diffeology
• 2.3 Smooth Maps and Diffeological Spaces
• 2.4 Examples of Diffeological Spaces
• 2.5 Comparison with Other Approaches to Geometry

3. Diffeology of Topological Spaces

• 3.1 Topological Spaces and Smooth Structures
• 3.2 The Diffeology of a Topological Space
• 3.3 Quasi-smooth and Smooth Maps
• 3.4 The Diffeology of a Set
• 3.5 Examples of Diffeologies on Topological Spaces

4. Differential Geometry and Diffeology

• 4.1 Differential Geometry Basics
• 4.2 The Role of Diffeology in Differential Geometry
• 4.3 Smooth Manifolds as Diffeological Spaces
• 4.4 Applications in Geometric Objects and Theorems

5. Smooth Maps and Smooth Functions

• 5.1 Definition of Smooth Maps
• 5.2 Properties of Smooth Functions in Diffeological Spaces
• 5.3 Open Maps and Diffeomorphisms
• 5.4 Composition of Smooth Maps

6. Structure of Diffeological Groups

• 6.1 Introduction to Diffeological Groups
• 6.2 Smooth Structures on Lie Groups
• 6.3 Diffeological Groups and Lie Algebras
• 6.4 Applications in Group Theory and Geometry

7. Diffeology and Category Theory

• 7.1 Introduction to Categories in Mathematics
• 7.2 Diffeological Spaces as Categories
• 7.3 Functors and Natural Transformations
• 7.4 Diffeological Objects in Category Theory

8. Examples and Applications

• 8.1 Euclidean Spaces and Diffeological Structures
• 8.2 The Circle and Other Simple Examples
• 8.3 Applications to Physics and Geometry
• 8.4 Computational Approaches in Diffeology

9. Advanced Topics

• 9.1 Diffeology and Singularities
• 9.2 Infinite-Dimensional Diffeological Spaces
• 9.3 Diffeology in Complex Geometry and Analysis
• 9.4 Higher-Dimensional Generalizations of Diffeology

10. Conclusion and Future Directions

• 10.1 Summary of Key Results
• 10.2 Open Problems and Research Directions
• 10.3 The Future of Diffeology in Mathematics

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