Differential topology with a view to applications Research notes in mathematics 1st Edition by David Chillingworth by DRJ Chillingworth ISBN 027300283X 9780273002833

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ISBN 10: 027300283X 
ISBN 13: 9780273002833
Author: DRJ Chillingworth

Differential topology is a branch of mathematics that studies the properties and structures of differentiable manifolds, focusing on the smoothness of maps and the topological properties of spaces that allow for differentiability. The discipline is a cornerstone of modern mathematics, providing the foundational tools necessary to understand complex geometric and topological structures. However, its scope extends far beyond pure mathematical theory. The methods and insights of differential topology are instrumental in tackling a wide range of problems in areas such as physics, engineering, computer science, and biology.

The goal of this book is to provide both a comprehensive introduction to the fundamental concepts of differential topology and a clear view of its practical applications. While the mathematical framework of differential topology involves abstract ideas, it is the power of these ideas to model real-world phenomena that makes this area of study so compelling. By emphasizing both theory and application, this book aims to bridge the gap between abstract mathematical concepts and their practical use in various scientific and technological domains.

Differential topology with a view to applications Research notes in mathematics 1st Table of contents:

• Chapter One: Basics of Topology and Differential Topology

  • Key Concepts in Topology: Sets, Spaces, and Continuity
  • Manifolds: Definition and Examples
  • Tangent Spaces and Differentiable Maps
  • Smooth Functions and Their Properties
  • Topological Invariants and Their Importance

• Chapter Two: Smooth Manifolds and Structures

  • Definition and Basic Properties of Smooth Manifolds
  • Charts and Atlases: Understanding Local Coordinates
  • Smooth Functions and Mappings Between Manifolds
  • Partition of Unity: A Fundamental Tool in Differential Topology
  • Immersions, Submersions, and Embeddings

• Chapter Three: The Fundamental Group and Higher Homotopy Groups

  • Introduction to Homotopy and Homology Theory
  • The Fundamental Group: Definition and Examples
  • Applications of the Fundamental Group in Classifying Manifolds
  • Higher Homotopy Groups and Their Role in Topology
  • Homotopy Invariant Properties and Theorems

• Chapter Four: Differential Forms and Stokes’ Theorem

  • Introduction to Differential Forms
  • Exterior Derivative and Cartan’s Formula
  • Integrating Differential Forms: Applications to Geometry and Physics
  • Stokes’ Theorem: Statement, Proof, and Applications
  • Applications to Fluid Dynamics, Electromagnetism, and More

• Chapter Five: Vector Bundles and Connections

  • Definition and Properties of Vector Bundles
  • Tangent and Cotangent Bundles
  • Connections and Covariant Derivatives
  • Curvature and Its Geometric Significance
  • Applications in Gauge Theory and General Relativity

• Chapter Six: Morse Theory and Critical Points

  • Basic Concepts in Morse Theory: Critical Points and Critical Values
  • The Morse Function: Definition and Examples
  • Morse Theory and Its Applications to Manifold Topology
  • Applications in Topological Invariants and Bifurcation Theory
  • Critical Points and Their Significance in Physics and Geometry

• Chapter Seven: Transversality and Its Applications

  • Definition of Transversality in Differential Topology
  • Transversality Theorems and Their Proofs
  • Applications to Intersection Theory and Singularities
  • Applications in Physics: Particle Interaction and Field Theory
  • Transversality in Algebraic Geometry and Topological Invariants

• Chapter Eight: Geometric Applications of Differential Topology

  • Applications to Geometry of Manifolds
  • The Role of Differential Topology in Geometrization Theorems
  • Applications to Symplectic Geometry and Differential Geometry
  • Topology of Moduli Spaces: Understanding Deformations and Singularities
  • Differential Topology and the Study of Curves and Surfaces in Higher Dimensions

• Chapter Nine: Differential Topology in Physics

  • General Relativity and the Role of Differential Topology
  • Topology of Spacetime and Black Hole Singularities
  • Quantum Field Theory: Applying Topological Concepts to Gauge Theories
  • Applications in Fluid Dynamics and Plasma Physics
  • Topological Defects in Condensed Matter Physics

• Chapter Ten: Global Analysis and Its Applications

  • Global Analysis on Manifolds: Key Results and Theorems
  • Applications to the Study of Differential Equations on Manifolds
  • The Atiyah-Singer Index Theorem and Its Implications
  • Applications to Quantum Mechanics and Quantum Field Theory
  • The Role of Global Analysis in Mathematical Physics

• Chapter Eleven: Singularities and Their Topological Analysis

  • Singularities of Smooth Maps: Classification and Analysis
  • The Role of Singularities in Dynamical Systems and Chaos Theory
  • Applications in Computer Vision, Image Recognition, and Robotics
  • Topological Methods in Understanding Critical Phenomena

• Chapter Twelve: Advanced Topics and Open Problems

  • Recent Developments in Differential Topology
  • Open Problems in the Field: Areas of Active Research
  • Connections with Other Fields: Algebraic Geometry, Topological K-Theory
  • Emerging Applications in Machine Learning, Data Analysis, and Robotics

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