PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD → Topology ↓


Differential Topology


Differential topology is a branch of mathematics that deals with smooth shapes and geometric structures. At its core, differential topology involves the study of differentiable functions on differentiable manifolds. To better understand this concept, let's break down the definition and examine each component carefully.

Basic concepts and foundations

In differential topology, the main concepts revolve around the concept of a manifold and how shapes can be easily transformed. A manifold is a space that locally resembles Euclidean space, which essentially means that at any small scale, it resembles our usual understanding of surface coordinates. The idea of dimension is important in a manifold. For example, a line is 1-dimensional, a plane is 2-dimensional, and so on.

Differentiable manifolds

A differentiable manifold is a manifold in which there is a globally consistent way to compute derivatives of functions. This means that you can talk about smooth functions on these manifolds, and that operations such as differentiation and integration are well-defined.

Take the surface of a sphere as an example of a 2-dimensional manifold. When you look closely at this sphere at any point, it looks like a flat plane - this is the Euclidean nature of manifolds. However, it is important to emphasize that beyond just being manifolds, differentiable manifolds have smoothly transitioning charts and atlases that allow differentiation.

Tangent space and derivative

Understanding in which directions we can move from a point in space is expressed in the concept of a tangent space. For a manifold, a tangent space at any point consists of all possible directions in which one can pass a tangent from that point. Formally, this can be understood using directional derivatives.

Tangent vector

A tangent vector is a vector that is tangent to the manifold at a given point. For example, if you are standing on the Earth (which is represented as a 2D surface), your shadow at noon (due to the position of the sun) essentially acts as a tangent vector - it touches the Earth's surface at only one point.

Tangent vector

Differentiable functions on manifolds

A function between manifolds is called differentiable if it behaves locally like a differentiable function between Euclidean spaces. If you imagine stretching, compressing, and folding a piece of paper without tearing it, the function that maps that transformation is a differentiable function.

Let ( f: M rightarrow N ) be a function where ( M ) and ( N ) are manifolds.
( f ) is differentiable if for every point ( p in M ), and the charts ( phi : U rightarrow mathbb{R}^n ) for ( M ) and ( psi : V rightarrow mathbb{R}^m ) for ( N ),
The map ( psi circ f circ phi^{-1} ) is differentiable.

Differences from traditional topology

Unlike standard topology, which is often concerned with arbitrary bending and shaping of objects, regardless of the mode of bending, differential topology emphasizes smoothness. This means that when discussing homeomorphisms (one-to-one and continuous functions between topological spaces), it focuses on diffeomorphisms, or differential bijections with differential inverses.

Diffeomorphism

Diffeomorphism is an extremely smooth and reversible map between manifolds. Imagine a sheet that you can smoothly twist and roll in space without tearing it or joining the edges together - this is similar to diffeomorphism.

A map ( f: M rightarrow N ) is a diffeomorphism if it is binary, differentiable, and its inverse ( f^{-1} ) is also differentiable.

Properties and applications of differential topology

An important property in differential topology is the concept of invariant, which refers to a characteristic that remains unchanged under transformations such as diffeomorphisms. For example, the Euler characteristic is an invariant that can classify surfaces such as spheres and tori.

Euler characteristic

The Euler characteristic is a topological invariant, a number that represents the shape or structure of a manifold. It is invariant under smooth transformations.

,
chi = v - e + f
,
where ( V ) is the number of vertices, ( E ) is the number of edges, and ( F ) is the number of faces.

Critical points and Morse theory

Morse theory is a branch of differential topology that provides a way to analyze the topology of a manifold by studying differentiable functions and their critical points on that manifold.

Important points

The critical point of a differentiable function is where the derivative of the function is zero.

critical point

Structure of differential topology

One of the strong results obtained from differential topology is that while many 2-dimensional manifolds (surfaces) can be classified hierarchically as described above, 3-dimensional manifolds are far more complex and are not yet completely classified.

Differential Topology not only delves deep into the visualization of abstract spaces, but also connects with physics, computing fields, and advanced theoretical frameworks due to its ability to introspect space, time, and dimensions that form the fabric of our reality — providing a bridge between abstract thought and the phenomena of the tangible universe.

Conclusion

In short, differential topology provides a broad path to explore manifolds and differential functions, helping to understand smooth transformations. The implications of these explorations are very broad, spanning mathematics, physics, computer science, and philosophy, as they relate to the analysis of spatial and temporal concepts.


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Differential Topology

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