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 ↓


Algebraic Topology


Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. Its primary goal is to find algebraic invariants that classify topological spaces up to homeomorphism, although more usually homotopy is classified up to equivalence. This area of mathematics combines techniques from both topology and algebra to investigate the properties of shapes and spaces.

To understand algebraic topology, it is useful to be familiar with some basic concepts of topology. A topological space is a set of points, each of which is associated with a neighborhood system, which helps define continuity and convergence, among other concepts. For example, a two-dimensional plane and a three-dimensional sphere are simple examples of topological spaces.

Basic concepts

Open sets and topological spaces

In topology, a set is open if, intuitively, it does not contain its boundary. For example, an open interval on the real line is an example of an open set. This concept can be generalized to any topological space.

A topological space is a set X together with a collection of open subsets T such that:

1. The empty set and X itself are in T
2. Any arbitrary union of members of T is also a member of T
3. The intersection of any finite number of members of T is also a member of T

Continuous work

A function between topological spaces is considered continuous if the pre-image of every open set is open. Continuity in topology generalizes the idea of a continuous function in calculus.

Homeomorphisms

A homeomorphism is a continuous function between topological spaces that has a continuous inverse function. If a homeomorphism between two topological spaces exists, the spaces are considered topologically equivalent. For example, a coffee cup and a donut are considered the same in topology because each can be transformed into the other without cutting or pasting.

Fundamental concepts of algebraic topology

Homotopy

Two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other. More formally, there exists a continuous map

H: X × [0,1] → Y

such that H(x,0) = f(x) and H(x,1) = g(x) for all x ∈ X The concept of homotopy forms the basis of the definition of homotopy equivalence.

Homotopy equivalence

Two spaces X and Y are isomorphically equivalent if there exist continuous maps

f: X → Y and g: Y → X

such that g ° f is isotopic to the identity map on X and f ° g is isotopic to the identity map on Y

Fundamental group

The fundamental group, denoted π 1 (X) , is the group of equivalence classes of loops based on a point in a space X , with the group operation being the composition of paths. This group provides information about the shape of a space. For example, it can show whether a space is simply connected, meaning that it has no "holes".

Example using fundamental group

Consider a circle S 1 Its fundamental group is isomorphic to the integers ℤ , since a loop around the circle can be characterized by the number of rotations around it.

Applications of algebraic topology

Mapping of class groups

Mapping class groups arise from the study of surfaces and provide valuable insights into both the topological and geometric structure of manifolds. They can be used to investigate complex geometric structures and their symmetries.

Morse theory

Morse theory is a branch of mathematics that analyzes the topology of a manifold by studying smooth functions on that manifold. Techniques of algebraic topology help to identify critical points and understand how they contribute to the overall structure.

Knot theory

Knot theory studies the embeddings of circles in 3-dimensional spaces. Algebraic topology provides tools, such as the fundamental group, to classify and distinguish knots.

Visual example: continuous distortion

The above example shows a continuous transformation (homotopy) from a circle to a figure-eight and back. This is an intuitive visualization for understanding homotopy concepts.

Conformity

Introduction to homology

While the fundamental group is a powerful tool, it is sometimes too complicated to work with directly. Homology theory simplifies the study of a space by associating a series of abelian groups or modules to it.

Simple homology

Simplicial homology is defined using simplicial complexes, which decompose spaces into collections of points, line segments, triangles, and higher-dimensional equivalences. Here's a simple example:

The above triangle is a basic representation of the 2-simplex. By decomposing a figure into such a simplex, we can define the boundary operator and calculate homology groups.

Betti numbers

The Betti number is an integer that tells us the maximum number of cuts that can be made without splitting the surface into two pieces. n th Betti number b n counts the number of n-dimensional holes in the surface.

For example, in a torus (doughnut shape):

b 0 = 1 (a connected component),
b 1 = 2 (two 1-dimensional holes, the core and perimeter of the donut),
b 2 = 1 (a 2-dimensional zero).

Example of homology groups

Consider a circle S 1 This circle has the following homology groups:

H0 ( S1 )=ℤ, which represents a single connected component.
H 1 ( S 1 ) = ℤ represents the loop around .
H n (S 1 ) = 0, for n > 1, since there are no higher-dimensional holes.

Cohomology

Cohomology is a theory that evolved from homology, in which some of the arrows in the construction of the theory have been reversed. This provides a powerful tool for differentiating topological spaces.

Cup products

The cup product is an operation in cohomology theory that allows to combine sections of different dimensions. This operation gives cohomology rings, which often provide more sophisticated invariants than symmetry groups alone.

Differential form

In differential geometry, differential forms are used to define cohomology for smooth manifolds. This approach allows algebraic topology to be applied to the field of calculus.

Borsuk–Ulam theorem and stationary points

The Borsuk–Ulam theorem states that any continuous function from the sphere to the plane results in at least one pair of antipodal points being mapped to the same point. Algebraic topology provides the tools for proving such fixed point theorems, which have applications in a wide variety of fields, including economics and physics.

Algebraic topology offers substantial depth and richness, serving as a bridge between abstract algebra and classical topology. The interplay between geometry, algebra, and analysis within this field provides both broad theoretical implications and practical applications across mathematical disciplines and beyond.


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