Graduate
  1. Real Analysis  1.1. Set Theory and Logic  1.1.1. Sets and Operations  1.1.2. Relations and Functions  1.1.3. Logic and Quantifiers  1.1.4. Cardinality of Sets  1.2. Metric Spaces  1.2.1. Open and Closed Sets  1.2.2. Convergence of Sequences  1.2.3. Continuous Functions in Metric Spaces  1.2.4. Compactness and Completeness in Metric Spaces in Real Analysis  1.3. Sequence and Series  1.3.1. Convergence of Sequences  1.3.2. Infinite Series  1.3.3. Uniform Convergence  1.3.4. Power Series  1.4. Differentiation  1.4.1. Mean Value Theorems  1.4.2. Taylor Series  1.4.3. Functions of Several Variables  1.4.4. Partial Derivatives  1.5. Integration  1.5.1. Riemann and Lebesgue Integration  1.5.2. Improper Integrals  1.5.3. Measure Theory  1.5.4. Fubini’s Theorem  1.6. Functional Analysis  1.6.1. Banach Spaces  1.6.2. Hilbert Spaces  1.6.3. Operators on Spaces  1.6.4. Spectral Theory
  2. Abstract Algebra  2.1. Groups  2.1.1. Definitions and Examples in Groups in Abstract Algebra  2.1.2. Group Homomorphisms  2.1.3. Normal Subgroups  2.1.4. Sylow Theorems  2.2. Rings and Fields  2.2.1. Ideals and Factor Rings  2.2.2. Field Extensions  2.2.3. Galois Theory  2.2.4. Polynomial Rings  2.3. Modules  2.3.1. Module Homomorphisms  2.3.2. Free Modules  2.3.3. Tensor Products  2.3.4. Exact Sequences  2.4. Linear Algebra  2.4.1. Vector Spaces  2.4.2. Eigenvalues and Eigenvectors  2.4.3. Inner Product Spaces  2.4.4. Diagonalization
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Compactness and Connectedness  3.1.3. Continuous Maps  3.1.4. Separation Axioms  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy and Homology  3.2.3. Simplicial Complexes  3.2.4. Exact Sequences in Homology  3.3. Differential Topology  3.3.1. Manifolds  3.3.2. Smooth Functions  3.3.3. Tangent and Cotangent Spaces  3.3.4. Vector Bundles
  4. Differential Equations  4.1. Ordinary Differential Equations  4.1.1. First-Order Equations  4.1.2. Second-Order Linear Equations  4.1.3. Systems of Differential Equations  4.1.4. Stability Analysis  4.2. Partial Differential Equations  4.2.1. Classification of PDEs  4.2.2. Fourier Series  4.2.3. Green's Functions  4.2.4. Method of Characteristics  4.3. Dynamical Systems  4.3.1. Stability Theory  4.3.2. Limit Cycles  4.3.3. Chaos Theory  4.3.4. Bifurcation Theory
  5. Probability and Statistics  5.1. Probability Theory  5.1.1. Random Variables  5.1.2. Probability Distributions  5.1.3. Law of Large Numbers  5.1.4. Central Limit Theorem  5.2. Statistical Inference  5.2.1. Hypothesis Testing  5.2.2. Confidence Intervals  5.2.3. Bayesian Methods  5.2.4. Maximum Likelihood Estimation  5.3. Stochastic Processes  5.3.1. Markov Chains  5.3.2. Brownian Motion  5.3.3. Poisson Processes  5.3.4. Martingales
  6. Numerical Analysis  6.1. Numerical Solutions of Equations  6.1.1. Root Finding  6.1.2. Iterative Methods  6.1.3. Convergence Analysis  6.1.4. Error Bounds in Numerical Solutions of Equations  6.2. Numerical Linear Algebra  6.2.1. Matrix Factorizations  6.2.2. Eigenvalue Computations  6.2.3. Sparse Matrices  6.2.4. Preconditioning Techniques  6.3. Numerical Integration and Differentiation  6.3.1. Quadrature Methods  6.3.2. Finite Differences  6.3.3. Error Analysis in Numerical Integration and Differentiation  6.3.4. Adaptive Methods
  7. Complex Analysis  7.1. Complex Numbers  7.1.1. Polar Form  7.1.2. Complex Conjugates  7.1.3. Exponential Form  7.1.4. Roots of Unity  7.2. Analytic Functions  7.2.1. Cauchy-Riemann Equations  7.2.2. Power Series  7.2.3. Conformal Mappings  7.2.4. Harmonic Functions  7.3. Integration in the Complex Plane  7.3.1. Cauchy’s Theorem  7.3.2. Residue Theorem  7.3.3. Contour Integration  7.3.4. Integral Transforms
  8. Mathematical Logic and Foundations  8.1. Propositional Logic  8.1.1. Truth Tables  8.1.2. Logical Equivalences  8.1.3. Proof Techniques in Propositional Logic  8.1.4. Resolution  8.2. Predicate Logic  8.2.1. Quantifiers in Predicate Logic  8.2.2. Formal Systems in Predicate Logic  8.2.3. Completeness and Soundness  8.2.4. Model Theory  8.3. Set Theory  8.3.1. Cardinality and Ordinality  8.3.2. Axiomatic Systems  8.3.3. Zermelo-Fraenkel Set Theory  8.3.4. Forcing
  9. Optimization  9.1. Linear Programming  9.1.1. Simplex Method  9.1.2. Duality in Linear Programming  9.1.3. Sensitivity Analysis  9.1.4. Interior Point Methods  9.2. Nonlinear Programming  9.2.1. Gradient Descent  9.2.2. Lagrange Multipliers  9.2.3. Convex Optimization  9.2.4. Quadratic Programming  9.3. Combinatorial Optimization  9.3.1. Graph Algorithms  9.3.2. Integer Programming  9.3.3. Network Flows  9.3.4. Approximation Algorithms
  10. Discrete Mathematics  10.1. Graph Theory  10.1.1. Graph Representations  10.1.2. Planarity and Coloring  10.1.3. Shortest Path Algorithms  10.1.4. Network Flows  10.2. Combinatorics  10.2.1. Permutations and Combinations  10.2.2. Generating Functions  10.2.3. Recurrence Relations  10.2.4. Inclusion-Exclusion Principle  10.3. Cryptography  10.3.1. Number Theory Applications in Cryptography  10.3.2. Symmetric and Asymmetric Cryptography  10.3.3. RSA and Elliptic Curve Cryptography  10.3.4. Post-Quantum Cryptography

Graduate → Topology ↓


Algebraic Topology


Algebraic topology is a fascinating field of mathematics that combines aspects of algebra with the study of topological spaces. Its main purpose is to understand the shape and structure of spaces by transforming topological problems into algebraic problems, which are often easier to solve. This branch of topology finds applications in many areas of mathematics and science, from geometry to data analysis.

Understanding space: A topographical perspective

Before diving into algebraic topology, it's important to know what a topological space is. In topology, we're concerned with properties of spaces that are preserved under continuous deformations, such as stretching or bending, but not tearing or sticking. Common examples include a spherical surface, a torus (think of a donut shape), and a Möbius strip.

Example 1: A Sphere and a Torus The surface of a sphere: a 2-dimensional object in 3D space, without any holes. A torus: a shape like a doughnut, which has one hole.

Basics of algebraic topology: Homotopy and homology

Homotopy

The concept of homotopy is central to algebraic topology. Intuitively, two shapes or spaces are homotopic if one of them can be continuously transformed into the other without cutting or pasting. Homotopy is an equivalence relation between continuous functions. This leads us to the notion of homotopy equivalence between spaces.

Example 2: Deforming Spaces Consider a coffee cup and a doughnut. In topology, these two shapes are considered equivalent because one can be deformed into the other (thanks to the hole in the middle).

Conformity

Homology is another important concept, providing a way to associate a sequence of algebraic objects, such as groups or modules, with a given space. These groups characterize the dimensionality of the space and are invariant under homeomorphism and homotopy equivalence, making them powerful tools for classifying topological spaces.

Example 3: Homology Groups Consider a circle, a filled-in circle (disk), and a torus. - The circle can be associated with homology groups that reflect it has one 1-dimensional loop. - The disk has trivial homology groups because it can be continuously shrunk to a point. - The torus has nontrivial homology due to its hole and the surface.

Fundamental group

One of the most basic but powerful algebraic structures in algebraic topology is the fundamental group. It captures information about the fundamental shape or "1-dimensional holes" of a topological space.

Formally, the fundamental group of a space X with base point x_0, denoted by π_1(X, x_0), consists of the equivalence classes of loops based on x_0 under the operation of path combination.

Example 4: Calculating a Fundamental Group The fundamental group of a circle (S¹) is the integers ℤ. Each loop can be mapped to an integer representing the number of times it winds around the center. Visual Example: Imagine a loop on the circle that winds around once: this corresponds to +1. If it winds in the opposite direction: this is -1.

Higher symmetry groups

While the fundamental group gives a great overview of the 1-dimensional aspects of a space, algebraic topology doesn't stop there. Higher homotopy groups, denoted as π_n(X) for n > 1, investigate higher-dimensional "holes".

Example 5: The 2-sphere and Higher Homotopy For a 2-sphere, the second homotopy group is nontrivial, while for simple connected spaces, π1(X) might be trivial.

Cohomology

Cohomology, though dual to homology, is an indispensable tool that provides rich algebraic structures. It gives a space a collection of algebraic invariants (groups), which helps to distinguish between different topological spaces.

Example 6: Cohomology of Simple Spaces Consider the real projective plane or the Klein bottle. Cohomology captures similarities and differences from a torus or a sphere.

Euler characteristic

The Euler characteristic is a topological invariant derived from homology, giving a single number that encapsulates information about the shape of a space. It is calculated from the homology groups of a space and can often simplify the classification of surfaces.

Example 7: Euler Characteristic Formula For a polygonal region, Euler's formula is often written as: χ = V - E + F Where V is vertices, E is edges, and F is faces. Calculate for a cube: χ = 8 - 12 + 6 = 2

Simplicial complex and CW complex

Algebraic topology often uses combinatorial methods, such as simplicial packages and CW packages, which make it easier to calculate topological invariants.

Example 8: Constructing a Simplicial Complex Consider a triangle with vertices A, B, and C. The simplicial complex consists of three 0-simplices (points), three 1-simplices (edges), and one 2-simplex (filled triangle).

CW complexes are a generalization of simplicial complexes, allowing greater flexibility in construction by joining cells together in higher dimensions, thus elegantly describing a larger class of topological spaces.

Mayer–Vietoris sequence and exact sequence

The Mayer–Vietoris sequence is an important computational tool for computing the homology (or co-homology) of a space that can be decomposed into simpler parts. It relates the homology groups of the whole space to those of its parts and their intersection.

Example 9: Decomposing a Space with Mayer-Vietoris For a circle (S¹) decomposed into two semicircles intersecting at two points, use the Mayer-Vietoris sequence to calculate homology.

Exact sequences are a general concept, sequences of algebraic objects connected by homeomorphisms, useful in topology and other fields for describing how different algebraic structures are related to one another.

Poincaré duality

This duality principle, named after Henri Poincaré, provides a deep connection between homology and cohomology on a closed, oriented manifold. This results in isomorphisms between homology and cohomology groups on different dimensions.

Example 10: Poincaré Duality on a Sphere For a 2-dimensional sphere, the duality shows a relation between 0th homology and 2nd cohomology, 1st homology and 1st cohomology.

Conclusion

Algebraic topology is a broad and rich field that connects various parts of mathematics and has far-reaching implications in understanding the fundamental shape and structure of space. It provides a language and toolkit for translating geometric problems into algebraic problems, providing new insights and solution techniques. By taking advantage of homotopy, homology, and related concepts, algebraic topology not only advances theoretical mathematics but also has practical applications in data analysis, robotics, and other fields.


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