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 ↓


Homotopy and Homology


Algebraic topology is an interesting branch of mathematics that uses algebra to study topological spaces. The central concept of this field involves understanding spaces by looking at their various properties. Two deep topics under this umbrella are homotopy and homology, which help us differentiate the properties of spaces by abstracting them into algebraic terms.

Homotopy

Homotopy is a concept that considers when two shapes can be continuously transformed into one another. To understand homotopy, let's consider what it means for two functions to be homotopic.

AB

Imagine two circles, one labeled A and the other B. If you can stretch, shrink, or fold circle A into circle B without tearing or sticking, then A and B are considered isotopes.

In a more formal sense, if we have two continuous functions f and g from a space X to a space Y, then they are isotopic if there exists a continuous function H: X × [0, 1] → Y such that:

H(x, 0) = f(x) and H(x, 1) = g(x) for all x in X.

Here, H is called the homotopy between f and g. The interval ([0, 1]) is considered as a time parameter, which shows how smoothly f is transformed into g.

Applications of homotopy

Homotopy is useful in many areas such as deformation of curves in calculus, analysis of shapes in computer graphics, and various problems in engineering and physics. A simple example using homotopy is when considering the fundamental group, which helps to classify spaces up to homotopy equivalence using loops.

Conformity

Homology is another tool developed in algebraic topology to study the topology of a space. Unlike homotopy, which focuses on transformations, homology identifies and captures structures in a space by immutable algebraic objects. Homology groups, which are sequences of abelian groups associated with a topological space, are fundamental in this concept. They are effective not only in detecting holes but also in detecting more complex structures in different dimensions.

Simple homology

Let us consider simplicial symmetry to understand the basic idea. Consider a simple structure made up of vertices, edges and faces known as a simplex. We connect these simplices together to form a simplicial complex. A space is then studied by looking at the interactions between these simplices.

ABC

Then the boundary operator (∂) is defined, which maps each n simplex to a combination of its n-1 simplices. Simplicial homology groups are formed as the quotient of the kernel (the simplices without boundary) by the image of these maps (themselves the boundaries).

H_n = Ker(∂_n) / Im(∂_{n+1})

Understanding homology with examples

Consider the triangle ABC as shown in the figure. If we look at its boundary, it is made up of edges AB, BC, and AC. The homology group in this case simplifies its properties into an abstract algebraic form. Triangular, polygonal, or even more complex structures can be reduced to this simple type of analysis.

Applications of homology

Homology is a powerful tool for identifying shapes and locations, useful in both pure and applied mathematics. It plays important roles in fields such as robotics (for finding locations), data analysis (in the form of permanent homology), and even genetics. For example, homology can help classify different DNA structures due to its effectiveness in separating locations by holes.

Connecting homotopy and homology

Both homotopy and homology frameworks arise from the desire to understand topological spaces through algebraic invariants. Homotopy deals with different versions of continuous transformations, while homology focuses more on the structural aspect, capturing holes and voids in spaces.

They work together in many analysis and practical applications, especially when complex manifolds or high-dimensional data are involved. Each provides a different lens of abstraction, which promotes our understanding of space and transformations within a broader algebraic and geometric framework. In mathematics, especially in topology, using them in combination helps solve complex classification problems of spaces that are homologically equivalent, while providing rich insights about their structure and behavior.

In short, these topics form the backbone of algebraic topology and inspire the study and interpretation of shapes, curves, spaces, and their dynamic transformations, which are crucial for research and applications in diverse scientific fields.


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