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 ↓


Simplicial Complexes


Simplicial complexes are a foundational concept in the field of algebraic topology, which connects algebra, geometry, and topology by helping mathematicians understand shapes and spaces through abstract constructions. In this lesson, we'll take a deep look at what simplicial complexes are, how they are constructed, and why they're useful in undergraduate-level mathematics.

Introduction to simplicial complexes

To understand simplicial complexes, it is important to first understand the idea of simplicial complexes. A simplicial complex is a generalization of the concept of a triangle or tetrahedron to arbitrary dimensions. Here are some visual examples:

Visual example

0-simplex (vertex) 1-simplex (edge) 2-simplex (triangle) 3-simplex (tetrahedron)

As is clear from the visualization:

  • The 0-simplex has only one point or vertex.
  • A 1-simplex is a line segment defined by two points (vertices).
  • The 2-simplex can be viewed as a triangle with three edge segments and three vertices.
  • The 3-simplex is a tetrahedron, with four triangular faces, six edges, and four vertices.

General definition of simplex

For arbitrary dimension, n-simplex is the convex cover of its n + 1 vertices. More formally:

        σ = {[t₀v₀ + t₁v₁ + ... + tₙvₙ] | 
              (t₀, t₁, ..., tₙ) are non-negative real numbers 
              such that t₀ + t₁ + ... + tₙ = 1}

where the points v₀, v₁, ..., vₙ are isomorphically independent, that is, none of these points can be represented as a combination of the others.

Construction of simplicial complexes

A simplicial complex is a collection of simplices that fit together in a precise way. Here are the rules:

  • In a simplicial complex each face of the simplex is also in the complex.
  • The intersection of any two simple numbers is either empty or has one face each.

Example of simplicial complex

Consider a structure formed by arranging several triangles in 2D space. Each triangle is a 2-simplex. If these triangles meet at their edges, and each edge or vertex belongs to at least one triangle, then you have a simplicial complex.

Mathematical properties and operations

Simplicial packages allow mathematicians to extract topological properties from geometric shapes by creating abstract 'skeletons'. Some key properties and operations include:

Conformity

Using simplicial packages, we can compute the homology groups of a space, which measure the 'holes' of different dimensions in that space. Homology provides an algebraic snapshot of the topology at many levels.

Barycentric subdivision

Barycentric subdivision is a process used to refine simplicial complexes into finer partitions. This is done by adding a barycenter (or centroid) to each simplex and then connecting it to existing vertices.

Neural lemma

The nerve of a collection of sets is a simplicial complex. The nerve lemma says that if the sets are good covers, then the nerve has the same topological type as the union of sets.

Applications of simplicial complexes

Simplicial complexes have wide applications in various areas of science and mathematics.

Topological data analysis

In the emerging field of topological data analysis (TDA), simplicial complexes are used to study the shape of data sets. By constructing simplicial complexes, one can analyze and visualize high-dimensional data and identify clusters or voids.

Computer graphics

In computer graphics, complex simplices are used to model 3D shapes and surfaces. Through techniques such as mesh simplification, they allow for the efficient rendering of complex surfaces.

Conclusion

Simplicial complexes serve as a powerful representation of geometric structures through an abstract algebraic framework. This abstraction allows spaces to be analyzed in ways that would be impossible with traditional geometry alone. By decomposing spaces into simpler, manageable pieces (simplices), we gain information about their intrinsic properties.


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Simplicial Complexes

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