Mathematics | 🗖 nosplit | ↑ parent "Ciro Santilli's Homepage" | 16, 218, 9k

🗖 nosplit | ⇓ toc | ↑ parent "Ciro Santilli's Homepage" | Wikipedia | 16, 218, 9k

The proper precise definition of mathematics can be found at: Section 2. "Formalization of Mathematics".

Table of contents | 16, 218, 9k
- 1. The beauty of mathematics | 243, 4, 520 | 🔗 link | 🗖 nosplit | ↑ parent "Mathematics"
  - 1.1. Lists of mathematical problems | 12, 3, 277 | 🔗 link | 🗖 nosplit | ↑ parent "The beauty of mathematics"
    - 1.1.1. Hilbert's problems | 21 | 🔗 link | 🗖 nosplit | ↑ parent "Lists of mathematical problems"
    - 1.1.2. Millennium Prize Problems | 233, 1, 244 | 🔗 link | 🗖 nosplit | ↑ parent "Lists of mathematical problems"
      - 1.1.2.1. Birch and Swinnerton-Dyer Conjecture | 11 | 🔗 link | 🗖 nosplit | ↑ parent "Millennium Prize Problems"
- 2. Formalization of Mathematics | 411, 11, 570 | 🔗 link | 🗖 nosplit | ↑ parent "Mathematics"
  - 2.1. Formal proof | 14, 2, 14 | 🔗 link | 🗖 nosplit | ↑ parent "Formalization of Mathematics"
    - 2.1.1. Axiom | 🔗 link | 🗖 nosplit | ↑ parent "Formal proof"
    - 2.1.2. Independence (mathematical logic) | 🔗 link | 🗖 nosplit | ↑ parent "Formal proof"
  - 2.2. Set (mathematics) | 85 | 🔗 link | 🗖 nosplit | ↑ parent "Formalization of Mathematics"
  - 2.3. Function (mathematics) | 22 | 🔗 link | 🗖 nosplit | ↑ parent "Formalization of Mathematics"
  - 2.4. Integer | 🔗 link | 🗖 nosplit | ↑ parent "Formalization of Mathematics"
  - 2.5. Real number | 3, 2, 3 | 🔗 link | 🗖 nosplit | ↑ parent "Formalization of Mathematics"
    - 2.5.1. Countable set | 0, 1, 0 | 🔗 link | 🗖 nosplit | ↑ parent "Real number"
      - 2.5.1.1. Cantor's diagonal argument | 🔗 link | 🗖 nosplit | ↑ parent "Countable set"
  - 2.6. Complex number | 35 | 🔗 link | 🗖 nosplit | ↑ parent "Formalization of Mathematics"
  - 2.7. Ordered pair | 🔗 link | 🗖 nosplit | ↑ parent "Formalization of Mathematics"
- 3. Algebra | 0, 63, 2k | 🔗 link | 🗖 nosplit | ↑ parent "Mathematics"
  - 3.1. Field (mathematics) | 🔗 link | 🗖 nosplit | ↑ parent "Algebra"
  - 3.2. Linear algebra | 0, 15, 147 | 🔗 link | 🗖 nosplit | ↑ parent "Algebra"
    - 3.2.1. Matrix | 0, 14, 147 | 🔗 link | 🗖 nosplit | ↑ parent "Linear algebra"
      - 3.2.1.1. Matrix multiplication | 10, 4, 80 | 🔗 link | 🗖 nosplit | ↑ parent "Matrix"
        
        3.2.1.1.1. Matrix multiplication algorithm | 1 | 🔗 link | 🗖 nosplit | ↑ parent "Matrix multiplication"
        
        3.2.1.1.2. Linear map (linear operator) | 56, 2, 69 | 🔗 link | 🗖 nosplit | ↑ parent "Matrix multiplication"
        
        3.2.1.1.2.1. Linear operator | 13 | 🔗 link | 🗖 nosplit | ↑ parent "Linear map"
        
        3.2.1.1.2.2. Self-adjoint operator | 🔗 link | 🗖 nosplit | ↑ parent "Linear map"
      - 3.2.1.2. Eigenvalues and eigenvectors | 0, 6, 20 | 🔗 link | 🗖 nosplit | ↑ parent "Matrix"
        
        3.2.1.2.1. Applications of eigenvalues and eigenvectors | 3 | 🔗 link | 🗖 nosplit | ↑ parent "Eigenvalues and eigenvectors"
        
        3.2.1.2.2. Eigenvalue | 8 | 🔗 link | 🗖 nosplit | ↑ parent "Eigenvalues and eigenvectors"
        
        3.2.1.2.3. Eigenvector | 8 | 🔗 link | 🗖 nosplit | ↑ parent "Eigenvalues and eigenvectors"
        
        3.2.1.2.4. Eigenvectors and eigenvalues of the identity matrix | 1 | 🔗 link | 🗖 nosplit | ↑ parent "Eigenvalues and eigenvectors"
        
        3.2.1.2.5. Spectral theorem | 0, 1, 0 | 🔗 link | 🗖 nosplit | ↑ parent "Eigenvalues and eigenvectors"
        
        3.2.1.2.5.1. Hermitian matrix | 🔗 link | 🗖 nosplit | ↑ parent "Spectral theorem"
      - 3.2.1.3. Orthogonal matrix | 11, 1, 47 | 🔗 link | 🗖 nosplit | ↑ parent "Matrix"
        
        3.2.1.3.1. Unitary matrix | 36 | 🔗 link | 🗖 nosplit | ↑ parent "Orthogonal matrix"
  - 3.3. Group (mathematics) | 0, 45, 2k | 🔗 link | 🗖 nosplit | ↑ parent "Algebra"
    - 3.3.1. Abelian group | 27 | 🔗 link | 🗖 nosplit | ↑ parent "Group"
    - 3.3.2. Symmetry | 25 | 🔗 link | 🗖 nosplit | ↑ parent "Group"
    - 3.3.3. Important mathematical groups | 0, 23, 826 | 🔗 link | 🗖 nosplit | ↑ parent "Group"
      - 3.3.3.1. Important discrete mathematical groups | 0, 4, 11 | 🔗 link | 🗖 nosplit | ↑ parent "Important mathematical groups"
        
        3.3.3.1.1. Cyclic group ( $Z_{n}$ ) | 🔗 link | 🗖 nosplit | ↑ parent "Important discrete mathematical groups"
        
        3.3.3.1.2. Alternating group ( $A_{n}$ ) | 🔗 link | 🗖 nosplit | ↑ parent "Important discrete mathematical groups"
        
        3.3.3.1.3. Wallpaper group | 3 | 🔗 link | 🗖 nosplit | ↑ parent "Important discrete mathematical groups"
        
        3.3.3.1.4. Space group | 8 | 🔗 link | 🗖 nosplit | ↑ parent "Important discrete mathematical groups"
      - 3.3.3.2. Lie group | 187, 17, 815 | 🔗 link | 🗖 nosplit | ↑ parent "Important mathematical groups"
        
        3.3.3.2.1. Applications of Lie groups to differential equations | 96 | 🔗 link | 🗖 nosplit | ↑ parent "Lie group"
        
        3.3.3.2.2. Lie algebra | 🔗 link | 🗖 nosplit | ↑ parent "Lie group"
        
        3.3.3.2.3. Continuous symmetry | 17, 1, 36 | 🔗 link | 🗖 nosplit | ↑ parent "Lie group"
        
        3.3.3.2.3.1. Local symmetry | 19 | 🔗 link | 🗖 nosplit | ↑ parent "Continuous symmetry"
        
        3.3.3.2.4. Important Lie groups | 0, 11, 434 | 🔗 link | 🗖 nosplit | ↑ parent "Lie group"
        
        3.3.3.2.4.1. General linear group ( $G L (n)$ ) | 42 | 🔗 link | 🗖 nosplit | ↑ parent "Important Lie groups"
        
        3.3.3.2.4.2. Special linear group ( $S L (n)$ ) | 17 | 🔗 link | 🗖 nosplit | ↑ parent "Important Lie groups"
        
        3.3.3.2.4.3. Orthogonal group ( $O (n)$ ) | 154, 1, 175 | 🔗 link | 🗖 nosplit | ↑ parent "Important Lie groups"
        
        3.3.3.2.4.3.1. Special orthogonal group ( $S O (n)$ ) | 21 | 🔗 link | 🗖 nosplit | ↑ parent "Orthogonal group"
        
        3.3.3.2.4.4. Unitary group ( $U (n)$ ) | 77, 3, 147 | 🔗 link | 🗖 nosplit | ↑ parent "Important Lie groups"
        
        3.3.3.2.4.4.1. Unitary group of dimension 2 ( $U (2)$ ) | 5 | 🔗 link | 🗖 nosplit | ↑ parent "Unitary group"
        
        3.3.3.2.4.4.2. Unit circle | 18 | 🔗 link | 🗖 nosplit | ↑ parent "Unitary group"
        
        3.3.3.2.4.4.3. Special unitary group ( $S U (n)$ ) | 47 | 🔗 link | 🗖 nosplit | ↑ parent "Unitary group"
        
        3.3.3.2.4.5. Poincare group | 21, 2, 53 | 🔗 link | 🗖 nosplit | ↑ parent "Important Lie groups"
        
        3.3.3.2.4.5.1. Lorentz group ( $S O (1, 3)$ ) | 15, 1, 32 | 🔗 link | 🗖 nosplit | ↑ parent "Poincare group"
        
        3.3.3.2.4.5.1.1. Indefinite orthogonal group ( $S O (m, n)$ ) | 17 | 🔗 link | 🗖 nosplit | ↑ parent "Lorentz group"
        
        3.3.3.2.5. Representation theory | 62 | 🔗 link | 🗖 nosplit | ↑ parent "Lie group"
    - 3.3.4. Group isomorphism | 0, 3, 118 | 🔗 link | 🗖 nosplit | ↑ parent "Group"
      - 3.3.4.1. Group homomorphism | 93, 2, 118 | 🔗 link | 🗖 nosplit | ↑ parent "Group isomorphism"
        
        3.3.4.1.1. Fundamental theorem on homomorphisms | 25 | 🔗 link | 🗖 nosplit | ↑ parent "Group homomorphism"
        
        3.3.4.1.2. Kernel (algebra) | 🔗 link | 🗖 nosplit | ↑ parent "Group homomorphism"
    - 3.3.5. Direct product of groups | 0, 1, 55 | 🔗 link | 🗖 nosplit | ↑ parent "Group"
      - 3.3.5.1. Semidirect product | 55 | 🔗 link | 🗖 nosplit | ↑ parent "Direct product of groups"
    - 3.3.6. Subgroup | 0, 12, 841 | 🔗 link | 🗖 nosplit | ↑ parent "Group"
      - 3.3.6.1. Subgroup generated by a group | 🔗 link | 🗖 nosplit | ↑ parent "Subgroup"
      - 3.3.6.2. Quotient group | 40, 10, 841 | 🔗 link | 🗖 nosplit | ↑ parent "Subgroup"
        
        3.3.6.2.1. Subquotient | 61 | 🔗 link | 🗖 nosplit | ↑ parent "Quotient group"
        
        3.3.6.2.2. Relationship between the quotient group and direct products | 297 | 🔗 link | 🗖 nosplit | ↑ parent "Quotient group"
        
        3.3.6.2.3. Normal subgroup | 135, 7, 443 | 🔗 link | 🗖 nosplit | ↑ parent "Quotient group"
        
        3.3.6.2.3.1. Simple group | 46 | 🔗 link | 🗖 nosplit | ↑ parent "Normal subgroup"
        
        3.3.6.2.3.2. Classification of finite groups | 64, 5, 262 | 🔗 link | 🗖 nosplit | ↑ parent "Normal subgroup"
        
        3.3.6.2.3.2.1. Classification of simple finite groups | 83, 3, 96 | 🔗 link | 🗖 nosplit | ↑ parent "Classification of finite groups"
        
        3.3.6.2.3.2.1.1. Monster group | 🔗 link | 🗖 nosplit | ↑ parent "Classification of simple finite groups"
        
        3.3.6.2.3.2.1.2. Jordan-Holder Theorem | 13 | 🔗 link | 🗖 nosplit | ↑ parent "Classification of simple finite groups"
        
        3.3.6.2.3.2.1.3. Composition series | 🔗 link | 🗖 nosplit | ↑ parent "Classification of simple finite groups"
        
        3.3.6.2.3.2.2. Group extension problem | 102 | 🔗 link | 🗖 nosplit | ↑ parent "Classification of finite groups"
- 4. Calculus | 10, 91, 4k | 🔗 link | 🗖 nosplit | ↑ parent "Mathematics"
  - 4.1. Limit (mathematics) | 44 | 🔗 link | 🗖 nosplit | ↑ parent "Calculus"
  - 4.2. Riemann integral | 🔗 link | 🗖 nosplit | ↑ parent "Calculus"
  - 4.3. Lebesgue integral | 23, 12, 821 | 🔗 link | 🗖 nosplit | ↑ parent "Calculus"
    - 4.3.1. Lebesgue integral vs Riemann integral | 211, 1, 274 | 🔗 link | 🗖 nosplit | ↑ parent "Lebesgue integral"
      - 4.3.1.1. Real world applications of the Lebesgue integral | 63 | 🔗 link | 🗖 nosplit | ↑ parent "Lebesgue integral vs Riemann integral"
    - 4.3.2. Lebesgue measurable | 🔗 link | 🗖 nosplit | ↑ parent "Lebesgue integral"
    - 4.3.3. Lebesgue integral of $L^{p}$ is complete but Riemann isn't | 83, 7, 516 | 🔗 link | 🗖 nosplit | ↑ parent "Lebesgue integral"
      - 4.3.3.1. Riesz-Fischer theorem | 56, 4, 287 | 🔗 link | 🗖 nosplit | ↑ parent "Lebesgue integral of $L^{p}$ is complete but Riemann isn't"
        
        4.3.3.1.1. $L^{p}$ is complete | 1 | 🔗 link | 🗖 nosplit | ↑ parent "Riesz-Fischer theorem"
        
        4.3.3.1.2. Fourier basis is complete for $L^{2}$ | 55, 2, 230 | 🔗 link | 🗖 nosplit | ↑ parent "Riesz-Fischer theorem"
        
        4.3.3.1.2.1. $L^{p}$ norm sequence convergence does not imply pointwise convergence | 41 | 🔗 link | 🗖 nosplit | ↑ parent "Fourier basis is complete for $L^{2}$ "
        
        4.3.3.1.2.2. Carleson's theorem | 134 | 🔗 link | 🗖 nosplit | ↑ parent "Fourier basis is complete for $L^{2}$ "
      - 4.3.3.2. $L^{p}$ | 28, 1, 146 | 🔗 link | 🗖 nosplit | ↑ parent "Lebesgue integral of $L^{p}$ is complete but Riemann isn't"
        
        4.3.3.2.1. $L^{2}$ | 118 | 🔗 link | 🗖 nosplit | ↑ parent " $L^{p}$ "
    - 4.3.4. Every Riemann integrable function is Lebesgue integrable | 8 | 🔗 link | 🗖 nosplit | ↑ parent "Lebesgue integral"
  - 4.4. Measure theory | 53 | 🔗 link | 🗖 nosplit | ↑ parent "Calculus"
  - 4.5. Fourier series | 125, 8, 358 | 🔗 link | 🗖 nosplit | ↑ parent "Calculus"
    - 4.5.1. Applications of the Fourier series | 0, 1, 65 | 🔗 link | 🗖 nosplit | ↑ parent "Fourier series"
      - 4.5.1.1. Solving partial differential equations with the Fourier series | 65 | 🔗 link | 🗖 nosplit | ↑ parent "Applications of the Fourier series"
    - 4.5.2. Discrete Fourier transform (DFT) | 🔗 link | 🗖 nosplit | ↑ parent "Fourier series"
    - 4.5.3. Fourier transform | 22, 3, 152 | 🔗 link | 🗖 nosplit | ↑ parent "Fourier series"
      - 4.5.3.1. Multidimensional Fourier transform | 40 | 🔗 link | 🗖 nosplit | ↑ parent "Fourier transform"
      - 4.5.3.2. Fourier inversion theorem | 38 | 🔗 link | 🗖 nosplit | ↑ parent "Fourier transform"
      - 4.5.3.3. Laplace transform | 52 | 🔗 link | 🗖 nosplit | ↑ parent "Fourier transform"
    - 4.5.4. History of the Fourier series | 16 | 🔗 link | 🗖 nosplit | ↑ parent "Fourier series"
  - 4.6. Topology | 12, 3, 55 | 🔗 link | 🗖 nosplit | ↑ parent "Calculus"
    - 4.6.1. Manifold | 🔗 link | 🗖 nosplit | ↑ parent "Topology"
    - 4.6.2. Complete metric space | 43 | 🔗 link | 🗖 nosplit | ↑ parent "Topology"
    - 4.6.3. Dense set | 🔗 link | 🗖 nosplit | ↑ parent "Topology"
  - 4.7. Distribution (mathematics) | 55, 1, 125 | 🔗 link | 🗖 nosplit | ↑ parent "Calculus"
    - 4.7.1. Dirac delta function | 70 | 🔗 link | 🗖 nosplit | ↑ parent "Distribution"
  - 4.8. Complex analysis | 14, 6, 273 | 🔗 link | 🗖 nosplit | ↑ parent "Calculus"
    - 4.8.1. Holomorphic function | 68 | 🔗 link | 🗖 nosplit | ↑ parent "Complex analysis"
    - 4.8.2. Analytic continuation | 50, 4, 191 | 🔗 link | 🗖 nosplit | ↑ parent "Complex analysis"
      - 4.8.2.1. Visualizing the Riemann hypothesis and analytic continuation by 3Blue1Brown | 21 | 🔗 link | 🗖 nosplit | ↑ parent "Analytic continuation"
      - 4.8.2.2. Identity theorem | 89, 2, 120 | 🔗 link | 🗖 nosplit | ↑ parent "Analytic continuation"
        
        4.8.2.2.1. Riemann zeta function | 0, 1, 31 | 🔗 link | 🗖 nosplit | ↑ parent "Identity theorem"
        
        4.8.2.2.1.1. Riemann hypothesis | 31 | 🔗 link | 🗖 nosplit | ↑ parent "Riemann zeta function"
  - 4.9. Hilbert space | 31 | 🔗 link | 🗖 nosplit | ↑ parent "Calculus"
  - 4.10. Differential equation | 0, 44, 2k | 🔗 link | 🗖 nosplit | ↑ parent "Calculus"
  - 4.11. Taylor series | 🔗 link | 🗖 nosplit | ↑ parent "Calculus"
  - 4.12. Gradient, Divergence, Curl, and Laplacian | 0, 4, 84 | 🔗 link | 🗖 nosplit | ↑ parent "Calculus"
  - 4.13. Infinitesimal | 22 | 🔗 link | 🗖 nosplit | ↑ parent "Calculus"
- 5. Functional equation | 0, 1, 12 | 🔗 link | 🗖 nosplit | ↑ parent "Mathematics"
  - 5.1. Cauchy's functional equation | 12 | 🔗 link | 🗖 nosplit | ↑ parent "Functional equation"
- 6. Game theory | 0, 4, 147 | 🔗 link | 🗖 nosplit | ↑ parent "Mathematics"
  - 6.1. Nash equilibrium | 139, 3, 147 | 🔗 link | 🗖 nosplit | ↑ parent "Game theory"
    - 6.1.1. Mixed strategy | 🔗 link | 🗖 nosplit | ↑ parent "Nash equilibrium"
    - 6.1.2. Prisoner's dilemma | 1 | 🔗 link | 🗖 nosplit | ↑ parent "Nash equilibrium"
    - 6.1.3. Bimatrix game | 7 | 🔗 link | 🗖 nosplit | ↑ parent "Nash equilibrium"
- 7. Geometry | 0, 8, 291 | 🔗 link | 🗖 nosplit | ↑ parent "Mathematics"
  - 7.1. Sphere | 0, 1, 10 | 🔗 link | 🗖 nosplit | ↑ parent "Geometry"
    - 7.1.1. 3-sphere ( $S^{3}$ ) | 10 | 🔗 link | 🗖 nosplit | ↑ parent "Sphere"
  - 7.2. Four-dimensional space | 9, 1, 37 | 🔗 link | 🗖 nosplit | ↑ parent "Geometry"
    - 7.2.1. Visualizing 4D | 28 | 🔗 link | 🗖 nosplit | ↑ parent "Four-dimensional space"
  - 7.3. Polytope | 0, 3, 244 | 🔗 link | 🗖 nosplit | ↑ parent "Geometry"
    - 7.3.1. Regular polytope | 5, 2, 244 | 🔗 link | 🗖 nosplit | ↑ parent "Polytope"
      - 7.3.1.1. Platonic solid | 34 | 🔗 link | 🗖 nosplit | ↑ parent "Regular polytope"
      - 7.3.1.2. Classification of regular polytopes | 205 | 🔗 link | 🗖 nosplit | ↑ parent "Regular polytope"
- 8. Mathematician | 110, 3, 582 | 🔗 link | 🗖 nosplit | ↑ parent "Mathematics"
  - 8.1. High flying bird vs gophers | 425 | 🔗 link | 🗖 nosplit | ↑ parent "Mathematician"
  - 8.2. List of mathematicians | 0, 1, 47 | 🔗 link | 🗖 nosplit | ↑ parent "Mathematician"
    - 8.2.1. James Harris Simons | 47 | 🔗 link | 🗖 nosplit | ↑ parent "List of mathematicians"
- 9. Number theory | 0, 11, 13 | 🔗 link | 🗖 nosplit | ↑ parent "Mathematics"
  - 9.1. Modular arithmetic | 0, 3, 13 | 🔗 link | 🗖 nosplit | ↑ parent "Number theory"
    - 9.1.1. Modular multiplication | 0, 1, 0 | 🔗 link | 🗖 nosplit | ↑ parent "Modular arithmetic"
      - 9.1.1.1. Modular multiplicative inverse | 🔗 link | 🗖 nosplit | ↑ parent "Modular multiplication"
    - 9.1.2. Modular exponentiation | 13 | 🔗 link | 🗖 nosplit | ↑ parent "Modular arithmetic"
  - 9.2. Prime number | 0, 6, 0 | 🔗 link | 🗖 nosplit | ↑ parent "Number theory"
    - 9.2.1. Prime number theorem | 🔗 link | 🗖 nosplit | ↑ parent "Prime number"
    - 9.2.2. Primality test | 🔗 link | 🗖 nosplit | ↑ parent "Prime number"
    - 9.2.3. Greatest common divisor | 0, 2, 0 | 🔗 link | 🗖 nosplit | ↑ parent "Prime number"
      - 9.2.3.1. Euclidean algorithm | 0, 1, 0 | 🔗 link | 🗖 nosplit | ↑ parent "Greatest common divisor"
        
        9.2.3.1.1. Extended Euclidean algorithm | 🔗 link | 🗖 nosplit | ↑ parent "Euclidean algorithm"
    - 9.2.4. Least common multiple (lcm) | 🔗 link | 🗖 nosplit | ↑ parent "Prime number"
- 10. Numerical analysis | 93, 1, 160 | 🔗 link | 🗖 nosplit | ↑ parent "Mathematics"
  - 10.1. Perturbation theory | 67 | 🔗 link | 🗖 nosplit | ↑ parent "Numerical analysis"
- 11. Polynomial | 0, 6, 53 | 🔗 link | 🗖 nosplit | ↑ parent "Mathematics"
  - 11.1. Algebraic equation | 0, 2, 13 | 🔗 link | 🗖 nosplit | ↑ parent "Polynomial"
    - 11.1.1. Algebraic number | 0, 1, 13 | 🔗 link | 🗖 nosplit | ↑ parent "Algebraic equation"
      - 11.1.1.1. Transcendental number | 13 | 🔗 link | 🗖 nosplit | ↑ parent "Algebraic number"
  - 11.2. Galois theory | 🔗 link | 🗖 nosplit | ↑ parent "Polynomial"
  - 11.3. Diophantine equation | 11, 1, 40 | 🔗 link | 🗖 nosplit | ↑ parent "Polynomial"
    - 11.3.1. Undecidable Diophantine equation problems | 29 | 🔗 link | 🗖 nosplit | ↑ parent "Diophantine equation"
- 12. Probability | 🔗 link | 🗖 nosplit | ↑ parent "Mathematics"
- 13. Mathematics bibliography | 33, 2, 137 | 🔗 link | 🗖 nosplit | ↑ parent "Mathematics"
  - 13.1. Mathematics YouTube channels | 0, 1, 104 | 🔗 link | 🗖 nosplit | ↑ parent "Mathematics bibliography"
    - 13.1.1. 3Blue1Brown | 104 | 🔗 link | 🗖 nosplit | ↑ parent "Mathematics YouTube channels"

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