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Epistemology - m a t h 4 w i s d o m - g m a i l
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*My work is in the Public Domain for all to share freely.*
- 读物 书 影片 维基百科
Introduction E9F5FC Questions FFFFC0 Software |
基本概念 - Orientation: Marked and unmarked opposites, outside and inside, antisymmetry, sign of permutation.
- Amounts: Summation. Identifying units (items, objects, areas, etc.) with numbers and adding them up.
- Smoothness: Change in one dimension is (extrinsically) comparable to (or even, intrinsically, of a much smaller order than) change in another dimension. Compare with difference between subconscious continuum and conscious discreteness.
- Generating functions relate: symmetry of analytic functions, algorithms, finite combinatorial symmetry. (Think of as a vector bundle - the infinite sequence ({$x_i$}) is the base space, and the coefficient is the fiber, and the fibers are related.)
- Linear operators - something you can have more of or less of (proportionately) - transformational action (like rotation or translation).
Math Companion Concepts I am trying to organize the 99 concepts below. Next to each concept I have assigned the branches of math where they are important, with the 26 branches taken from the same Companion, and referenced by the numbers given in the book, as follows:
22., 23. Zermelo-Fraenkel Axioms. Ways of constructing sets; of knowing if they are the same set; and of assuring that they don't have cycles. 22. Axiom of Choice. Given a set, you can select a representative element. 22. Axiom of Determinacy. The existence of a winning strategy for at least one of two players of an infinite game. 23., 22. Peano Axioms. Describes the natural numbers. - Natural numbers
11., 6., 26., 4. Pi 1., 22. Irrational and Transcendental Numbers - Solutions and nonsolutions to linear equations, polynomials equations, with integer coefficients.
9. Determinants. Sum of oriented permutations = oriented volume of matrix. 18., 6., 17., 10a., 10b., B1. Knot Polynomials 18b., 18a. Generating Functions 2., 11. Exponential and Logarithmic Functions X 11., 7., 15. Trigonometric Functions XX 11., 9., 15., 7. Spherical Harmonics 12., 11., 21. Linear and Nonlinear Waves and Solitons 2. Gamma Function 2., 5. L-Functions 2., 5., 7. Riemann Zeta Function 22., 15., 25. Measures 1., 4., 6., 15. Quadratic Forms 12. Distributions 19 b., 22., 11., 26. Probability Distributions 7., 4., BX1. Differential Forms and Integration X 9., 15., 23. Tensor Products XXX 9., 15., 11., 12. Linear Operators and Their Properties 4. Modular Forms 6., 7., 10a., 1. Universal Covers. A space Y and a continuous surjection from Y to X. 6., 7. Vector Bundles. A space E and a continuous map p from E to X such that the inverse image {$p^{-1}(x)$} is an (n-dimensional) vector space. 25. Ising Model
12. Euler and Navier-Stokes Equations XX 12., 11., 15., 7., 6. Heat Equation XX 12., 14., 15., 24., 25., 11. Schroedinger Equation 12., 18., 21., 11., 19b. Special Functions 4. Elliptic Curves
XX 11., 9., 12., 26. Fourier Transform - Use complex roots {$e^{ir\theta}$} as set of frequencies to identify sequence and function.
11. Fast Fourier Transform XXX 11., 15., 12., 19b., 18., 7., 2., 14., 9. Transforms - Identifies sequence (and its self-symmetry) and function (and its self-symmetry).
22. Countable and Uncountable Sets 14. Mandelbrot 22. Ordinals 22. Cardinals 23., 6. Categories 19a. Graphs X 18., 20., 11., 9. Expanders 10a. Buildings 10a., 9. Leech Lattice 4., 5. Varieties 20., 9., 18. Matroids 3., 10b. Modular Arithmetic 10a., 12., 6. Fuchsian Groups X 5., 4., 15., 7., 6., 11., 23. Quantum Groups - Algebras of polynomials up to equivalence on a variety - where the points are group elements.
- Map from X to Y gives rise to adjoint map from C[Y] to C[X].
- Algebras has coproduct. Group structure yields Hopf algebra or quantum group (which has additional structure analogous to that of a Lie group).
- Makes sense over any field.
- There exist noncommutative Hopf algebras based on deformations q, thus grounding noncommutative geometry.
- Hopf algebras are the next simplest categories after abelian groups that admit a Fourier transform.
- Axioms of Hopf algebra are symmetric under arrow reversal.
- Bicrossproduct quantum groups are simultaneously coordinate algebras and symmetry algebras.
1. Galois Groups 6., 4. Braid Groups 1. Ideal Class Group 10b., 9., 17. Monster Group 10b., 9., 19b. Permutation Groups 1., 4., 9. Rings, Ideals, and Modules 4., 1., 5. Schemes 1. Number Fields X 9., 15. Representations X 9., 15., 10a. Quaternions, Octonions and Normed Division Algebras 23., 22. Models of Set Theory
4., 6. Projective Space 7., 6., 4., 8. Manifolds 7., 4., 16. Calabi-Yau Manifolds X 7., 22., 18., 20., 13., 15. Metric Spaces 7., 6., 11., 10a. Riemann Surfaces X 7., 6., 15., 14., 24. Symplectic Manifolds 4., 7., 16. Orbifolds 15. C*-Algebras 15. Function Spaces 15., 11. Hilbert Spaces 6., 7., 4. Topological Spaces 8., 6., 7., 4., 1., 11. Moduli Spaces XX 15., 12., 11., 9. Normed Spaces and Banach Spaces 15. Von Neumann Algebras
7., 14. Dimension 13., 7. Curvature 7. Compactness and Compactification 6., 7., 4. Genus XX 14., 12., 15., 9., 17., 16., 7. Hamiltonians X 15., 9., 7. Spectrum 9., 10 b. Jordan Normal Form 14., 12. Dynamical Systems and Chaos 25. Phase Transitions
2. Euclidean Algorithm and Continued Fractions 14., 7., 12., 11., 24. Variational Methods X 20., 11., 15., 19b., 26. Quantum Computation 20., 9., 14., 10a., 18. Simplex Algorithm 21., 11. Wavelets 20. Computational Complexity Classes 7., 6., 12. Ricci Flow. Replaces a manifold with a smoother manifold. 14., 12., 7. Optimization and Lagrange Multipliers
10a., 4., 6., 16. Duality 1., 2., 4. Local and Global in Number Theory
X 9., 6., 10a., 10b., 12., 17., 7. Lie Theory 6. Homology and Cohomology 6., 15. K-Theory |

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