Epistemology m a t h 4 w i s d o m - g m a i l +370 607 27 665 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 I'm organizing the concepts in The Princeton Companion to Mathematics. What is the place for geometric constructions? Mathematical answers Quantify answers N Exact counting (Listing) S= 18a. Enumerative combinatorics (polynomial time algorithms for computation) 18., 20., 11., 9. Expanders 18., 6., 17., 10a., 10b., B1. Knot Polynomials N An Extremal problems S= 19a. Extremal Combinatorics 19a. Graphs N Averages S= 19b. Probabilistic combinatorics 19 b., 22., 11., 26. Probability Distributions Explain coincidences S= 18b. Algebraic combinatorics (interpreting formulas) 18b., 18a. Generating Functions Manage discrepancies N An Estimates S= 2. Analytic number theory 2. Euclidean Algorithm and Continued Fractions 2., 11. Exponential and Logarithmic Functions 2. Gamma Function 2., 5. L-Functions 2., 5., 7. Riemann Zeta Function Algo An Approximations S= 21. Numerical analysis (algorithms for approximating the continuum) 21., 11. Wavelets Al An Predictions S= 24. Stochastic processes (model the evolution of random phenomena) Formulate intuition as constraints on equations G An E= 13. General relativity and the Einstein equations (expressing, interpreting and validating a theory of physics) 13., 7. Curvature Solve equations. Any solutions? Unique solution? Constraints on solutions? Al G Linear equations. S= 9. Representation Theory 9. Determinants 9., 10 b. Jordan Normal Form 9., 6., 10a., 10b., 12., 17., 7. Lie Theory 9., 15., 11., 12. Linear Operators and Their Properties 9., 15. Representations 9., 15., 10a. Quaternions, Octonions and Normed Division Algebras 9., 15., 23. Tensor Products N Al Polynomial equations. S= 1. Algebraic numbers 1. Galois Groups 1. Ideal Class Group 1., 22. Irrational and Transcendental Numbers 1., 2., 4. Local and Global in Number Theory 1., 4., 6., 15. Quadratic Forms 1. Number Fields 1., 4., 9. Rings, Ideals, and Modules G Al Polynomial equations in several variables. S= 4. Algebraic geometry 4. Elliptic Curves 4. Modular Forms 4., 6. Projective Space 4., 7., 16. Orbifolds 4., 1., 5. Schemes 4., 5. Varieties N G Diophantine equations. S= 5. Arithmetic geometry 5., 4., 15., 7., 6., 11., 23. Quantum Groups An Al Differential equations. S= 12. Partial differential equations 12. Distributions 12. Euler and Navier-Stokes Equations 12., 11., 15., 7., 6. Heat Equation 12., 11., 21. Linear and Nonlinear Waves and Solitons 12., 14., 15., 24., 25., 11. Schroedinger Equation 12., 18., 21., 11., 19b. Special Functions Articulate instructions. Find explicit proofs and algorithms Algo N S= 20. Computational complexity (what can be computed efficiently or not) 20. Computational Complexity Classes 20., 9., 18. Matroids 20., 11., 15., 19b., 26. Quantum Computation 20., 9., 14., 10a., 18. Simplex Algorithm F Al S= 23. Logic and model theory (formal languages about mathematical structures, whether a proof exists or not) 23., 6. Categories 23., 22. Models of Set Theory 23., 22. Peano Axioms Discover patterns Al N S= Groups (symmetries), 10b. Combinatorial group theory (groups in terms of their generators and relations) 10b., 9., 17. Monster Group 10b., 9., 19b. Permutation Groups Classify structures. N Algo Building blocks and combinations. E= 3. Computational number theory (identifying primes as components or in totality) 3., 10b. Modular Arithmetic Families and exceptions. E= Algebraic topology G An Transformation demonstrates equivalence. E= 7. Differential topology (classifying smooth manifolds - list all smooth structures on any topological manifold and be able to identify them - a certain set of discrete subgroups of the isometry group of any one of the eight model spaces determines a compact manifold with the corresponding geometric structure) 7., 6., 4., 8. Manifolds 7., 4., BX1. Differential Forms and Integration 7., 4., 16. Calabi-Yau Manifolds 7., 14. Dimension 7. Compactness and Compactification 7., 22., 18., 20., 13., 15. Metric Spaces 7., 6., 12. Ricci Flow 7., 6., 11., 10a. Riemann Surfaces 7., 6., 15., 14., 24. Symplectic Manifolds G Al Invariant demonstrates nonequivalence. E= 6. Algebraic topology 6., 4. Braid Groups 6., 7., 4. Genus 6. Homology and Cohomology 6., 15. K-Theory 6., 7., 4. Topological Spaces 6., 7., 10a., 1. Universal Covers 6., 7. Vector Bundles G An Map to a structure E= 8. Moduli spaces (give a geometric structure to the totality of the objects we are trying to classify) 8., 6., 7., 4., 1., 11. Moduli Spaces Improve results An N Weaken hypotheses. E= 15. Operator algebras (expanding from finite-dimensional equations to integral equations) 15. C*-Algebras 15. Function Spaces 15., 11. Hilbert Spaces 15., 12., 11., 9. Normed Spaces and Banach Spaces 15., 9., 7. Spectrum 15. Von Neumann Algebras An G Strengthen conclusions. E= 11. Harmonic analysis (determining the properties of functions that are not explicitly describable, for example, the effect of operators on the boundedness of functions) 11., 9., 12., 26. Fourier Transform 11. Fast Fourier Transform 11., 6., 26., 4. Pi 11., 9., 15., 7. Spherical Harmonics 11., 15., 12., 19b., 18., 7., 2., 14., 9. Transforms 11., 7., 15. Trigonometric Functions Prove a more abstract result. E= Category theory Suspend rigor. Work with arguments that are not fully rigorous. An Algo E=Conditional results = 14. Dynamics (how systems evolve in time) 14., 12. Dynamical Systems and Chaos 14., 12., 15., 9., 17., 16., 7. Hamiltonians 14. Mandelbrot 14., 12., 7. Optimization and Lagrange Multipliers 14., 7., 12., 11., 24. Variational Methods Al An E=Numerical evidence. = 25. Probabilistic models of critical phenomena (modeling thresholds for divergent outcomes) 25. Ising Model 25. Phase Transitions An G E="Illegal" calculations. = 16. Mirror symmetry (reformulating a physical theory's information in a mirror theory) Determine compatibility. Whether different mathematical properties are compatible. An Al E= 17. Vertex operator algebras (formulating perspective: relating quantum data and space-time manifold) Reintrepret ideas. Al G Identify characteristic properties. E= 10a. Geometric group theory (groups in terms of their actions expressed geometrically) 10a. Buildings 10a., 4., 6., 16. Duality 10a., 12., 6. Fuchsian Groups 10a., 9. Leech Lattice N F Generalize after reformulation P= 22. Set theory (distinguishing between cardinals-sets and ordinals-lists and relating the two) 22. Axiom of Choice 22. Axiom of Determinacy 22. Cardinals 22. Countable and Uncountable Sets 22., 15., 25. Measures 22. Ordinals 22., 23. Zermelo-Fraenkel Axioms G N Higher dimensions and several variables. E= 26. High-dimensional geometry and its probabilistic analogues (most efficient boundary for volume, the sphere, models random distributions) N Numbers, G Geometry, Al Algebra, Algo Algorithms, An Analysis, P Proof, F Foundations B1. Chemistry B2. Biology B3. Wavelets and Applications B4. Traffic in Networks B5. Algorithm Design B6. Reliable Transmission of Information B7. Cryptography B8. Economic Reasoning B9. Money B10. Statistics B10. Bayesian Analysis B10., B6., Designs B11. Medical Statistics B12. Philosophical Analysis B13. Music B14. Art BX1. Physics Consider underlying assumptions. Counting, averaging, extremes - all suppose "many". Are the math answers expressing the 12 topologies? Should there be two more kinds? Ideas Bifurcation of topics Continuity vs. Topology (properties that are not affected by continuous transformations) Factorization vs. Primes Consciousness - matching the unconscious and the conscious - is like solving an equation.
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