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• 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.