ENC Proposal [a modification of the iLumina Proposal]
1.
Elementary
Mathematics
1.1. Numbers
1.1.1. Natural
1.1.2. Integers
1.1.3. Rational
1.1.4. Irrational
1.1.4.1. Algebraic
1.1.4.2.
p
1.1.4.3. e
1.1.5. Real
1.1.6. Complex
1.2. Arithmetic
1.1.1.
Operations
1.1.1.1. Addition
1.1.1.2. Subtraction
1.1.1.3. Multiplication
1.1.1.4. Division
1.1.1.5. Square
Roots
1.1.1.6. Factorials
1.1.2.
Fractions
1.1.2.1. Equivalent
Fractions
1.1.2.2. Addition
1.1.2.3. Subtraction
1.1.2.4. Multiplication
1.1.2.5. Division
1.1.2.6. Ratio
and Proportion
1.1.3.
Decimals
1.1.3.1. Addition
1.1.3.2. Subtraction
1.1.3.3. Multiplication
1.1.3.4. Division
1.1.3.5. Percents
1.1.4.
Estimation
1.1.5.
Comparison of numbers
1.1.6.
Exponents
1.1.6.1. Multiplication
1.1.6.2. Division
1.1.6.3. Powers
1.2. Patterns and Sequences
1.2.1.
Geometric Patterns
1.2.2.
Tilings and Tessellations
1.2.3.
Golden Ratio
1.2.4.
Fibonacci Sequence
1.2.5.
Arithmetic Sequence
1.2.6.
Geometric Sequence
1.3. Shapes and Figures
1.3.1.
Plane shapes
1.3.2.
Spatial Sense
1.3.3.
Symmetry
1.3.4.
Similar Figures
1.3.5.
Solid Shapes
1.4. Measurement
1.4.1.
Units of Measurement
1.4.1.1. Metric
System
1.4.1.2. Standard
Units
1.4.2.
Linear Measure
1.4.2.1. Distance
1.4.2.2. Circumference
1.4.2.3. Perimeter
1.4.2.4. Scale
1.4.3.
Area
1.4.3.1. Area
of Polygons
1.4.3.2. Area
of Circles
1.4.3.3. Surface
Area
1.4.4.
Volume
1.5.
Data
1.5.1.
Data Representation
1.5.1.1. Bar
graph
1.5.1.2. Box-and-whiskers
plot
1.5.1.3. Circle
graph/pie graph
1.5.1.4. Graphing
1.5.1.5. Histogram
1.5.1.6. Line-of-best-fit
1.5.1.7. Line
plot
1.5.1.8. Pictograph
1.5.1.9. Scatter
plot
1.5.1.10. Stem-and-leaf
plot
1.5.1.11. Table
1.5.2.
Data Collection
1.5.2.1. Experiment
1.5.2.2. Hypothesis
1.5.2.3. Sampling
1.5.2.4. Survey
1.5.3.
Data Analysis
1.5.3.1. Measures
of Central Tendency
1.5.3.1.1. Mean
1.5.3.1.2. Median
1.5.3.1.3. Mode
1.5.3.2. Correlation
1.5.3.3. Distribution
2.
Logic and
Foundations
2.1. Logic
2.1.1.
Venn Diagrams
2.1.2.
Propositional and Predicate Logic
2.1.3.
Induction
2.1.4.
Methods of Proof
2.2. Set Theory
2.2.1.
Sets and Set Operations
2.2.2.
Relations and Functions
2.2.3.
Cardinality
2.2.4.
Axiom of Choice
2.3. Computability, Decidability and Recursion
2.4. Model Theory
3.
Algebra and
Discrete Mathematics
3.1. Algebra
3.1.1.
Graphing
3.1.2.
Functions
3.1.2.1. Linear
3.1.2.2. Quadratic
3.1.2.3. Polynomial
3.1.2.4. Rational
3.1.2.5. Exponential
3.1.2.6. Logarithmic
3.1.2.7. Piece-wise
3.1.2.8. Step
3.1.3.
Equations
3.1.3.1. Linear
3.1.3.2. Quadratic
3.1.3.3. Polynomial
3.1.3.4. Rational
3.1.3.5. Exponential
3.1.3.6. Logarithmic
3.1.3.7. Systems
3.1.4.
Inequalities
3.1.5.
Matrices
3.1.6.
Sequences and Series
3.1.7.
Algebraic Proof
3.2. Linear Algebra
3.2.1.
Systems of Linear Equations
3.2.2.
Matrix algebra
3.2.3.
Vectors in R3
3.2.4.
Vector Spaces
3.2.5.
Linear Transformations
3.2.6.
Eigenvalues and Eigenvectors
3.2.7.
Inner Product Spaces
3.3. Abstract Algebra
3.3.1.
Groups
3.3.2.
Rings and Ideals
3.3.3.
Fields
3.3.4.
Galois Theory
3.3.5.
Multilinear Algebra
3.4. Number Theory
3.4.1.
Integers
3.4.2.
Primes
3.4.2.1. Divisibility
3.4.2.2. Factorization
3.4.2.3. Distributions
of Primes
3.4.3.
Congruences
3.4.4.
Diophantine Equations
3.4.5.
Irrational Numbers
3.4.6.
Famous Problems
3.4.7.
Coding Theory
3.4.8.
Cryptography
3.5. Discrete Mathematics
3.5.1.
Cellular Automata
3.5.2.
Combinatorics
3.5.3.
Game Theory
3.5.4.
Algorithms
3.5.5.
Graph Theory
3.5.6.
Linear Programming
3.5.7.
Order and Lattices
3.5.8.
Theory of Computation
3.6. Modular Arithmetic
3.7. Category Theory
3.8. K-Theory
3.9. Homological Algebra
4.
Geometry
4.1.
Plane Geometry
4.1.1.
Measurement
4.1.2.
Geometric Proof
4.1.3.
Parallel and Perpendicular Lines
4.1.4.
Angles
4.1.5.
Triangles
4.1.5.1. Pythagorean
Theorem
4.1.5.2. Properties
of Right Triangles
4.1.6.
Congruence
4.1.7.
Similarity
4.1.8.
Polygons
4.1.8.1. Rectangles
4.1.8.2. Squares
4.1.8.3. Trapezoids
4.1.8.4. Pentagons
4.1.8.5. Hexagons
4.1.8.6. Regular
Polygons
4.1.9.
Circles
4.2. Solid Geometry
4.2.1.
Lines and Planes
4.2.2.
Angles
4.2.3.
Spheres
4.2.4.
Cones
4.2.5.
Cylinders
4.2.6.
Pyramids
4.2.7.
Prisms
4.2.8.
Polyhedra
4.3. Analytic Geometry
4.3.1.
Cartesian Coordinates
4.3.2.
Lines
4.3.3.
Circles
4.3.4.
Planes
4.3.5.
Conics
4.3.6.
Polar Coordinates
4.3.7.
Parametric Curves
4.3.8.
Surfaces
4.3.9.
Curvilinear Coordinates
4.3.10. Distance
Formula
4.4. Projective Geometry
4.5. Differential Geometry
4.6. Algebraic Geometry
4.7. Topology
4.7.1.
Point Set Topology
4.7.2.
General Topology
4.7.3.
Differential Topology
4.7.4.
Algebraic Topology
4.8.
Trigonometry
4.8.1.
Angles
4.8.2.
Trigonometric Functions
4.8.3.
Inverse Trigonometric Functions
4.8.4.
Trigonometric Identities
4.8.5.
Trigonometric Equations
4.8.6.
Roots of Unity
4.8.7.
Spherical Trigonometry
5.
Calculus
5.1. Single Variable
5.1.1.
Functions
5.1.2.
Limits
5.1.3.
Continuity
5.1.4.
Differentiation
5.1.5.
Integration
5.1.6.
Series
5.2. Several Variables
5.2.1.
Functions of Several Variables
5.2.2.
Limits
5.2.3.
Continuity
5.2.4.
Partial Derivatives
5.2.5.
Multiple integrals
5.2.6.
Taylor Series
5.3. Advanced Calculus
5.3.1.
Vector Valued Functions
5.3.2.
Line Integrals
5.3.3.
Surface Integrals
5.3.4.
Stokes Theorem
5.3.5.
Linear spaces
5.3.6.
Fourier Series
5.3.7.
Orthogonal Functions
5.4. Tensor Calculus
5.5. Calculus of Variations
5.6. Operational Calculus
6.
Analysis
6.1. Real Analysis
6.1.1.
Metric Spaces
6.1.2.
Convergence
6.1.3.
Continuity
6.1.4.
Differentiation
6.1.5.
Integration
6.1.6.
Measure Theory
6.2. Complex Analysis
6.2.1.
Convergence
6.2.2.
Infinite Series
6.2.3.
Analytic Functions
6.2.4.
Integration
6.2.5.
Contour Integrals
6.2.6.
Conformal Mappings
6.2.7.
Several Complex Variables
6.3. Numerical Analysis
6.3.1.
Computer Arithmetic
6.3.2.
Solutions of Equations
6.3.3.
Solutions of Systems
6.3.4.
Interpolation
6.3.5.
Numerical Differentiation
6.3.6.
Numerical Integration
6.3.7.
Numerical Solutions of
ODEs
6.3.8.
Numerical Solutions of
PDEs
6.3.9.
Miscellaneous
6.4. Signal Analysis
6.4.1.
Fourier Series
6.4.2.
Fourier Transforms
6.4.3.
Filters
6.4.4.
Noise
6.4.5.
Sampling Theory
6.4.6.
Wavelet Analysis
6.4.7.
Data Compression
6.4.8.
Image Processing
6.5. Functional Analysis
6.5.1.
Hilbert Spaces
6.5.2.
Banach Spaces
6.5.3.
Topological Spaces
6.5.4.
Locally Convex Spaces
6.5.5.
Bounded Operators
6.5.6.
Spectral Theorem
6.5.7.
Unbounded Operators
6.6. Harmonic Analysis
6.7. Global Analysis
7.
Differential
Equations
7.1. Ordinary Differential Equations
7.1.1.
First Order
7.1.2.
Second Order
7.1.3.
Linear Oscillations
7.1.4.
Nonlinear Oscillations
7.1.5.
Systems of Differential Equations
7.1.6.
Sturm - Liouville Problems
7.1.7.
Special Functions
7.1.8.
Power Series Methods
7.1.9.
Laplace Transforms
7.2. Partial Differential Equations
7.2.1.
First Order
7.2.2.
Elliptic
7.2.3.
Parabolic
7.2.4.
Hyperbolic
7.2.5.
Integral Transforms
7.2.6.
Integral Equations
7.2.7.
Potential Theory
7.2.8.
Nonlinear Equations
7.2.9.
Symmetries and Integrability
7.3. Difference Equations
7.3.1.
First Order
7.3.2.
Second Order
7.3.3.
Linear Systems
7.3.4.
Z-Transforms
7.3.5.
Orthogonal Polynomials
7.4. Dynamical Systems
7.4.1.
1D Maps
7.4.2.
2D Maps
7.4.3.
Lyapunov
Exponents
7.4.4.
Bifurcations
7.4.5.
Fractals
7.4.6.
Differential
Dynamics
7.4.7.
Conservative
Dynamics
7.4.8.
Chaos
7.4.9.
Complex
Dynamical Systems
8.
Statistics
and Probability
8.1. Statistics
8.1.1.
Sampling
8.1.2.
Expectation Value and Variance
8.1.3.
Linear Regression
8.1.4.
Nonlinear Regression
8.1.5.
Queuing Theory
8.1.6.
Bayesian Statistics
8.2. Probability
8.2.1.
Brownian Motion
8.2.2.
Random Variables
8.2.2.1. Discrete
Distributions
8.2.2.2. Continuous
Distributions
8.2.2.3. Expectation
Value
8.2.3.
Central Limit Theorem
8.2.4.
Markov Chains
8.2.5.
Probability Measures
8.2.6.
Stochastic Processes
9.
Applied
Mathematics
9.1. Mathematical Physics
9.2. Mathematical Economics
9.3. Mathematical Biology
9.4. Mathematics for Business
9.5. Engineering Mathematics
9.6. Mathematical Sociology
9.7. Mathematics for Social Sciences
9.8. Mathematics for Computer Science
10. Mathematics History
10.1.
Famous Problems
10.2.
Biographies of Mathematicians