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math.md

Level 1: Foundational Mathematics (1-3 months)

Algebra

Equations and Inequalities

• Linear equations and systems of linear equations
• Quadratic equations and formulas
• Inequalities and systems of inequalities

Functions

• Domain and range
• Composition and inverse functions
• Graphing functions

Calculus

Limits

• Basic limit theorems (sum, product, chain rule)
• Squeeze theorem and Sandwich theorem

Derivatives

• Rules of differentiation (power rule, product rule, quotient rule)
• Geometric interpretation of derivatives

Integrals

• Basic integration rules (substitution, integration by parts)
• Fundamental Theorem of Calculus

Geometry and Trigonometry

Points, Lines, and Planes

• Distance and midpoint formulas
• Slope and equation of a line

Triangles and Circles

• Angles and trigonometric functions (sine, cosine, tangent)
• Pythagorean identity and trigonometric identities

Number Theory

Divisibility and Prime Numbers

• Divisibility rules and prime factorization
• Euclidean algorithm and greatest common divisors

Congruences and Diophantine Equations

• Modular arithmetic and congruences
• Linear Diophantine equations and solutions

Level 2: Intermediate Mathematics (3-6 months)

Linear Algebra

Vector Operations

• Vector addition and scalar multiplication
• Dot product and cross product

Matrix Operations

• Matrix addition and multiplication
• Inverse and determinant of a matrix

Linear Transformations

• Matrix representation and composition
• Eigenvalues and eigenvectors

Differential Equations

First-Order ODEs

• Separable and linear first-order ODEs
• Bernoulli and Riccati equations

Higher-Order ODEs

• Linear homogeneous and nonhomogeneous ODEs
• Undetermined coefficients method

Combinatorics and Graph Theory

Permutations and Combinations

• Fundamental counting principle and permutations
• Combinations and the binomial theorem

Graph Theory

• Basic graph concepts (vertices, edges, degrees)
• Graph traversals and connectivity

Probability and Statistics

Basic Concepts

• Events, probability measures, and conditional probability
• Random variables and probability distributions

Statistical Inference

• Hypothesis testing and confidence intervals
• Regression analysis and correlation

Level 3: Advanced Mathematics (6-12 months)

Real Analysis

Sequences and Series

• Convergence tests and series convergence
• Power series and Taylor series

Continuity and Differentiability

• Continuity and uniform continuity
• Differentiability and mean value theorem

Abstract Algebra

Group Theory

• Group axioms and group homomorphisms
• Subgroups and quotient groups

Ring Theory

• Ring axioms and ring homomorphisms
• Ideals and quotient rings

Topology

Point-Set Topology

• Topological spaces and topological invariants
• Connectedness and compactness

Algebraic Topology

• Fundamental group and homotopy
• Betti numbers and Euler characteristic

Measure Theory and Functional Analysis

Measure Theory

• Lebesgue measure and measurable functions
• Lebesgue-Stieltjes integral and Fubini's theorem

Functional Analysis

• Normed vector spaces and operator norms
• Banach spaces and Hahn-Banach theorem

Additional Tips and Resources

• Practice solving problems on platforms like Project Euler, Brilliant, and AoPS
• Read mathematics textbooks and online resources: MIT OpenCourseWare, Wolfram Alpha, and Math StackExchange
• Join online communities: Math StackExchange, Reddit's r/math, and Math Overflow
• Participate in math competitions: AMC, AIME, and Putnam Competition
• Read research papers and articles on arXiv, ResearchGate, and Academia.edu

Remember, mastering mathematics takes time and dedication. Focus on building a strong foundation, and gradually move on to more advanced topics. With persistence and practice, you'll be well-equipped to tackle a significant portion of math questions on StackExchange.