## Mathematics Chapters

**Counting and numbers:**- Natural numbers and their properties
- Whole numbers and their properties
- Integers and their properties
- Rational numbers and their properties
- Irrational numbers and their properties
- Real numbers and their properties
- Complex numbers and their properties

**Arithmetic:**- Addition and subtraction of numbers
- Multiplication and division of numbers
- Prime numbers and their properties
- Divisibility and factorization
- Greatest common divisor and least common multiple
- Modular arithmetic
- Basic number theory

**Algebra:**- Variables and expressions
- Equations and inequalities
- Linear equations and systems of linear equations
- Quadratic equations and systems of quadratic equations
- Polynomials and their properties
- Rational expressions and their properties
- Exponential and logarithmic functions

**Geometry:**- Euclidean geometry and its axioms
- Triangles and their properties
- Circles and their properties
- Quadrilaterals and their properties
- Similarity and congruence
- Trigonometry and its applications
- Coordinate geometry

**Trigonometry:**- Trigonometric functions and their properties
- Trigonometric identities and equations
- Trigonometric ratios and applications
- Inverse trigonometric functions
- Trigonometric limits and derivatives
- Trigonometric integrals

**Calculus:**- Limits and continuity
- Derivatives and their applications
- Differentiation rules and formulas
- Mean value theorem and its applications
- Optimization and related rates
- Antiderivatives and integration
- Integration techniques

**Vectors:**- Vector algebra and its properties
- Scalar and vector products
- Lines and planes in space
- Vector calculus and its applications
- Gradient, divergence, and curl
- Vector fields and their properties

**Matrices:**- Matrix algebra and its properties
- Determinants and their properties
- Systems of linear equations and matrices
- Eigenvalues and eigenvectors
- Diagonalization and similar matrices
- Applications of matrices in geometry and physics

**Probability:**- Probability spaces and events
- Conditional probability and Bayes’ theorem
- Discrete and continuous random variables
- Probability distributions and their properties
- Expected value and variance
- Central limit theorem and its applications
- Markov chains and their applications

**Statistics**:- Descriptive statistics and measures of central tendency
- Measures of dispersion and skewness
- Probability distributions and their properties
- Point estimation and interval estimation
- Hypothesis testing and significance levels
- Chi-square test and analysis of variance
- Regression and correlation analysis

**Complex numbers:**- Introduction to complex numbers
- Operations on complex numbers
- Polar form and De Moivre’s theorem
- Complex functions and their properties
- Complex analysis and its applications
- Complex integration and Cauchy’s theorem
- Laurent series and singularities

**Differential equations:**- Introduction to differential equations
- First-order differential equations and their solutions
- Second-order differential equations and their solutions
- Higher-order differential equations and their solutions
- Systems of differential equations and their solutions
- Laplace transforms and their applications
- Fourier series and their applications

**Discrete mathematics:**- Set theory and its properties
- Relations and functions
- Graph theory and its applications
- Trees and their properties
- Combinatorics and counting principles
- Recurrence relations and their solutions
- Boolean algebra and its applications

**Numerical methods:**- Root finding and optimization
- Interpolation and extrapolation
- Numerical differentiation and integration
- Matrix and linear equation solvers
- Differential equation solvers
- Monte Carlo methods and simulations
- Computer programming and algorithms

**Linear algebra:**- Vector spaces and subspaces
- Linear transformations and matrices
- Basis and dimension
- Orthogonality and inner product spaces
- Eigenvalues, eigenvectors, and diagonalization
- Positive definite matrices and quadratic forms
- Applications of linear algebra in science and engineering

**Multivariable Calculus:**- Vector-valued functions and curves in space
- Partial derivatives and gradients
- Multiple integrals and their applications
- Line integrals and Green’s theorem
- Surface integrals and the divergence theorem
- Stokes’ theorem and applications
- Vector calculus and its applications

**Real Analysis:**- Set theory and topology of the real line
- Limits, continuity, and differentiability
- Mean value theorem and Taylor’s theorem
- Riemann integration and its properties
- Sequences and series of functions
- Fourier series and Fourier transforms
- Lebesgue integration and measure theory

**Group Theory:**- Introduction to groups and group axioms
- Subgroups and cosets
- Group homomorphisms and isomorphisms
- Normal subgroups and quotient groups
- Group actions and applications
- Sylow theorems and finite groups
- Simple groups and their classification

**Ring Theory:**- Introduction to rings and ring axioms
- Ideals and quotient rings
- Ring homomorphisms and isomorphisms
- Integral domains and fields
- Polynomial rings and their properties
- Euclidean domains and unique factorization domains
- Applications of ring theory in algebra and geometry

**Topology:**- Introduction to topology and topological spaces
- Continuity and convergence in topological spaces
- Separation axioms and connectedness
- Compactness and its properties
- Metric spaces and completeness
- Homotopy and fundamental groups
- Applications of topology in geometry and analysis

**Abstract Algebra:**- Groups, rings, and fields
- Homomorphisms and isomorphisms
- Subgroups, subrings, and subfields
- Quotient groups, quotient rings, and quotient fields
- Group actions and permutation groups
- Galois theory and field extensions
- Applications of abstract algebra in number theory and geometry

**Differential Geometry:**- Curves and surfaces in Euclidean space
- Tangent vectors and normal vectors
- Curvature and torsion of curves
- Gaussian and mean curvature of surfaces
- Riemannian manifolds and their properties
- Geodesics and parallel transport
- Applications of differential geometry in physics and engineering

**Complex Analysis:**- Analytic functions and their properties
- Cauchy-Riemann equations and harmonic functions
- Complex integration and Cauchy’s theorem
- Laurent series and singularities
- Residue theorem and its applications
- Conformal mappings and the Riemann mapping theorem
- Applications of complex analysis in engineering and physics

**Algebraic Geometry:**- Projective and affine varieties
- Algebraic curves and surfaces
- Bezout’s theorem and intersection theory
- Divisors and line bundles
- Riemann-Roch theorem and cohomology
- Schemes and their properties
- Applications of algebraic geometry in number theory and cryptography

**Partial Differential Equations:**- Classification of PDEs and their properties
- First-order PDEs and characteristics
- Second-order linear PDEs and their solutions
- Nonlinear PDEs and their solutions
- Numerical methods for solving PDEs
- Applications of PDEs in physics and engineering
- Analytical methods for solving PDEs

**Number Theory:**- Prime numbers and their distribution
- Congruences and modular arithmetic
- Diophantine equations and Fermat’s last theorem
- Quadratic residues and reciprocity laws
- Arithmetic functions and their properties
- Continued fractions and their applications
- Cryptography and number theory

**Algebraic Number Theory:**- Algebraic integers and rings of algebraic integers
- Unique factorization domains and ideals
- Class groups and the class number problem
- Dirichlet’s unit theorem and its applications
- Cyclotomic fields and Kummer theory
- Galois representations and the inverse Galois problem
- Applications of algebraic number theory in cryptography and coding theory

**Representation Theory:**- Group representations and their properties
- Character theory and orthogonality relations
- Maschke’s theorem and complete reducibility
- Schur’s lemma and tensor products
- Lie algebras and their representations
- Applications of representation theory in physics and chemistry
- Symmetry and its role in representation theory

**Algebraic Topology:**- Homotopy groups and their properties
- Homology groups and their properties
- Mayer-Vietoris sequence and applications
- Cohomology groups and their properties
- Cup product and Poincaré duality
- K-theory and vector bundles
- Applications of algebraic topology in geometry and physics

**Analytic Number Theory:**- Dirichlet series and Euler products
- Prime number theorem and Riemann hypothesis
- Distribution of prime numbers and zeroes of zeta function
- Modular forms and L-functions
- Tate’s thesis and the Birch and Swinnerton-Dyer conjecture
- Applications of analytic number theory in cryptography and coding theory
- Circle method and sieve methods

## Higher Mathematics

- Group theory
- Ring theory
- Field theory
- Abstract algebra
- Number fields and Galois theory
- Real analysis
- Complex analysis
- Measure theory
- Topological vector spaces
- Functional analysis
- Nonlinear dynamics and chaos
- Numerical analysis
- Optimization theory
- Control theory
- Probability theory (including stochastic processes)
- Game theory (in more depth)
- Cryptography
- Coding theory
- Computational complexity theory
- Graph algorithms
- Discrete optimization
- Algebraic topology
- Differential topology
- Manifolds and differential geometry (in more depth)
- Lie groups and Lie algebras
- Algebraic geometry
- Category theory
- Homological algebra
- Representation theory
- Set theory (in more depth)
- Algebraic number theory
- Analytic number theory
- Modular forms and elliptic curves
- p-adic numbers
- Arakelov theory
- Diophantine equations
- Algebraic combinatorics
- Enumerative combinatorics
- Design theory
- Coding theory (in more depth)
- Computational algebraic geometry
- Computational number theory
- Computational group theory
- Computational topology
- Cryptography (in more depth)
- Geometric measure theory
- Geometric analysis
- Nonlinear partial differential equations
- Harmonic analysis
- Operator algebras
- Quantum groups and Hopf algebras
- Quantum mechanics and quantum field theory
- Algebraic quantum field theory
- Knot theory and topological quantum field theory
- Representation theory (in more depth)
- Quantum computation
- Homotopy theory
- Stable homotopy theory
- Algebraic topology (in more depth)
- Algebraic K-theory
- Topological K-theory
- Spectral sequences
- Derived categories
- Homological mirror symmetry
- Noncommutative geometry
- Motivic homotopy theory
- Arithmetic geometry
- Langlands program
- Category theory (in more depth)
- Higher category theory

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