Mathematics Chapters

Mathematics Chapters

  1. Counting and numbers:
  2. 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
  3. 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
  4. 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
  5. Trigonometry:
    • Trigonometric functions and their properties
    • Trigonometric identities and equations
    • Trigonometric ratios and applications
    • Inverse trigonometric functions
    • Trigonometric limits and derivatives
    • Trigonometric integrals
  6. 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
  7. 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
  8. 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
  9. 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
  10. 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
  11. 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
  12. 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
  13. 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
  14. 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
  15. 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
  16. 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
  17. 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
  18. 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
  19. 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
  20. 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
  21. 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
  22. 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
  23. 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
  24. 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
  25. 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
  26. 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
  27. 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
  28. 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
  29. 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
  30. 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

  1. Group theory
  2. Ring theory
  3. Field theory
  4. Abstract algebra
  5. Number fields and Galois theory
  6. Real analysis
  7. Complex analysis
  8. Measure theory
  9. Topological vector spaces
  10. Functional analysis
  11. Nonlinear dynamics and chaos
  12. Numerical analysis
  13. Optimization theory
  14. Control theory
  15. Probability theory (including stochastic processes)
  16. Game theory (in more depth)
  17. Cryptography
  18. Coding theory
  19. Computational complexity theory
  20. Graph algorithms
  21. Discrete optimization
  22. Algebraic topology
  23. Differential topology
  24. Manifolds and differential geometry (in more depth)
  25. Lie groups and Lie algebras
  26. Algebraic geometry
  27. Category theory
  28. Homological algebra
  29. Representation theory
  30. Set theory (in more depth)
  31. Algebraic number theory
  32. Analytic number theory
  33. Modular forms and elliptic curves
  34. p-adic numbers
  35. Arakelov theory
  36. Diophantine equations
  37. Algebraic combinatorics
  38. Enumerative combinatorics
  39. Design theory
  40. Coding theory (in more depth)
  41. Computational algebraic geometry
  42. Computational number theory
  43. Computational group theory
  44. Computational topology
  45. Cryptography (in more depth)
  46. Geometric measure theory
  47. Geometric analysis
  48. Nonlinear partial differential equations
  49. Harmonic analysis
  50. Operator algebras
  51. Quantum groups and Hopf algebras
  52. Quantum mechanics and quantum field theory
  53. Algebraic quantum field theory
  54. Knot theory and topological quantum field theory
  55. Representation theory (in more depth)
  56. Quantum computation
  57. Homotopy theory
  58. Stable homotopy theory
  59. Algebraic topology (in more depth)
  60. Algebraic K-theory
  61. Topological K-theory
  62. Spectral sequences
  63. Derived categories
  64. Homological mirror symmetry
  65. Noncommutative geometry
  66. Motivic homotopy theory
  67. Arithmetic geometry
  68. Langlands program
  69. Category theory (in more depth)
  70. Higher category theory

Leave a Comment

Your email address will not be published. Required fields are marked *

Scroll to Top