scientiax.com

Home

Topics of Math

Basic Level

Here are the key topics in basic-level mathematics:

  1. Numbers and Arithmetic:
    • Types of Numbers: Whole numbers, integers, rational numbers, irrational numbers, real numbers
    • Basic Operations: Addition, subtraction, multiplication, division
    • Place Value and Number System
    • Fractions, Decimals, and Percentages
    • Factors and Multiples
    • Prime Numbers and Composite Numbers
    • LCM and HCF (Least Common Multiple and Highest Common Factor)
  2. Algebra:
    • Basic Algebraic Expressions
    • Solving Simple Linear Equations
    • Basic Identities
    • Simplification of Expressions
    • Substitution
  3. Geometry:
    • Basic Geometric Shapes: Circle, triangle, square, rectangle, polygon
    • Perimeter, Area, and Volume of Basic Shapes
    • Angles: Types of angles (acute, right, obtuse), angle sum property
    • Symmetry and Reflection
    • Coordinate Geometry (Introduction)
  4. Measurement:
    • Units of Measurement (Length, Mass, Time, Volume)
    • Conversion of Units
    • Perimeter, Area, and Volume of Simple Objects
    • Time: Reading clocks, time calculations, understanding time intervals
  5. Data Handling and Statistics:
    • Collecting and Organizing Data
    • Mean, Median, Mode
    • Bar Graphs and Pictograms
    • Probability (Basic)
  6. Number Patterns and Sequences:
    • Number Series: Arithmetic sequences
    • Patterns in Numbers: Even, odd, square numbers
    • Basic Concepts of Series and Sequences
  7. Ratio and Proportion:
    • Understanding Ratios and Proportions
    • Direct and Inverse Proportions
  8. Basic Trigonometry:
    • Introduction to Trigonometric Ratios (Sine, Cosine, Tangent) in Right Angled Triangles

These topics provide a foundation for more advanced concepts and problem-solving.

Matric Level

Here are the key topics in mathematics at the matriculation (secondary school) level:

1. Number System:

  • Types of Numbers: Natural numbers, Whole numbers, Integers, Rational numbers, Irrational numbers, Real numbers
  • Real Number System and Operations
  • Laws of Exponents
  • Surds and Indices
  • Absolute Value

2. Algebra:

  • Algebraic Expressions and Identities
  • Factorization (Methods: Grouping, Quadratic trinomials, Difference of squares)
  • Linear Equations in One Variable
  • Quadratic Equations (Roots, Factorization, Completing the square, and Quadratic formula)
  • Polynomials and Division Algorithm
  • Arithmetic Progression (AP) and Geometric Progression (GP)
  • Simplification and Expansion of Algebraic Expressions

3. Geometry:

  • Basic Geometrical Shapes and Properties
  • Properties of Triangles (Pythagoras theorem, Congruence, Similarity)
  • Circles (Theorems on chords, tangents, and secants)
  • Angle Properties (Angle sum property, Parallel lines, and Transversals)
  • Coordinate Geometry: Distance formula, Section formula, Mid-point formula
  • Areas and Perimeter of Geometrical Figures (Triangles, Squares, Rectangles, Parallelograms, Circles)

4. Mensuration:

  • Perimeter and Area of 2D Shapes (Circle, Rectangle, Square, Triangle, Parallelogram)
  • Volume and Surface Area of 3D Shapes (Cube, Cuboid, Cone, Cylinder, Sphere, Hemisphere)
  • Frustum of a Cone

5. Trigonometry:

  • Trigonometric Ratios (Sine, Cosine, Tangent, Cosecant, Secant, Cotangent)
  • Trigonometric Identities (Basic and Compound)
  • Height and Distance Problems (Applications of Trigonometry)
  • Trigonometric Ratios of Angles (30°, 45°, 60°, etc.)

6. Statistics:

  • Collection and Organization of Data
  • Mean, Median, Mode
  • Frequency Distribution
  • Probability (Basic concepts, Simple events)

7. Probability:

  • Understanding Probability: Sample space, Events, Probability of an event
  • Simple Probability Problems

8. Linear Equations and Simultaneous Equations:

  • Solutions of Linear Equations in Two Variables (Graphical Method, Substitution, Elimination)
  • Systems of Linear Equations (Solving using methods like Substitution, Elimination, and Cramer’s Rule)

9. Graphs:

  • Cartesian Plane and Plotting Points
  • Graphs of Linear Equations (y = mx + c)
  • Graphs of Quadratic Functions (y = ax² + bx + c)
  • Interpretation of Graphs

10. Set Theory:

  • Types of Sets: Finite, Infinite, Equal, Null, Singleton, Universal Set
  • Operations on Sets: Union, Intersection, Difference, Complement
  • Venn Diagrams

11. Logarithms:

  • Definition and Properties of Logarithms
  • Laws of Logarithms (Product, Quotient, Power laws)
  • Solving Logarithmic Equations

These topics form a comprehensive framework for the matric level of mathematics and are foundational for higher education in mathematics, engineering, and sciences.

Intermediate Level

Here are the key topics in mathematics at the intermediate (higher secondary school) level:

1. Number System and Real Numbers:

  • Types of Numbers: Natural numbers, Integers, Rational and Irrational numbers
  • Real Number System and Operations
  • Properties of Real Numbers
  • Laws of Exponents and Radicals
  • Absolute Value and Modulus Function
  • Complex Numbers (Introduction, Imaginary Numbers, Real and Complex Parts)

2. Algebra:

  • Polynomials and Their Types (Degree, Coefficients, Roots)
  • Factorization of Polynomials (Methods: Grouping, Long Division, Synthetic Division)
  • Quadratic Equations and Their Solutions (Roots, Factorization, Completing the square, Quadratic formula)
  • Arithmetic Progression (AP) and Geometric Progression (GP)
  • Binomial Theorem (Expansion and Applications)
  • Permutations and Combinations
  • Matrices and Determinants (Basic operations, Inverse, Adjoint, Solving systems of linear equations)

3. Coordinate Geometry:

  • Cartesian Coordinate System
  • Distance Formula, Section Formula, Midpoint Formula
  • Equation of a Line (Slope, Point-Slope form, Intercept form)
  • Conic Sections (Parabolas, Ellipses, Hyperbolas)
  • Circles (Equation, Chords, Tangents, and Secants)

4. Calculus:

  • Limits and Continuity
  • Differentiation (Basic Rules: Power, Product, Quotient, Chain Rule)
  • Derivatives of Standard Functions (Polynomials, Trigonometric, Logarithmic, Exponential)
  • Applications of Derivatives (Tangent, Normal, Rate of Change, Maxima/Minima)
  • Integration (Indefinite and Definite Integrals, Methods of Integration: Substitution, Partial Fractions, Integration by Parts)
  • Area under a Curve (Definite Integrals)
  • Differential Equations (Introduction to First Order, Separable Equations)

5. Trigonometry:

  • Trigonometric Ratios and Identities
  • Trigonometric Functions of Angles (0°, 30°, 45°, 60°, 90°, etc.)
  • Inverse Trigonometric Functions
  • Trigonometric Equations
  • Height and Distance (Applications in Surveying)
  • Trigonometric Identities (Sum, Difference, Double Angle, Half Angle)
  • Solutions of Triangles (Sine Rule, Cosine Rule)

6. Vectors:

  • Vector Operations (Addition, Subtraction, Scalar Multiplication)
  • Dot Product and Cross Product
  • Scalar Triple Product and Vector Triple Product
  • Applications of Vectors in Geometry and Physics

7. Matrices and Determinants:

  • Types of Matrices (Row, Column, Square, Diagonal, Identity)
  • Operations on Matrices (Addition, Multiplication, Transpose)
  • Determinants and Their Properties
  • Inverse of a Matrix (Adjoint method, Properties)
  • Cramer’s Rule for Solving Linear Equations

8. Probability:

  • Basic Concepts of Probability (Sample Space, Events, Probability of an Event)
  • Conditional Probability
  • Addition and Multiplication Theorems
  • Bayes’ Theorem
  • Random Variables and Probability Distributions
  • Binomial Distribution, Poisson Distribution

9. Statistics:

  • Measures of Central Tendency (Mean, Median, Mode)
  • Measures of Dispersion (Range, Variance, Standard Deviation)
  • Skewness and Kurtosis
  • Correlation and Regression Analysis
  • Probability Distributions (Binomial, Normal, Poisson)

10. Differential Equations:

  • Introduction to Differential Equations
  • First-Order Differential Equations (Separable, Linear, Homogeneous)
  • Second-Order Differential Equations (Homogeneous, Non-Homogeneous)

11. Linear Programming:

  • Formulation of Linear Programming Problems
  • Graphical Method of Solving Linear Programming Problems
  • Simplex Method (Basic Introduction)

12. Dynamics and Kinematics:

  • Motion in One, Two, and Three Dimensions
  • Laws of Motion (Newton’s Laws)
  • Work, Energy, and Power
  • Motion under Gravity
  • Projectile Motion

These topics provide a strong foundation in mathematics at the intermediate level, preparing students for higher studies in fields such as engineering, physics, economics, and advanced mathematics.

Undergraduate Level

Here are the key topics in mathematics at the undergraduate (college/university) level, categorized into different branches of mathematics:

1. Calculus:

  • Limits and Continuity: Definition of limits, L’Hopital’s rule, Continuity of functions
  • Differentiation: Rules of differentiation (product, quotient, chain rule), Derivatives of standard functions (polynomials, trigonometric, exponential, logarithmic), Applications of differentiation (Maxima, Minima, Rate of Change)
  • Integration: Definite and indefinite integrals, Methods of integration (substitution, partial fractions, by parts, trigonometric substitution), Area under curves, Volume of solids of revolution, Integration by reduction formulae
  • Multivariable Calculus: Partial derivatives, Gradient, Directional derivative, Chain rule, Taylor and Maclaurin series, Double and triple integrals, Line integrals, Green’s Theorem, Divergence and Curl, Surface integrals
  • Differential Equations: First-order differential equations, Second-order linear differential equations, Applications, Laplace transforms, Series solutions, Partial differential equations

2. Linear Algebra:

  • Matrices and Determinants: Types of matrices, Operations on matrices, Inverse of a matrix, Determinants and properties, Cramer’s rule, Adjoint and cofactor, Rank of a matrix
  • Vector Spaces: Definition of vector spaces, Subspaces, Basis, and Dimension, Linear dependence and independence, Row space, Column space, Null space
  • Linear Transformations: Matrix representation of linear transformations, Eigenvalues and eigenvectors, Diagonalization, Characteristic polynomial, Inner product spaces
  • Systems of Linear Equations: Gaussian elimination, Homogeneous and non-homogeneous systems, Row echelon form, Matrix inverse methods

3. Abstract Algebra:

  • Groups: Group definition, Subgroups, Cyclic groups, Lagrange’s Theorem, Cosets, Normal subgroups, Quotient groups, Group homomorphisms and isomorphisms
  • Rings: Ring properties, Subrings, Ideals, Ring homomorphisms, Quotient rings, Integral domains, Fields, Field extensions
  • Vector Spaces and Modules: Module theory, Linear maps, Dual spaces, Eigenvalues and Eigenvectors in vector spaces
  • Galois Theory: Galois groups, Field extensions, Solvability by radicals

4. Real Analysis:

  • Sequences and Series: Convergence of sequences, Cauchy sequences, Limits, Series convergence tests (comparison test, ratio test, root test), Power series
  • Continuity: Continuous functions, Types of discontinuities, Intermediate value theorem, Extreme value theorem
  • Differentiability: Differentiability of functions, Mean value theorem, Taylor’s theorem, Rolle’s theorem
  • Integrability: Riemann integration, Fundamental Theorem of Calculus, Improper integrals, Lebesgue integration (in advanced courses)

5. Complex Analysis:

  • Complex Numbers: Basic operations on complex numbers, Polar and exponential form, Argand plane, Complex conjugate
  • Analytic Functions: Cauchy-Riemann equations, Power series, Laurent series, Singularities, Residue theorem, Contour integration
  • Complex Integration: Cauchy’s integral theorem, Cauchy’s integral formula, Applications to real integrals
  • Conformal Mapping: Mapping properties, Applications to potential theory and fluid mechanics

6. Topology:

  • Topological Spaces: Open and closed sets, Basis of a topology, Subspace topology, Continuous functions
  • Compactness and Connectedness: Compact sets, Heine-Borel theorem, Connected sets, Path-connectedness
  • Separation Axioms: Hausdorff space, Regular spaces, Normal spaces
  • Fundamental Group: Loop space, Homotopy, Fundamental group, Applications in algebraic topology

7. Probability and Statistics:

  • Probability Theory: Axioms of probability, Conditional probability, Bayes’ Theorem, Random variables, Probability distributions (discrete and continuous), Expectation and variance, Moment generating functions
  • Statistical Inference: Point estimation, Interval estimation, Hypothesis testing, Confidence intervals, P-values, Chi-square tests, ANOVA, Regression analysis, Correlation
  • Random Processes: Markov chains, Poisson processes, Queuing theory, Stochastic processes

8. Differential Geometry:

  • Curves and Surfaces: Curvature, Frenet-Serret formulas, Geodesics, Surfaces in Euclidean space, Metric tensors
  • Differential Forms: Exterior derivatives, Stokes’ theorem, Manifolds, Riemannian geometry

9. Numerical Methods:

  • Numerical Solution of Equations: Root-finding algorithms (Bisection, Newton-Raphson, Secant), Interpolation (Lagrange, Newton)
  • Numerical Integration: Trapezoidal rule, Simpson’s rule, Romberg integration
  • Numerical Linear Algebra: Matrix factorizations (LU, QR), Eigenvalue algorithms, Least squares
  • Finite Differences and Numerical Solutions to Differential Equations

10. Mathematical Logic:

  • Propositional and Predicate Logic: Logical connectives, Truth tables, Logical equivalences, Quantifiers, Arguments, Proof methods (Direct, Indirect, Contradiction)
  • Set Theory: Set operations, Power sets, Cardinality, Cantor’s theorem, Zermelo-Fraenkel set theory, Axiom of choice
  • Combinatorics: Permutations, Combinations, Pigeonhole principle, Inclusion-exclusion principle

11. Combinatorics and Graph Theory:

  • Basic Combinatorics: Permutations and combinations, Binomial coefficients, Partitions of numbers
  • Graph Theory: Graphs, Digraphs, Trees, Planarity, Connectivity, Eulerian and Hamiltonian paths, Coloring, Graph algorithms (Dijkstra, Kruskal, Prim)

12. Mathematical Modelling:

  • Optimization: Linear programming, Convex optimization, Integer programming, Game theory, Network flows
  • Mathematical Biology: Models of population growth, Epidemic models, Predator-prey models
  • Economics and Finance: Mathematical modeling in economics, Financial mathematics, Game theory applications in economics

13. Operations Research:

  • Linear Programming: Simplex method, Duality theory, Linear programming in economics and business
  • Network Theory: Graph theory in operations research, Maximum flow problems, Minimal cost flow problems
  • Queuing Theory: Models of queues, Applications in telecommunications, manufacturing systems
  • Inventory Theory: Models for managing inventory and stock levels

These topics provide a solid foundation for further exploration into more advanced fields in mathematics and its applications across diverse domains such as engineering, economics, physics, data science, and operations research.

Postgraduate Level

At the postgraduate level, mathematics becomes highly specialized and advanced. The following are the major areas and topics covered in postgraduate mathematics:

1. Advanced Calculus (Real Analysis):

  • Measure Theory: Sigma-algebras, Measures, Lebesgue measure, Measurable sets and functions, Integration with respect to a measure (Lebesgue integral), Dominated convergence theorem, Fubini’s theorem
  • Functional Analysis: Banach spaces, Hilbert spaces, Linear operators, Spectral theorem, Eigenvalues and eigenvectors in infinite-dimensional spaces, L2 spaces, Bounded operators
  • Integration Theory: Lebesgue integration, Riemann-Stieltjes integrals, Convergence theorems, Applications in probability and statistics

2. Complex Analysis:

  • Advanced Topics in Complex Functions: Analytic continuation, Laurent series, Singularities, Residue theorem and its applications
  • Riemann Surfaces: Concept of Riemann surfaces, Meromorphic functions, Mapping class group
  • Complex Dynamics: Julia sets, Mandelbrot set, Iterative processes in the complex plane
  • Conformal Mappings: Schwarz-Christoffel transformation, Conformal invariants, Applications in physics and engineering

3. Abstract Algebra:

  • Group Theory: Representation theory, Sylow theorems, Simple groups, Solvable groups, Jordan-Hölder theorem
  • Ring Theory: Commutative and non-commutative rings, Ideal theory, Noetherian and Artinian rings, Polynomial rings, Field extensions, Galois theory
  • Module Theory: Modules over commutative and non-commutative rings, Tensor products of modules, Exact sequences, Free modules
  • Homological Algebra: Chain complexes, Homology groups, Exact sequences, Derived categories, Applications in topology and algebraic geometry

4. Topology:

  • Point-Set Topology: Compactness (Tietze extension theorem, Ascoli-Arzelà theorem), Connectedness, Continuity, Hausdorff spaces, Separation axioms, Urysohn’s lemma, Tychonoff theorem
  • Algebraic Topology: Fundamental group, Homotopy, Covering spaces, Simplicial complexes, Homology, Cohomology, Betti numbers
  • Differential Topology: Manifolds, Smooth maps, Tangent spaces, Differential forms, Stokes’ theorem, Critical points and Morse theory
  • Topological Groups and Lie Groups: Lie algebra, Structure of Lie groups, Applications in physics and geometry

5. Differential Geometry:

  • Riemannian Geometry: Riemannian manifolds, Geodesics, Curvature tensors (Ricci curvature, Riemann curvature), Einstein equations in general relativity
  • Differential Forms and Exterior Calculus: Exterior derivative, Stokes’ theorem, Poincaré lemma, Applications in physics (electromagnetism)
  • Geometry of Curves and Surfaces: Frenet-Serret formulas, Gaussian curvature, Surface integrals, Minimal surfaces, The Gauss-Bonnet theorem

6. Mathematical Logic:

  • Model Theory: Structures, Definability, Completeness and soundness, Compactness theorem, Löwenheim-Skolem theorem
  • Set Theory: Axiom of choice, Zermelo-Fraenkel set theory (ZF), Large cardinals, Ordinals, Forcing, Consistency and independence results
  • Proof Theory: Formal systems, Natural deduction, Incompleteness theorems (Gödel’s incompleteness), Lambda calculus, Computability and decidability

7. Probability and Stochastic Processes:

  • Advanced Probability Theory: Probability spaces, Conditional probability, Random variables, Expectation, Laws of large numbers, Central limit theorem
  • Stochastic Processes: Markov chains, Poisson processes, Brownian motion, Wiener process, Ergodicity, Stochastic differential equations (SDEs)
  • Queueing Theory: Advanced models in queuing, Network theory, Service models
  • Martingales and Stochastic Calculus: Martingale convergence theorems, Stochastic integration, Ito’s Lemma, Black-Scholes model in finance

8. Mathematical Physics:

  • Partial Differential Equations (PDEs): Elliptic, Parabolic, Hyperbolic equations, Fourier analysis, Green’s functions, Boundary value problems
  • Quantum Mechanics: Operator theory, Schrödinger equation, Quantum field theory, Path integrals, Symmetry in quantum systems
  • Statistical Mechanics: Thermodynamic limit, Partition functions, Boltzmann distribution, Entropy, Applications to condensed matter systems
  • Relativity and Cosmology: General and special relativity, Einstein’s field equations, Black holes, Cosmological models

9. Functional Analysis:

  • Banach Spaces: Banach fixed-point theorem, Hahn-Banach theorem, Baire’s category theorem
  • Hilbert Spaces: Orthonormal sets, Fourier series, Spectral theory, Unbounded operators, Operator algebras
  • Operator Theory: Bounded and unbounded operators, Spectral theory, Compact operators, Functional calculus, Applications to PDEs and quantum mechanics

10. Combinatorics and Graph Theory:

  • Advanced Combinatorics: Ramsey theory, Extremal combinatorics, Designs, Graph coloring, Block designs, Catalan numbers
  • Graph Theory: Advanced topics in connectivity, Planarity, Graph algorithms, Network flows, Matching theory, Spectral graph theory
  • Random Combinatorics: Random graphs, Random walks, Probabilistic methods in combinatorics

11. Algebraic Geometry:

  • Affine and Projective Varieties: Polynomial equations, Algebraic sets, Zariski topology
  • Sheaf Theory and Cohomology: Sheaves, Čech cohomology, Applications to complex geometry
  • Moduli Spaces: Parameter spaces for algebraic varieties, Moduli of curves, Moduli of vector bundles, Applications to string theory
  • Singularity Theory: Resolution of singularities, Classifying singularities, Applications in physics

12. Number Theory:

  • Analytic Number Theory: Prime number theorem, Riemann zeta function, Dirichlet series, Modular forms
  • Algebraic Number Theory: Rings of integers, Ideal theory, Class numbers, Algebraic integers, Cyclotomic fields
  • Diophantine Equations: Solving polynomial equations in integers, Methods of descent, Applications in cryptography

13. Optimization and Operations Research:

  • Linear and Nonlinear Optimization: Convex optimization, Duality theory, KKT conditions, Interior-point methods, Linear programming algorithms
  • Integer Programming: Branch-and-bound, Cutting-plane methods, Applications in scheduling and logistics
  • Dynamic Programming: Bellman equations, Optimal control theory, Applications in economics and engineering
  • Game Theory: Nash equilibrium, Cooperative and non-cooperative games, Extensive form games, Auction theory

14. Numerical Analysis:

  • Numerical Methods for ODEs and PDEs: Finite difference methods, Finite element methods, Spectral methods
  • Numerical Linear Algebra: Eigenvalue problems, Iterative methods, Krylov subspaces
  • Numerical Optimization: Gradient-based methods, Conjugate gradient method, Quasi-Newton methods
  • Error Analysis: Numerical stability, Convergence analysis

These topics represent a high level of specialization in mathematics, often requiring significant prior knowledge in undergraduate mathematics. Postgraduate mathematics offers opportunities for deep theoretical exploration, with applications in physics, engineering, economics, computer science, cryptography, data science, and many other fields.

Ph.D. Level

At the PhD level, mathematics becomes even more specialized and research-oriented. Below are the broad areas of mathematics that are typically explored at this level, with an emphasis on advanced, cutting-edge topics:

1. Advanced Real Analysis:

  • Measure Theory and Integration: Lebesgue measure, Lebesgue integration, Lp spaces, Signed measures, Fubini-Tonelli theorem, Vitali covering lemma, L∞ spaces
  • Functional Analysis: Banach and Hilbert spaces, Compact operators, Unbounded operators, Spectral theory, Operator algebras, Dual spaces, Weak topology, Reflexivity
  • Banach Algebras and C-Algebras*: Gelfand theory, Applications in quantum mechanics
  • Dynamical Systems: Ergodic theory, Attractors, Chaos theory, Bifurcations, Lyapunov exponents, Topological entropy

2. Advanced Complex Analysis:

  • Riemann Surfaces and Complex Geometry: Conformal mapping, Teichmüller theory, Moduli space of Riemann surfaces, Uniformization theorem
  • Analytic Number Theory: Zeta functions, Modular forms, L-functions, Langlands program, Automorphic representations
  • Asymptotic Methods: Asymptotic expansions, Saddle point method, Large deviations, Resurgence
  • Quantum Field Theory (QFT): Complex analysis techniques in QFT, Feynman diagrams, Conformal field theory

3. Algebraic Geometry:

  • Schemes and Sheaves: Sheaf theory, Affine and projective schemes, Localization, Quasi-coherent sheaves
  • Coherent Sheaves and Derived Categories: Derived functors, Grothendieck’s six operations, Spectral sequences, Moduli spaces of bundles
  • Singularity Theory: Resolution of singularities, Singularities of algebraic varieties, Minimal model program, Classification of surfaces
  • Intersection Theory: Cohomology of varieties, Chern classes, Gromov-Witten invariants, K-theory
  • Homotopy Theory: Algebraic topology applied to algebraic varieties, Applications in string theory and physics

4. Advanced Algebra:

  • Representation Theory: Lie groups and Lie algebras, Weyl’s character formula, Semisimple Lie algebras, Kac-Moody algebras, Modular representation theory
  • Homotopy and Homology: Spectral sequences, Stable homotopy theory, K-homology, Topological groups, Applications to topology and geometry
  • Group Theory: Structure of infinite groups, Solvable and nilpotent groups, Free groups, Classifying groups, Group cohomology, Torsion elements
  • Commutative Algebra: Local algebra, Primary decomposition, Hilbert functions, Noetherian rings, Cohen-Macaulay rings, Grobner bases
  • Advanced Field Theory: Galois theory in higher dimensions, Algebraic geometry techniques, Field extensions

5. Topology and Geometric Topology:

  • Higher-Dimensional Topology: Knot theory, 3-manifolds, Surgery theory, Topological quantum field theory
  • Homotopy Theory: Homotopy groups, Higher homotopy groups, Topological groups, Category theory
  • Algebraic Topology: Persistent homology, Floer homology, Categorification, K-homology, Topological invariants
  • Low-Dimensional Topology: 4-manifolds, Seiberg-Witten invariants, Knot invariants, Chern-Simons theory
  • Differential Topology: Smooth manifolds, Morse theory, Characteristic classes, Minimal surfaces, Calabi-Yau manifolds

6. Differential Geometry:

  • Advanced Riemannian Geometry: Ricci flow, Einstein manifolds, Geometric flows, Kähler geometry, Calabi conjecture, Gromov-Hausdorff convergence
  • Geometric Analysis: Minimal surfaces, Hamiltonian mechanics, Geodesics, Geometric invariants
  • Global Analysis: Index theory, Atiyah-Singer Index Theorem, Heat kernel techniques, Elliptic partial differential equations
  • Symplectic Geometry: Symplectic manifolds, Contact geometry, Floer theory, Hamiltonian dynamics, Poisson geometry, Symplectic topology
  • Algebraic and Symplectic Geometry: Mirror symmetry, String theory, Topological strings, Fukaya categories

7. Advanced Number Theory:

  • Analytic Number Theory: The Riemann Hypothesis, Distribution of primes, Sieve methods, Multiplicative number theory, Dirichlet series
  • Algebraic Number Theory: Class field theory, Cyclotomic fields, Modular forms, Iwasawa theory, Galois representations
  • Diophantine Geometry: Mordell’s conjecture, Birch and Swinnerton-Dyer conjecture, Rational points, Heights and convexity
  • Transcendental Number Theory: Liouville’s theorem, Baker’s theory, Algebraic independence
  • Homotopy Theoretic Methods: Algebraic topology applied to number theory, Motivic cohomology

8. Mathematical Logic and Set Theory:

  • Model Theory: Stability theory, Classification theory, Minimal structures, Ultrafilters, Definability
  • Set Theory: Large cardinals, Forcing, Constructible sets, Determinacy, Axiom of choice, ZFC and independence results
  • Computability Theory: Turing degrees, Recursion theory, Gödel’s incompleteness theorems, Computability over different models
  • Category Theory: Sheaves, Functors, Natural transformations, Monoidal categories, Topos theory, Higher categories, Derived categories
  • Proof Theory: Proof systems, Cut-elimination, Proofs of consistency, Logical foundations of mathematics

9. Quantum Computing and Cryptography:

  • Quantum Algorithms: Shor’s algorithm, Grover’s algorithm, Quantum Fourier transform, Quantum entanglement, Quantum parallelism
  • Quantum Information Theory: Quantum entropy, Quantum error correction, Quantum cryptography, Bell inequalities
  • Cryptographic Protocols: Public-key cryptography, Zero-knowledge proofs, RSA, Elliptic curve cryptography, Secure multi-party computation
  • Post-Quantum Cryptography: Lattice-based cryptography, Quantum resistance, Multivariate polynomial-based cryptography
  • Quantum Error Correction: Stabilizer codes, Surface codes, Fault tolerance in quantum computing

10. Stochastic Processes and Probability:

  • Advanced Stochastic Calculus: Stochastic differential equations (SDEs), Ito calculus, Stochastic processes in finance, Filtering theory
  • Stochastic Analysis: Malliavin calculus, Stochastic integrals, Martingales, Brownian motion, Feynman-Kac formula
  • Random Walks and Markov Chains: Markov processes, Ergodic theory, Random matrix theory, Large deviations
  • Measure-Valued Processes: Applications to mathematical finance, Stochastic optimal control, Stochastic differential games

11. Optimization Theory:

  • Convex Optimization: Duality theory, Interior-point methods, Large-scale optimization, KKT conditions, Linear programming
  • Optimal Control Theory: Hamilton-Jacobi-Bellman equations, Pontryagin’s maximum principle, Stochastic optimal control
  • Nonlinear Optimization: Global optimization, Heuristic methods, Multi-objective optimization, Evolutionary algorithms
  • Mathematical Programming: Integer programming, Combinatorial optimization, Network flows, Game theory in optimization

12. Advanced Topics in Partial Differential Equations (PDEs):

  • Elliptic PDEs: Variational methods, Sobolev spaces, Regularity theory, Elliptic regularity, Green’s function
  • Hyperbolic and Parabolic PDEs: Stability theory, Wave equations, Heat equations, Well-posedness
  • Nonlinear PDEs: Korteweg-de Vries equation, Navier-Stokes equations, Nonlinear analysis, Pattern formation, Solitons
  • Stochastic PDEs: Random dynamical systems, Pathwise solutions, Applications in finance and physics

13. Mathematical Finance:

  • Stochastic Calculus in Finance: Black-Scholes model, Hedging, Risk-neutral pricing, Exotic options
  • Asset Pricing Models: Capital Asset Pricing Model (CAPM), Arbitrage pricing theory, Market microstructure
  • Mathematical Models in Finance: Portfolio optimization, Interest rate models, Credit risk, Pricing of derivative securities

14. Mathematical Modelling:

  • Numerical Simulations: Finite element methods, Multiscale modeling, Monte Carlo methods, Computational fluid dynamics (CFD)
  • Biomathematics: Mathematical models in biology, Epidemic models, Neural networks, Evolutionary dynamics
  • Climate Modelling and Mathematical Physics: Weather prediction models, Climate change simulation, Earth system modeling

At the PhD level, mathematics is primarily research-driven, with a deep focus on specialized theories, cutting-edge problems, and new mathematical concepts that extend beyond

the established framework of earlier mathematical education. Advanced topics and current research are often integrated into a dissertation that contributes new knowledge to the field.