1. Real Analysis:

• Sequences and Series: Convergence, Divergence, Tests for Convergence, Power Series.
• Functions of Real Variables: Continuity, Differentiability, Mean Value Theorems.
• Riemann Integration: Riemann-Stieltjes Integral, Improper Integrals.
• Metric Spaces: Open and Closed Sets, Compactness, Completeness.
• Lebesgue Measure: Lebesgue Integral, Measure Theory Basics.

2. Complex Analysis:

• Complex Numbers: Algebra of Complex Numbers, Complex Functions.
• Analytic Functions: Cauchy-Riemann Equations, Complex Integration.
• Contour Integration: Cauchy’s Integral Theorems, Residue Theory.
• Conformal Mapping: Mapping Properties, Applications.

3. Algebra:

• Groups: Group Actions, Sylow Theorems, Normal Subgroups.
• Rings and Fields: Polynomial Rings, Euclidean Domains, Field Extensions.
• Linear Algebra: Vector Spaces, Linear Transformations, Eigenvalues, Diagonalization.
• Module Theory: Module Homomorphisms, Free Modules, Finitely Generated Modules.
• Matrix Theory: Eigenvalues, Eigenvectors, Cayley-Hamilton Theorem.

4. Topology:

• Basic Topological Spaces: Topological Spaces, Basis, Subspaces, Product Spaces.
• Metric Spaces: Open and Closed Sets, Completeness, Compactness.
• Continuity: Homeomorphisms, Connectedness, Path Connectedness.
• Compactness: Sequential Compactness, Properties of Compact Spaces.

5. Differential Equations:

• Ordinary Differential Equations: Existence and Uniqueness Theorems, Linear Equations.
• Partial Differential Equations: Separation of Variables, Laplace Equation, Wave Equation, Heat Equation.
• Method of Solutions: Green’s Function, Fourier Series, Sturm-Liouville Theory.

6. Numerical Analysis:

• Interpolation: Lagrange Polynomials, Newton’s Divided Difference.
• Numerical Differentiation and Integration: Trapezoidal Rule, Simpson’s Rule, Newton-Cotes Formula.
• Root Finding: Newton-Raphson Method, Secant Method, Fixed Point Iteration.
• Numerical Linear Algebra: Gaussian Elimination, LU Decomposition, Iterative Methods.
• Error Analysis: Absolute and Relative Error, Taylor Series Approximation.

7. Linear Programming:

• Formulation: Linear Programming Problems, Duality Theorems.
• Graphical Method: Feasible Solutions, Basic Feasible Solutions.
• Simplex Method: Optimality Conditions, Tableau Form.
• Duality: Dual Problems, Dual Simplex Method.

8. Probability and Statistics:

• Random Variables: Discrete and Continuous Random Variables, Probability Distributions.
• Sampling Distributions: Normal, Binomial, Poisson Distributions.
• Estimation: Point Estimation, Interval Estimation, Maximum Likelihood Estimation.
• Testing of Hypothesis: Null and Alternative Hypotheses, Type I and Type II Errors.
• Regression Analysis: Simple Linear Regression, Multiple Linear Regression.