Semester-wise Syllabus for MSc Mathematics

Semester 1: Core Foundations

1. Advanced Calculus & Real Analysis

   • Limits, continuity, sequences, series, Riemann integration

   • Metric spaces, uniform convergence

2. Abstract Algebra

   • Groups, rings, fields, homomorphisms, quotient structures

   • Sylow theorems, polynomial rings

3. Linear Algebra

   • Vector spaces, linear transformations, eigenvalues

   • Matrix decompositions (QR, SVD)

4. Differential Equations

   • ODEs (exact, linear, Bernoulli), PDEs (wave, heat equations)

5. Computer Applications in Mathematics

   • Programming with Python/Matlab for numerical methods

6. Lab/ Practicals

   • Solving ODEs/PDEs numerically, coding algorithms

Semester 2: Advanced Topics

1. Complex Analysis

   • Analytic functions, Cauchy’s theorem, residue calculus

2. Topology

   • Basis, compactness, connectedness, quotient topology

3. Discrete Mathematics

   • Graph theory, combinatorics, Boolean algebra

4. Numerical Methods

   • Interpolation, numerical integration, root-finding algorithms

5. Probability & Statistics

   • Distributions, Bayes’ theorem, hypothesis testing

6. Lab/ Practicals

   • Simulations in MATLAB/R, graph theory applications

Semester 3: Specializations & Electives

Core Subjects

1. Functional Analysis

   • Banach and Hilbert spaces, spectral theory

2. Number Theory

   • Prime numbers, modular arithmetic, cryptography applications

Electives (Choose 2–3)

• Operations Research: Linear programming, game theory

• Fluid Dynamics: Navier-Stokes equations

• Algebraic Geometry: Varieties, ideals

• Financial Mathematics: Black-Scholes model, stochastic calculus

1. Seminar/Project Work

   • Research paper review or mini-project (e.g., cryptanalysis, fluid simulation)

Semester 4: Research & Dissertation

1. Thesis/Dissertation

   • Original research in pure/applied mathematics (e.g., modeling, proofs)

2. Advanced Electives

   • Machine Learning Math: Optimization, gradient descent

   • Quantum Computing: Linear algebra applications

3. Viva Voce

   • Defense of dissertation before faculty