Science
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M.Math
2 Years
1 College
M.Math Syllabus 2026
The M.Math (M.Math) syllabus covers a structured programme spanning 2 Years designed to build both foundational knowledge and specialised expertise. Below is the detailed semester-wise subject breakdown and programme structure.
M.Math Semester-wise Subjects
M.Math Syllabus & Subjects
The M.Math curriculum at ISI Kolkata comprises 20 courses - 13 compulsory and 7 electives chosen from a pool of 22 options. The programme is heavily weighted toward pure mathematics, with the flexibility to specialise in areas like algebraic geometry, number theory, or differential topology through elective choices.
Compulsory Subjects (13 Courses)
| Subject | Key Topics |
|---|---|
| Algebra I | Group theory, Sylow theorems, ring theory, module theory |
| Algebra II | Galois theory, field extensions, commutative algebra, Noetherian rings |
| Real Analysis | Measure theory, Lebesgue integration, L^p spaces |
| Complex Analysis | Analytic functions, Riemann mapping theorem, entire functions, analytic continuation |
| Functional Analysis | Banach spaces, Hilbert spaces, bounded operators, spectral theory |
| Topology I | Topological spaces, compactness, connectedness, fundamental group |
| Topology II (Algebraic Topology) | Covering spaces, homology, cohomology, exact sequences |
| Differential Geometry | Manifolds, tangent bundles, differential forms, de Rham cohomology |
| Ordinary Differential Equations | Existence and uniqueness theorems, stability theory, boundary value problems |
| Partial Differential Equations | Laplace equation, heat equation, wave equation, Sobolev spaces |
| Probability Theory | Probability measures, random variables, limit theorems, martingales |
| Statistics | Estimation theory, hypothesis testing, sufficiency, Bayesian methods |
| Combinatorics | Enumerative combinatorics, graph theory, Ramsey theory |
Elective Subjects (Choose 7 from 22)
Algebra & Number Theory
- Algebraic Number Theory
- Commutative Algebra
- Representation Theory
- Algebraic Geometry
Analysis & Geometry
- Harmonic Analysis
- Operator Theory
- Riemannian Geometry
- Several Complex Variables
Topology & Dynamics
- Differential Topology
- Algebraic Topology II
- Dynamical Systems
- Ergodic Theory
Applied & Computational
- Numerical Analysis
- Mathematical Logic
- Stochastic Processes
- Advanced Probability
- Coding Theory
- Cryptography
Preparation Resources
| Subject | Recommended Textbooks |
|---|---|
| Algebra | Dummit & Foote; Artin; Lang's Algebra |
| Real Analysis | Rudin (Principles + Real & Complex); Folland; Royden |
| Topology | Munkres; Hatcher's Algebraic Topology |
| Complex Analysis | Ahlfors; Conway; Stein & Shakarchi |
| Functional Analysis | Kreyszig; Rudin (Functional Analysis); Brezis |
M.Math Programme Structure & Credit Distribution
M.Math Year-wise Curriculum
The M.Math programme is structured across 4 semesters over 2 years. Year 1 builds the core mathematical foundations, while Year 2 allows deep specialisation through electives.
Year 1 - Core Foundations
| Semester 1 | Semester 2 |
|---|---|
| Algebra I (Groups, Rings, Modules) | Algebra II (Galois Theory, Commutative Algebra) |
| Real Analysis (Measure Theory) | Complex Analysis |
| Topology I (General Topology) | Topology II (Algebraic Topology) |
| Ordinary Differential Equations | Functional Analysis |
| Probability Theory | Statistics |
Year 2 - Specialisation & Electives
| Semester 3 | Semester 4 |
|---|---|
| Differential Geometry (Compulsory) | Partial Differential Equations (Compulsory) |
| Combinatorics (Compulsory) | Elective V |
| Elective I | Elective VI |
| Elective II | Elective VII |
| Elective III | - |
| Elective IV | - |
Popular Elective Combinations
Algebraic Path
For students aiming for PhD in Algebra/Number Theory:
- Algebraic Number Theory
- Commutative Algebra
- Representation Theory
- Algebraic Geometry
- Algebraic Topology II
- Homological Algebra
- Mathematical Logic
Analysis/Geometry Path
For students aiming for PhD in Analysis/Geometry:
- Harmonic Analysis
- Operator Theory
- Riemannian Geometry
- Several Complex Variables
- Differential Topology
- Dynamical Systems
- Ergodic Theory
Assessment Pattern
| Component | Weightage |
|---|---|
| Mid-Semester Examination | 30% |
| End-Semester Examination | 50% |
| Assignments & Problem Sets | 20% |
A minimum cumulative grade point average (CGPA) of 5.0 on a 10-point scale is required for degree conferral. Students falling below this threshold after any semester are placed on academic probation.
Skills Developed in M.Math
Skills Required & Acquired in M.Math
M.Math demands strong foundational skills at entry and develops world-class mathematical capabilities that are valued across academia and industry.
Skills Required Before Joining
Essential Mathematical Skills
- Proof Writing: Ability to construct rigorous mathematical proofs - epsilon-delta arguments, induction, contradiction, contrapositive
- Real Analysis: Deep understanding of sequences, series, continuity, differentiability, Riemann integration
- Linear Algebra: Vector spaces, linear transformations, eigenvalues, diagonalization, inner product spaces
- Abstract Algebra: Group theory (Lagrange, Cauchy, Sylow theorems), ring theory, field extensions
- Metric Space Topology: Open/closed sets, compactness, connectedness, completeness
Helpful Background
- Complex Analysis: Analytic functions, Cauchy's theorem, residue calculus
- Measure Theory Basics: Lebesgue measure, measurable functions, integration
- Elementary Topology: Topological spaces, continuous maps, quotient topology
- Mathematical Maturity: Comfort with abstraction, ability to read and understand advanced textbooks independently
- Problem-Solving: Experience with competition-style and proof-style problems (Olympiad, NBHM, TIFR background helps)
Skills Acquired During M.Math
Advanced Mathematical Skills
- Algebraic Topology: Fundamental groups, homology, cohomology, exact sequences
- Functional Analysis: Banach/Hilbert space theory, spectral theory, distributions
- Differential Geometry: Manifolds, vector bundles, differential forms, Riemannian metrics
- Galois Theory: Field extensions, splitting fields, fundamental theorem
Research & Analytical Skills
- Research Methodology: Literature review, problem formulation, conjecture development
- Advanced Proof Techniques: Cohomological methods, category-theoretic arguments, spectral sequences
- Mathematical Writing: LaTeX proficiency, clear exposition of complex ideas, paper writing
- Seminar Presentation: Communicating deep results to expert audiences
Industry-Transferable Skills
- Abstract Reasoning: Pattern recognition and structural analysis across domains
- Quantitative Modelling: Probabilistic and measure-theoretic frameworks for financial/data models
- Algorithmic Thinking: Combinatorial and algebraic approaches to computational problems
- Problem Decomposition: Breaking complex systems into tractable mathematical components