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M-SC in Mathematics at Indian Institute of Technology Kanpur
Kanpur Nagar, Uttar Pradesh
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About the Specialization
What is Mathematics at Indian Institute of Technology Kanpur Kanpur Nagar?
This M.Sc. Mathematics program at IIT Kanpur focuses on building a strong foundation in core mathematical disciplines, including Algebra, Analysis, Topology, and Probability. It emphasizes rigorous theoretical understanding coupled with problem-solving skills, preparing students for advanced research and diverse analytical roles in India''''s evolving tech and finance sectors. The program''''s interdisciplinary nature allows students to explore applications in various scientific and engineering fields.
Who Should Apply?
This program is ideal for highly motivated B.Sc. graduates with a strong aptitude for mathematics, seeking entry into research careers or advanced analytics. It also suits individuals aspiring for faculty positions in academia or those looking to pivot into quantitative roles in finance, data science, or scientific computing within Indian and global firms. A valid JAM score and strong academic record in undergraduate mathematics are prerequisite.
Why Choose This Course?
Graduates of this program can expect to pursue Ph.D. studies at top global universities or secure lucrative positions as quantitative analysts, data scientists, research scientists, or academicians in India. Entry-level salaries typically range from INR 8-15 LPA, with experienced professionals earning significantly more (INR 20-40+ LPA). The strong analytical foundation also prepares them for roles in emerging fields like AI/ML research in Indian startups and MNC R&D centers.
Student Success Practices
Foundation Stage
Master Core Mathematical Concepts- (Semester 1-2)
Focus intensely on understanding the foundational principles of Algebra, Analysis, and Topology. Regularly solve problems from standard textbooks (e.g., Dummit & Foote for Algebra, Rudin for Analysis) and actively participate in tutorial sessions to solidify understanding.
Tools & Resources
NPTEL videos for conceptual clarity, Standard reference books (e.g., Serge Lang, Walter Rudin, Munkres), Peer study groups, departmental tutorials
Career Connection
A robust theoretical base is critical for cracking Ph.D. entrance exams (e.g., GATE, CSIR NET, GRE Subject Test) and excelling in quantitative roles that demand deep analytical reasoning and proofs.
Develop Advanced Problem-Solving Acumen- (Semester 1-2)
Beyond theoretical understanding, practice solving a wide variety of challenging problems from textbooks and competition archives. Engage in mathematical contests or puzzles to sharpen logical thinking and develop innovative problem-solving strategies, crucial for research.
Tools & Resources
Online platforms like AoPS (Art of Problem Solving), Previous year question papers for competitive exams (JAM, GATE, NET), Math clubs and problem-solving workshops within IIT Kanpur
Career Connection
Essential for competitive exams, success in research, and any analytical industry role where complex mathematical issues need structured and creative solutions.
Cultivate Early Research Curiosity- (Semester 1-2)
Actively attend departmental seminars, colloquia, and guest lectures to expose yourself to diverse research areas in mathematics and its applications. Engage with professors to understand their research and identify potential areas of interest for your future project or Ph.D.
Tools & Resources
Departmental seminar schedules and notices, arXiv.org for preprints in mathematics, Faculty research pages on the IITK website
Career Connection
Early exposure to research helps in selecting a relevant M.Sc. project, finding a suitable Ph.D. advisor, and identifying specialization areas in academia or R&D roles.
Intermediate Stage
Strategic Elective Selection & Specialization- (Semester 3-4)
Carefully choose elective courses that align with your long-term career goals, whether in pure mathematics, applied mathematics, or interdisciplinary areas like financial mathematics or data science. Consult faculty advisors for personalized guidance on course selection based on your aspirations.
Tools & Resources
IITK course catalogs and elective lists, Faculty advisors and academic mentors, Alumni network for career insights and industry trends
Career Connection
Specializing early helps build a strong, focused profile for specific Ph.D. topics or quantitative industry roles, making you a more attractive candidate in a competitive job market.
Engage in Advanced Research Projects- (Semester 3-4)
Work closely with a faculty mentor on your compulsory M.Sc. Project (MA699). This involves formulating a novel problem, conducting a thorough literature review, developing appropriate mathematical methodologies, implementing solutions, analyzing results, and writing a comprehensive thesis. Aim for publishable quality work.
Tools & Resources
IITK library resources (journals, databases), LaTeX for typesetting academic papers, Research software like MATLAB, Python with SciPy/NumPy, R
Career Connection
A strong research project demonstrates your ability to conduct independent research, which is crucial for Ph.D. admissions and R&D positions in both academia and industry.
Network and Professional Development- (Semester 3-4)
Participate in national and international workshops, conferences, or summer schools relevant to your specialization. Network with peers, researchers, and potential collaborators to broaden your academic and professional horizons and explore advanced opportunities.
Tools & Resources
Conference announcements (IMS, AMS, local workshops), Professional mathematical societies (e.g., Indian Mathematical Society), IITK alumni portal and career development center
Career Connection
Networking can lead to Ph.D. opportunities, postdoctoral positions, research collaborations, or industry contacts that facilitate placements and career transitions post-M.Sc.
Advanced Stage
Program Structure and Curriculum
Eligibility:
- B.Sc. degree with Mathematics as one of the subjects having 60% marks/6.0 CPI (for General/OBC candidates) and 55% marks/5.5 CPI (for SC/ST/PD candidates) and a valid JAM score.
Duration: 2 years (4 semesters)
Credits: 68 Credits
Assessment: Assessment pattern not specified
Semester-wise Curriculum Table
Semester 1
| Subject Code | Subject Name | Subject Type | Credits | Key Topics |
|---|---|---|---|---|
| MA511 | Algebra I | Core | 8 | Group Theory (Sylow''''s theorems), Ring Theory (PID, UFD, Modules, Tensor products), Field Theory (Galois Theory, Finite fields), Cyclotomic extensions, Radical extensions |
| MA512 | Real Analysis | Core | 8 | Measure Theory (Lebesgue measure, measurable functions), Lebesgue integration (modes of convergence, Lp spaces), Abstract Integration (measure spaces, signed measures), Radon-Nikodym theorem, Functional Analysis (Hahn-Banach, Open mapping theorem) |
| MA513 | Topology | Core | 8 | Topological spaces (basis, continuity, product topology), Connectedness, compactness (separation axioms), Countability axioms, Metrization theorems, Urysohn Lemma, Tychonoff theorem, Homotopy, fundamental groups |
Semester 2
| Subject Code | Subject Name | Subject Type | Credits | Key Topics |
|---|---|---|---|---|
| MA514 | Complex Analysis | Core | 8 | Analytic functions (Cauchy-Riemann, power series), Complex integration (Cauchy theorems, Morera’s theorem), Residue theorem, Maximum modulus principle, Conformal mappings, Harmonic and Entire functions |
| MA515 | Linear Algebra | Core | 8 | Vector spaces, Linear transformations, Eigenvalues and eigenvectors, Cayley-Hamilton theorem, Canonical forms (Jordan and Rational), Bilinear and Quadratic forms, Inner product spaces (Gram-Schmidt, Orthogonal transformations) |
| MA516 | Probability and Statistics | Compulsory | 8 | Probability spaces, Random variables, Distributions, Joint distributions, Conditional probability, Expectation, Convergence types, Law of large numbers, Central limit theorem, Estimation theory (point and interval estimation), Hypothesis testing, Regression and correlation |
Semester 3
| Subject Code | Subject Name | Subject Type | Credits | Key Topics |
|---|---|---|---|---|
| MA521 | Discrete Mathematics | Elective | 8 | Combinatorics (Pigeonhole, Inclusion-Exclusion), Graph Theory (Connectivity, Matching, Trees), Recurrence Relations, Generating Functions, Logic (Propositional, Predicate), Boolean Algebra, Lattices |
| MA522 | Measure Theory | Elective | 8 | Sigma-algebras, Outer measures, Caratheodory extension, Measurable functions, Lebesgue integration, Convergence theorems, Fubini''''s theorem, Lp spaces, Radon-Nikodym theorem |
| MA523 | Applied Probability | Elective | 8 | Random walks, Branching processes, Martingales, Queueing theory, Renewal theory, Markov chains, Hidden Markov models, Applications in finance and biology |
| MA524 | Stochastic Processes | Elective | 8 | Markov Chains (discrete & continuous time), Poisson processes, Renewal processes, Martingales, Random walks, Brownian Motion, Ito''''s lemma, Stochastic Differential Equations |
| MA525 | Numerical Analysis | Elective | 8 | Error analysis (floating-point arithmetic), Interpolation (Lagrange, Newton), Numerical differentiation and integration, Solutions of linear and non-linear equations, Eigenvalue problems, Numerical solutions of ODEs/PDEs |
| MA526 | Mathematical Modelling | Elective | 8 | Principles of modelling, Dimensional analysis, Scaling, Perturbation methods, Optimization, Differential equations, Difference equations, Compartmental models, Case studies (biology, physics, economics) |
| MA527 | Partial Differential Equations | Elective | 8 | First-order PDEs (method of characteristics), Classification of second-order PDEs, Wave equation, Heat equation, Laplace equation, Green''''s functions, Energy methods, Fourier series and transform methods |
| MA528 | Functional Analysis | Elective | 8 | Normed linear spaces, Banach spaces, Hilbert spaces, Orthonormal bases, Bounded linear operators, Dual spaces, Hahn-Banach Theorem, Open Mapping Theorem, Closed Graph Theorem, Uniform Boundedness Principle |
| MA529 | Operator Theory | Elective | 8 | Compact operators, Fredholm operators, Self-adjoint operators, Normal operators, Spectral theorem for compact self-adjoint operators, Unbounded operators, Closed operators, Symmetric operators, Self-adjoint extensions |
| MA531 | Commutative Algebra | Elective | 8 | Rings, Ideals (prime, maximal, radical), Modules, Noetherian rings, Artinian rings, Primary decomposition, Localization, Integral Extensions, Hilbert Nullstellensatz, Dimension theory |
| MA532 | Algebraic Topology | Elective | 8 | Fundamental group, Covering spaces, Simplicial homology, Singular homology, Excision theorem, Mayer-Vietoris sequence, Cohomology theory, CW complexes |
| MA533 | Differential Geometry | Elective | 8 | Manifolds, Tangent spaces, Vector fields, Flows, Lie derivatives, Differential forms, Exterior derivatives, Integration on manifolds, Riemannian metrics, Geodesics, Curvature (Gaussian, Principal, Sectional) |
| MA534 | Number Theory | Elective | 8 | Divisibility, Congruences, Modular arithmetic, Quadratic residues, Reciprocity law, Arithmetic functions (Euler phi, Mobius), Diophantine equations, Continued fractions, Algebraic number theory (basic concepts) |
| MA535 | Cryptography | Elective | 8 | Classical ciphers (Caesar, Vigenere), Symmetric key cryptography (DES, AES), Public key cryptography (RSA, ElGamal), Hash functions, Digital signatures, Key exchange (Diffie-Hellman), Elliptic Curve Cryptography |
| MA536 | Coding Theory | Elective | 8 | Basic concepts of error control codes, Linear codes (Hamming, Golay), Cyclic codes, BCH codes, Reed-Solomon codes, Decoding algorithms (syndrome decoding) |
| MA537 | Graph Theory | Elective | 8 | Paths, Cycles, Trees, Bipartite graphs, Matching (Hall''''s theorem), Connectivity (vertex and edge), Planarity (Kuratowski''''s theorem), Coloring (vertex, edge, chromatic number) |
| MA538 | Optimisation Techniques | Elective | 8 | Linear programming (Simplex method, Duality), Non-linear programming (convexity, KKT conditions), Gradient methods, Lagrangian duality, Dynamic programming (Bellman equation), Network flow problems |
| MA539 | Financial Mathematics | Elective | 8 | No-arbitrage principle, Risk-neutral pricing, Brownian motion, Ito''''s lemma, Stochastic calculus, Black-Scholes model for option pricing, Hedging strategies, Interest rate models (Vasicek, CIR) |
| MA541 | Advanced Real Analysis | Elective | 8 | Abstract measure theory, Lp spaces, Sobolev spaces, Distributions, Harmonic analysis (Fourier transforms on Lp), Weak convergence, Compactness methods, Calculus of variations (Euler-Lagrange equations) |
| MA542 | Fluid Dynamics | Elective | 8 | Kinematics of fluids, Continuity equation, Euler''''s and Navier-Stokes equations, Ideal fluids, Viscous fluids, Potential flow, Boundary layers, Compressible flow (shock waves) |
| MA543 | Elasticity | Elective | 8 | Stress and Strain tensors, Hooke''''s Law, Elastic constants, Equations of equilibrium and motion, Plane stress and plane strain problems, Torsion of non-circular sections, Bending of beams |
Semester 4
| Subject Code | Subject Name | Subject Type | Credits | Key Topics |
|---|---|---|---|---|
| MA699 | M.Sc. Project | Project | 12 | Research methodology, Literature review and problem formulation, Development of mathematical models/solutions, Data analysis and interpretation, Thesis writing and presentation of results |
| MA601 | Wavelets | Elective | 6 | Fourier Analysis (transforms, series), Continuous and Discrete Wavelet Transforms, Multiresolution Analysis, Filter Banks, Daubechies Wavelets, Coiflets, Applications (signal/image processing, compression) |
| MA602 | Ergodic Theory | Elective | 6 | Measure-preserving transformations, Birkhoff and von Neumann ergodic theorems, Mixing properties (weak, strong), Entropy (Kolmogorov-Sinai, topological), Spectral properties, Applications to dynamical systems |
| MA603 | Finite Element Methods | Elective | 6 | Variational formulations (Ritz, Galerkin), Weak solutions of PDEs, Interpolation functions (shape functions), Element types (triangular, quadrilateral), Assembly process, Numerical integration, Error analysis |
| MA604 | Geometric Quantization | Elective | 6 | Symplectic manifolds, Darboux theorem, Prequantization (line bundles, connections), Polarizations (real, complex, Kahler), Quantization map (Dirac operator), Connection to quantum mechanics |
| MA605 | Advanced Partial Differential Equations | Elective | 6 | Weak solutions, Distributions, Sobolev spaces, Trace theorems, Elliptic, Parabolic, Hyperbolic equations (existence, regularity), Maximum principle, Dispersive equations |
| MA606 | Topics in Non-linear Functional Analysis | Elective | 6 | Monotone operators, Maximal monotone operators, Fixed point theory (Brouwer, Schauder, Leray-Schauder), Variational methods, Mountain pass theorem, Critical point theory, Applications to Non-linear PDE |
| MA607 | Topics in Commutative Algebra | Elective | 6 | Valuation theory, Discrete valuation rings, Local rings, Completions, Dimension theory (Krull dimension), Regular local rings, Cohen-Macaulay rings |
| MA608 | Topics in Ring Theory | Elective | 6 | Non-commutative rings, Radical of a ring, Semisimple rings, Jacobson radical, Group rings, Representations of rings, Homological algebra (Hochschild cohomology), Matrix rings, Skew polynomial rings |
| MA609 | Topics in Operator Algebras | Elective | 6 | C*-algebras, GNS construction, Von Neumann algebras, Factors, States and representations, K-theory (K 0 and K 1), Non-commutative geometry (basic ideas) |
| MA610 | Algebraic Number Theory | Elective | 6 | Number fields, Algebraic integers, Rings of integers, Ideal factorization, Dedekind domains, Ideal class group, Units in number fields, Dirichlet''''s unit theorem, Ramification theory, Valuations |
| MA611 | Elliptic Curves | Elective | 6 | Weierstrass equations, Group law on elliptic curves, Mordell-Weil theorem, Torsion points, Isogenies, Endomorphism rings, L-functions of elliptic curves, Applications in cryptography |
| MA612 | Probability Theory | Elective | 6 | Axiomatic probability, Probability spaces, Random variables, Distribution functions, Martingales, Optional stopping theorem, Central Limit Theorem (Lindeberg-Feller), Characteristic functions, Conditional expectation |
| MA613 | Theory of Distributions | Elective | 6 | Test functions, Spaces of distributions, Operations on distributions (derivatives, products), Fourier transform of distributions, Sobolev spaces (definitions, embeddings), Applications to PDE (fundamental solutions) |
| MA614 | Advanced Matrix Theory | Elective | 6 | Generalized inverses (Moore-Penrose), Matrix norms, Condition numbers, Perturbation theory for eigenvalues, Matrix equations (Sylvester, Lyapunov), Eigenvalue bounds and inequalities |
