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B-S-M-S 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 Mathematics B.S.-M.S. dual degree program at IIT Kanpur offers a rigorous and comprehensive education in pure and applied mathematics, fostering deep analytical and problem-solving skills. The curriculum is designed to produce highly skilled mathematicians and scientists equipped to tackle complex challenges in academia, research, and various industries. India''''s growing R&D sector and burgeoning tech industry demand strong mathematical foundations, making this program highly relevant and in demand for quantitative roles.
Who Should Apply?
This program is ideal for high-achieving fresh graduates from 10+2 with a strong aptitude for mathematics, seeking a deep theoretical and practical understanding. It also caters to individuals aspiring for research careers in mathematics, data science, quantitative finance, or scientific computing. Candidates should possess excellent analytical abilities, a passion for abstract reasoning, and a keen interest in applying mathematical principles to real-world problems.
Why Choose This Course?
Graduates of this program can expect to pursue advanced research (Ph.D.) in India or abroad, or secure high-impact roles in diverse sectors. Career paths in India include data scientist, quantitative analyst in finance, research scientist, academician, or software development engineer with an analytical focus. Entry-level salaries can range from INR 8-15 LPA for fresh graduates, growing significantly with experience, especially in analytics and financial modeling roles.
Student Success Practices
Foundation Stage
Master Core Mathematical Foundations- (Semester 1-2)
Dedicate significant effort to thoroughly understand MTH101 (Linear Algebra & Diff. Eq.) and MTH102 (Complex Analysis & Diff. Geometry). Actively solve problems beyond textbook exercises and seek help from TAs/professors during office hours and tutorial sessions.
Tools & Resources
NPTEL courses for conceptual clarity, Standard textbooks (e.g., Strang, Apostol, Courant), Peer study groups, IITK Mathematics Department''''s tutorial and doubt-clearing sessions
Career Connection
A strong foundation in these core areas is crucial for advanced mathematics, data science, engineering, and quantitative finance roles. It builds logical reasoning and analytical skills highly valued in competitive placements.
Develop Programming and Computational Skills- (Semester 1-2)
Beyond ESC101, actively practice coding (Python/C++) to solve mathematical problems. Participate in coding competitions on platforms like CodeChef and explore open-source projects related to scientific computing or data analysis.
Tools & Resources
HackerRank, CodeChef for competitive programming, NumPy, SciPy, Matplotlib libraries for Python, GitHub for collaborative projects, IITK''''s computing facilities and hackathons
Career Connection
Essential for roles in data science, quantitative finance, and software development, particularly in India''''s tech sector. Computational skills enable practical application of theoretical mathematical knowledge.
Engage with HSS Electives for Holistic Growth- (Semester 1-2)
Choose HSS electives strategically to broaden perspectives beyond STEM. Actively participate in discussions, presentations, and writing assignments to improve communication, critical thinking, and understanding of societal issues.
Tools & Resources
Institute''''s various HSS department offerings, Writing Center for academic writing support, Debate clubs, literary societies, and student forums for public speaking
Career Connection
Enhances crucial soft skills, cultural awareness, and ethical reasoning, which are vital for leadership roles, effective team collaboration, and navigating diverse professional environments in India and globally.
Intermediate Stage
Explore Research Opportunities and Projects- (Semester 3-5)
Seek out professors for UGRPs (Undergraduate Research Projects) or summer internships within the department or other IITs. This exposes you to research methodology and helps identify specific areas of interest for advanced study.
Tools & Resources
Departmental research project listings, IITK Summer Undergraduate Research Program (SURP), Research paper databases (e.g., MathSciNet, arXiv), Direct faculty mentorship and guidance
Career Connection
Crucial for building a strong profile for MS/Ph.D. admissions, especially in top Indian and international universities, and provides hands-on experience for R&D roles in industry.
Specialize through Departmental Electives- (Semester 3-5)
Carefully select Departmental Electives that align with your long-term career goals, whether in pure mathematics (e.g., Algebraic Geometry), applied mathematics (e.g., Financial Mathematics), or interdisciplinary fields like theoretical computer science.
Tools & Resources
Departmental course catalog and faculty expertise areas, Career counseling services from IITK CDC, Networking with seniors and alumni working in chosen specializations
Career Connection
Develops specialized expertise, making you a more attractive candidate for specific roles or research areas in diverse sectors like finance, analytics, and software development.
Participate in Mathematical Competitions & Workshops- (Semester 3-5)
Engage in national and international math competitions (e.g., Putnam, local Olympiads) or attend workshops/conferences focused on specific mathematical fields. This enhances problem-solving under pressure and facilitates networking.
Tools & Resources
Problem-solving books for competitions (e.g., Problem-Solving Strategies by Engel), Online math communities like Art of Problem Solving, IITK clubs for competitive math (e.g., Math Club), Announcements for conferences and symposia
Career Connection
Boosts analytical skills, fosters a competitive spirit, and provides networking opportunities valuable for both academia and high-tech industry roles in India.
Advanced Stage
Undertake Impactful MS Thesis Research- (Semester 6-10)
Utilize the MS Thesis Project (MTH599) to delve deep into a complex mathematical problem. Aim for original contributions, meticulous documentation, and high-quality thesis writing. Actively present findings at departmental seminars and conferences.
Tools & Resources
Faculty advisors for sustained guidance, Literature review tools (e.g., Zotero, Mendeley), LaTeX for professional thesis writing, Departmental seminar series and research symposia
Career Connection
The thesis is a capstone project demonstrating independent research capability, critical for Ph.D. admissions in top global universities, R&D positions, and roles requiring advanced problem-solving in leading Indian companies.
Build a Professional Network and Seek Internships- (Semester 6-10)
Actively network with alumni, faculty, and industry professionals through conferences, guest lectures, and LinkedIn. Seek relevant long-term internships in areas like quantitative finance, data science, or core research labs to gain practical exposure.
Tools & Resources
IITK Career Development Centre (CDC) for placement/internship support, LinkedIn for professional networking and alumni connections, Alumni association events and mentorship programs, Leading job portals for internships in India (e.g., Internshala, Naukri)
Career Connection
Internships provide invaluable real-world experience, practical skills, and often lead to pre-placement offers. Networking opens doors to diverse career opportunities and mentorship crucial for career progression in India.
Prepare for Placements and Higher Studies- (Semester 6-10)
Tailor your resume and cover letters for specific roles. Practice technical interviews, aptitude tests, and group discussions rigorously. For higher studies, prepare for GRE/GATE, identify suitable universities, and secure strong letters of recommendation from faculty.
Tools & Resources
CDC workshops on resume building, interview techniques, and GDs, Online platforms for aptitude and technical interview preparation (e.g., LeetCode, GeeksforGeeks), Faculty members for recommendation letters and academic guidance, University websites and portals (e.g., for Ph.D. applications abroad)
Career Connection
Systematic and targeted preparation ensures a successful transition to desired career paths, whether in top companies in India (e.g., major banks, tech giants) or leading international universities for advanced degrees.
Program Structure and Curriculum
Eligibility:
- Admission is based on JEE Advanced rank, requiring 10+2 (or equivalent) with Physics, Chemistry, and Mathematics from a recognized board. Specific cut-offs are determined annually.
Duration: 5 years (10 semesters)
Credits: 204 Credits
Assessment: Internal: undefined, External: undefined
Semester-wise Curriculum Table
Semester 1
| Subject Code | Subject Name | Subject Type | Credits | Key Topics |
|---|---|---|---|---|
| MTH101 | Linear Algebra & Differential Equations | Core | 9 | Matrices and vector spaces, Linear transformations and eigenvalues, First and higher order ordinary differential equations, Series solutions and Laplace transforms, Introduction to Partial Differential Equations |
| PHY101 | Concepts of Physics | Core | 9 | Classical Mechanics (Newton''''s laws, energy, momentum), Oscillations and Waves, Electromagnetism fundamentals, Optics (interference, diffraction), Special Relativity and basics of Quantum Physics |
| CHM101 | Chemistry-I | Core | 9 | Atomic structure and chemical bonding, Molecular geometry and intermolecular forces, Thermodynamics and chemical kinetics, Electrochemistry and redox reactions, Fundamentals of organic chemistry |
| ESC101 | Fundamentals of Computing | Core | 8 | Introduction to programming (e.g., C/Python), Data types, operators, and control flow, Functions, arrays, and pointers, File I/O and basic data structures, Algorithm design and problem-solving |
| TA101 | Engineering Graphics | Core | 7 | Orthographic projections and multi-view drawings, Isometric and perspective views, Sectional views and auxiliary views, Dimensioning and tolerancing standards, Introduction to CAD software and visualization |
| LFS101 | Introduction to Life Sciences | Core | 6 | Cell biology and molecular biology, Genetics and inheritance patterns, Evolutionary biology and biodiversity, Ecology and environmental science principles, Human physiology and basic immunology |
| HSS-I | Humanities & Social Sciences Elective - I | Elective | 7 | Varies based on chosen elective from broad areas such as Philosophy, Psychology, Sociology, Economics, History, Literature, or Fine Arts., Focuses on critical thinking, societal understanding, and human values in an Indian context. |
Semester 2
| Subject Code | Subject Name | Subject Type | Credits | Key Topics |
|---|---|---|---|---|
| MTH102 | Complex Analysis & Differential Geometry | Core | 9 | Complex numbers and analytic functions, Cauchy-Riemann equations and contour integration, Residue theorem and conformal mappings, Curves and surfaces in three dimensions, First and second fundamental forms, Gaussian curvature |
| PHY102 | Introduction to Quantum Physics | Core | 9 | Blackbody radiation and photoelectric effect, Wave-particle duality and uncertainty principle, Schrödinger equation and its applications, Hydrogen atom structure and quantum numbers, Introduction to spin and quantum statistics |
| CHM102 | Chemistry-II | Core | 9 | Coordination chemistry and organometallics, Spectroscopic techniques (UV-Vis, IR, NMR), Polymer chemistry and materials science, Catalysis and reaction mechanisms, Introduction to environmental chemistry |
| ESC102 | Engineering Graphics and Drawing | Core | 7 | Advanced orthographic and isometric projections, Sectional and auxiliary views with complex geometries, Intersections and developments of surfaces, Computer-aided design (CAD) fundamentals, Assembly drawings and industrial drawing practices |
| TA102 | Introduction to Manufacturing Processes | Core | 7 | Casting, forming, and machining processes, Welding and other joining techniques, Additive manufacturing (3D printing), Metrology and quality control, Process selection and economic considerations |
| HSS-II | Humanities & Social Sciences Elective - II | Elective | 7 | Varies based on chosen elective from broad areas such as Philosophy, Psychology, Sociology, Economics, History, Literature, or Fine Arts., Aims to broaden perspectives and enhance communication skills relevant to Indian societal contexts. |
| OE-I | Open Elective - I | Elective | 7 | Varies based on student''''s choice from a diverse range of courses offered across different departments., Focuses on interdisciplinary learning or personal interest development to round out the technical education. |
Semester 3
| Subject Code | Subject Name | Subject Type | Credits | Key Topics |
|---|---|---|---|---|
| MTH203 | Probability and Statistics | Core | 9 | Axioms of probability, conditional probability, Bayes'''' theorem, Random variables and probability distributions (discrete and continuous), Moment generating functions and characteristic functions, Sampling theory, estimation, and confidence intervals, Hypothesis testing, correlation, and regression analysis |
| MTH204 | Algebra | Core | 9 | Group theory (subgroups, normal subgroups, homomorphisms), Quotient groups, Sylow theorems, group actions, Ring theory (ideals, integral domains, fields), Polynomial rings and unique factorization domains, Introduction to field extensions and Galois theory |
| MTH205 | Ordinary Differential Equations | Core | 9 | Existence and uniqueness of solutions, Picard''''s theorem, Qualitative theory of ODEs, phase plane analysis, Linear systems of ODEs and stability theory, Boundary value problems and Sturm-Liouville theory, Green''''s functions for ODEs and series solutions |
| HSS-III | Humanities & Social Sciences Elective - III | Elective | 7 | Varies based on chosen elective, continuing to develop critical understanding of humanities and social dynamics., May include advanced topics in economics, psychology, sociology, or public policy relevant to India. |
| OE-II | Open Elective - II | Elective | 7 | Varies based on student''''s choice from a diverse range of courses offered across different departments., Opportunity to explore subjects outside the core specialization, such as management, design, or interdisciplinary studies. |
| OE-III | Open Elective - III | Elective | 7 | Varies based on student''''s choice from a diverse range of courses offered across different departments., Allows for flexibility in academic interests and skill acquisition relevant to emerging fields. |
| DE-I | Departmental Elective - I | Elective | 9 | Topics vary based on chosen elective from advanced mathematics subjects like Cryptography, Coding Theory, Graph Theory, or Combinatorics., Focus on building specialized knowledge within core mathematical disciplines based on student interests. |
Semester 4
| Subject Code | Subject Name | Subject Type | Credits | Key Topics |
|---|---|---|---|---|
| MTH206 | Real Analysis | Core | 9 | Metric spaces and topological properties, Sequences and series of real numbers and functions, Continuity, uniform continuity, differentiation, Riemann integral and its properties, Introduction to Lebesgue measure and integration |
| MTH207 | Topology | Core | 9 | Topological spaces, open and closed sets, basis, Continuous functions and homeomorphisms, Product and quotient topologies, Compactness, connectedness, path connectedness, Separation axioms and countability axioms |
| MTH208 | Numerical Methods | Core | 9 | Error analysis and root finding techniques (Newton-Raphson), Numerical solutions of linear and non-linear algebraic equations, Interpolation and approximation techniques (Lagrange, splines), Numerical differentiation and integration (quadrature rules), Numerical solutions to ordinary differential equations (Runge-Kutta) |
| MTH209 | Operations Research | Core | 9 | Linear programming: Simplex method, duality theory, Transportation and assignment problems, Network flow problems (Max-flow min-cut), Queuing theory and inventory control models, Dynamic programming and introduction to game theory |
| HSS-IV | Humanities & Social Sciences Elective - IV | Elective | 7 | Varies based on chosen elective, providing a well-rounded education in social sciences or humanities., May include topics like public policy, ethics, or advanced linguistic studies, enhancing critical perspective. |
| DE-II | Departmental Elective - II | Elective | 9 | Advanced topics from areas such as algebraic number theory, representation theory, or advanced combinatorics., Further specialization in theoretical or applied mathematics, tailored to research interests. |
Semester 5
| Subject Code | Subject Name | Subject Type | Credits | Key Topics |
|---|---|---|---|---|
| MTH301 | Functional Analysis | Core | 9 | Normed linear spaces, Banach spaces, Hilbert spaces, Bounded linear operators and functionals, Dual spaces and Hahn-Banach theorem, Compact operators and spectral theory, Applications in differential equations and quantum mechanics |
| MTH302 | Partial Differential Equations | Core | 9 | First order PDEs and method of characteristics, Classification of second order PDEs (wave, heat, Laplace), Separation of variables and Fourier series, Green''''s functions and integral transforms, Distribution theory and weak solutions |
| MTH303 | Number Theory | Core | 9 | Divisibility, congruences, modular arithmetic, Prime numbers and fundamental theorem of arithmetic, Quadratic residues and reciprocity laws, Diophantine equations and algebraic number theory introduction, Distribution of primes and zeta function |
| OE-IV | Open Elective - IV | Elective | 7 | Varies based on student''''s choice from a diverse range of courses offered across different departments., Provides an opportunity to gain skills or knowledge in an area of personal interest or career relevance. |
| DE-III | Departmental Elective - III | Elective | 9 | Specialized topics in areas such as discrete mathematics, mathematical logic, or financial mathematics., Allows for deep dive into specific mathematical applications or theoretical constructs. |
| DE-IV | Departmental Elective - IV | Elective | 9 | Advanced concepts in pure or applied mathematics, e.g., advanced topics in topology, geometry, or analysis., Tailored to student research interests or specific career aspirations in advanced fields. |
Semester 6
| Subject Code | Subject Name | Subject Type | Credits | Key Topics |
|---|---|---|---|---|
| MTH304 | Measure Theory | Core | 9 | Sigma-algebras, measures, outer measures, Lebesgue measure on the real line, Measurable functions and Lebesgue integral, Convergence theorems (Monotone, Dominated, Fatou''''s Lemma), Product measures and Fubini''''s theorem |
| MTH305 | Group Theory | Core | 9 | Sylow theorems and classification of finite groups, Soluble and nilpotent groups, Group actions and permutation groups, Representations of finite groups, Free groups and group presentations |
| DE-V | Departmental Elective - V | Elective | 9 | Advanced modules in algebra, analysis, or applied mathematics such as advanced probability theory or stochastic processes., Designed to cater to diverse student interests and emerging research areas in mathematics. |
| DE-VI | Departmental Elective - VI | Elective | 9 | Further specialized topics in mathematical physics, computational mathematics, or scientific computing., Provides an opportunity to integrate mathematical knowledge with other scientific or engineering domains. |
| DE-VII | Departmental Elective - VII | Elective | 9 | Elective covering topics like advanced combinatorics, fuzzy set theory, or game theory., Aims to deepen expertise in specific advanced mathematical areas and their applications. |
| MTH491 | Project - I | Project | 7 | Independent study and literature review on a chosen mathematical topic, Problem identification and preliminary research design, Methodology exploration and initial data collection/analysis, Oral presentation and written report on project progress, Mentored research experience, fostering independent inquiry |
Semester 7
| Subject Code | Subject Name | Subject Type | Credits | Key Topics |
|---|---|---|---|---|
| MTH401 | Differential Geometry | Core | 9 | Manifolds, tangent spaces, vector fields, Connections and covariant derivatives, Riemannian metrics and curvature, Geodesics and completeness, Introduction to Lie groups and Lie algebras |
| MTH402 | Abstract Algebra | Core | 9 | Rings and modules, Noetherian rings, Prime and maximal ideals, factorization in rings, Field theory, Galois theory in depth, Tensor products and exterior algebra, Introduction to homological algebra concepts |
| DE-VIII | Departmental Elective - VIII | Elective | 9 | Specialized topics in applied mathematics like computational fluid dynamics, inverse problems, or mathematical modeling., Focus on practical applications and computational tools in addressing real-world challenges. |
| DE-IX | Departmental Elective - IX | Elective | 9 | Advanced concepts in pure mathematics such as algebraic geometry, algebraic topology, or advanced number theory., Prepares students for higher-level research and theoretical contributions in mathematics. |
| OE-V | Open Elective - V | Elective | 7 | Varies based on student''''s choice, providing breadth to their academic profile., Could include courses in management, entrepreneurship, data science, or social sciences, complementing mathematical skills. |
| MTH492 | Project - II | Project | 7 | Continuation of Project I with deeper analysis and problem-solving, Refinement of methodology and advanced research techniques, Data analysis, interpretation of results, and critical evaluation, Interim report and presentation of findings to departmental faculty, Strengthening independent research capabilities and scientific communication |
Semester 8
| Subject Code | Subject Name | Subject Type | Credits | Key Topics |
|---|---|---|---|---|
| MTH403 | Algebraic Topology | Core | 9 | Homotopy theory, fundamental group, covering spaces, Simplicial and singular homology, Homology and cohomology groups, Euler characteristic and classification of surfaces, Introduction to fiber bundles and characteristic classes |
| DE-X | Departmental Elective - X | Elective | 9 | High-level elective in pure mathematics, such as category theory, logic, or advanced combinatorics., Designed for students aiming for advanced theoretical research and academic careers. |
| DE-XI | Departmental Elective - XI | Elective | 9 | Applied mathematics elective, e.g., topics in fluid dynamics, mathematical biology, or control theory., Provides skills for solving complex real-world problems using advanced mathematical tools and modeling. |
| OE-VI | Open Elective - VI | Elective | 7 | Varies based on student''''s choice, further diversifying their academic and professional skill set., Could be advanced programming, foreign language, or social entrepreneurship, enhancing interdisciplinary appeal. |
| MTH493 | Project - III | Project | 7 | Advanced stage of project work, focusing on significant contribution and originality, Finalization of research methodology and rigorous analysis of findings, Comprehensive report writing, including results, discussion, and conclusions, Defense of project work and preparation of publication-ready manuscript, Showcase of advanced problem-solving and research acumen |
Semester 9
| Subject Code | Subject Name | Subject Type | Credits | Key Topics |
|---|---|---|---|---|
| MTH501 | Advanced Linear Algebra | Core | 9 | Canonical forms (Jordan, rational), invariant subspaces, Bilinear forms, quadratic forms, symmetric operators, Tensor products and multilinear algebra, Matrix decompositions (SVD, QR) and numerical linear algebra, Applications in scientific computing, data analysis, and quantum information |
| DE-XII | Departmental Elective - XII | Elective | 9 | Master''''s level advanced mathematics elective, possibly in areas like machine learning theory, wavelet analysis, or advanced differential geometry., Aimed at providing cutting-edge knowledge in selected mathematical fields, fostering research capabilities. |
| DE-XIII | Departmental Elective - XIII | Elective | 9 | Another master''''s level elective focusing on current research trends in pure or applied mathematics., Fosters specialized expertise and prepares for doctoral studies or advanced research careers. |
| OE-VII | Open Elective - VII | Elective | 7 | Final open elective, potentially aligning with career aspirations such as advanced computing, economics, or humanities for a well-rounded profile., Contributes to holistic development and cross-disciplinary understanding for complex problem-solving. |
| MTH599 (Part 1) | MS Thesis Project (Part 1) | Project | 14 | Initiation of Master''''s thesis research, comprehensive literature survey, Formulation of research problem and precise objectives, Development of theoretical framework and methodologies, Preliminary results and critical evaluation of approaches, Regular interaction with advisor and engagement with research group activities |
Semester 10
| Subject Code | Subject Name | Subject Type | Credits | Key Topics |
|---|---|---|---|---|
| DE-XIV | Departmental Elective - XIV | Elective | 9 | Highly advanced elective, possibly exploring research frontiers in specific mathematical sub-disciplines like quantum computing or advanced topology., Aims to refine expertise in a chosen area of specialization within mathematics for high-impact careers. |
| DE-XV | Departmental Elective - XV | Elective | 9 | Another advanced elective chosen from a wide array of master''''s level courses in pure or applied mathematics., Provides flexibility for students to tailor their expertise towards specific academic, industrial, or government applications. |
| OE-VIII | Open Elective - VIII | Elective | 7 | Last open elective to round off the dual degree, potentially focusing on ethical aspects of science, intellectual property, or advanced communication skills., Complements technical knowledge with broader professional competencies for a holistic graduate profile. |
| MTH599 (Part 2) | MS Thesis Project (Part 2) | Project | 14 | Completion of Master''''s thesis research, in-depth analysis of results and findings, Writing and rigorous editing of the thesis manuscript adhering to academic standards, Preparation for thesis defense, incorporating feedback and refining arguments, Presentation of novel findings and contributions to academic community, Demonstration of independent scholarship, critical thinking, and advanced expertise |
