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PH-D in Mathematics at Indian Institute of Science
Bengaluru, Karnataka
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About the Specialization
What is Mathematics at Indian Institute of Science Bengaluru?
This Ph.D. in Mathematics program at Indian Institute of Science, Bengaluru focuses on developing rigorous research capabilities in core and applied mathematical areas. Recognised globally for its research output, IISc''''s program prepares scholars for advanced academic and research roles. It addresses the critical need for high-end mathematical talent in India''''s burgeoning R&D sectors, contributing to scientific advancement and innovation within the country.
Who Should Apply?
This program is ideal for highly motivated individuals holding a Master''''s degree in Mathematics or a strong Bachelor''''s degree in a relevant scientific or engineering discipline. It attracts fresh graduates with an aptitude for theoretical and abstract thinking, seeking to contribute original research. Academics looking to deepen their expertise, and professionals aiming for research and development positions in cutting-edge industries, will find this program intellectually stimulating and career-advancing.
Why Choose This Course?
Graduates of this program can expect to pursue distinguished careers in academia, research institutions, and R&D divisions across diverse industries in India. Potential career paths include university professors, research scientists in government labs (e.g., DRDO, ISRO), data scientists, quantitative analysts in finance, and cryptographers. Starting salaries for PhDs in India can range from INR 8-15 LPA in industry, with academic positions offering competitive research grants and long-term stability, fostering a continuous growth trajectory.
Student Success Practices
Foundation Stage
Master Core Mathematical Concepts Rigorously- (Semester 1-2)
Dedicate significant time to thoroughly understand foundational courses like Analysis, Algebra, Topology, and Differential Equations. Focus on grasping proofs, definitions, and theorems deeply rather than rote learning. Actively participate in seminars, discussion groups, and problem-solving sessions.
Tools & Resources
Textbooks by standard authors (e.g., Rudin for Analysis, Dummit & Foote for Algebra), Departmental lecture notes, Online resources like NPTEL and MIT OpenCourseware
Career Connection
A strong conceptual foundation is critical for clearing the comprehensive examination and forms the bedrock for advanced research, enabling innovative problem-solving in any future role.
Proactively Engage with Faculty and Research Areas- (Semester 1-2)
Attend departmental research colloquia and faculty talks to identify potential research interests and advisors. Initiate discussions with professors about their work and possible PhD projects. This early engagement helps in choosing a research area aligning with your aptitude and the department''''s strengths.
Tools & Resources
Faculty profiles on the department website, Research group pages, Departmental seminar schedules, One-on-one meetings with potential advisors
Career Connection
Early identification of a research focus and advisor ensures a smoother transition into thesis work, accelerating your research output and networking within your chosen field.
Develop Advanced Problem-Solving and Presentation Skills- (Semester 1-2)
Beyond coursework, regularly practice solving challenging problems from various mathematical domains. Form study groups to discuss complex topics and present solutions to peers. Develop clear and concise presentation skills for seminars and comprehensive exam preparations.
Tools & Resources
Problem books in advanced mathematics, Online platforms like StackExchange (Mathematics), Departmental peer-led study groups, Practice comprehensive exam papers
Career Connection
Strong problem-solving abilities are essential for original research, while effective communication skills are vital for disseminating findings, securing collaborations, and teaching in academic or industrial settings.
Intermediate Stage
Successfully Navigate the Comprehensive Examination- (End of Semester 3 / Beginning of Semester 4)
Prepare systematically for the comprehensive examination by reviewing all core coursework and practicing past papers. Seek guidance from faculty on exam patterns and common pitfalls. This milestone validates your foundational knowledge and readiness for independent research.
Tools & Resources
Comprehensive exam syllabus, Previous year question papers (if available), Faculty office hours, Peer study groups, Focused revision of key textbooks
Career Connection
Passing the comprehensive exam is a crucial step towards becoming an independent researcher, demonstrating your mastery of the field and paving the way for thesis work.
Initiate and Deepen Research in Your Chosen Area- (Semester 3-5)
Once an advisor and research topic are finalized, begin an in-depth literature review, identify open problems, and start formulating concrete research questions. Regularly meet with your advisor for guidance, feedback, and to discuss progress and challenges.
Tools & Resources
Academic databases (MathSciNet, JSTOR, arXiv), Reference management software (Zotero, Mendeley), Research journals in your area, Regular advisor meetings
Career Connection
This stage is where you start building your research portfolio, producing original results that will form the basis of your thesis and future publications, essential for academic and R&D careers.
Engage with the Broader Research Community- (Semester 3-5)
Attend national and international conferences, workshops, and summer/winter schools relevant to your research area. Present preliminary findings through posters or short talks. Network with other researchers and faculty, seeking feedback and exploring potential collaborations.
Tools & Resources
Conference websites, Department travel grants, IISc-funded workshops, Professional societies (e.g., Indian Mathematical Society)
Career Connection
Exposure to external research perspectives refines your work, opens doors to collaborations, and builds your professional network, which is vital for future job searches and academic growth.
Advanced Stage
Publish and Disseminate Research Findings- (Semester 6 onwards)
Focus on writing and submitting your research results to peer-reviewed international journals. Aim for multiple publications, ensuring clarity, rigor, and originality. Actively engage in the peer-review process for your own and others'''' work.
Tools & Resources
LaTeX document preparation system, Reputable mathematical journals, Pre-print servers like arXiv, Feedback from advisor and peers
Career Connection
A strong publication record is paramount for securing post-doctoral positions, faculty roles, and high-level R&D jobs, demonstrating your capability as an independent and impactful researcher.
Prepare a High-Quality Thesis and Defend Effectively- (Semester 7-8 and beyond)
Systematically compile your research findings into a coherent, well-structured Ph.D. thesis. Work closely with your advisor on multiple drafts, refining arguments and presentation. Practice your viva voce (thesis defense) presentation thoroughly, anticipating questions from examiners.
Tools & Resources
Thesis writing guidelines from IISc, LaTeX templates, Feedback from advisor and thesis committee members, Mock defense sessions
Career Connection
A well-defended thesis is the culmination of your PhD journey, leading to degree conferral and serving as a primary credential for all subsequent academic and research career opportunities.
Strategize for Post-Ph.D. Career Paths- (Semester 7-8 and final year)
Explore various career options – academia (post-docs, faculty), industry R&D, data science, quantitative finance – and tailor your applications. Attend career workshops, prepare a compelling CV and research statement, and seek recommendation letters from your advisor and mentors.
Tools & Resources
IISc Career Cell, University job boards, Academic job market websites, LinkedIn, Networking contacts, Mentorship
Career Connection
Proactive career planning ensures a smooth transition post-PhD, helping you secure a position that aligns with your research expertise and long-term professional aspirations, maximizing the impact of your advanced degree.
Program Structure and Curriculum
Eligibility:
- Master''''s degree in Mathematics or an allied discipline (e.g., Statistics, Computer Science, Engineering with strong Math background), or a Bachelor''''s degree in Engineering/Technology/Science/Mathematics/Statistics/Computer Science (or equivalent) with a specified minimum aggregate/CGPA. Specific qualifying examination (e.g., GATE, NET, NBHM) scores are generally required.
Duration: Minimum 3 years, typically 4-5 years including coursework and research
Credits: Minimum 24 credits (M.Sc. entry) or 36 credits (B.E./B.Tech. entry) for coursework Credits
Assessment: Internal: undefined, External: undefined
Semester-wise Curriculum Table
Semester 1
| Subject Code | Subject Name | Subject Type | Credits | Key Topics |
|---|---|---|---|---|
| MA 211 | Analysis I | Core | 3 | Real Analysis Fundamentals, Metric Spaces, Continuity and Differentiation, Riemann Integration, Sequences and Series of Functions |
| MA 215 | Complex Analysis | Core | 3 | Complex Numbers and Functions, Analytic Functions, Cauchy-Riemann Equations, Contour Integration, Residue Theory |
| MA 223 | Linear Algebra | Core | 3 | Vector Spaces, Linear Transformations, Eigenvalues and Eigenvectors, Inner Product Spaces, Canonical Forms |
| MA 224 | Algebra I | Core | 3 | Group Theory, Rings and Fields, Homomorphisms, Ideals, Polynomial Rings |
| MA 243 | Differential Equations | Core | 3 | First Order ODEs, Second Order Linear ODEs, Systems of ODEs, Stability Analysis, Existence and Uniqueness Theorems |
| MA 253 | Probability | Core | 3 | Axioms of Probability, Random Variables, Probability Distributions, Expectation and Variance, Law of Large Numbers, Central Limit Theorem |
| MA 261 | Numerical Analysis | Core | 3 | Error Analysis, Solution of Nonlinear Equations, Interpolation and Approximation, Numerical Integration and Differentiation, Numerical Solution of ODEs |
| MA 271 | Introduction to Topology | Core | 3 | Topological Spaces, Continuity and Homeomorphisms, Connectedness, Compactness, Quotient Spaces |
| MA 291 | Logic | Core | 3 | Propositional Logic, First-Order Logic, Gödel''''s Completeness Theorem, Model Theory Basics, Axiomatic Set Theory |
| MA 312 | Topics in Functional Analysis | Core | 3 | Banach Spaces, Hilbert Spaces, Bounded Linear Operators, Spectral Theory, Distributions |
| MA 313 | Measure Theory | Core | 3 | Lebesgue Measure, Measurable Functions, Lebesgue Integral, Convergence Theorems, Product Measures |
| MA 315 | Several Complex Variables | Core | 3 | Holomorphic Functions in Multiple Variables, Cauchy-Riemann System, Domains of Holomorphy, Sheaves and Cohomology, Complex Manifolds |
| MA 322 | Algebraic Number Theory | Core | 3 | Number Fields, Rings of Integers, Prime Factorization, Class Group, Dirichlet''''s Unit Theorem |
| MA 325 | Topics in Representation Theory | Core | 3 | Group Representations, Modules, Characters, Lie Algebra Representations, Applications in Physics |
| MA 332 | Differential Geometry | Core | 3 | Manifolds, Tangent Spaces, Vector Fields, Differential Forms, Connections and Curvature |
| MA 333 | Geometric PDE | Core | 3 | Elliptic PDEs, Parabolic PDEs, Hyperbolic PDEs, Geometric Flows, Harmonic Maps |
| MA 342 | Dynamical Systems | Core | 3 | Flows and Maps, Stability Theory, Limit Cycles, Chaos Theory, Bifurcations |
| MA 352 | Stochastic Processes | Core | 3 | Markov Chains, Poisson Processes, Random Walks, Martingales, Brownian Motion |
| MA 354 | Advanced Probability | Core | 3 | Measure Theoretic Probability, Conditional Expectation, Martingale Theory, Weak Convergence, Stochastic Integrals |
| MA 361 | Advanced Numerical Methods | Core | 3 | Numerical Linear Algebra, Iterative Methods, Finite Difference Methods, Finite Element Methods, Spectral Methods |
| MA 373 | Introduction to Algebraic Topology | Core | 3 | Homotopy, Fundamental Group, Covering Spaces, Simplicial Homology, Cell Complexes |
| MA 381 | Cryptography and Coding Theory | Core | 3 | Classical Ciphers, Public-Key Cryptography (RSA), Elliptic Curve Cryptography, Error-Correcting Codes, Finite Fields |
| MA 391 | Set Theory | Core | 3 | Axioms of Set Theory (ZFC), Ordinals and Cardinals, Axiom of Choice, Continuum Hypothesis, Large Cardinals |
Semester 2
| Subject Code | Subject Name | Subject Type | Credits | Key Topics |
|---|---|---|---|---|
| MA 212 | Analysis II | Core | 3 | Lebesgue Theory, Lp Spaces, Fourier Series, Distribution Theory, Functional Analysis Basics |
| MA 216 | Topology | Core | 3 | Topological Spaces, Connectedness and Compactness, Separation Axioms, Product and Quotient Spaces, Metrization Theorems |
| MA 222 | Algebra II | Core | 3 | Module Theory, Field Extensions, Galois Theory, Tensor Products, Noetherian Rings |
| MA 226 | Commutative Algebra | Core | 3 | Rings and Modules, Ideals and Prime Ideals, Localization, Noetherian and Artinian Rings, Dimension Theory |
| MA 231 | ODE | Core | 3 | Existence and Uniqueness, Linear Systems, Stability Theory, Phase Plane Analysis, Boundary Value Problems |
| MA 241 | PDE | Core | 3 | First Order PDEs, Classification of Second Order PDEs, Wave Equation, Heat Equation, Laplace Equation |
| MA 252 | Statistics | Core | 3 | Sampling Distributions, Point Estimation, Hypothesis Testing, Confidence Intervals, Linear Regression |
| MA 262 | Linear Programming | Core | 3 | Simplex Method, Duality Theory, Sensitivity Analysis, Transportation Problem, Network Flow Problems |
| MA 272 | Algebraic Topology | Core | 3 | Homotopy Theory, Fundamental Group, Covering Spaces, Homology Theory, Cohomology Theory |
| MA 281 | Combinatorics | Core | 3 | Counting Principles, Generating Functions, Recurrence Relations, Graph Theory Basics, Combinatorial Designs |
| MA 292 | Set Theory | Core | 3 | Axiomatic Set Theory (ZFC), Transfinite Induction, Cardinals and Ordinals, Constructible Universe, Forcing Method |
| MA 311 | Functional Analysis | Core | 3 | Normed and Banach Spaces, Hilbert Spaces, Bounded Linear Operators, Spectral Theory, Compact Operators |
| MA 314 | Harmonic Analysis | Core | 3 | Fourier Series, Fourier Transform, Distributions, Hardy Spaces, Wavelets |
| MA 321 | Representation Theory | Core | 3 | Representations of Finite Groups, Characters, Schur''''s Lemma, Induced Representations, Representations of Lie Algebras |
| MA 323 | Group Theory | Core | 3 | Finite Groups, Sylow Theorems, Solvable and Nilpotent Groups, Group Actions, Free Groups |
| MA 324 | Lie Algebras | Core | 3 | Definitions and Examples, Ideals and Homomorphisms, Solvable and Nilpotent Lie Algebras, Structure Theory, Root Systems |
| MA 331 | Algebraic Geometry | Core | 3 | Affine and Projective Varieties, Nullstellensatz, Schemes, Coherent Sheaves, Riemann-Roch Theorem |
| MA 334 | Differential Geometry II | Core | 3 | Riemannian Metrics, Geodesics, Curvature Tensors, Ricci Curvature, Einstein Manifolds |
| MA 341 | Control Theory | Core | 3 | State Space Models, Controllability and Observability, Stabilization, Optimal Control, Lyapunov Stability |
| MA 351 | Probability II | Core | 3 | Convergence of Random Variables, Conditional Probability and Expectation, Martingales, Ergodic Theory, Stochastic Differential Equations |
| MA 362 | Optimization | Core | 3 | Linear Programming, Nonlinear Programming, Convex Optimization, Lagrange Multipliers, KKT Conditions |
| MA 371 | Riemannian Geometry | Core | 3 | Riemannian Manifolds, Connections and Curvature, Geodesics, Jacobi Fields, Comparison Theorems |
| MA 372 | Algebraic Topology II | Core | 3 | Homotopy Groups, Fibre Bundles, Spectral Sequences, K-Theory, Characteristic Classes |
| MA 382 | Graph Theory | Core | 3 | Connectivity, Matching and Factors, Coloring, Planar Graphs, Random Graphs |
