Ph.D. PROGRAMME IN MATHEMATICS offers an exciting and unique opportunity to students for pursuing research in the following areas:

- Pure Mathematics
- Applied Mathematics & Scientific Computing
- Probability & Statistics
- Theoretical Computer Science

Under this programme students undergo a substantial amount of relevant course work consisting of advance topics in Aglebra, Analysis, Topology, Numerical Analysis, Solid & Fluid Mechanics and Statistics, followed by research work under the supervision of an Advisor, who is decided by the Department, taking into account the aptitude, the needs and the preferences of the student.

- Nurtures creative and analytical abilities of scholars.
- Provides flexible approach for self-study and seminar courses in different areas.
- Encourages interdisciplinary research for bridging the gap between theoretical and applied areas.
- Cultivates communication skills through seminars and tutorials.
- Offers opportunity to engage in developmental research work.

The Department of Mathematics is well recognised for teaching and research. It has a large faculty with research interests covering a wide range of fields:

- Algebra, Combinatorics, Topology and Geometry
- Functional Analysis, Harmonic Analysis, Partial Differential Equations and Systems & Control.
- Numerical Analysis & Scientific Computing.
- Statistical Inference, Applied Probability, Statistical Quality Control and Biostatistics.
- Algorithms & Complexity and Combinatorial Optimization,

Collabortive research with other Science and Engineering Departments of the Institute is encouraged. Faculty members undertake projects sponsored by organizations such as National Board for Higher Mathematics, Indian National Science Academy, Board of Research in Nuclear Science, Council of Scientific & Industrial Research, Department of Science and Technology, Department of Bio-Technology and Indian Council of Medical Research etc. A strong group of Industrial Mathematics has evolved in the Department for providing Industry-Academic linkage.

The graduates of this Department have been placed in various academic institutions - IITs, IISc and several universities in India and abroad. The Training and Placement Office of the Institute arranges Campus Interviews for the students with prospective employer in Industry and R & D Organisations with strong inputs from the Department. In recent years our students have been hired by prestigious organisations like TCS, ISRO, Infotech, TRDDC, CDAC, ORG etc.

The Institute offers Teaching Assistantships requiring eight hours of work per week. Students can also be supported by scholarships / fellowships of other organizations such as National Board for Higher Mathematics, Council of Scientific & Industrial Research, University Grants Commission, Department of Science & Technology. Admissions take pace twice a year in June and in December on the basis of a test and an interview. The candidates should have obtained first class at the Masters degree in Mathematics/Statistics/Computer Science and must have a valid GATE score or an award of NBHM/CSIR/UGC Research Fellowship.

For students who have joined the Ph.D. program before July 2017, the following rules apply.

- 1. Each student in the Ph.D. programme in the Department of Mathematics is required to take a Qualifying Examination to be administered by the DPGC, Department of Mathematics.
- 2. The qualifying examination shall consist of written examinations on any two of the following six topics: (i) Algebra, (ii) Analysis, (iii) Geometry and Topology, (iv) Differential Equations, (v) Probability and (vi) Statistics.
- 3. The qualifying examination shall be conducted in the week before the beginning of the Autumn semester as well as the Spring semester. The results of the qualifying examination shall be made known before the date of registration.
- 4. Each student is permitted at most two attempts for each of the two chosen topics. It is expected that every student aspiring to do Ph.D. will successfully complete the qualifying examination requirement by the end of the first academic year. In any case, the maximum time allotted for a student to pass the qualifying examination shall be three semesters. If a student fails to pass the examination in one or both of the chosen topics, then he/she may be permitted a change of topic. However, the ceiling of maximum two attempts and the overall ceiling of three semesters to complete the qualifying examination requirement shall always remain in force.
- 5. A student may choose a Ph.D. thesis advisor after the successful fulfillment of the qualifying examination requirement. Until that time, the Head, Department of Mathematics shall be the guide for all official purposes.
- 6. In case a student fails the qualifying examination requirement, then he/she may be offered the option of transferring to M.Phil. programme by continuing for about a semester so as to complete the requirements for the degree of M.Phil. degree.
- 7. Notwithstanding the qualifying examination requirement, the course requirement stipulated by the Academic Office for the Ph.D. (or the M.Phil., as the case may be) programme will have to be satisfied. It may be noted that currently the course requirement for Ph.D. students consist of completing at least 34 course credits. However, students may register for up to two 500-level M.Sc. courses (for example, those that are relevant for the topic of qualifiers) to partially satisfy the credit requirement.

For students joining the Ph.D. program from July 2017 onwards, the following rules apply.

- 1. Each student in the Ph.D. program of the Department of Mathematics is required to take Qualifying Examinations (QE) to be administered by the DPGC, Department of Mathematics.
- 2. The QE shall consist of written examinations on seven subjects each of 100 marks, namely, (1) Algebra, (2) Analysis, (3) Geometry and Topology, (4) Differential Equations, (5) Probability, (6) Statistics and (7) Combinatorics and Theoretical Computer Science. A PhD candidate must score a minimum of 60 in at least two of these subjects to fulfil the QE requirements.
- 3. The Q.E. will be conducted during 1-15 July and 15-31 December each year and the results will be declared by 21st July and 7th January, respectively.
- 4. The student needs to pass the QE in two out of seven subjects (scoring a minimum of 60 marks out of 100 marks in each of the subjects) within three semesters of their joining the Ph.D. program. There are NO further restrictions on the number of attempts allowed to complete the QE requirement.
- 5. A student may choose a Ph.D. thesis advisor after the successful fulfilment of the coursework and QE requirement. Until that time, the Head, Department of Mathematics shall be the guide for all official purposes.
- 6. In case a student fails the QE requirement, then he/she may be offered the option of transferring to M.Phil. program by continuing for about a semester so as to complete the requirements for the degree of M.Phil.degree.
- 7. Notwithstanding the QE requirement, the course requirement stipulated by the Academic Office for the Ph.D. (or the M.Phil., as the case may be) program will have to be satisfied. The Ph.D. students are required to complete the coursework with a minimum of 34 credits and a minimum 6.0 CPI. However, the students may register for up to two 500-level M.Sc. courses (for example, those that are relevant for the topic of qualifiers) to partially satisfy the credit requirement.

For students joining the Ph.D. program from July 2017 onwards, the syllabus for qualifying examinations can be found here.

For students who have joined the Ph.D. program before July 2017, the syllabus is the following:

Groups, Simple groups and solvable groups, nilpotent groups, simplicity of alternating groups, composition series, Jordan-Holder Theorem. Semidirect products. Free groups, free abelian groups.

Rings, Examples (including polynomial rings, formal power series rings, matrix rings and group rings), ideals, prime and maximal ideals, rings of fractions, Chinese Remainder Theorem for pairwise comaximal ideals. Euclidean Domains, Principal Ideal Domains and Unique Factorizations Domains. Poly-nomial rings over UFD's.

Fields, Characteristic and prime subfields, Field extensions, Finite, algebraic and finitely generated field extensions, Classical ruler and compass constructions, Splitting fields and normal extensions, algebraic closures. Finite fields, Cyclotomic fields, Separable and inseparable extensions.

Galois groups, Fundamental Theorem of Galois Theory, Composite extensions, Examples (including cyclotomic extensions and extensions of finite fields).

Norm, trace and discriminant.

Solvability by radicals, Galois' Theorem on solvability.

Cyclic extensions, Abelian extensions, Transcendental extensions.

Modules, submodules, quotient modules and module homomorphisms.

Generation of modules, direct sums and free modules. Tensor products of modules. Exact sequences, projective modules.

Tensor algebras, symmetric and exterior algebras.

Finitely generated modules over principal ideal domains, invariant factors, elementary divisors, rational canonical forms. Applications to finitely generated abelian groups and linear trans-formations.

Noetherian rings and modules, Hilbert basis theorem, Primary decomposition of ideals in noetherian rings.

Integral extensions, Going-up and Going-down theorems, Extension and contraction of prime ideals, Noether's Normalization Lemma, Hilbert's Nullstellensatz.

Localization of rings and modules. Primary decompositions of modules.

Semi-algebra, Algebra, Monotone class, Sigma-algebra, Monotone class theorem. Measure spaces.

Extension of measures from algebras to the generated sigma-algebras: Measurable sets; Lebesgue Measure and its properties.

Measurable functions and their properties; Integration and Convergence theorems.

Introduction to Lp-spaces, Riesz-Fischer theorem; Riesz Representation theorem for L2 spaces.

Absolute continuity of measures, Radon-Nikodym theorem. Duals of Lp-spaces.

Product measure spaces, Fubini's theorem.

Fundamental Theorem of Calculus for Lebesgue Integrals.

Normed spaces. Continuity of linear maps. Hahn-Banach Extension and Separation Theorems. Banach spaces. Dual spaces and transposes.

Uniform Boundedness Principle and its applications. Closed Graph Theorem, Open Mapping Theorem and their applications. Spectrum of a bounded operator. Examples of compact operators on normed spaces.

Inner product spaces, Hilbert spaces. Orthonormal basis. Projection theorem and Riesz Representation Theorem for Hilbert spaces.

Functions on Euclidean spaces, continuity, differentiability; partial and directional derivatives, Chain Rule, Inverse Function Theorem, Implicit Function Theorem.

Riemann Integral of real-valued functions on Euclidean spaces, measure zero sets, Fubini's Theorem.

Partition of unity, change of variables.

Integration on chains, tensors, differential forms, Poincare Lemma, singular chains, integration on chains, Stokes' Theorem for integrals of differential forms on chains. (general version). Fundamental theorem of calculus.

Differentiable manifolds (as subspaces of Euclidean spaces), differentiable functions on manifolds, tangent spaces, vector fields, differential forms on manifolds, orientations, integration on manifolds, Stokes' Theorem on manifolds.

Paths and homotopy, homotopy equivalence, contractibility, deformation retracts.

Basic constructions: cones, mapping cones, mapping cylinders, suspension.

Cell complexes, subcomplexes, CW pairs.

Fundamental groups. Examples (including the fundamental group of the circle) and applications (including Fundamental Theorem of Algebra, Brouwer Fixed Point Theorem and Borsuk-Ulam Theorem, both in dimension two). Van Kampen's Theorem, Covering spaces, lifting properties, deck transformations. Universal coverings (existence theorem optional).

Simplicial complexes, barycentric subdivision, stars and links, simplicial approximation. Simplicial Homology. Singular Homology. Mayer-Vietoris Sequences. Long exact sequence of pairs and triples.

Homotopy invariance and excision (without proof).

Methods for solving first order as well as second order equations, Power Series methods with properties of Bessel functions and Legendre polynomials.

Existence and Uniqueness of Initial Value Problems: Picard's and Peano's Theorems, Gronwall's inequality, continuation of solutions and maximal interval of existence, continuous dependence.

Higher order linear differential equations and linear systems: fundamental solutions, Wronskian, variation of constants, matrix exponential solution, behaviour of solutions.

Two Dimensional Autonomous Systems and Phase Space Analysis: critical points, proper and improper nodes, spiral points and saddle points.

Asymptotic Behavior: stability (linearized stability and Lyapunov methods).

Boundary Value Problems for Second Order Equations: Green's function, Sturm comparision theorems and oscillations, eigenvalue problems

Cauchy Problems for First Order Hyperbolic Equations: method of characteristics, Monge cone.

Classification of Second Order Partial Differential Equations: normal forms and characteristics.

Initial and Boundary Value Problems: Lagrange-Green's identity and uniqueness by energy methods.

Stability theory, energy conservation and dispersion equation: mean value property, weak and strong maximum principle, Green's function, Poisson's formula, Dirichlet's principle, existence of solution using Perron's method (without proof).

Heat equation: initial value problem, fundamental solution, weak and strong maximum principle and uniqueness results.

Wave equation: uniqueness, D'Alembert's method, method of spherical means and Duhamel's principle.

Methods of separation of variables for heat, and wave equations.

Probability measure, probability space, construction of Lebesgue measure, extension theorems, limit of events, Borel-Cantelli lemma.

Random variables, Random vectors, distributions, multidimensional distributions, independence.

Expectation, change of variable theorem, convergence theorems.

Sequence of random variables, modes of convergence. Moment generating function and characteristics functions, inversion and uniqueness theorems, continuity theorems, Weak and strong laws of large number, central limit theorem.

Radon-Nikodym theorem, definition and properties of conditional expectation, conditional distributions.

Stochastic processes: Description and definition. Markov chains with finite and countably infinite state spaces. Classification of states, irreducibility, ergodicity. Basic limit theorems.

Markov processes with discrete and continu-ous state spaces. Poisson process, pure birth process, birth and death process. Brownian motion.

Applications to queueing models and reliability theory.

Basic theory and applications of renewal processes, stationary processes. Branching processes. Markov Renewal and semi-Markov processes, regenerative processes.

Distribution of functions of random variables, Order Statistics. Estimation - loss function, risk, minimum risk unbiased estimators, maximum likelihood estimation, method of moments, Bayes estimation.

Sufficient Statistics, completeness, Basu's Theorem, exponential families, invariance and maximal invariant statistics.

Testing of Hypotheses - parametric and non-parametric problems, examples with data analytic applications.

Confidence Intervals.

Simple and multiple linear regression models, estimation, tests and confidence regions. Check for normality assumption. Likelihood ratio test, confidence intervals and hypotheses tests; tests for distributional assumptions. Collinearity, outliers; analysis of residuals, Selecting the best fit, regression equation, transformation of response variables. Ridge's regression.

The following courses are likely to be offered in each of the corresponding semesters.

Course Code | Name of the Course | L | T | P | C |
---|---|---|---|---|---|

MA 811 | Algebra I | 3 | 0 | 0 | 6 |

MA 813 | Measure Theory | 3 | 0 | 0 | 6 |

MA 815 | Differential Topology | 3 | 0 | 0 | 6 |

MA 817 | Partial Differential Equations I | 3 | 0 | 0 | 6 |

MA 833 | Weak Convergence and Martingale Theory | 3 | 0 | 0 | 6 |

MA 821 | Theory of Estimation | 3 | 0 | 0 | 6 |

MA 861 | Combinatorics-I | 3 | 0 | 0 | 6 |

Course Code | Name of the Course | L | T | P | C |
---|---|---|---|---|---|

MA 812 | Algebra II | 3 | 0 | 0 | 6 |

MA 814 | Complex Analysis | 3 | 0 | 0 | 6 |

MA 816 | Algebraic Topology | 3 | 0 | 0 | 6 |

MA 818 | Partial Differential Equations II | 3 | 0 | 0 | 6 |

MA 820 | Stochastic Processes | 3 | 0 | 0 | 6 |

MA 822 | Testing of Hypothesis | 3 | 0 | 0 | 6 |

MA 824 | Functional Analysis | 3 | 0 | 0 | 6 |

MA 862 | Combinatorics-II | 3 | 0 | 0 | 6 |

1. The credit requirements for students having M.Sc. or equivalent qualification shall be 34 to 46 credits.

2. Credits acquired through PG level courses shall be 24 or more.

3. Students may earn upto a maximum of 8 credits through seminars which should be spread over two semesters.

4. Each student must credit at least 3 of the 13 courses MA 813 – 822 & MA 824.

PhD students are also required to take two courses on communication skills, viz, HS791 (Communication Skills I) offered by the HSS department and MA792 (Communication Skills II) offered by the Mathematics department

a) These courses are compulsory for all Ph.D. students.

b) Ph.D. students are normally required to clear the Communication skills course within the first two semesters.

c) These courses are an addition to the minimum course credit requirement prescribed by the DPGCs/IDPCs.

d) Students having a qualifying degree from IIT who have cleared the ‘Communication Skills’ course during their M.Tech. Programme are exempted from this requirement.

In addition to the above courses, some or all of the following courses may be offered subject to sufficient demand from the students and availability of faculty.

Course Code | Name of the Course | L | T | P | C |
---|---|---|---|---|---|

MA 841 | Topics in Algebra I | 3 | 0 | 0 | 6 |

MA 843 | Topics in Analysis I | 3 | 0 | 0 | 6 |

MA 845 | Topics in Combinatorics I | 3 | 0 | 0 | 6 |

MA 847 | Topics in Geometry I | 3 | 0 | 0 | 6 |

MA 849 | Topics in Topology I | 3 | 0 | 0 | 6 |

MA 851 | Topics in Number Theory I | 3 | 0 | 0 | 6 |

MA 853 | Topics in Differential Equations I | 3 | 0 | 0 | 6 |

MA 855 | Topics in Numerical Analysis I | 3 | 0 | 0 | 6 |

MA 857 | Topics in Probability I | 3 | 0 | 0 | 6 |

MA 859 | Topics in Statistics I | 3 | 0 | 0 | 6 |

MAS 801 | Seminar | 0 | 0 | 0 | 4 |

Course Code | Name of the Course | L | T | P | C |
---|---|---|---|---|---|

MA 842 | Topics in Algebra II | 3 | 0 | 0 | 6 |

MA 844 | Topics in Analysis II | 3 | 0 | 0 | 6 |

MA 846 | Topics in Combinatorics II | 3 | 0 | 0 | 6 |

MA 848 | Topics in Geometry II | 3 | 0 | 0 | 6 |

MA 850 | Topics in Topology II | 3 | 0 | 0 | 6 |

MA 852 | Topics in Number Theory II | 3 | 0 | 0 | 6 |

MA 854 | Topics in Differential Equations II | 3 | 0 | 0 | 6 |

MA 856 | Topics in Numerical Analysis II | 3 | 0 | 0 | 6 |

MA 858 | Topics in Probability II | 3 | 0 | 0 | 6 |

MA 860 | Topics in Statistics II | 3 | 0 | 0 | 6 |

MAS 802 | Seminar | 0 | 0 | 0 | 4 |

**Note:** *Each course is of 6 credits with the structure of 3-0-0-6. A prerequisite for an even numbered course is exposure to the preceding odd numbered course, except in the case of MA 824 for which exposure to MA 819 shall be the prerequisite.*