The LNM Institute of Information Technology

Doctoral Admissions - Department of Mathematics

Important Dates

Portal opening date01-03-2024
Portal Closing date30-06-2024
Date of Dispatch of Call letter08-07-2024
Date of Examination18/19-07-2024
Date of Declaration of Result25-07-2024
Reporting & Joining date / verification of the documents01-08-2024

Eligibility Criteria

Master’s degree in mathematics or in the relevant subject with 55% marks or equivalent CPI/CGPA.


Bachelor’s degree in science and engineering (typically a 4-year programme) relevant to Mathematics with 90% marks or equivalent CPI/CGPA and a valid GATE of 99 percentile or above in the relevant area of application.


Note: The students awaiting their qualifying examination results are also eligible to apply. They need to submit the qualifying mark sheets and/or certificates at the time of admission.

For Part-Time PhD program

  1. The minimum qualification for these candidates is the same as for full-time candidates except for experience requirement.
  2. Experience required for admission to part-time Ph.D. Programmes (Mathematics) is

Experience required for admission to part-time Ph.D. Programmes


Work Experience (Post Qualification)

Master’s degree in relevant subject

1 Year

Bachelor’s degree in science or engineering (a 4- year programme) relevant to the field in which the candidate is applying.

3 Years

  3. The part-time candidates should submit, a No Objection Certificate (NOC) from the organization, in which s/he is employed, at the time of admissions giving an undertaking that s/he would be released from the normal duties to fulfill the course-work requirement (and qualifier examination, if applicable).

Admission Procedure

Admission to the Ph.D. Programme is offered to eligible candidates on the basis of an entrance examination and an interview. Consequently, all applicants who qualify in the written examination have to appear for the personal interview on the same day or very next day. Candidates who have qualified for national level fellowships like CSIR/UGC-NET-JRF, DST-INSPIRE, NBHM Ph.D. scholarship or equivalent valid fellowship in the relevant subject/discipline may be exempted from the written examination but not from the interview. No request for remote interviews would be entertained.


Note: DST-Inspire candidates will have two choices:

Choice 1 (Default choice): To be exempted from the written test: In this case, if finally admitted, they will not be getting any Institute fellowship and their fellowship will be subject to acceptance and approval by DST, whenever it happens. Their admission will be with fellowship status as DST-Inspire Fellowship, from the beginning.

Choice 2: NOT to be exempted from the written test: In this case, if finally admitted, they will be under Institute fellowship category until they get DST-Inspire fellowship. If they get DST-Inspire fellowship, then their fellowship status will be changed from Institute Fellowship to DST-Inspire Fellowship.

Syllabus for Ph.D. Entrance Exam

The entrance examination shall contain both objective and subjective type of questions. The syllabus for the written entrance examination shall consist of 50% of research aptitude/methodology and 50% shall be subject-specific. Research aptitude/methodology part shall be of generic nature, intended to assess the research aptitude of the candidate. This part primarily shall contain questions to test research aptitude, reasoning ability, graphical analysis, analytical and numerical ability, data interpretation, quantitative aptitude of the candidate. The subject specific syllabus is as follows:

  1. Linear Algebra:Linear transformations; finite dimensional vector spaces and their matrix representations, rank; systems of linear equations, eigenvectors and eigenvalues, minimal polynomial, Cayley-Hamilton theorem, Hermitian, skew-Hermitian and unitary matrices, diagonalization; Gram-Schmidt orthonormalization process, finite dimensional inner product spaces, self-adjoint operators.
  2. Complex Analysis:Analytic functions, bilinear transformations; conformal mappings; complex integration: Cauchy’s integral theorem and formula; Liouville’s theorem, maximum modulus principle; Laurent’s and Taylor’s series; residue theorem and applications for evaluating real integrals.
  3. Real Analysis:Sequences and series of functions, power series, uniform convergence, Fourier series, functions of several variables, maxima, minima; multiple integrals, Riemann integration, surface, line and volume integrals; theorems of Green, Gauss and Stokes; metric spaces, completeness, Weierstrass approximation theorem, Lebesgue measure, compactness; measurable functions, Fatou’s lemma, Lebesgue integral, dominated convergence theorem.
  4. Ordinary Differential Equations:First order ordinary differential equations, existence and uniqueness theorems, linear ordinary differential equations of higher order with constant coefficients; systems of linear first order ordinary differential equations; linear second order ordinary differential equations with variable coefficients; method of Laplace transforms for solving ordinary differential equations, Legendre and Bessel functions and their orthogonality; series solutions.
  5. Algebra:Normal subgroups and automorphisms; homomorphism theorems; group actions, Sylow’s theorems and their applications; Euclidean domains, unique factorization domains and principal ideal domains; prime ideals and maximal ideals in commutative rings; fields, finite fields.
  6. Functional Analysis:Hahn-Banach extension theorem, Banach spaces, open mapping and closed graph theorems, Hilbert spaces, principle of uniform boundedness, orthonormal bases, Riesz representation theorem, bounded linear operators.
  7. Numerical Analysis: Numerical solution of algebraic and transcendental equations: secant method, bisection, Newton-Raphson method, fixed point iteration; interpolation: Lagrange error of polynomial interpolation, Newton interpolations; numerical differentiation; numerical integration: Trapezoidal and Simpson rules, Gauss Legendre quadrature, least square polynomial approximation; method of undetermined parameters; numerical solution of systems of linear equations: direct methods (LU decomposition, Gauss elimination), iterative methods (Gauss-Seidel and Jacobi); matrix eigenvalue problems: power method, numerical solution of ordinary differential equations and initial value problems: Euler’s method, Runge-Kutta methods, Taylor series methods.
  8. Partial Differential Equations:Linear and quasi linear first order partial differential equations, method of characteristics; second order linear equations in two variables and their classification; Dirichlet, Cauchy and Neumann problems; solutions of Laplace, diffusion and wave equations in two variables; Fourier transform, Fourier series and Laplace transform methods of solutions for the above equations.
  9. Topology:Basic concepts of topology, connectedness, product topology, compactness, countability and separation axioms, Urysohn’s lemma.
  10. Probability and Statistics: Probability space, Bayes theorem, conditional probability, independence, joint and conditional distributions, random variables, standard probability distributions and their properties, conditional expectation, expectation, moments; strong and weak law of large numbers, sampling distributions, central limit theorem, UMVU estimators, maximum likelihood estimators, standard parametric tests based on normal, testing of hypotheses, X2, t, F–distributions; linear regression; interval estimation.
  11. Linear Programming:Linear programming problem and its formulation, convex sets and their properties, basic feasible solution, graphical method, simplex method, big-M and two phase methods; unbounded LPP’s and infeasible, alternate optima; dual problem and duality theorems, dual simplex method and its application in post optimality analysis; unbalanced and balanced transportation problems, Hungarian method for solving assignment problems, u-v method for solving transportation problems.


  • Students may avail financial assistantship from external funding agencies (such as UGC/CSIR/NBHM/DST/DAE etc.) as well as from industries.
  • Financial assistance is available from the Institute to the meritorious students in the form of teaching and research assistantship as per Institute norms, for those candidates who may not have any external fellowship.
  • Partial financial support is available to the meritorious students to attend the workshops, short term courses and for paper presentation in refereed quality conferences as per Institute norms.
  • The institute fellowship for full time Ph.D. candidates is as follows:
    • INR 31,000 per month for initial two years
    • INR 35,000 per month for the next three years
  • Part time candidates are not eligible for the institute fellowship.


  • Students are encouraged to avail financial assistantship from external funding agencies also (such as UGC/CSIR/NBHM/DST/DAE etc.) as well as from industries. They may have to be involved in Department as per rules.
  • All Ph.D. students must work as TAs/RAs if they are being funded by the Institute.

Major Research Areas

  1. Numerical Analysis, Numerical Solution of ODEs and PDEs, Finite Difference Methods, Finite Element Methods, Optimal Control Problems, Domain Decomposition Methods, Singularly Perturbed Problems, Computational Finance, Layer Adapted Meshes, B-Spline Collocation Methods, Fractional Differential Equations, Theoretical aspects of nonlinear elliptic and parabolic PDEs.
  2. Topology and Function Spaces, Functional Analysis.
  3. Cryptography, Post Quantum Cryptography, Boolean Functions.
  4. Probability and Statistics.
  5. Algebra, Non-commutative algebras, Lie Algebras, Representation Theory.

(Applications are invited in the research area of Topology and Function Spaces, Cryptography, Boolean functions, Algebra, Representation Theory, Functional Analysis, Quantum Information Theory, Probability & Statistics, Theoretical PDEs)