phdadmissions@lnmiit.ac.in

1800-180-6566

**The schedule mentioned below are tentative: **

Last date of application to be considered for Ph.D admission in odd semester 2021-2022 | 30th June (Wednesday) |

Date of Entrance Examination and Interview in all department for PhD admission odd semester 2021-2022 | 15 July (Thursday) |

Result of Ph.D admission process will be declared | 28th July (Wednesday) |

Reporting and joining date | 16th August (Monday) |

*The dates may change based on the Corona pandemic situation

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

OR

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

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.

**Please note that the qualifying degree of applicant should be from an accredited university /deemed-to-be-university or an institution of higher learning of comparable standing. The eligibility criteria given above are the absolute minimum. Institute may prescribe any requirements over and above these. In case the candidates are awarded grades/CPI/CGPA instead of marks, applicant would have to produce the grade-percentage conversion criterion or formula provided by the university/institution from where the applicant has obtained the degree. In case, where, a university/institute does not provide any scheme for converting CPI/CGPA into equivalent marks, UGC and AICTE guidelines would be applicable.**

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.

**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:**

**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.**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.**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.**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.**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.**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.**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.**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.**Topology:**Basic concepts of topology, connectedness, product topology, compactness, countability and separation axioms, Urysohn’s lemma.**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.**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.

Eligibility | Fellowship: Research and/or Teaching assistantship |
---|---|

Registered students in Ph.D. Program in Sciences, Humanities and Social Sciences Departments (if not receiving external fellowship or support via funded project) | Rs. 23,850/- per month for initial two years and Rs. 27,300/- per month for next three years subject to satisfactory progress. |

Registered students in Ph.D. Program in Sciences, Humanities and Social Sciences Departments with NET qualification (who are not getting external fellowship or funded project support) | Rs. 27,500/- per month for initial two years and Rs. 30,800/- per month for next three years subject to satisfactory progress. |

**Note:**

(a) 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.

(b) All Ph.D. students must work as TAs/RAs if they are being funded by the Institute.

- 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.
- Topology and Function Space, Functional Analysis.
- Cryptography, Post Quantum Cryptography, Boolean Functions.
- Probability Theory.
- Probability Theory.
- Commutative algebra, Non-commutative algebras, Lie algebras, Number Theory.