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.