
Mathematics
Sequences and Series of Real Numbers: Sequences and series of real numbers, Convergent and divergent sequences, bounded and monotone sequences, Convergence criteria for sequences of real numbers, Cauchy sequences, absolute and conditional convergence; Tests of convergence for series of positive terms  comparison test, ratio test, root test; Leibnitz test for convergence of alternating series.  Functions of One Variable: Limit, continuity, differentiation, RolleÃÆÂ¢ÃâÃ¢âÂ¬ÃâÃ¢âÂ¢s Theorem, Mean value theorem, Taylor's theorem, maxima and minima.
 Functions of Two Real Variables: Limit, continuity, partial derivatives, differentiability, maxima and minima.
Method of Lagrange multipliers, Homogeneous functions including EulerÃÆÂ¢ÃâÃ¢âÂ¬ÃâÃ¢âÂ¢s theorem.  Integral Calculus: Integration as the inverse process of differentiation, definite integrals and their properties, Fundamental theorem of integral calculus. Double and triple integrals, change of order of integration. Calculating surface areas and volumes using double integrals and applications. Calculating volumes using triple integrals and applications.
 Differential Equations: Ordinary differential equations of the first order of the form yÃÆÂ¢ÃâÃ¢âÂ¬ÃâÃÅ=f(x,y). BernoulliÃÆÂ¢ÃâÃ¢âÂ¬ÃâÃ¢âÂ¢s equation, exact differential equations, integrating factor, Orthogonal trajectories, Homogeneous differential equationsseparable solutions, Linear differential equations of second and higher order with constant coefficients, method of variation of parameters. CauchyEuler equation.
 Vector Calculus: Scalar and vector fields, gradient, divergence, curl and Laplacian. Scalar line integrals and vector line integrals, scalar surface integrals and vector surface integrals, Green's, Stokes and Gauss theorems and their applications.
 Group Theory: Groups, subgroups, Abelian groups, non abelian groups, cyclic groups, permutation groups; Normal subgroups, LagrangeÃÆÂ¢ÃâÃ¢âÂ¬ÃâÃ¢âÂ¢s Theorem for finite groups, group homomorphisms and basic concepts of quotient groups (only group theory).
 Linear Algebra: Vector spaces, Linear dependence of vectors, basis, dimension, linear transformations, matrix representation with respect to an ordered basis, Range space and null space, ranknullity theorem; Rank and inverse of a matrix, determinant, solutions of systems of linear equations, consistency conditions. Eigenvalues and eigenvectors. CayleyHamilton theorem. Symmetric, skewsymmetric, hermitian, skewhermitian, orthogonal and unitary matrices.
 Real Analysis: Interior points, limit points, open sets, closed sets, bounded sets, connected sets, compact sets; completeness of R, Power series (of real variable) including TaylorÃÆÂ¢ÃâÃ¢âÂ¬ÃâÃ¢âÂ¢s and MaclaurinÃÆÂ¢ÃâÃ¢âÂ¬ÃâÃ¢âÂ¢s, domain of convergence, termWise differentiation and integration of power series.
ISI, CMI, TIFR M. Sc / Ph. D in Mathematics Entrance Exam Syllabus
Here is the ISI, CMI, TIFR Postgraduate in Mathematics Entrance Syllabus & Previous Year Question Paper 1. ISI M.MATH ENTRANCE EXAM SYLLABUS:
TEST CODE: MMA (Objective type)
SYLLABUS  Algebra Arithmetic, geometric and harmonic progression. Continued fractions. Elementary combinatorics: Permutations and combinations, Binomial theorem. Theory of equations. Inequalities. Complex numbers and De MoivreÃÆÂ¢ÃâÃ¢âÂ¬ÃâÃ¢âÂ¢s theorem. Elementary set theory. Functions and relations. Elementary number theory: Divisibility, Congruences, Primality. Algebra of matrices. Determinant, rank and inverse of a matrix. Solutions of linear equations. Eigenvalues and eigenvectors of matrices.
Simple properties of a group.  Coordinate geometry Straight lines, circles, parabolas, ellipses and hyperbolas.
 Calculus / Sequences and series Power series, Taylor and Maclaurin series.
 Limits and continuity of functions of one variable. Differentiation and integration of functions of one variable with applications. De nite integrals. Maxima and minima. Functions of several variables  limits, continuity, differentiability. Double integrals and their applications. Ordinary linear differential equations. and their applications. Ordinary linear differential equations.
 Elementary discrete probability theory Combinatorial probability, Conditional probability, Bayes theorem. Binomial and Poisson distributions.
2. TEST CODE: PMB (Descriptive type)
SYLLABUS Convergence and divergence of sequence and series;Cauchy sequence and completeness; BolzanoWeierstrass theorem; continuity, uniform continuity, differentiability, directional derivatives, Jacobians, Taylor Expansion; integral calculus of one variable, existence of Riemann integral, Fundamental theorem of calculus, change of variable;elementary topological notions for metric space: open, closed and compact sets, connectedness; elements of ordinary differential equations. Equivalence relations and partitions; vector spaces, subspaces, basis, dimension, direct sum; matrices, systems of linear equations, determinants; diagonalization, triangular forms; linear transformations and their representation as matrices; groups, subgroups, quotients, homomorphisms, products, Lagrange's theorem, Sylow's theorems; rings, ideals, maximal ideals, prime ideals, quotients, integral domains, unique factorization domains, polynomial rings; fields, algebraic extensions, separable and integral domains, unique factorization domains, polynomial rings; fields, algebraic extensions, separable and normal extensions, finite fields. 3. CMI M.SC {MATHS} ENTRANCE EXAM
SYLLABUS  Algebra
 (a) Groups, homomorphisms, cosets, Lagrange's Theorem, Sylow Theorems, symmetric group Sn, conjugacy class, rings, ideals, quotient by ideals, maximal and prime ideals, fields, algebraic extensions, finite fields
 (b) Matrices, determinants, vector spaces, linear transformations, span, linear independence, basis, dimension, rank of a matrix, characteristic polynomial, eigenvalues, eigenvectors, upper triangulation, diagonalization, nilpotent matrices, scalar (dot) products, angle, rotations, orthogonal matrices, Gln, SLn, On, SO2, SO3.
 Complex Analysis
 Holomorphic functions, CauchyRiemann equations, integration, zeroes of analytic functions, Cauchy
 formulas, maximum modulus theorem, open mapping theorem, Louville's theorem, poles and singularities, residues and contour integration, conformal maps, Rouche's theorem, MoreraÃÆÂ¢ÃâÃ¢âÂ¬ÃâÃÅs theorem, scalar (dot) products, angle, rotations, orthogonal matrices, Gln, SLn, On, SO2, SO3.
 Calculus and Real Analysis
 (a) Real Line: Limits, continuity, differentiability, Reimann integration, sequences, series, limsup, liminf, pointwise and uniform convergence, uniform continuity, Taylor expansions,
 (b) Multivariable: Limits, continuity, partial derivatives, chain rule, directional derivatives,total derivative, Jacobian, gradient, line integrals, surface integrals, vector elds, curl, divergence, Stoke's theorem
 (c) General: Metric spaces, Heine Borel theorem, Cauchy sequences, completeness, Weierstrass approximation.
Topology. Topological spaces, base of open sets, product topology, accumulation points, boundary, continuity, connectedness, path connectedness, compactness, Hausdor spaces, normal spaces, UrysohnÃÆÂ¢ÃâÃ¢âÂ¬ÃâÃÅs lemma, Tietze extension, TychonoÃÆÂ¢ÃâÃ¢âÂ¬ÃâÃ¢âÂ¢s theorem. GATE 2016: Syllabus for Mathematics (MA)
Linear Algebra: Finite dimensional vector spaces; Linear transformations and their matrix representations, rank; systems of linear equations, eigen values and eigen vectors, minimal polynomial, CayleyHamilton Theorem, diagonalisation, Hermitian, SkewHermitian and unitary matrices; Finite dimensional inner product spaces, GramSchmidt orthonormalization process, selfadjoint operators.
Complex Analysis: Analytic functions, conformal mappings, bilinear transformations; complex integration Cauchy's integral theorem and formula Liouville's theorem maximum modulus principle Taylor and Laurent's series residue theorem and applications for evaluating real integrals.
Real Analysis: Sequences and series of functions, uniform convergence, power series, Fourier series, functions of several variables, maxima, minima; Riemann integration, multiple integrals, line, surface and volume integrals, theorems of Green, Stokes and Gauss; metric spaces, completeness, Weierstrass approximation theorem, compactness; Lebesgue measure measurable functions Lebesgue integral Fatou's lemma dominated convergence theorem.
Ordinary Differential Equations: First order ordinary differential equations, existence and uniqueness theorems, systems of linear first order ordinary differential equations, linear ordinary differential equations of higher order with constant coefficients; linear second order ordinary differential equations with variable coefficients; method of Laplace transforms for solving ordinary differential equations, series solutions; Legendre and Bessel functions and their orthogonality.
Algebra: Normal subgroups and homomorphism theorems, automorphisms; Group actions, Sylow's theorems and their applications; Euclidean domains, Principle ideal domains and unique factorization domains. Prime ideals and maximal ideals in commutative rings; Fields, finite fields.
Functional Analysis: Banach spaces, HahnBanach extension theorem, open mapping and closed graph theorems, principle of uniform boundedness; Hilbert spaces, orthonormal bases, Riesz representation theorem, bounded linear operators.
Numerical Analysis: Numerical solution of algebraic and transcendental equations: bisection, secant method, NewtonRaphson method, fixed point iteration; interpolation: error of polynomial interpolation, Lagrange, Newton interpolations; numerical differentiation; numerical integration: Trapezoidal and Simpson rules, Gauss Legendrequadrature, method of undetermined parameters; least square polynomial approximation; numerical solution of systems of linear equations: direct methods (Gauss elimination, LU decomposition); iterative methods (Jacobi and GaussSeidel); matrix eigenvalue problems: power method, numerical solution of ordinary differential equations: initial value problems: Taylor series methods Euler's method RungeKutta methods.
Partial Differential Equations: Linear and quasilinear first order partial differential equations, method of characteristics; second order linear equations in two variables and their classification; Cauchy, Dirichlet and Neumann problems; solutions of Laplace, wave and diffusion equations in two variables; Fourier series and Fourier transform and Laplace transform methods of solutions for the above equations.
Mechanics: Virtual work Lagrange's equations for holonomic systems Hamiltonian equations.
Topology: Basic concepts of topology, product topology, connectedness, compactness, countability and separation axioms Urysohn's Lemma
Probability and Statistics: Probability space, conditional probability, Bayes theorem, independence, Random variables, joint and conditional distributions, standard probability distributions and their properties, expectation, conditional expectation, moments; Weak and strong law of large numbers,central limit theorem; Sampling distributions, UMVU estimators, maximum likelihood estimators, Testing of hypotheses, standard parametric tests based on normal, X2 , t, F ÃÆÃ¢â¬Å¡ÃâÂ distributions; Linear regression; Interval estimation.
Linear programming: Linear programming problem and its formulation, convex sets and their properties, graphical method, basic feasible solution, simplex method, bigM and two phase methods infeasible and unbounded LPP's alternate optima Dual problem and duality theorems, dual simplex method and its application in post optimality analysis; Balanced and unbalanced transportation problems, u u method for solving transportation problems; Hungarian method for solving assignment problems.
Calculus of Variation and Integral Equations: Variation problems with fixed boundaries; sufficient conditions for extremum, linear integral equations of Fredholm and Volterra type, their iterative solutions. CSIRUGC National Eligibility Test (NET) for Junior Research Fellowship and Lecturership COMMON SYLLABUS FOR PART `B' AND `C' MATHEMATICAL SCIENCES UNIT ÃÆÃ¢â¬Å¡ÃâÂ 1
Analysis: Elementary set theory, finite, countable and uncountable sets, Real number system as a complete ordered field, Archimedean property, supremum, infimum. Sequences and series, convergence, limsup, liminf. Bolzano Weierstrass theorem, Heine Borel theorem. Continuity, uniform continuity, differentiability, mean value theorem.
Sequences and series of functions, uniform convergence.
Riemann sums and Riemann integral, Improper Integrals.
Monotonic functions, types of discontinuity, functions of bounded variation, Lebesgue measure, Lebesgue integral.
Functions of several variables, directional derivative, partial derivative, derivative as a linear transformation, inverse and implicit function theorems.
Metric spaces, compactness, connectedness. Normed linear Spaces. Spaces of continuous functions as examples.
Linear Algebra: Vector spaces, subspaces, linear dependence, basis, dimension, algebra of linear transformations.
Algebra of matrices, rank and determinant of matrices, linear equations.
Eigenvalues and eigenvectors, CayleyHamilton theorem.
Matrix representation of linear transformations. Change of basis, canonical forms, diagonal forms, triangular forms, Jordan forms.
Inner product spaces, orthonormal basis.
Quadratic forms, reduction and classification of quadratic forms
UNIT ÃÆÃ¢â¬Å¡ÃâÂ 2
Complex Analysis: Algebra of complex numbers, the complex plane, polynomials, power series, transcendental functions such as exponential, trigonometric and hyperbolic functions. Analytic functions, CauchyRiemann equations. Contour integral, Cauchy's theorem, Cauchy's integral formula, Liouville's theorem, Maximum modulus principle, Schwarz lemma, Open mapping theorem.
Taylor series, Laurent series, calculus of residues.
Conformal mappings, Mobius transformations.
Algebra: Permutations, combinations, pigeonhole principle, inclusionexclusion principle, derangements.
Fundamental theorem of arithmetic, divisibility in Z, congruences, Chinese Remainder Theorem, Euler's ÃÆÃâÃâÃÅ function, primitive roots.
Groups, subgroups, normal subgroups, quotient groups, homomorphisms, cyclic groups, permutation groups, Cayley's theorem, class equations, Sylow theorems.
Rings, ideals, prime and maximal ideals, quotient rings, unique factorization domain, principal ideal domain, Euclidean domain.
Polynomial rings and irreducibility criteria.
Fields, finite fields, field extensions, Galois Theory.
Topology: basis, dense sets, subspace and product topology, separation axioms, connectedness and compactness.
UNIT ÃÆÃ¢â¬Å¡ÃâÂ 3
Ordinary Differential Equations (ODEs):
Existence and uniqueness of solutions of initial value problems for first order ordinary differential equations, singular solutions of first order ODEs, system of first order ODEs.
General theory of homogenous and nonhomogeneous linear ODEs, variation of parameters, SturmLiouville boundary value problem, Green's function.
Partial Differential Equations (PDEs):
Lagrange and Charpit methods for solving first order PDEs, Cauchy problem for first order PDEs.
Classification of second order PDEs, General solution of higher order PDEs with constant coefficients, Method of separation of variables for Laplace, Heat and Wave equations.
Numerical Analysis :
Numerical solutions of algebraic equations, Method of iteration and NewtonRaphson method, Rate of convergence, Solution of systems of linear algebraic equations using Gauss elimination and GaussSeidel methods, Finite differences, Lagrange, Hermite and spline interpolation, Numerical differentiation and integration, Numerical solutions of ODEs using Picard, Euler, modified Euler and RungeKutta methods.
Calculus of Variations:
Variation of a functional, EulerLagrange equation, Necessary and sufficient conditions for extrema. Variational methods for boundary value problems in ordinary and partial differential equations.
Linear Integral Equations:
Linear integral equation of the first and second kind of Fredholm and Volterra type, solutions with separable kernels. Characteristic numbers and eigenfunctions, resolvent kernel.
Classical Mechanics:
Generalized coordinates, Lagrange's equations, Hamilton's canonical equations, Hamilton's principle and principle of least action, Twodimensional motion of rigid bodies, Euler's dynamical equations for the motion of a rigid body about an axis, theory of small oscillations.
UNIT ÃÆÃ¢â¬Å¡ÃâÂ 4
Descriptive statistics, exploratory data analysis
Sample space, discrete probability, independent events, Bayes theorem. Random variables and distribution functions (univariate and multivariate); expectation and moments. Independent random variables, marginal and conditional distributions. Characteristic functions. Probability inequalities (Tchebyshef, Markov, Jensen). Modes of convergence, weak and strong laws of large numbers, Central Limit theorems (i.i.d. case).
Markov chains with finite and countable state space, classification of states, limiting behaviour of nstep transition probabilities, stationary distribution, Poisson and birthanddeath processes.
Standard discrete and continuous univariate distributions. sampling distributions, standard errors and asymptotic distributions, distribution of order statistics and range.
Methods of estimation, properties of estimators, confidence intervals. Tests of hypotheses: most powerful and uniformly most powerful tests, likelihood ratio tests. Analysis of discrete data and chisquare test of goodness of fit. Large sample tests.
Simple nonparametric tests for one and two sample problems, rank correlation and test for independence. Elementary Bayesian inference.
GaussMarkov models, estimability of parameters, best linear unbiased estimators, confidence intervals, tests for linear hypotheses. Analysis of variance and covariance. Fixed, random and mixed effects models.
Simple and multiple linear regression. Elementary regression diagnostics. Logistic regression.
Multivariate normal distribution, Wishart distribution and their properties. Distribution of quadratic forms. Inference for parameters, partial and multiple correlation coefficients and related tests. Data reduction techniques: Principle component analysis, Discriminant analysis, Cluster analysis, Canonical correlation. Simple random sampling, stratified sampling and systematic sampling. Probability proportional to size sampling. Ratio and regression methods.
Completely randomized designs, randomized block designs and Latinsquare designs. Connectedness and orthogonality of block designs, BIBD. 2Kfactorial experiments: confounding and construction.
Hazard function and failure rates, censoring and life testing, series and parallel systems.
Linear programming problem, simplex methods, duality. Elementary queuing and inventory models. Steadystate solutions of Markovian queuing models: M/M/1, M/M/1 with limited waiting space, M/M/C, M/M/C with limited waiting space, M/G/1.

