CSIR NET 2026 Mathematics Candidates Appeared

PROPOSED SYLLABUS FOR ‘Mathematical Science’ – CSIR-UGC NET for JRF and Lectureship
UNIT – 1
Real 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, Cayley-Hamilton 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 – 2
Complex Analysis:
Algebra of complex numbers, the complex plane, polynomials, power series, transcendental functions (exponential, trigonometric, hyperbolic)
Analytic functions, Cauchy-Riemann 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, pigeon-hole principle, inclusion-exclusion principle, derangements
Fundamental theorem of arithmetic, divisibility in Z, congruences, Chinese Remainder Theorem, Euler’s Ø-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 – 3
Ordinary Differential Equations (ODEs):
Existence and uniqueness of solutions of initial value problems for first order ODEs, singular solutions of first order ODEs, system of first order ODEs
General theory of homogeneous and non-homogeneous linear ODEs, variation of parameters
Sturm-Liouville 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 Newton-Raphson method, Rate of convergence
Solution of systems of linear algebraic equations using Gauss elimination and Gauss-Seidel methods
Finite differences, Lagrange, Hermite and spline interpolation
Numerical differentiation and integration
Numerical solutions of ODEs using Picard, Euler, modified Euler and Runge-Kutta methods
Calculus of Variations:
Variation of a functional, Euler-Lagrange 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
Two-dimensional motion of rigid bodies
Euler’s dynamical equations for the motion of a rigid body about an axis
Theory of small oscillations

UNIT – 4
Statistics and Probability:
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 n-step transition probabilities, stationary distribution, Poisson and birth-and-death processes
Distributions and Inference:
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 chi-square 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
Regression and ANOVA:
Gauss-Markov 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 Analysis:
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
Sampling and Design of Experiments:
Simple random sampling, stratified sampling and systematic sampling
Probability proportional to size sampling, ratio and regression methods
Completely randomized designs, randomized block designs and Latin-square designs
Connectedness and orthogonality of block designs, BIBD
2K factorial experiments: confounding and construction
Reliability and Operations Research:
Hazard function and failure rates, censoring and life testing, series and parallel systems
Linear programming problem, simplex methods, duality
Elementary queuing and inventory models
Steady-state 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

All students are expected to answer questions from Unit I. Students in mathematics are expected to answer additional questions from Unit II and III. Students with statistics are expected to answer additional questions from Unit IV.