CSIR NET 2026 Mathematics Candidates Appeared

The CSIR NET Mathematics syllabus covers undergraduate and postgraduate topics. It emphasizes theoretical understanding, logical application, and analytical ability.

UNIT – 1
Analysis:
Real number system, sequences, series, convergence.
Continuity, differentiability, mean value theorem.
Riemann and Lebesgue integrals.
Functions of several variables, directional and partial derivatives.
Metric spaces, compactness, connectedness.
Linear Algebra:
Vector spaces, linear transformations, matrices, rank, determinant.
Eigenvalues, eigenvectors, Cayley-Hamilton theorem.
Canonical forms (Diagonal, Triangular, Jordan).
Inner product spaces, orthonormal basis, quadratic forms.
UNIT – 2
Complex Analysis:
Complex numbers, analytic functions, Cauchy-Riemann equations.
Contour integrals, Cauchy’s theorems, residue theorem.
Power series (Taylor, Laurent), conformal mappings.
Algebra:
Number theory (divisibility, congruences, Chinese Remainder Theorem).
Groups (subgroups, normal subgroups, quotient groups, homomorphisms, Sylow theorems).
Rings (ideals, quotient rings, UFD, PID, ED, polynomial rings).
Fields, field extensions, Galois Theory.
Topology:
Basis, dense sets, subspace and product topology.
Separation axioms, connectedness, and compactness.
UNIT – 3
Ordinary Differential Equations (ODEs):
First-order ODEs (existence, uniqueness, singular solutions, systems).
Linear ODEs (homogeneous/non-homogeneous), variation of parameters, Sturm-Liouville BVP, Green’s function.
Partial Differential Equations (PDEs):
First-order PDEs (Lagrange, Charpit, Cauchy problem).
Classification of second-order PDEs.
General solution of higher-order PDEs, separation of variables (Laplace, Heat, Wave).
Numerical Analysis:
Numerical solutions of algebraic equations (Iteration, Newton-Raphson, convergence).
Solutions of linear systems (Gauss elimination, Gauss-Seidel).
Interpolation (Lagrange, Hermite, Spline), numerical differentiation and integration.
Numerical solutions of ODEs (Picard, Euler, Runge-Kutta).
Calculus of Variations:
Variation of a functional, Euler-Lagrange equation, extrema conditions.
Linear Integral Equations:
Fredholm and Volterra type (first and second kind), separable kernels, resolvent kernel.
Classical Mechanics:
Generalized coordinates, Lagrange’s and Hamilton’s equations, Hamilton’s principle.
Motion of rigid bodies, theory of small oscillations.
UNIT – 4 (Focus for Statistics Students)
Probability & Statistics:
Descriptive statistics, probability, Bayes theorem, random variables (univariate/multivariate).
Distributions, expectation, moments, characteristic functions, probability inequalities.
Modes of convergence, laws of large numbers, Central Limit Theorem.
Markov chains, Poisson, and birth-and-death processes.
Sampling distributions, standard errors, order statistics.
Statistical Inference:
Methods of estimation, properties of estimators, confidence intervals.
Tests of hypotheses (MP, UMP, likelihood ratio, chi-square, large sample, non-parametric).
Elementary Bayesian inference.
Linear Models & Regression:
Gauss-Markov models, linear unbiased estimators, ANOVA, ANCOVA.
Fixed, random, and mixed effects models.
Simple and multiple linear regression, logistic regression.
Multivariate Analysis & Data Reduction:
Multivariate normal distribution, Wishart distribution.
Inference for parameters, correlation coefficients.
PCA, Discriminant analysis, Cluster analysis, Canonical correlation.
Sampling and Design of Experiments:
Sampling techniques (SRS, stratified, systematic, PPS), ratio and regression methods.
Experimental designs (CRD, RBD, LSD, BIBD), factorial experiments.
Reliability, Queuing & Inventory Models:
Hazard function, failure rates, censoring.
Linear Programming (simplex, duality).
Elementary queuing (M/M/1, M/M/C) and inventory models.
Note on Structure:
All students must answer questions from Unit I.
Mathematics students answer additional questions from Unit II and III.
Statistics students answer additional questions from Unit IV.