CSIR NET — Mathematical Sciences Preparation

Sequences, series, continuity, differentiability, Riemann & Lebesgue integration, measure theory, L^p spaces.

Vector spaces, linear transformations, eigenvalues, Jordan canonical form, bilinear forms, inner product spaces.

Analytic functions, Cauchy's theorem, power series, residue calculus, conformal mappings, Riemann mapping theorem.

Groups, rings, fields, modules, Galois theory, Sylow theorems, field extensions.

Topological spaces, compactness, connectedness, separation axioms, fundamental group, covering spaces.

Existence & uniqueness, linear systems, stability theory, Sturm-Liouville problems, dynamical systems.

First & second order PDE, wave/heat/Laplace equations, Green's functions, Fourier analysis.

Interpolation, quadrature, iterative methods, finite differences, numerical solutions of ODE/PDE.

Probability spaces, random variables, distributions, limit theorems, estimation theory, Bayesian inference.

Lagrangian & Hamiltonian mechanics, variational principles, rigid body dynamics, small oscillations.

Banach & Hilbert spaces, bounded operators, spectral theory, Hahn-Banach theorem, open mapping theorem.

Curves, surfaces, curvature, geodesics, Gauss-Bonnet theorem, manifolds, differential forms.

Simplex method, duality, game theory, convex optimisation, integer programming.

Divisibility, primes, congruences, quadratic residues, arithmetic functions, applications in cryptography.

Sufficiency, UMVUE, Neyman-Pearson, UMP tests, LRT, Bayesian inference, linear regression, ANOVA.

SRS, stratified, systematic, PPS sampling, CRD, RBD, LSD, BIBD, 2^k factorial experiments.

Hazard function, censoring, life testing, series/parallel systems, M/M/1, M/M/c, M/G/1, inventory models.