CSIR NET Mathematical Sciences — syllabus checklist

1.1 The Real Line

• Elements of set theory: operations, De Morgan and difference
• Completeness, sup/inf, Archimedean property
• Sequences: convergence, monotone, Bolzano–Weierstrass, Cauchy
• limsup, liminf and subsequential limits
• Series: comparison, ratio, root, Raabe, condensation, alternating, rearrangements

1.2 Continuity and Differentiation

• Continuity, uniform continuity, Lipschitz
• Differentiability, mean value theorems, Taylor, L'Hôpital

1.3 Integration

• Riemann integration and criteria
• Improper integrals and convergence tests

1.4 Sequences and Series of Functions

• Pointwise vs uniform convergence, M-test, Dini
• Power series, radius of convergence, Abel's theorem
• Arzelà–Ascoli and equicontinuity

1.5 Functions of Several Variables

• Partial derivatives, differentiability, chain rule
• Inverse and implicit function theorems, extrema

1.6 Metric Spaces

• Open/closed sets, limit points, closure, interior
• Compactness: open covers, sequential, Heine–Borel
• Completeness and Baire category
• Connectedness and path-connectedness

1.7 Lebesgue Measure and Integration

• Measurable sets and functions
• Lebesgue integral, MCT, DCT, Fatou
• L^p spaces essentials

1.8 Vector Spaces and Linear Maps

• Bases, dimension, rank–nullity
• Linear transformations, matrix representation, change of basis

1.9 Eigenvalues and Canonical Forms

• Eigenvalues, characteristic & minimal polynomials, Cayley–Hamilton
• Diagonalisability criteria
• Jordan canonical form
• Rational canonical form

1.10 Inner Product Spaces and Forms

• Gram–Schmidt, orthogonal/unitary/normal matrices, spectral theorem
• Quadratic forms, positive definiteness, Sylvester's law

1.11 Determinants and Matrix Tricks

• Determinants, trace, block matrices, rank inequalities

2.1 Analytic Functions

• Cauchy–Riemann equations, harmonic functions
• Power series and analyticity

2.2 Cauchy Theory

• Cauchy's theorem and integral formula
• Liouville, Morera, maximum modulus principle

2.3 Singularities and Residues

• Laurent series, classification of singularities, Casorati–Weierstrass
• Residue theorem and standard contour integrals

2.4 Zeros and Mappings

• Argument principle, Rouché's theorem, open mapping
• Conformal maps, Möbius transformations, Schwarz lemma

2.5 Groups

• Subgroups, cosets, Lagrange, cyclic groups
• Normal subgroups, quotients, isomorphism theorems
• Permutation groups: cycles, sign, conjugacy in S_n and A_n
• Group actions, class equation, p-groups
• Sylow theorems and groups of small order
• Finite abelian groups

2.6 Rings and Fields

• Ideals, quotient rings, prime & maximal ideals, CRT
• Euclidean, PID, UFD hierarchy
• Polynomial rings and irreducibility tests
• Field extensions, splitting fields, finite fields
• Galois theory essentials

2.7 Topology

• Topological spaces, bases, subspace/product/quotient
• Continuity, homeomorphism, separation axioms
• Compactness and Tychonoff
• Connectedness and components
• Standard spaces: cofinite, cocountable, Sorgenfrey, Cantor set

3.1 Ordinary Differential Equations

• Existence–uniqueness, Picard, Lipschitz
• Linear ODE, Wronskian, variation of parameters, systems
• Sturm–Liouville problems and Green's functions
• Stability and phase portraits

3.2 Partial Differential Equations

• First-order PDE: Lagrange, Charpit, characteristics
• Classification and canonical forms
• Laplace, heat and wave equations: separation of variables

3.3 Numerical Analysis

• Root finding: bisection, Newton–Raphson, fixed point, order of convergence
• Interpolation and numerical integration with error terms
• Numerical ODE: Euler, Runge–Kutta

3.4 Calculus of Variations

• Euler–Lagrange equation and standard functionals
• Isoperimetric problems

3.5 Linear Integral Equations

• Fredholm and Volterra equations
• Separable kernels and resolvent kernels

3.6 Classical Mechanics

• Lagrangian formalism and generalised coordinates
• Hamiltonian formalism and conservation laws

4.1 Probability

• Axioms, conditional probability, independence, Bayes
• Random variables, distributions, moments, MGF
• Standard discrete and continuous distributions
• Joint distributions, transformations, order statistics

4.2 Limit Theorems and Markov Chains

• Modes of convergence, WLLN, SLLN, CLT
• Markov chains: classification of states, stationary distributions

4.3 Estimation

• Sufficiency, completeness, UMVUE, Cramér–Rao
• MLE and method of moments

4.4 Hypothesis Testing

• Neyman–Pearson lemma and UMP tests
• Likelihood ratio and standard tests

4.5 Linear Models and Multivariate

• Gauss–Markov, regression, ANOVA basics
• Multivariate normal distribution

4.6 Sampling and Design of Experiments

• SRS, stratified and systematic sampling
• CRD, RBD, LSD essentials

4.7 Linear Programming

• Linear programming, simplex and duality