Syllabus for IIT JAM Mathematics Exam:

Real Analysis:
- Real number system, the least upper bound property, countable and uncountable sets.
- Sequences and series of real numbers, convergence of sequences, Cauchy sequences, and subsequences.
- Continuity and differentiability of functions, mean value theorem, intermediate value theorem, Taylor's theorem, and L'Hospital's rule.
- Riemann integrals, improper integrals, and their convergence.
- Functions of several variables, partial derivatives, total derivative, and Jacobian.
- Uniform continuity, uniform convergence, and power series.

Complex Analysis:
- Analytic functions, Cauchy-Riemann equations, and harmonic functions.
- Complex integration, Cauchy's theorem, and Cauchy's integral formula.
- Taylor and Laurent series expansions, residues, and contour integration.
- Conformal mappings, Mobius transformations, and the Riemann mapping theorem.
- Maximum modulus principle and Schwarz lemma.

Linear Algebra:
- Vector spaces, subspaces, linear dependence, basis, and dimension.
- Linear transformations, matrix representation, rank, and nullity.
- Eigenvalues, eigenvectors, and diagonalization.
- Inner product spaces, orthogonality, and orthogonal projections.
- Determinants, eigenvalues, and eigenvectors of matrices.

Abstract Algebra:
- Groups, subgroups, cyclic groups, normal subgroups, and quotient groups.
- Homomorphisms, isomorphisms, and group actions.
- Rings, ideals, quotient rings, and homomorphisms.
- Integral domains, fields, polynomial rings, and factorization of polynomials.
- Euclidean domains, principal ideal domains, and unique factorization domains.

Ordinary Differential Equations:
- First-order ordinary differential equations, exact differential equations, and integrating factors.
- Second-order linear equations with constant coefficients, and homogeneous and non-homogeneous equations.
- Systems of linear first-order ordinary differential equations, and matrix representation.
- Laplace transforms and their applications to ordinary differential equations.

Probability:
- Sample space, events, and probability axioms.
- Conditional probability, independence of events, and Bayes' theorem.
- Random variables, probability distributions, and joint distributions.
- Moments, moment-generating functions, and characteristic functions.
- Limit theorems: weak law of large numbers and central limit theorem.

Statistics:
- Descriptive statistics: measures of central tendency and dispersion.
- Probability distributions: binomial, Poisson, normal, and exponential.
- Estimation: point and interval estimation of parameters.
- Hypothesis testing: types of errors, power, and confidence intervals.
- Chi-square test, t-test, and F-test.

Linear Programming:
- Linear programming problems, feasible solutions, and basic feasible solutions.
- Graphical method, simplex method, and duality in linear programming.
- Sensitivity analysis, transportation and assignment problems.

Mathematical Logic:
- Propositional logic, first-order logic, and predicate calculus.
- Logical connectives, truth tables, and tautologies.
- Inference rules, logical equivalences, and quantifiers.

Differential Equations:
- First-order ordinary differential equations, exact differential equations, and integrating factors.