• Abstract vector spaces. Linear transformations, inner product spaces. spectral theorems, orthogonal bases.
• Sequences of real numbers; supremum, continuity, Riemann integral, differentiation, sequences and series of functions, uniform continuity and uniform convergence.
• Analytic functions, Cauchy's theorem, Cauchy's integral formula, residues, zeroes of analytic functions; Laurent series, the maximum principle.

• Systems of ordinary differential equations, autonomous systems in the plane, phase portraits, linear systems, stability, Lyapunov's method, Poincare-Bendixson theorem, applications.
• First order equations, well-posedness, characteristics, wave equation, heat equation, Laplace equation, boundary conditions, Fourier series, applications.

SAMPLE EXAMS
(APPLIED MATH - PRELIMINARY)

• Group theory, Sylow theorems and structure of finitely generated Abelian groups, applications of group theory.
• Ring and module theory, principal ideal domains, unique factorization domains, Euclidean rings, field theory and Galois theory.

SAMPLE EXAMS
(PURE MATH-PRELIMINARY)

• Combinatorics, independence, conditioning; Poisson-process; discrete and continuous distributions with statistical applications; expectation, transformations moment-generating functions; introduction to statistical inference.
• Sampling distributions, limiting distributions; maximum likelihood methods; sufficiency and its statistical inference implications; pivotal quantities; interval estimation; tests of hypotheses, optimality.

SAMPLE EXAMS
(PROBABILITY & STATISTICS-PRELIMINARY)