Pınar Ongan

Complex differentiation, Cauchy-Riemann equations, holomorphic functions, conformal mappings, contour integration, Cauchy’s theorem, Taylor and Laurent series, open mapping theorem, maximum modulus principle, applications of the residue theorem.

Prepositional and quantificational logic, formal grammar, semantical interpretation, formal deduction, completeness theorems, selected topics from model theory and proof theory.

Sets and counting, probability and relative frequency, conditional probability, Bayes’ theorem, independence, discrete and continuous random variables, binomial, Poisson and normal distributions, functions of random variables, law of large numbers, generating functions, characteristic functions, moments, compound distributions, central limit theorems, Markov chains and their limiting probabilities.

Solutions of nonlinear equations, Newton’s method, fixed points and functional iterations, LU factorization, pivoting, norms, analysis of errors, orthogonal factorization and least square problems, polynomial interpolation, spline interpolation, numerical differentiation, Richardson extrapolation, numerical integration, Gaussian quadratures, error analysis.

Topics from the theory of integration, Fourier series, Dirichlet kernel and convergence tests, orthogonal families, convergence in the mean, Parseval’s equation, Fejér and Poisson kernels, applications.

Sequences and series of functions, Arzelà-Ascoli theorem, Stone-Weierstrass theorem, Fourier series, inverse and implicit function theorems, integration.

Metric spaces, convergence, completeness, continuity, compactness, connectedness, contraction mapping principle.

Divisibility theory, Euclidean algorithm, congruences, solutions of polynomial congruences, primitive roots, power residues, quadratic reciprocity law, arithmetical functions, distribution of prime numbers, Pell’s equation, quadratic forms, some Diophantine equations.

Vector spaces over an arbitrary field, linear independence and bases, linear transformations and matrices, fields, field extensions, algebraic extensions, Kronecker’s theorem, finite fields.

Introduction to group theory, subgroups, Lagrange’s theorem, factor groups, permutation groups, group homomorphisms, isomorphism theorems, introduction to ring theory, ideals, ring homomorphisms, divisibility, polynomial rings, field of rational functions.