Pınar Ongan

Lebesgue measure and integration on the line. Convergence theorems. General measure and integration. Lp spaces. Decomposition of measures. Radon Nikodym theorem. Product measures and Fubini’s theorem.

Introduction to Algebraic Geometry, Euclidean and non-Euclidean Geometry.

Student presentation on an area not covered in classes under the supervision of an instructor.

Review of vector spaces, normed vector spaces, lP and LP spaces, Banach and Hilbert spaces, duality, bounded linear operators and functionals.

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.