Topics in real analysis.

Courses that cover these topics count towards your Master’s Degree application in the area “Real analysis”, only. This means they count up to at most 16 ECTS credits. Furthermore, applicants are expected to have taken courses covering both, Riemann–Stieltjes integral and Lebesgue integration.

1. Logic. Sets, equivalence relation, inequalities and orderings, maps.
2. The real numbers. Supremum and infimum. Monotone sequences. Convergence of sequences, Chauchy sequences. Real numbers are complete.
3. Continuity of real functions. Intermediate value theorem. Inverse functions. Convergence and uniform convergence of function sequences. Direct comparison test.
4. Differentiability of real functions. Mean value theorem. Taylor’s Theorem. Find minimum and maximum values.
5. Series, absolute and conditional convergence. Geometric series, harmonic series. Critieria for convergence. Series of functions, component wise differentiation and integration. Powerseries, radius of convergence.
6. Elementary functions: exponential map, trigonometric functions, their inverses.
7. Integrals Fundamental theorem of calculus. Antiderivative. Improper integrals. Tools: partial integration, substitution. Leibniz’ Rule for integrals.
8. Topological spaces, continuity, compactness. Metric spaces, completeness. Banach fixed point theorem. Normed vector spaces.
9. Differential maps, partial differentiation. Taylor ‘s Theorem. Inverse function theorem. Implicit function theorem. Extreme values, Lagrange multipliers.
10. Riemann–Stieltjes integral, Lebesgue measure, Lebesgue integral. Dominated convergence theorem. Transformation formula. Fubini’s Theorem.
11. Submanifolds of R^n. Integration over submanifolds. Gaussian integrals.
12. Ordinary differential equations. Local existence and uniqueness. Linear differential equations and systems of linear differential equations. Fundamental solutions. Variation of constants. Linear differential equations with constant coefficients.

Topics in analytical geometry and linear algebra.

Courses that cover these topics count towards your Master’s Degree application in the area “Analytical geometry and linear algebra”, only. This means they count up to at most 16 ECTS credits.

1. Basic knowledge: Sets and maps, proofs: Proof by contradiction, Induction. Basics of groups, rings (in particular polynomial rings), fields; Introduction of complex numbers an residue fields.
2. Structures of vector spaces: linear dependence, basis, dimension; linear maps and fundamental theorem of homomorphisms.
3. Matrices I: Gaussian algorithm, trace and determinant, permutations, Cramer rule, solving of linear systems.
4. Eigenvalues: Characteristic polynomial, diogonalisability, Cayley-Hamilton theorem.
5. Euclidean geometry, geometry of unitary transformations Scalar products and norms, orthogonality, normal maps, euclidean and unitary vector spaces.
6. Quadratic and hermitian forms, Principal axis theorem, Sylvester's law of inertia
7. Affine and projective geometry
8. Matrices II: Jordan normal form, matrix exponentials
9. Multilinear algebra: tensor products and tensor algebras, exterior product.

Topics in numerical mathematics.

Courses that contain these topics will be counted towards your Master’s Degree application in the area “Numerical mathematics”.

1. Conditioning and stability: Error analysis, error propagation, numerical stability, ill/well-conditioned.
2. Matrix decomposition: LU decomposition, QR decomposition, Cholesky decomposition.
3. Non-linear systems: Fixed-point iteration, Banach fixed-point theorem, Newton's method, Gauss–Newton algorithm.
4. Linear systems: Iterative algorithms, fixed point iteration, Gauss-Seidel method, gradient descent.
5. Eigenvalue problems: QR algorithm, singular value decomposition, Lanczos algorithm.
6. Numerical integration and interpolation: Polynomial interpolation, spline interpolation, Bézier curve, Newton-Cotes formulas, Gaussian quadrature rule, Gauss–Legendre quadrature.
7. Initial value problem: Initial condition, single-step method, implicit and explicit methods, Runge–Kutta method, adaptive step size, Fehlberg's method.
8. Boundary value problem: Boundary conditions, elliptic differential equations, finite difference method.

Topics in measure and probability theory.

Courses that contain these topics will be counted towards your Master’s Degree application in the area “Measure and probability”. Furthermore, applicants are expected to have taken courses covering all these topics.

1. Foundations: Sigma-algebra, measure, measurable space, measurable function.
2. Lebesgue theory: Lebesgue sigma-algebra, Lebesgue measure, Lebesgue integral.
3. Measure-theoretic probability theory: Axiomatic theory of probability, probability measure, probability space, continuous/discrete probability distribution.
4. Convergence of random variables: Convergence in measure, convergence in probability, weak convergence.
5. Weak law and strong law of large numbers (incl. proofs).
6. Central limit theorem and theorem of Lindeberg- Lindeberg-Lévy (incl. proofs).