# Application

# Application

## Key Information

- Name:
- Mathematics
- Degree:
- Master of Science
- Teaching language:
- English, German (see FAQ)
- Standard period of studies:
- 4 semesters
- Start of studies:
- Winter and summer semester
- Admission:
- Restricted

(Application to the faculty)

## Requirements

- Bachelor's Degree in Mathematics or related subject with a total of at least 180 ECTS credits
- Eligibility:
- at least 90 ECTS credits in mathematics, including:
- at least 16 ECTS credits in real analysis
- at least 16 ECTS credits in analytical geometry and linear algebra
- a total of at least 8 ECTS credits in the following topics:
- analysis on manifolds
- functional analysis
- complex analysis
- advanced algebra (group-,ring-, field theory, Galois theory)

- at least 8 ECTS credits in measure and probability theory
- at least 8 ECTS credits in numerical mathematics
- further 34 ECTS credits in mathematics at a level exceeding the above mentioned courses

- at least 90 ECTS credits in mathematics, including:
- Language requirements (proof must not be older than 2 years): English proficiency at level C1 or higher (e.g. IELTS Band 7,0 or TOEFL iBT 94), alternatively German proficiency at level C1 or higher (e.g. German Abitur certificate).

Details: Real analysis

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.

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

- 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.
- Structures of vector spaces: linear dependence, basis, dimension; linear maps and fundamental theorem of homomorphisms.
- Matrices I: Gaussian algorithm, trace and determinant, permutations, Cramer rule, solving of linear systems.
- Eigenvalues: Characteristic polynomial, diogonalisability, Cayley-Hamilton theorem.
- Euclidean geometry, geometry of unitary transformations Scalar products and norms, orthogonality, normal maps, euclidean and unitary vector spaces.
- Quadratic and hermitian forms, Principal axis theorem, Sylvester's law of inertia
- Affine and projective geometry
- Matrices II: Jordan normal form, matrix exponentials
- 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”.

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

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

## Application procedure

The Master's Degree programme begins each summer and winter semester. Applications must be uploaded via the application portal.

The portal is expected to open mid-October 2024 for submission of applications for April intake.

The application is only valid for the semester applied for.

The application must include the following documents:

- Short Curriculum Vitae briefly describing your education career (please upload a PDF document; in German or English). Please include your date of birth and your name as given in your passport. Preferably, use the following form: Europass CV (extern link)
- Copy of your degree certificates for all academic degrees earned (please upload a PDF document; original plus translation into German or English)
- Syllabus/course descriptions for all degree programmes resp. mathematics lecture courses completed or not yet completed as PDF or alternatively hyperlinks to syllabus/course descriptions for all degree programmes resp. mathematics lecture courses completed or not yet completed
- Transcripts of records for all academic studies completed or not yet completed (please upload a PDF document; original plus translation into German or English)
- Proof of language proficiency (please upload a PDF document)

If you are already enrolled at the University of Göttingen, please consider the following:

- As soon as it is likely that you will complete the Bachelor's Degree (in mathematics) in the current semester, you should apply.
- Please re-register regularly for the programme that you are currently enrolled in. The enrollment to the Master's Degree programme in mathematics is being done in form of a change of the degree course after being admitted successfully.

### Online Application System

(closed)## Frequently Asked Questions (FAQ)

## Further Questions

If you need help or have any questions about the application process, please feel free to contact us by email: