### Degree Programmes at the Department of Mathematics and Statistics

#### Bachelor Degree Programmes

#### Master Degree Programmes

### Courses

The Department of Mathematics and Statistics offers a wide range of courses, most of which are held on a regular basis. The recent and past course offers can be found on our websites, as well as in the official course catalogue of the university.

**If required, all Master courses can be held in English**

Furthermore, it is possible to deliver all exams in a language other than German if all parties agree.

In the following you can find English course descriptions of our courses.

#### Brief descriptions of our courses (in progress)

The English course descriptions for our current courses can also be found in ZEuS.

**Algorithmic Algebraic Geometry**

We will start by recalling basic notions from commutative algebra, focusing on prime ideals. We will present interactions among algebraic objects (like a ring) and geometric objects (like a variety). We will show how to compute basic invariants of ideals/varieties, like the degree and dimension. For this we introduce Groebner basis and present Buchberger's algorithm to compute them. As an application we present elimination theory, which allows to compute images of polynomial maps. Practical examples based on current computer algebra software, like Macaulay2, will be given.

**Form Methods for Partial Differential Equations**

This course gives an introduction into quadratic forms, which are a very efficient tool to define differential operators and to study the (parabolic) partial differential equations in which they appear. These techniques appear frequently in mathematical physics, where they are used to define Schrödinger operators, and in the theory of stochastic processes, where special classes of quadratic forms, the so-called Dirichlet forms, can be used to define certain Markov processes.

In this course, we will study the relationship between sectorial forms and the associated operators which turn out to generate holomorphic contraction semigroups. We then study important properties of these semigroups such as positivity, extrapolation to the L^p-scale and ultracontractivity. If time permits, we will then study the spectrum of the associated operators in more detail.

**Hyperbolic Forms and Linear Matrix Inequalities**

Some of the topics we plan to discuss in this lecture (time permitting): Semidefinite representations for convex sets with strict curvature

conditions on the boundary (following Helton-Nie); counterexamples to the Helton-Nie conjecture; proof of the Lax conjecture (following

Hanselka); semidefinite representation for smooth hyperbolicity cones (following Netzer-Sanyal); multiaffine hyperbolic forms and

matroids; cones of interlacers and their boundaries; (possibly) hyperbolic forms and the Kadison-Singer conjecture.

**Introduction to Riemannian Geometry**

In a surface in three space it is a nontrivial problem to find the shortest path between two given points, the line segment joining them not lying in that surface in general. We shall attack that problem from a quite abstract point of view: Surfaces are replaced with "Riemannian manifolds", spaces endowed with an infinitesimal metric allowing to measure the length of "smooth" paths. If time allows there also will be some comments on basics of general relativity (space times etc.).

**Mathematical Statistics II**

In the first part we focus on weak convergence in metric spaces, with applications to stochastic processes and functional limit theorems. In the second part, a systematic introduction to the theory of statistical estimation is given. Using a decision theoretic framework, mathematical methods for comparing and constructing statistical methods are discussed.

**Mathematics and Infectious Diseases**

Modeling and mathematical analysis plays an essential role in the understanding of the spreading of diseases. The lecture

presents an introduction both for ODE and for PDE models. The presentation mainly follows the book [2] and the lecture notes [3],

supplemented by some parts of the collection [1]. Mathematical parts will be presented with more details and proofs.

[1] Brauer, F., van den Driessche, P., Wu, J. (Eds): Mathematical Epidemiology. Lec.\ Notes Math. 1945. Mathematical Biosciences Subseries. Springer, Berlin (2008).

[2] Li, M.Y.: An Introduction to Mathematical Modeling of Infectious Diseases. Mathematics of Plane Earth 2. Springer, Cham (2018).

[3] Sallet, G.: Mathematical Epidemiology. Lecture Notes, www.iecl.univ-lorraine.fr/~Gauthier.Sallet/Lecture-Notes-Pretoria-2018.pdf

Audience aimed at: students of mathematics and physics starting from their third year of bachelor studies (including the bachelor of education), and master or Ph.D. students of other subjects being interested in mathematical aspects.

**Multivariate Statistics**

An introduction to the theory of the multivariate normal distribution is given. In the first part, we focus on parameter estimation and testing for the main parameters of multivariate normal distributions, and selected topics in the analysis of variance and regression. In the second part, an introduction to the the theory of functional data analysis is given.

**Numerical Analysis of Hyperbolic Differential Equations**

The chacteristic feature of nonlinear hyperbolic conservation laws is that even for arbitrarily smooth initial data, discontinuous solutions can arise in finite time. In this lecture, appropriate numerical approximation of such solutions requires special techniques which will be presented (monotone schemes, TV-Stability, Riemann solver, entropy properties, limited reconstruction).

**Numerical Analysis of Stochastic Differential Equations** (winter term)

Stochastic differential equations play an important role in economics and science.

Quite often they are used as mathematical models to display the time course of processes governd by deterministic and random effects. Applications are the price trend of stocks or the pricing of options. The lecture introduces into the numerical analysis of stochastic and parabolic differential equations and focuses on interesting applications.

**Optimization II **(winter term)

The course deals with constrained non-linear optimization and related numerical methods, like interior-point, penalty, augmented Lagrange, SQP and barrier methods. These will be analysed and their numerical implementation discussed.

**Real Algebraic Geometry I**

The study of systems of polynomial inequalities leads to the study of rings which are endowed with something that resembles an order. This additional structure raises many new questions that have to be clarified. These questions arise already at a very basic level so that we need as prerequisites only basic linear algebra, algebra and analysis. The course will be divided into four chapters: Ordered fields, Hilbert’s 17th problem, Prime cones and real Stellensätze, Schmüdgen’s Positivstellensatz.

**Real Algebraic Geometry II**

We first study archimedean quadratic modules or preorderings and prove the Positivstellensatz. It has plenty of applications to sum-of-squares type representations of positive polynomials, and we'll discuss some of those. After a few complements in commutative algebra and algebraic geometry (power series rings, regular and singular points of varieties) we next treat the archimedean local-global principle and give applications to Nichtnegativstellensätze. Further topics will include moment problem, stability questions and applications of sums of squares to optimization, e.g. Lasserre relaxation.

**Risk Measures**

We give an introduction to the theory of risk measures and nonlinear expectations. The lecture includes (dual) representation results, the capital allocation problem, optimal risk sharing, model uncertainty, dynamic risk measures and nonlinear semigroups.

**Stochastic Analysis **(winter term)

We give an introduction to Stochastic Analysis. The lecture includes the construction of stochastic processes (in particular the Brownian motion), Ito’s integral, Ito’s formula, stochastic differential equations, the martingal representation theorem, and Girsanov’s change of measure theorem.

**Theory and Numerics of Partial Differential Equations I **(winter term)

In the first part of the lecture, various important types of partial differential equations are introduced, and typical questions about elliptic, hyperbolic and parabolic boundary and initial values problems are studied. The second part delivers an overview about numerical methods for elliptic, parabolic and hyperbolic partial differential equations with a focus on finite difference, finite volume and finite element methods.

**Theory of Partial Differential Equations II **(winter term)

In this course, evolution equations are studied with modern methods as semigroup theory, Sobolev Spaces for the solutions, and energy methods.

**Theory of Partial Differential Equations III**

As a continuation of the lecture PDE II, methods to deal with nonlinear evolution equations are presented. Initial value problems are considered where the spatial domain is all of R^n. Existence theorems and an analysis of the asymptotic behavior of solutions will be discussed. For details see the book mentioned below.

**Time Series Analysis **(summer term)

A systematic introduction to time series analysis is given, with an emphasis on understanding mathematical foundations and their implications for data analysis. The spectral representation of stationary processes leads to an elegant theory in the Hilbert space of square integrable variables. Parametric and nonparametric statistical inference and forecasting are discussed in the time and frequency domain.

**Topological Algebras**

The aim of this course is to give an overview on the theory of topological algebras and of the standard tools used in tackling problems involving them. Normed and Banach algebras will be introduced but we will mainly investigate how far one can go beyond this classical framework while still retaining substantial results. Particular attention will be given to locally multiplicatively convex algebras and tensor algebras, but also other special classes of topological algebras will be closely studied, e.g., Frechet algebras, locally bounded algebras, projective and inductive limit algebras, topological division algebras, etc. We will also introduce the spectrum of a given topological algebra and study its relation to maximal ideals. Topological algebras play an important role also in some problems appearing in real algebraic geometry, that will be outlined in this course and could be a starting point for a master thesis within the *Schwerpunkt Reelle Geometrie und Algebra*.