Differential Geometry II – Geometry of Submanifolds
Lecturer: Guofang Wang
Assistant: Xuwen Zhang
Language: in German
Lecture: Mo, Mi, 12-14h, SR 404, Ernst-Zermelo-Str. 1
Tutorial: 2 hours, date to be determined
In the lecture, we discuss the geometry of the sub-manifolds of Euclidian spaces. Examples of such sub-manifolds are curves in the plane and surfaces in the 3-dimensional space. In the 1st part we introduce the external geometry of the sub-manifold, e.g. the second fundamental form, the mean curvature, the first variation of the area, the equations of Gauss, Codazzi and Ricci. In the 2nd part we examine tminimal hypersurfaces (minimal surfaces), the hypersurface with constant mean curvature and the geometric inequalities, the isoperimetric inequality and its generalisations.
Analysis III and "Differential Geometry" or "Curves and Surfaces"
Pure Mathematics
Elective
Mathematics
Concentration Module
Lecturer: Wolfgang Soergel
Assistant: Xier Ren
Language: in German
Lecture: Di, Do, 8-10h, HS II, Albertstr. 23b
Tutorial: 2 hours, date to be determined
In linear algebra you studied linear systems of equations. In commutative algebra, we study polynomial equation systems such as \(x^2+y^2 = \) 1 and their solution sets, the algebraic varieties. It will turn out that such a variety is closely related to the ring of the restrictions of polynomial functions on that variety, and that we can extrapolate this relationship to a geometric understanding of any commutative rings, in particular the ring of the integers. Commutative algebra, algebraic geometry, and number theory grow together in this conceptual building. The lecture aims to introduce into this conceptual world. We will especially focus on the dimension of algebraic varieties and their cutting behavior, which generalizes the phenomena known from the linear algebra on the case of polynomial equation systems.
necessary: Linear Algebra I+II
useful: Algebra and Number Theory
Pure Mathematics
Elective
Mathematics
Concentration Module
Lecturer: David Criens
Assistant: Samuel Adeosun
Language: in English
Lecture: Mi, 14-16h, HS II, Albertstr. 23b, Do, 10-12h, SR 404, Ernst-Zermelo-Str. 1
Tutorial: 2 hours, date to be determined
This lecture builds the foundation of one of the key areas of probability theory: stochastic analysis. We start with a rigorous construction of the It^o integral that integrates against a Brownian motion (or, more generally, a continuous local martingale). In this connection, we learn about It^o's celebrated formula, Girsanov’s theorem, representation theorems for continuous local martingales and about the exciting theory of local times. Then, we discuss the relation of Brownian motion and Dirichlet problems. In the final part of the lecture, we study stochastic differential equations, which provide a rich class of stochastic models that are of interest in many areas of applied probability theory, such as mathematical finance, physics or biology. We discuss the main existence and uniqueness results, the connection to the martingale problem of Stroock-Varadhan and the important Yamada-Watanabe theory.
Probability Theory I and II (Stochastic Processes)
Applied Mathematics
Elective
Mathematics
Concentration Module
Reading courses
Lecturer: All professors and 'Privatdozenten' of the Mathematical Institute
Language: Talk/participation possible in German and English
In a reading course, the material of a four-hour lecture is studied in supervised self-study. In rare cases, this may take place as part of a course; however, reading courses are not usually listed in the course catalog. If you are interested, please contact a professor or a private lecturer before the start of the course; typically, this will be the supervisor of your Master's thesis, as the reading course ideally serves as preparation for the Master's thesis (both in the M.Sc. and the M.Ed. programs).
The content of the reading course, the specific details, and the coursework requirements will be determined by the supervisor at the beginning of the lecture period. The workload should be equivalent to that of a four-hour lecture with exercises.
Elective
Mathematics
Concentration Module
Lecturer: Sören Bartels
Assistant: Tatjana Schreiber
Language: in English
Lecture: Mi, 10-12h, SR 226, Hermann-Herder-Str. 10
Tutorial: 2 hours, date to be determined
The lecture addresses algorithmic aspects in the practical realization of mathematical methods in big data analytics and machine learning. The first part will be devoted to the development of recommendation systems, clustering methods and sparse recovery techniques. The architecture and approximation properties as well as the training of neural networks are the subject of the second part. Convergence results for accelerated gradient descent methods for nonsmooth problems will be analyzed in the third part of the course. The lecture is accompanied by weekly tutorials which will involve both, practical and theoretical exercises.
Lectures "Numerik I, II" or lecture "Basics in Applied Mathematics"
Applied Mathematics
Elective
Mathematics
Concentration Module
Lecturer: Mikhail Tëmkin
Language: in English
Lecture: Mo, 14-16h, SR 404, Ernst-Zermelo-Str. 1
Tutorial: 2 hours, date to be determined
The notion of a manifold is fundamental importance. On one hand, it is a common ground for many branches of pure and applied mathematics, as well as mathematical physics. On the other hand, it itself is a lush source of elegant, unexpected and structural results. Next, algebraic topology is to mathematics what the periodic table is to chemistry: it offers order to what seems to be chaotic (more precisely, to topological spaces of which manifolds is an important example). Finally, differential topology studies smooth manifolds using topological tools. As it turns out, narrowing the scope to manifolds provides many new beautiful methods, structure and strong results, that are applicable elsewhere -- as we will see in the course. Necessary notions from algebraic topology will be covered in the beginning.
Point-set topology (e.g. "Topology" from summer semester of 2024)
Pure Mathematics
Elective
Mathematics
Concentration Module
Lecturer: Amador Martín Pizarro
Assistant: Charlotte Bartnick
Language: in German
Lecture: Mi, 14-16h, SR 125, Ernst-Zermelo-Str. 1
Tutorial: 2 hours, date to be determined
Groups without any non-trivial normal subgroup are called simple groups. Similar to prime numbers for the natural numbers, simple groups form the building blocks for finite groups. It is easy to see that Abelian finite simple groups are cyclic. Non-Abelian examples are alternating groups and Lie-type groups.
The classification of finite simple groups is far beyond the scope of this course. However, we will illustrate some of the recurring ideas of classification and, in particular, prove the following result of Brauer and Fowler:
Theorem: Let G be a finite group of even order such that the centre is of odd order. Then there is an element \(g \neq 1_G\) with \(|G| < |C_G (g)|^3\) .
This theorem had a particularly large impact on the classification of finite simple groups, as it suggests that these could be classified by examining the centralisers of elements of order 2.
Algebra and Number Theory
Pure Mathematics
Elective
Mathematics
Concentration Module
Lecturer: Ernst August v. Hammerstein
Assistant: Sebastian Hahn
Language: in English
Lecture: Mo, 14-16h, HS II, Albertstr. 23b
Exercise session: Mi, 16-18h, SR 403, Ernst-Zermelo-Str. 1
Lévy processes are the continuous-time analogues of random walks in discrete time as they possess, by definition, independent and stationary increments. They form a fundamental class of stochastic processes which has widespread applications in financial and insurance mathematics, queuing theory, physics and telecommunication. The Brownian motion and the Poisson process, which may already be known from other lectures, also belong to this class. Despite their richness and flexibility, Lévy processes are usually analytically and numerically very tractable because their distributions are generated by a single univariate distribution which has the property of infinite divisibility.
The lecture starts with an introduction into infinitely divisible distributions and the derivation of the famous Lévy-Khintchine formula. Then it will be explained how the Lévy processes emerge from these distributions and how the characteristics of the latter influence the path properties of the corresponding processes. Finally, after a short look at the method of subordination, option pricing in Lévy-driven financial models will be discussed.
necessary: Probability Theory I
useful: Probability Theory II (Stochastic Processes)
Applied Mathematics
Elective
Mathematics
Concentration Module
Lecturer: Thorsten Schmidt
Assistant: Simone Pavarana
Language: in English
Lecture: Mi, 12-14h, HS II, Albertstr. 23b
Tutorial: 2 hours, date to be determined
In this lecture we will study new and highly efficient tools from machine learning which are applied to stochastic problems. This includes neural SDEs as a generalisation of stochastic differential equations relying on neural networks, transformers as a versatile tool not only for languages but also for time series, transformers and GANs as generator of time series and a variety of applications in Finance and insurance such as (robust) deep hedging, signature methods and the application of reinforcement learning.
The prerequisites are stochastics, for some parts we will require a good understanding of stochastic processes. A (very) short introduction will be given in the lectures – so for fast learners it would be possible to follow the lectures even without the courses on stochastic processes.
Applied Mathematics
Elective
Mathematics
Concentration Module
Lecturer: Patrick Dondl
Assistant: Eric Trébuchon
Language: in English
Lecture: Mi, 14-16h, SR 226, Hermann-Herder-Str. 10
Tutorial: 2 hours, date to be determined
This course provides a comprehensive introduction to mathematical modeling. We will learn the systematic process of translating real-world problems into mathematical frameworks, analyzing them using appropriate mathematical tools, and interpreting the results in practical contexts. The course covers both discrete and continuous modeling approaches, with emphasis on differential equations, variational problems, and optimization techniques. Through case studies in physics, biology, engineering, and economics, students will develop skills in model formulation, validation, and refinement. Special attention is given to dimensional analysis, stability theory, and numerical methods necessary for implementing solutions with a focus on numerical methods for ordinary differential equations. The course combines theoretical foundations with hands-on experience in computational tools for model simulation and analysis.
Analysis I, II, Linear Algebra I, II, Numerics I, II
Applied Mathematics
Elective
Mathematics
Concentration Module
Numerical Optimization
Lecturer: Moritz Diehl
Assistant: Léo Simpson
Language: in English
Tutorial / flipped classroom: Di, 14-16h, HS II, Albertstr. 23b
Sir-in Exam: Date to be announced
The aim of the course is to give an introduction into numerical methods for the solution of optimization problems in science and engineering. The focus is on continuous nonlinear optimization in finite dimensions, covering both convex and nonconvex problems. The course divided into four major parts:
The course is organized as inverted classroom based on lecture recordings and a lecture manuscript, with weekly alternating Q&A sessions and exercise sessions. The lecture is accompanied by intensive computer exercises offered in Python (6 ECTS) and an optional project (3 ECTS). The project consists in the formulation and implementation of a self-chosen optimization problem or numerical solution method, resulting in documented computer code, a project report, and a public presentation. Please check the website for further information.
necessary: Analysis I–II, Linear Algebra I–II
useful: Introduction to Numerics
Applied Mathematics
Elective
Mathematics
Concentration Module
Advanced Course in Schemes
Lecturer: Andreas Demleitner
Language: in German
Lecture: Mo, 12-14h, SR 403, Ernst-Zermelo-Str. 1
Exercise session: Do, 14-16h, -, -
Pure Mathematics
Elective
Mathematics
Concentration Module