Detailed information can be found in the course descriptions and in the module handbooks (in German only).
Lecturer: Michael Růžička
Assistant: Alexei Gazca
Language: in German
Lecture: Di, Mi, 8-10h, HS Rundbau, Albertstr. 21
Tutorial: 2 hours, various dates
11.02., 09:00-12:00, HS Rundbau, Albertstr. 21, HS Weismann-Haus, Albertstr. 21a, GHS Physik, Hermann-Herder-Str. 3a, GHS Biologie, Schänzlestr. 1
Sit-in exam (resit) 15.04., 09:00-12:00, HS Rundbau, Albertstr. 21, HS Weismann-Haus, Albertstr. 21a
Analysis I is one of the two basic lectures in the mathematics course. It deals with concepts based on the notion of limit. The central topics are: induction, real and complex numbers, convergence of sequences and series, completeness, exponential function and trigonometric functions, continuity, derivation of functions of one variable and regulated integrals.
Required: High school mathematics. \ Attendance of the preliminary course (for students in mathematics) is recommended.
Analysis
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Linear Algebra I
Lecturer: Stefan Kebekus
Assistant: Marius Amann
Language: in German
Lecture: Mo, Do, 8-10h, HS Rundbau, Albertstr. 21
Tutorial: 2 hours, various dates
27.02., 09:00-12:00, HS Rundbau, Albertstr. 21, GHS Physik, Hermann-Herder-Str. 3a, GHS Biologie, Schänzlestr. 1
Sit-in exam (resit) 26.04., 09:00-13:00, HS Rundbau, Albertstr. 21
Linear Algebra I is one of the two introductory lectures in the mathematics degree program that form the basis for further courses. Topics covered include: fundamental concepts (in particular fundamental concepts of set theory and equivalence relations), groups, fields, vector spaces over arbitrary fields, basis and dimension, linear mappings and transformation matrix, matrix calculus, linear systems of equations, Gaussian elimination, linear forms, dual space, quotient vector spaces and homomorphism theorem, determinant, eigenvalues, polynomials, characteristic polynomial, diagonalizability, affine spaces. The background to the mathematical content is explained in terms of ideas and the history of mathematics.
Required: High school mathematics. \ Attendance of the preliminary course (for students in mathematics) is recommended.
Linear Algebra
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Lecturer: Sören Bartels
Assistant: Tatjana Schreiber
Language: in German
Lecture: Mi, 14-16h, HS Weismann-Haus, Albertstr. 21a
Tutorial: 2 hours, every other week, various dates
Numerics is a sub-discipline of mathematics that deals with the practical solution of mathematical problems. As a rule, problems are not solved exactly but approximately, for which a sensible compromise between accuracy and computational effort must be found. The first part of the two-semester course focuses on questions of linear algebra such as solving linear systems of equations and determining the eigenvalues of a matrix. Attendance at the accompanying practical exercises ({\em Praktische Übung zur Numerik}) is recommended. These take place every 14 days, alternating with the lecture's tutorial.
Required: Linear Algebra~I \ Recommended: Linear Algebra~II and Analysis~I (required for Numerics~II)
Numerics
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Lecturer: Angelika Rohde
Assistant: Johannes Brutsche
Language: in German
Lecture: Fr, 10-12h, HS Weismann-Haus, Albertstr. 21a
Tutorial: 2 hours, every other week, various dates
07.03., 09:00-10:00
Sit-in exam (resit) 05.09.
Stochastic is, to put it loosely, the “mathematics of chance”, about which---possibly contrary to first impressions---many precise and not at all random statements can be formulated and proven. The aim of the lecture is to give an introduction to stochastic modeling, to explain some basic concepts and results of Stochastic and to illustrate them with examples. It is also intended as a motivating preparation for the lecture “Probability Theory” in the summer semester, especially for students in the B.Sc. in Mathematics. Topics covered include: Discrete and continuous random variables, probability spaces and measures, combinatorics, expected value, variance, correlation, generating functions, conditional probability, independence, weak law of large numbers, central limit theorem. The lecture Elementary Probability Theory~II in the summer semester will mainly be devoted to statistical topics. If you are interested in a practical, computer-supported implementation of individual lecture contents, participation in the regularly offered practical excercise “Praktischen Übung Stochastik" is also recommended (in parallel or subsequently).
Required: Linear Algebra~I, Analysis~I and II. \ Note that Linear Algebra~I can be attended in parallel.
Elementary Probabilty Theory
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Lecturer: Wolfgang Soergel
Assistant: Damian Sercombe
Language: in German
Lecture: Di, Do, 10-12h, HS Weismann-Haus, Albertstr. 21a
Tutorial: 2 hours, various dates
25.02., 13:00-16:00, HS Rundbau, Albertstr. 21
Sit-in exam (resit) 09.04., 09:00-12:00
This lecture continues the linear algebra courses. It treats groups, rings, fields and applications in the number theory and geometry. The highlights of the lecture are the classification of finite fields, the impossibility of the trisection of angles with circle and ruler, the non-existence of a solution formula for the general equations of fifth degree and the quadratic reciprocity law.
Required: Linear Algebra~I and II
Algebra and Number Theory
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Algebraic Number Theory
Lecturer: Abhishek Oswal
Assistant: Andreas Demleitner
Language: in English
Lecture: Di, Do, 12-14h, HS II, Albertstr. 23b
Tutorial: 2 hours, date to be determined
Short description of topics: Number fields, Prime decomposition in Dedekind domains, Ideal class groups, Unit groups, Dirichlet's unit theorem, local fields, valuations, decomposition and inertia groups, introduction to class field theory.
Required: Algebra and Number Theory
Elective (Option Area)
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Lecturer: Patrick Dondl
Assistant: Oliver Suchan
Language: in German
Lecture: Mo, 12-14h, HS Rundbau, Albertstr. 21, Mi, 10-12h, HS Weismann-Haus, Albertstr. 21a
Tutorial: 2 hours, various dates
19.02., 10:15-11:45, HS Rundbau, Albertstr. 21
Sit-in exam (resit) 28.04., 10:00-11:30, SR 226, Hermann-Herder-Str. 10
Lebesgue measure and measure theory, Lebesgue integral on measure spaces and Fubini's theorem, Fourier series and Fourier transform, Hilbert spaces. Differential forms, their integration and outer derivative. Stokes' theorem and Gauss' theorem.
Required: Analysis I and II, Linear Algebra I
Elective (Option Area)
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Lecturer: Sebastian Goette
Assistant: Mikhail Tëmkin
Language: in German
Lecture: Mo, Mi, 14-16h, HS II, Albertstr. 23b
Tutorial: 2 hours, date to be determined
Differential geometry, especially Riemannian geometry, deals with the geometric properties of curved spaces. Such spaces also occur in other areas of mathematics and physics, for example in geometric analysis, theoretical mechanics and the general theory of relativity.
Required: Analysis~I–III, Lineare Algebra~I and II \ Recommended: Analysis of Curves and Surfaces ("Kurven und Flächen"), Topology
Elective (Option Area)
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Lecturer: Guofang Wang
Assistant: Christine Schmidt, Xuwen Zhang
Language: in German
Lecture: Mo, Mi, 12-14h, HS II, Albertstr. 23b
Tutorial: 2 hours, date to be determined
A large number of different problems from the natural sciences and geometry lead to partial differential equations. Consequently, there can be no talk of an all-encompassing theory. Nevertheless, there is a clear picture for linear equations, which is based on three prototypes: the potential equation \(-\Delta u = f\), the heat equation \(u_t - \Delta u = f\) and the wave equation \(u_{tt} - \Delta u = f\), which we will examine in the lecture.
Required: Analysis III \ Recommended: Complex Analysis ({\em Funktionentheorie})
Elective (Option Area)
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Lecturer: Sören Bartels
Assistant: Vera Jackisch
Language: in English
Lecture: Di, Do, 10-12h, SR 226, Hermann-Herder-Str. 10
Tutorial: 2 hours, date to be determined
The aim of this course is to give an introduction into theory of linear partial differential equations and their finite difference as well as finite element approximations. Finite element methods for approximating partial differential equations have reached a high degree of maturity, and are an indispensable tool in science and technology. We provide an introduction to the construction, analysis, and implementation of finite element methods for different model problems. We will address elementary properties of linear partial differential equations along with their basic numerical approximation, the functional-analytical framework for rigorously establishing existence of solutions, and the construction and analysis of basic finite element methods.
Required: Analysis~I and II, Linear Algebra~I and II as well as knowledge about higher-dimensional integration (e.g. from Analysis~III or Extensions of Analysis) \ Recommended: Numerics for differential equations, Functional analysis
Elective (Option Area)
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Lecturer: David Criens
Assistant: Eric Trébuchon
Language: in German
Lecture: Di, Mi, 16-18h, HS II, Albertstr. 23b
Tutorial: 2 hours, date to be determined
24.02., 14:00-16:00, HS II, Albertstr. 23b
Complex analysis deals with functions \(f : \mathbb C \to \mathbb C\) , which map complex numbers to complex numbers. Many concepts of Analysis~I can be directly transferred to this case, e.\,g. the definition of differentiability. One might expect that this would lead to a theory analogous to Analysis~I but much more is true: in many respects you get a more elegant and simpler theory. For example, complex differentiability on an open set implies that a function is even infinitely often differentiable, and this is further consistent with analyticity. For real functions, all these notions are different. However, some new ideas are also necessary: For real numbers \(a\), \(b\) one integrates for \[\int_a^b f(x) \mathrm dx\] over the elements of the interval \([a, b]\) or \([b, a]\). However, if \(a\), \(b\) are complex numbers, it is no longer so clear clear how such an integral is to be calculated. One could, for example, in the complex numbers along the line that connects \(a, b \in \mathbb C\), or along another curve that leads from \(a\) to \(b\). Does this lead to a well-defined integral term or does such a curve integral depend on the choice of the curve?
Required: Analysis I+II, Linear Algebra I
Elective (Option Area)
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Lecturer: Ernst August v. Hammerstein
Assistant: Sebastian Stroppel
Language: in English
Lecture: Mo, Mi, 14-16h, SR 404, Ernst-Zermelo-Str. 1
The lecture builds on basic knowledge about Probability Theory. The fundamental problem of statistics is to infer from a sample of observations as precise as possible statements about the data-generating process or the underlying distributions of the data. For this purpose, the most important methods from statistical decision theory such as test and estimation methods are introduced in the lecture. \\ Key words hereto include Bayes estimators and tests, Neyman-Pearson test theory, maximum likelihood estimators, UMVU estimators, exponential families, linear models. Other topics include ordering principles for reducing the complexity of models (sufficiency and invariance). Statistical methods and procedures are used not only in the natural sciences and medicine, but in almost all areas in which data is collected and analyzed This includes, for example, economics (“econometrics”) and the social sciences (especially psychology). However, in the context of this lecture, we will focus less on applications, but---as the name suggests---more on the mathematical justification of the methods.
Required: Probability Theory (in particular measure theory and conditional probabilities/expectations)
Elective (Option Area)
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Set Theory – Independence Proofs
Lecturer: Maxwell Levine
Assistant: Hannes Jakob
Language: in English
Lecture: Di, Do, 12-14h, SR 404, Ernst-Zermelo-Str. 1
Tutorial: 2 hours, date to be determined
How does one prove that something cannot be proved? More precisely, how does one prove that a particular statement does not follow from a particular collection of axioms?
These questions are often asked with respect to the axioms most commonly used by mathematicians: the axioms of Zermelo-Fraenkel set theory, or ZFC for short. In this course, we will develop the conceptual tools needed to understand independence proofs with respect to ZFC. On the way we will develop the theory of ordinal and cardinal numbers, the basics of inner model theory, and the method of forcing. In particular, we will show that Cantor's continuum hypothesis, the statement that \(2^{\aleph_0}=\aleph_1\), is independent of ZFC.
Required: Mathematical Logic
Elective (Option Area)
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Lecturer: Annette Huber-Klawitter, Amador Martín Pizarro
Assistant: Christoph Brackenhofer
Language: in German
Lecture: Di, Do, 10-12h, HS II, Albertstr. 23b
Tutorial: 2 hours, date to be determined
Semi-algebraic geometry is about properties of subsets of \(**R**^n\), which are given by inequalities of the form [ f(x1, \dots, xn)\geq 0] for polynomials \(f\in**R**[X_1,\dots,X_n]\).
The theory has many different facets. On the one hand, it can be seen as a version of algebraic geometry over \(\mathbf{R}\) (or even more generally over so-called real closed fields). On the other hand, the properties of these fields are a central tool for the model-theoretic proof of Tarski-Seidenberg's theorem on quantifier elimination in real closed fields. Geometrically, this is interpreted as a projection theorem.
From this theorem, a proof of Hilbert's 17th problem easily follows, which was solved by Artin in 1926.
\textit{Is every real polynomial \(P \in \mathbf{R}[x_1, \dots, x_n]\), which takes a non-negative value for every \(n\)-tuple in \(\mathbf{R}^n\), a sum of squares of rational functions (i.e., quotients of polynomials)?}
In the lecture, we will explore both aspects. Necessary tools from commutative algebra or model theory will be discussed according to the prior knowledge of the audience.
Required: Algebra and Number Theory \ Recommended: Knowledge in commutative algebra and algebraic geometry (cf. Kommutative Algebra und Einführung in die algebraische Geometrie), model theory
Elective (Option Area)
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Theory and Numerics for Partial Differential Equations – Nonlinear Problems
Lecturer: Sören Bartels, Patrick Dondl
Language: in English
The lecture addresses the development and analysis of numerical methods for the approximation of certain nonlinear partial differential equations. The considered model problems include harmonic maps into spheres, total-variation regularized minimization problems, and nonlinear bending models. For each of the problems, a suitable finite element discretization is devised, its convergence is analyzed and iterative solution procedures are developed. The lecture is complemented by theoretical and practical lab tutorials in which the results are deepened and experimentally tested.
Required: Introduction to Theory and Numerics for PDEs or Introduction to PDEs
Elective (Option Area)
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Lecturer: Peter Pfaffelhuber
Assistant: Samuel Adeosun
Language: in English
Questions sesssion / flipped classroom: Mo, 10-12h, HS II, Albertstr. 23b
Letcure (4 hours): asynchronous videos
A stochastic process \((X_t)_{t\in I}\) is nothing more than a family of random variables, where \(I\) is some index set modeling time. Simple examples are random walks, Markov chains, Brownian motion and derived processes. The latter play a particularly important role in the modeling of financial mathematics or questions from the sciences. We will first deal with martingales, which describe fair games. After constructing the Poisson process and Brownian motion, we will focus on properties of Brownian motion. Infinitesimal characteristics of a Markov process are described by generators, which allows a connection to the theory of partial differential equations. Finally, a generalization of the law of large numbers is discussed with the ergodic theorem for stationary stochastic processes. Furthermore, insights are given into a few areas of application, such as biomathematics or random graphs.
Required: Probability Theory I
Elective (Option Area)
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Lecturer: Thorsten Schmidt
Assistant: Moritz Ritter
Language: in English
Lecture: Mo, Mi, 12-14h, SR 404, Ernst-Zermelo-Str. 1
Tutorial: 2 hours, date to be determined
This lecture marks the culmination of our series on probability theory, achieving the ultimate goal of this series: the combination of stochastic analysis and financial mathematics---a field that has yielded an amazing wealth of fascinating results since the 1990s. The core is certainly the application of semimartingale theory to financial markets culminating in the fundamental theorem of asset pricing. This results is used everywhere in financial markets for arbitrage-free pricing.
After this we look into modern forms of stochastic analysis covering neural SDEs, signature methods, uncertainty and term structure models. The lecture will conclude with an examination of the latest applications of machine learning in financial markets and the reciprocal influence of stochastic analysis on machine learning.
Required: Probability Theory II (Stochastic Processes)
Elective (Option Area)
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Lecturer: Xuwen Zhang
Language: in English
Lecture: Mo, 14-16h, SR 127, Ernst-Zermelo-Str. 1
Tutorial: 2 hours, date to be determined
We will study functions of bounded variation, which are functions whose weak first partial derivatives are Radon measures. This is essentially the weakest definition of a function to be differentiable in the measure-theoretic sense. After discussing the basic properties of them, we move on to the study of sets of finite perimeter, which are Lebesgue measurable sets in the Euclidean space whose indicator functions are BV functions. Sets of finite perimeter are fundamental in the modern Calculus of Variations as they generalize in a natural measure-theoretic way the notion of sets with regular boundaries and possess nice compactness, thus appearing in many Geometric Variational problems. If time permits, we will discuss the (capillary) sessile drop problem as one important application.
Required: Basic knowledge in measure theory and analysis is required.
Elective (Option Area)
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Lecturer: Eva Lütkebohmert-Holtz
Assistant: Hongyi Shen
Language: in English
Lecture: Mo, 10-12h, HS 3042, KG III
Exercise session: Di, 8-10h, HS 1015, KG I
Sit-in exam (resit) 14.08., 15:00-18:00
This course covers an introduction to financial markets and products. Besides futures and standard put and call options of European and American type we also discuss interest-rate sensitive instruments such as swaps.
For the valuation of financial derivatives we first introduce financial models in discrete time as the Cox--Ross--Rubinstein model and explain basic principles of risk-neutral valuation. Finally, we will discuss the famous Black--Scholes model which represents a continuous time model for option pricing.
Required: Elementary Probability Theory~I
Elective (Option Area)
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Lecturer: Maximilian Stegemeyer
Language: in English
Lecture: Do, 14-16h, SR 404, Ernst-Zermelo-Str. 1
Exercise session: Di, 8-10h, SR 127, Ernst-Zermelo-Str. 1
Lie groups and operations of Lie groups play a central role in geometry and topology. They can be used to describe continuous symmetries, one of the most important concepts of mathematics and physics. Exploiting symmetries, e.g. when describing homogeneous spaces, makes it easier to solve many specific problems and often provides a deeper insight into the structures examined. In addition, the geometry and topology of Lie groups and homogeneous spaces is of great interest.
In this lecture, we start with introducing the basic theory of Lie groups and Lie algebras, especially with insights into the structure theory of Lie algebras. In the second part we will look at homogeneous spaces with a special focus on Riemannian symmetric spaces. The latter form an important class of examples of Riemannian manifolds. In addition to the Lie-theoretical aspects, a special focus will always be on the homogeneous Riemannian metrics of the respective spaces.
Required: Differential geometry~I
Elective (Option Area)
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Lecturer: David Criens
Assistant: Dario Kieffer
Language: in English
Lecture: Do, 12-14h, SR 226, Hermann-Herder-Str. 10
Tutorial: 2 hours, date to be determined
The class of Markov chains is an important class of (discrete-time) stochastic processes that are used frequently to model for example the spread of infections, queuing systems or switches of economic scenarios. Their main characteristic is the Markov property, which roughly means that the future depends on the past only through the current state. In this lecture we provide the mathematical foundation of the theory of Markov chains. In particular, we learn about path properties, such as recurrence and transience, state classifications and discuss convergence to the equilibrium. We also study extensions to continuous time. On the way we discuss applications to biology, queuing systems and resource management. If the time allows, we also take a look at Markov chains with random transition probabilities, so-called random walks in random environment, which is a prominent model in the field of random media.
Required: Elementary Probability Theory~I \ Recommended: Analysis~III, Probability Theory~I
Elective (Option Area)
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Mathematical Physics
Lecturer: Wolfgang Soergel
Language: in German
Lecture: Di, 16-18h, SR 403, Ernst-Zermelo-Str. 1
Introduction to classic mechanics from the point of view of mathematics. We start with the mathematical modelling of space and time. Then we discuss Newton's equations of movement, physical systems with compulsory conditions, the D'Alembert principle, the Hamilton formalism and its derivation from the Newton's equations and applications of Hamilton formalism.
Required: Analysis III
Elective (Option Area)
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Students in the B.Sc. Mathematics programme can choose to give a talk, in which case the course counts as a seminar. Usability and requirements as for the seminar ‘Theory of Non-Commutative Algebras’.
Lecturer: Diyora Salimova
Assistant: Ilkhom Mukhammadiev
Language: in English
Lecture: Di, Fr, 12-14h, SR 226, Hermann-Herder-Str. 10
Tutorial: 2 hours, date to be determined
Computer exercise: 2 hours, date to be determined
Oral exam 06.12.
This course takes only place in the first half of the semester, until end of November.
The aim of this course is to enable the students to carry out simulations and their mathematical analysis for stochastic models originating from applications such as mathematical finance and physics. For this, the course teaches a decent knowledge on stochastic differential equations (SDEs) and their solutions. Furthermore, different numerical methods for SDEs, their underlying ideas, convergence properties, and implementation issues are studied.
Required: Probability and measure theory, basic numerical analysis and basics of MATLAB programming.
Elective (Option Area)
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Lecturer: Moritz Diehl
Assistant: Florian Messerer
Language: in English
Tutorial / flipped classroom: Di, 14-16h, HS II, Albertstr. 23b
The aim of the course is to give an introduction to numerical methods for the solution of optimal control problems in science and engineering. The focus is on both discrete time and continuous time optimal control in continuous state spaces. It is intended for a mixed audience of students from mathematics, engineering and computer science.
The course covers the following topics:
The lecture is accompanied by intensive weekly computer exercises offered both in MATLAB and Python (6~ECTS) and an optional project (3~ECTS). The project consists in the formulation and implementation of a self-chosen optimal control problem and numerical solution method, resulting in documented computer code, a project report, and a public presentation.
Required: Analysis~I and II, Linear Algebra~I and II \ Recommended: Numerics I, Ordinary Differential Equations, Numerical Optimization
Elective (Option Area)
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Introduction to Mathematics Education
Lecturer: Katharina Böcherer-Linder
Language: in German
Mo 10-12h, SR 226, Hermann-Herder-Str. 10, Fr, 8-10h, SR 127, Ernst-Zermelo-Str. 1
Fr, 14-16h, SR 127, Ernst-Zermelo-Str. 1
10.02., 08:15-10:00
Mathematics didactic principles and their learning theory foundations and possibilities of teaching implementation (also e.g. with the help of digital media). \\ Theoretical concepts on central mathematical thinking activities such as concept formation, modeling, problem solving and reasoning. \\ Mathematics didactic constructs: Barriers to understanding, pre-concepts, basic ideas, specific difficulties with selected mathematical content. \\ Concepts for dealing with heterogeneity, taking into account subject-specific characteristics particularities (e.g. dyscalculia or mathematical giftedness).\\ Levels of conceptual rigour and formalization as well as their age-appropriate implementation.
Required: Analysis~I, Linear Algebra~I
(Introduction to) Mathematics Education
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Organisation: Susanne Knies
Language: in German
What characterizes a good tutorial? This question will be discussed in the first workshop and tips and suggestions will be given. Experiences will be shared in the second workshop.
Elective (Option Area)
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Computer exercises for Introduction to Theory and Numerics of Partial Differential Equations
Lecturer: Sören Bartels
Assistant: Vera Jackisch
Language: in English
The computer tutorial accompanies the lecture with programming exercises.
See the lecture – additionally: programming knowledge.
Elective (Option Area)
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Computer exercises in Numerics
Lecturer: Sören Bartels
Assistant: Tatjana Schreiber
Language: in German
In the computer tutorial accompanying the Numerics (first term) lecture the algorithms developed and analyzed in the lecture are put into practice and and tested experimentally. The implementation is carried out in the programming languages Matlab, C++ and Python. Elementary programming knowledge is assumed.
See the lecture {\em Numerics I} (which should be attended in parallel or should already have been completed). \ Additionally: Elementary programming knowledge.
Computer Exercise
Elective (Option Area)
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Please note the registration modalities for the individual proseminars published in the comments to the course catalog: As a rule, places are allocated after pre-registration by e-mail at the preliminary meeting at the end of the lecture period of the summer semester. You must then register online for the examination; the registration period runs from August 1, 2024 to October 9, 2024; if you would like to attend a proseminar but have not been allocated a place, please contact the degree program coordinator immediately.
Lecturer: Susanne Knies, Ludwig Striet
Language: in German
Seminar: Do, 12-14h, SR 125, Ernst-Zermelo-Str. 1
Preliminary Meeting 15.07., 13:00, SR 403, Ernst-Zermelo-Str. 1
Preparation meetings for talks: Dates by arrangement
Numerous dynamic processes in the natural sciences can be modeled by ordinary differential equations. In this proseminar we will deal with explicit solution methods for differential equations as well as the application situations (reaction kinetics, predator-prey models, mathematical pendulum, different growth processes, . . . ) which can be described by them.
Analysis~I and II, Lineare Algebra~I and II
Undergraduate Seminar
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Lecturer: Angelika Rohde
Assistant: Johannes Brutsche
Language: in German
Seminar: Mi, 12-14h, SR 125, Ernst-Zermelo-Str. 1, Fr, 13-15h, SR 404, Ernst-Zermelo-Str. 1
Preregistration:
Preliminary Meeting 16.07., 10:15, Raum 232, Ernst-Zermelo-Str. 1
Preparation meetings for talks: Dates by arrangement
Paul Erd\H{o}s liked to talk about the BOOK in which God keeps the \textit{perfect} proofs of mathematical theorems, according to the famous quote by G. H. Hardy that "there is no permanent place for ugly mathematics" ([1], Preface). In an attempt at a best approximation to this BOOK, Aigner and Ziegler have published a large number of sentences with elegant, sophisticated, and sometimes surprising evidence. In this proseminar, a selection of these results will be presented. The spectrum of topics covers all different areas of mathematics, from number theory, geometry, analysis, and combinatorics to graph theory and includes well-known results, such as Littlewood and Offord's lemma, the Dinitz problem, Hilbert's third problem (of his 23 problems presented at the International Congress of Mathematicians in Paris in 1900), the Borsuk conjecture, and many more.
Linear Algebra~I and II, Analysis~I and II
Undergraduate Seminar
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Lecturer: Wolfgang Soergel
Assistant: Damian Sercombe
Language: in German
Seminar: Di, 14-16h, SR 127, Ernst-Zermelo-Str. 1
Preregistration:
Preparation meetings for talks: Dates by arrangement
In this proseminar we will discuss topics that are found in various textbooks and scripts for basic lectures in linear algebra but which are not part of the standard material. The lectures build on each other only slightly.
Linear Algebra ~I and II, Analysis~I and II.
Undergraduate Seminar
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Please note the registration modalities for the individual seminars published in the comments to the course catalog: As a rule, places are allocated after pre-registration by e-mail at the preliminary meeting at the end of the lecture period of the summer semester. You must then register online for the exam; the registration period runs from August 1, 2024 to October 9, 2024.
Lecturer: Ernst August v. Hammerstein
Language: in German
A knot can be mathematically defined relatively simply as a closed curve in the three-dimensional space \(\mathbb{R}^3\). From everyday life, one is certainly already familiar with different types of knots, e.g, surgeons knot, sailors knots, and many more. The aim of mathematical knot theory is to find characteristic quantities for the description and classification of knots and thus possibly also to be able to decide whether two knots are equivalent, i.e., if they can be transformed into one another through certain operations.
Ropes, cords or wires can be used to illustrate knots as well as interlacings. Prospective teachers can use these not only in this seminar, but perhaps also later in the classroom to display different results in a very practical way.
Required: Basic Mathematics courses. \ Possibly a little knowledge in topology in addition.
Undergraduate Seminar
Elective (Option Area)
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Lecturer: Thorsten Schmidt
Assistant: Moritz Ritter
Language: Talk/participation possible in German and English
Seminar: Fr, 10-12h, SR 125, Ernst-Zermelo-Str. 1
Preregistration:
Preliminary Meeting 18.10.
Preparation meetings for talks: Dates by arrangement
This seminar will focus on theoretical machine learning results, including modern universal approximation theorems, approximation of filtering methods through transformes, application of machine learning methods in financial markets and possibly other related topics. Moreover, we will cover topics in stochastic analysis, like fractional Ito calculus, uncertainty, filtering and optimal transport. You are also invited to suggest related topics.
Required: Basic Probability and either Machine Learning or Probability Theory II (Stochastic Processes).
Elective (Option Area)
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Machine-Learning Methods in the Approximation of PDEs
Lecturer: Sören Bartels
Assistant: Tatjana Schreiber
Language: Talk/participation possible in German and English
Machine-learning methods have recently been used to approximate solutions of partial differential equations. While in some cases they lead to advantages over classical approaches, their general superiority is widely open. In the seminar we will review the main concepts and recent developments.
Introduction to Theory and Numerics for PDEs
Elective (Option Area)
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Medical Data Science
Lecturer: Harald Binder
Language: Talk/participation possible in German and English
Seminar: Mi, 10-11:30h, HS Medizinische Biometrie, 1. OG, Stefan-Meier-Str. 26
Preregistration:
Preliminary Meeting 17.07., HS Medizinische Biometrie, 1. OG, Stefan-Meier-Str. 26
To answer complex biomedical questions from large amounts of data, a wide range of analysis tools is often necessary, e.g. deep learning or general machine learning techniques, which is often summarized under the term ``Medical Data Science''. Statistical approaches play an important rôle as the basis for this. A selection of approaches is to be presented in the seminar lectures that are based on recent original work. The exact thematic orientation is still to be determined.
Good knowledge of probability theory and mathematical statistics.
Elective (Option Area)
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Lecturer: Guofang Wang
Assistant: Xuwen Zhang
Language: Talk/participation possible in German and English
Seminar: Mi, 16-18h, SR 125, Ernst-Zermelo-Str. 1
Preliminary Meeting 17.07., 16:00
Preparation meetings for talks: Dates by arrangement
Minimal surfaces are surfaces in space with a “minimal” area and can be described using holomorphic functions. They occur, for example in the investigation of soap skins and the construction of stable objects (e.g. in architecture). In the investigation of minimal surfaces elegant methods from various mathematical fields such as function theory, calculus of variations, differential geometry and partial differential equations. are applied.
Required: Analysis III or knowledge about multidimensional integration and complex analysis. \ Recommended: Elementary knowledge about differential geometry.
Elective (Option Area)
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Lecturer: Sebastian Goette
Assistant: Mikhail Tëmkin
Language: Talk/participation possible in German and English
Seminar: Di, 14-16h, SR 125, Ernst-Zermelo-Str. 1
Preliminary Meeting 16.07., SR 125, Ernst-Zermelo-Str. 1
Preparation meetings for talks: Dates by arrangement
We will discuss advanced topics in algebraic topology. Depending on the interest of the participants we could work on one of the following topics---if you have other topic suggestions, please contact the lecturer.
Algebraic Topology~I and II
Elective (Option Area)
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.
Lecturer: Annette Huber-Klawitter
Assistant: Xier Ren
Language: Talk/participation possible in German and English
Seminar: Fr, 8-10h, SR 404, Ernst-Zermelo-Str. 1
Preregistration:
Preliminary Meeting 15.07., 11:00, SR 318, Ernst-Zermelo-Str. 1
Preparation meetings for talks: Dates by arrangement
In this seminar, we are going to study finite dimensional (unital, possibly non-commutative) algebras over a (commutative) field \(k\). Prototypes are the rings of square matrices over \(k\), finite field extensions, or the algebra \(k^n\) with diagonal multiplication.
We will concentrate on path algebras of finite quivers (German: Köcher). Modules over them are equivalently described as representations of the quiver. Many algebraic properties can be directly understood from properties of the quiver.
Required: Linear Algebra \ Recommended: Algebra and Number Theory, Commutative Algebra and Introduction to Algebraic Geometry
Elective (Option Area)
Requirement for coursework, assessments, and examinations are described in the current supplements of the module handbooks.