Further Chapters in Analysis
Lecture: Mi, 8-10h, HS Weismann-Haus, Albertstr. 21a
Tutorial: 2 hours, various dates
Sit-in exam: date to be announced
Requirements on examinations, assessments and coursework will be described in the supplements of the module handbooks to be published as part of the course cataloque by end of October.
Teacher: Ernst August v. Hammerstein
Language: in German
This compulsory lecture for teacher training students in the M.Ed. builds on the basic lectures Analysis I and II and supplements them with the following two main topics:
\textit{Multidimensional integration:} The one-dimensional Riemann integral known from Analysis I is generalized to real-valued functions of several variables, for which a suitable instrument for measuring the content/volume of multidimensional sets is first introduced with the Jordan content. Then the classical integral theorems (transformation theorem, Fubini's theorem) are derived, and path and surface integrals are considered. With the help of the divergence and rotation of vector fields, the two aforementioned integral types can be related to each other using the integral theorems of Gauß and Stokes, which considerably simplifies the calculations in practical applications.
\textit{Complex Analysis:} In contrast to Analysis I, here the (complex) differentiability of functions of a complex variable is examined. As will be shown, complex differentiable, so-called holomorphic functions are subject to much stricter rules and laws than their real-valued counterparts, which leads to both beautiful and surprising results. To this end, we will prove Cauchy's intergal theorem and Cauchy's integral formula and take a closer look at applications and conclusions from these.
Analysis~I and II, Linear Algebra~I and II
Further Chapters in Analysis (MEd18, MEH21, MEdual24)
Mathematical Statistics
Lecture: Di, Do, 14-16h, SR 404, Ernst-Zermelo-Str. 1
Tutorial: 2 hours, date to be determined
Requirements on examinations, assessments and coursework will be described in the supplements of the module handbooks to be published as part of the course cataloque by end of October.
Teacher: Ernst August v. Hammerstein
Language: in English
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.
Probability Theory (in particular measure theory and conditional probabilities/expectations)
Elective (Option Area) (2HfB21)
Compulsory Elective in Mathematics (BSc21)
Applied Mathematics (MSc14)
Mathematics (MSc14)
Concentration Module (MSc14)
Elective (MSc14)
Advanced Lecture in Stochastics (MScData24)
Elective in Data (MScData24)
Mathematical Methods for Economics and Finance
Lecture: Di, 10-12h, HS 1221, KG I
Exercise session: Fr, 10-12h, -, -
Teacher: Ernst August v. Hammerstein
Language: in English
Lecture: Mo, 14-16h, HS II, Albertstr. 23b
Exercise session: Mi, 16-18h, SR 403, Ernst-Zermelo-Str. 1
Teacher: Ernst August v. Hammerstein
Assistant: Sebastian Hahn
Language: in English
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)
Elective (Option Area) (2HfB21)
Compulsory Elective in Mathematics (BSc21)
Supplementary Module in Mathematics (MEd18)
Applied Mathematics (MSc14)
Mathematics (MSc14)
Concentration Module (MSc14)
Elective (MSc14)
Elective in Data (MScData24)
Lecture: Mo, Mi, 10-12h, HS 00-026, Georges-Köhler-Allee 101
Tutorial: 2 hours, various dates
Teacher: Ernst August v. Hammerstein
Assistant: Damian Sercombe
Lecture: Mo, Mi, 14-16h, SR 404, Ernst-Zermelo-Str. 1
Teacher: Ernst August v. Hammerstein
Assistant: Sebastian Stroppel
Language: in English
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) (2HfB21)
Compulsory Elective in Mathematics (BSc21)
Applied Mathematics (MSc14)
Mathematics (MSc14)
Concentration Module (MSc14)
Elective (MSc14)
Advanced Lecture in Stochastics (MScData24)
Elective in Data (MScData24)
Teacher: 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.
Supplementary Module in Mathematics (MEd18)
Undergraduate Seminar (2HfB21, BSc21, MEH21, MEB21)
Elective (Option Area) (2HfB21)
Teacher: Ernst August v. Hammerstein
Assistant: Hongyi Shen
Language: in English
Lecture: Fr, 10-12h, HS Weismann-Haus, Albertstr. 21a
Teacher: Ernst August v. Hammerstein
Assistant: Timo Enger
Compulsory Elective in Mathematics (BSc21)
Mo Mi, 10-12h, HS 00-026, Georges-Köhler-Allee 101
Teacher: Ernst August v. Hammerstein
Assistant: Saskia Glaffig
Di, 14-16h, PC-Pool Raum -100, Hermann-Herder-Str. 10
Teacher: Ernst August v. Hammerstein
Computer Exercise (2HfB21, MEH21, MEB21)
Supplementary Module in Mathematics (MEd18)
Lecture: Fr, 10-12h, HS Weismann-Haus, Albertstr. 21a
Sit-in exam 13.02., 10:00-12:00
Teacher: Ernst August v. Hammerstein
Assistant: Timo Enger
general: ,
Elementary Probability Theory I (BSc21, MEB21, MEdual24)
Lecture: Di, 10-12h, HS 3042, KG III
Teacher: Ernst August v. Hammerstein
Assistant: Sven Knaust
general:
Compulsory Elective in Mathematics (BSc21)
Supplementary Module in Mathematics (MEd18)
Lecture: Mo, 10-12h, HS 1098, KG I
Teacher: Ernst August v. Hammerstein
Assistant: Samuel Adeosun
Mo, 14-16h, SR 125, Ernst-Zermelo-Str. 1
Teacher: Ernst August v. Hammerstein
Assistant: Ernst August v. Hammerstein
Undergraduate Seminar (2HfB21, BSc21, MEH21, MEB21)