Time and place

Lecture: Mo, Mi, 14-16h, HS II, Albertstr. 23b
Tutorial: 2 hours, date to be determined

Teacher: Angelika Rohde
Assistant: Johannes Brutsche
Language: in English

A stochastic process \((X_t)_{t\in T}\) is a family of random variables, where mostly the situation \(T = \mathbb{N}\) or \(T = [0, 1]\) is studied. Basic examples include stationary time series, the Poisson process and Brownian motion as well as processes derived from those. The lecture includes ergodic theory and its applications, Brownian motion and especially the study of its path properties, the elegant concept of weak convergence on Polish spaces as well as functional limit theorems. Finally, we introduce stochastic integration with respect to local martingales, based on the continuous time version of the martingale transform.

Probability Theory I

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)

Time and place

Seminar: Mo, 16-18h, SR 127, Ernst-Zermelo-Str. 1
Preregistration: If your are interested in the seminar, please write an email to Johannes Brutsche listing your prerequisites in probability and note if you plan to attend the Probability Theory II.
Preliminary Meeting 22.07., 14:00, Raum 232, Ernst-Zermelo-Str. 1
Preparation meetings for talks: Dates by arrangement

Teacher: Angelika Rohde
Assistant: Johannes Brutsche
Language: Talk/participation possible in German and English

Random walks are stochastic processes (in discrete time) formed by successive summation of independent, identically distributed random variables and are one of the most studied topics in probability theory. Many results that are part of this seminar also carry over to Brownian motion and related processes in continuous time. In particular, the theory for random walks contains many central and elegant proof ideas which can be extended to various other settings. We start the theory at the very beginning but quickly move on to proving local central limit theorems, study Green's function and recurrence properties, hitting times and the Gambler's ruin estimate. Further topics may include a dyadic coupling with Brownian motion, Dirichlet problems, random walks that are not indexed in \(\mathbb{N}\) but the lattice \(\mathbb{Z}^d\), and intersection probabilities for multidimensional random walks (which are processes \(X:\mathbb{N}\rightarrow\mathbb{R}^d\)). Here, we will see that in dimension \(d=1,2,3\) two paths hit each other with positive probability, while for \(d\geq 4\) they avoid each other almost surely.

Probability Theory I \
Some talks only require knowledge of Stochastics I, so if you are interested in the seminar and have not taken part in the probability theory I class, do not hesitate to reach out to us regarding a suitable topic.

Elective (Option Area) (2HfB21)
Mathematical Seminar (BSc21)
Compulsory Elective in Mathematics (BSc21)
Supplementary Module in Mathematics (MEd18)
Mathematical Seminar (MSc14)
Elective (MSc14)
Mathematical Seminar (MScData24)
Elective in Data (MScData24)

Time and place

Lecture: Fr, 10-12h, HS Weismann-Haus, Albertstr. 21a
Tutorial: 2 hours, every other week, various dates
Sit-in exam 05.09., 10:00-12:00, HS Rundbau, Albertstr. 21, GHS Physik, Hermann-Herder-Str. 3a

Teacher: Johannes Brutsche
Assistant: Dario Kieffer
Language: in German

After gaining an insight into the basics and various methods and questions of stochastics and probability theory in the Stochastics I lecture, this lecture will mainly focus on statistical topics, especially those that are relevant for students studying to become secondary school teachers. However, the lecture can also be a (hopefully) useful supplement and a good basis for later attendance of the course lecture ‘Mathematical Statistics’ for students in the B.Sc. in Mathematics with an interest in stochastics.

After clarifying the term ‘statistical model’, methods for constructing estimators (e.g. maximum likelihood principle, method of moments) and quality criteria for these (reliability of expectations, consistency) are discussed. Confidence intervals and hypothesis tests are also introduced. Linear models are considered as further applications and, if time permits, other statistical methods. The properties of exponential families and multivariate normal distributions, which are useful for many test and estimation methods, are also introduced.

Linear Algebra I+II and Analysis I+II

Elementary Probabilty Theory (2HfB21, MEH21)
Elementary Probability Theory II (MEdual24)
Compulsory Elective in Mathematics (BSc21)

Time and place

Lecture: Di, Do, 10-12h, HS Weismann-Haus, Albertstr. 21a
Tutorial: 2 hours, date to be determined
Sit-in exam 22.09., 10:00-12:00
Sit-in exam (resit) 31.10.

Teacher: Angelika Rohde
Assistant: Johannes Brutsche
Language: in English

The problem of axiomatising probability theory was solved by Kolmogorov in 1933: a probability is a measure of the set of all possible outcomes of a random experiment. From this starting point, the entire modern theory of probability develops with numerous references to current applications.

The lecture is a systematic introduction to this area based on measure theory and includes, among other things, the central limit theorem in the Lindeberg-Feller version, conditional expectations and regular versions, martingales and martingale convergence theorems, the strong law of large numbers and the ergodic theorem as well as Brownian motion.

necessary: Analysis I+II, Linear Algebra I, Elementary Probability Theory I

useful: Analysis III

Elective (Option Area) (2HfB21)
Compulsory Elective in Mathematics (BSc21)
Mathematical Concentration (MEd18, MEH21)
Applied Mathematics (MSc14)
Elective (MSc14)
Advanced Lecture in Stochastics (MScData24)
Elective in Data (MScData24)

Time and place

Lecture: Fr, 10-12h, HS Weismann-Haus, Albertstr. 21a
Tutorial: 2 hours, every other week, various dates
Sit-in exam 07.03., 09:00-10:00
Sit-in exam (resit) 05.09., 10:00-12:00, HS Rundbau, Albertstr. 21, GHS Physik, Hermann-Herder-Str. 3a

Teacher: Angelika Rohde
Assistant: Johannes Brutsche
Language: in German

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 (2HfB21, MEH21)
Elementary Probability Theory I (BSc21, MEB21, MEdual24)

Time and place

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

Teacher: Angelika Rohde
Assistant: Johannes Brutsche
Language: in German

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 (2HfB21, BSc21, MEH21, MEB21)

Time and place

Lecture: Di, Do, 8-10h, HS Rundbau, Albertstr. 21

Teacher: Angelika Rohde
Assistant: Johannes Brutsche

Linear Algebra II (BScInfo19, BScPhys20)

Teacher: Angelika Rohde
Assistant: Johannes Brutsche

Supplementary Module in Mathematics (MEd18)
Compulsory Elective in Mathematics (BSc21)

Time and place

Lecture: Mo, Do, 8-10h, HS Rundbau, Albertstr. 21
Sit-in exam 19.03., 08:00-12:00

Teacher: Angelika Rohde
Assistant: Johannes Brutsche
general: ,

Linear Algebra I (BScInfo19, BScPhys20)

Time and place

Do, 12-14h, SR 226, Hermann-Herder-Str. 10

Teacher: Johannes Brutsche, Jakob Stiefel
general:

Computer Exercise (2HfB21, MEH21, MEB21)
Supplementary Module in Mathematics (MEd18)

Time and place

Lecture: Mo, Mi, 14-16h, SR 404, Ernst-Zermelo-Str. 1

Teacher: Angelika Rohde
Assistant: Johannes Brutsche
general:

Time and place

Lecture: Do, 14-16h, SR 226, Hermann-Herder-Str. 10

Teacher: Johannes Brutsche
Assistant: Saskia Glaffig
general:

Time and place

Lecture: Mo, Mi, 14-16h, HS II, Albertstr. 23b

Teacher: Angelika Rohde
Assistant: Johannes Brutsche
general:

Compulsory Elective in Mathematics (BSc21)

Time and place

Lecture: Mo, 12-14h, SR 127, Ernst-Zermelo-Str. 1, Mi, 12-14h, HS II, Albertstr. 23b

Teacher: Angelika Rohde
Assistant: Johannes Brutsche
general:

Compulsory Elective in Mathematics (BSc21)

Time and place

Mo, 12-14h, HS Weismann-Haus, Albertstr. 21a
Sit-in exam 11.02., 14:00-00:00

Teacher: Ernst August von Hammerstein, Johannes Brutsche
general:

Computer Exercise (2HfB21, MEH21, MEB21)
Supplementary Module in Mathematics (MEd18)

Time and place

Di Do, 10-12h, , online

Teacher: Angelika Rohde
Assistant: Johannes Brutsche
general:

Compulsory Elective in Mathematics (BSc21)

Time and place

Lecture: Mo, Mi, 12-14h, genauere Informationen zum Ablauf durch Dozenten nach Belegung in HISinOne, -
Sit-in exam 09.03., 08:00-11:00
Sit-in exam (resit) 21.05., 09:00-11:00

Teacher: Angelika Rohde
Assistant: Johannes Brutsche
general:

Applied Mathematics (MSc14)
Mathematical Concentration (MEd18, MEH21)
Compulsory Elective in Mathematics (BSc21)

Time and place

Di Mi, 16-18h, HS II, Albertstr. 23b

Teacher: Ernst August von Hammerstein
Assistant: Johannes Brutsche

Undergraduate Seminar (2HfB21, BSc21, MEH21, MEB21)

Time and place

Lecture: Mo, Mi, 10-12h, HS Weismann-Haus, Albertstr. 21a

Teacher: Angelika Rohde
Assistant: Johannes Brutsche
general:

Analysis III (BSc21)
Mathematical Concentration (MEd18, MEH21)

Time and place

Lecture: Mo, Mi, 8-10h, HS Rundbau, Albertstr. 21
Sit-in exam 08.08., 09:00-12:00
Sit-in exam (resit) 08.01., 14:00-17:00

Teacher: Angelika Rohde
Assistant: Johannes Brutsche
general: , ,

Analysis II (BScInfo19, BScPhys20)

Time and place

Di 14-16h, SR 127, Ernst-Zermelo-Str. 1, Mo, 10-12h, SR 125, Ernst-Zermelo-Str. 1

Teacher: Angelika Rohde
Assistant: Johannes Brutsche

Undergraduate Seminar (2HfB21, BSc21, MEH21, MEB21)

Time and place

Lecture: Di, Mi, 8-10h, HS Rundbau, Albertstr. 21
Sit-in exam 22.02., 14:00-16:00
Sit-in exam (resit) 25.04., 13:00-16:00

Teacher: Angelika Rohde
Assistant: Johannes Brutsche
general: , ,