Attention: There are some short-term room changes!
Differential Geometry II – Geometry of Submanifolds
Lecture: Mo, Mi, 12-14h, SR 404, Ernst-Zermelo-Str. 1
Tutorial: 2 hours, date to be determined
Teacher: Guofang Wang
Assistant: Xuwen Zhang
Language: in German
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"
Elective
Lecture: Di, Do, 8-10h, HS II, Albertstr. 23b
Tutorial: 2 hours, date to be determined
Teacher: Wolfgang Soergel
Assistant: Xier Ren
Language: in German
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
Elective
Lecture: Di, Do, 12-14h, HS II, Albertstr. 23b
Tutorial: 2 hours, date to be determined
Sir-in Exam: Date to be announced
Teacher: Amador Martín Pizarro
Assistant: Stefan Ludwig
Language: in German
This introductory course in mathematical logic consists of several parts. It the basics of predicate logic and a brief introduction to model theory and the axiom system
as well as the axiom system of set theory. The aim of the lecture is to explain the
recursion-theoretical content of the predicate calculus, in particular the so-called Peano-
arithmetic and Gödel's incompleteness theorems.
Basic knowledge of mathematics from first semester lectures
Elective
Lecture: Di, Do, 10-12h, HS II, Albertstr. 23b
Tutorial: 2 hours, date to be determined
22.07.
Teacher: Heike Mildenberger
Assistant: Hannes Jakob
Language: in German
A topological space consists of a basic set \(X\) and a family of open subsets of the basic set, which is called topology on \(X\). Examples over the basic sets \(\mathbb R\) and \({\mathbb R}^n\) are given in the analysis lectures. The mathematical subject \glqq{}Topology\grqq\ is the study of topological spaces and the investigation of topological spaces. Our lecture is an introduction to set-theoretic and algebraic topology.
Analysis I and II, Linear Algebra I
Elective
Lecture: Mo, 14-16h, SR 404, Ernst-Zermelo-Str. 1
Tutorial: 2 hours, date to be determined
Teacher: Mikhail Tëmkin
Language: in English
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)
Elective
Lecture: Mi, 14-16h, SR 125, Ernst-Zermelo-Str. 1
Tutorial: 2 hours, date to be determined
Teacher: Amador Martín Pizarro
Assistant: Charlotte Bartnick
Language: in German
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
Elective
Advanced Course in Schemes
Lecture: Mo, 12-14h, SR 403, Ernst-Zermelo-Str. 1
Exercise session: Do, 14-16h, -, -
Teacher: Andreas Demleitner
Language: in German
Elective
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
Computer exercises in Statistics
Mo, 14-16h, PC-Pool Raum 201, Hermann-Herder-Str. 10
Teacher: Sebastian Stroppel
Language: in German
This computer exercise course is aimed at students who have already attended the lectures Elementary Probability Theory I and II or are attending the second part this semester. Computer-based methods will be discussed to deepen the understanding of the lecture material and demonstrate further application examples. For this purpose, the programming language python is used. After an introduction to python, methods of descriptive statistics and graphical analysis of data will be considered, the numerical generation of random numbers will be explained and parametric and non-parametric tests and linear regression methods will be discussed. Previous knowledge of python and/or programming skills are not required.
Analysis I+II, Linear Algebra I+II, Elementary Probability Theory I+II (part II can be followed in parallel).
Elective
Do, 14-16h, PC-Pool Raum -100, Hermann-Herder-Str. 10
Teacher: Carola Heinzel
Assistant: Samuel Adeosun
Language: in English
This course introduces the foundational concepts and practical skills necessary for understanding and implementing machine learning models, with a particular focus on deep learning and neural networks. Students will progress from basic programming skills in Python , with a focus on the PyTorch library, to advanced topics such as training multi-layer perceptrons, optimization techniques, and transformer architectures. By the end of the course, participants will have the ability to implement and analyze neural networks, apply optimization strategies, and understand modern transformer-based models for tasks such as text generation and time series analysis.
Basic knowledge in programming and basic knowledge in stochastics.
Elective
Di, 12-14h, SR 404, Ernst-Zermelo-Str. 1
Teacher: Peter Pfaffelhuber
Assistant: Sebastian Stroppel
Language: in English
Lean4 is both, a programming language and an interactive theorem prover. By the latter, we mean software that is able to check mathematical proofs. It is interactive since the software tells you what remains to be proven after every line of code. The course is an introduction to this technique, with examples from various fields of mathematics. Lean4 is special since researchers all over the world are currently building a library of mathematical theories, which contains at the moment around 1.5 million lines of code. I aim to cover basics from calculus, algebra, topology and measure theory in Lean4.
Analysis 1, 2, Linear algebra 1
Elective
Seminar: Do, 10-12h, SR 403, Ernst-Zermelo-Str. 1
Preregistration: by e-mail to Wolfgang Soergel
Preliminary Meeting 28.01., 14:15, SR 127, Ernst-Zermelo-Str. 1
Preparation meetings for talks: Dates by arrangement
Teacher: Wolfgang Soergel
Assistant: Damian Sercombe
Language: Talk/participation possible in German and English
This seminar is intended to introduce to the theory of linear algebraic groups. Linear algebraic groups are generalizations of the matrix groups known from linear algebra.
I imagine a format in which I or Sercombe lecture and in between the seminar participants give talks. The seminar is a reasonable addition to the commutative algebra lecture. Reference to that lecture increase in the course of the seminar.
Algebra and Number Theory (where the details of Galois theory and field theory are less relevant than the general theory of groups and rings) and Linear Algebra.
Elective
Seminar: Di, 14-16h, SR 125, Ernst-Zermelo-Str. 1
Preliminary Meeting 04.02., 12:15, SR 218, Ernst-Zermelo-Str. 1
Preparation meetings for talks: Dates by arrangement
Teacher: Ernst Kuwert
Analysis I–III
Elective
Seminar: Di, 12-14h, SR 226, Hermann-Herder-Str. 10
Preliminary Meeting 04.02., 10:00, SR 318, Ernst-Zermelo-Str. 1
Preparation meetings for talks: Dates by arrangement
Teacher: Nadine Große
Assistant: Maximilian Stegemeyer
Language: Talk/participation possible in German and English
Differential Geometry I
Elective
Seminar: Di, 16-18h, SR 403, Ernst-Zermelo-Str. 1
Preliminary Meeting 29.01., 13:15, Raum 313, Ernst-Zermelo-Str. 1
Preparation meetings for talks: Dates by arrangement
Teacher: Heike Mildenberger
Assistant: Maxwell Levine
Language: Talk/participation possible in German and English
Set Theory
Elective
Seminar on p-adic Geometry
Seminar: Mo, 10-12h, SR 404, Ernst-Zermelo-Str. 1
Preliminary Meeting 13.02., 14:30, SR 404, Ernst-Zermelo-Str. 1, Please email Abhisehk Oswal, and Ben Snodgrass if you are interested in the seminar but cannot make it to the preliminary meeting.
Teacher: Abhishek Oswal
Assistant: Ben Snodgrass
Language: in English
It has become clear over the last several decades that \(p\)-adic techniques play an indispensable role in arithmetic geometry. At an elementary level, \(p\)-adic numbers provide a compact and convenient language to talk about congruences between integers. Concretely, just as the field of real numbers \(\mathbb R\) arise as the completion of the field \(\mathbb Q\) of rational numbers with respect to the usual notion of distance on \(\mathbb Q\), the field \(\mathbb Q_p\) of \(p\)-adic numbers arise as the completion of \(\mathbb Q\) with respect to an equally natural \(p\)-adic metric. Roughly, in the \(p\)-adic metric, an integer \(n\) is closer to \(0\), the larger the power of the prime number \(p\) that divides it. A general philosophy in number theory is then to treat all these completions \(\mathbb R\), \(\mathbb Q_p\) of the field \(\mathbb Q\) on an equal footing. As we shall see in this course, familiar concepts from real analysis (i.e. notions like analytic functions, derivatives, measures, integrals, Fourier analysis, real and complex manifolds, Lie groups...), have completely parallel notions over the \(p\)-adic numbers.
While the Euclidean topology of \(\mathbb R^n\) is rather well-behaved (so one may talk meaningfully about paths, fundamental groups, analytic continuation, ...), the \(p\)-adic field \(\mathbb Q_p\) on the other hand is totally disconnected. This makes the task of developing a well-behaved notion of global \(p\)-adic analytic manifolds/spaces rather difficult. In the 1970s, John Tate’s introduction of the concept of rigid analytic spaces, solved these problemsand paved the way for several key future developments in \(p\)-adic geometry.
The broad goal of this course will be to introduce ourselves to this world of \(p\)-adic analysis and rigid analytic geometry (due to Tate). Along the way, we shall see a couple of surprising applications of this circle of ideas to geometry and arithmetic. Specifically, we plan to learn Dwork’s proof of the fact that the zeta function of an algebraic variety over a finite field is a rational function.
Field theory, Galois theory and Commutative algebra.
Some willingness to accept unfamiliar concepts as black boxes. Prior experience with algebraic number theory, or algebraic geometry will be beneficial but not necessary.
Elective