Lecturer: Wolfgang Soergel
Assistant: Damian Sercombe
Language: in German

Time and place

Lecture: Di, Do, 10-12h, HS Weismann-Haus, Albertstr. 21a
Tutorial: 2 hours, various dates
Sit-in exam: date to be announced

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.

Linear Algebra I and II

Pure Mathematics
Elective

Lecturer: Maximilian Stegemeyer
Language: in German

Time and place

Lecture: Di, Do, 10-12h, SR 404, Ernst-Zermelo-Str. 1
Exercise session: Mi, 14-16h, SR 403, Ernst-Zermelo-Str. 1

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.

Algebraic topology studies topological spaces by assigning algebraic objects, e.g. groups, vector spaces or rings, to them in a particular way. This assignment is usually done in a way which is invariant under homotopy equivalences. Therefore one often speaks of homotopy invariants and algebraic topology can be seen as the study of the construction and the properties of homotopy invariants.

In this lecture we will first recall the notion of the fundamental group of a space and study its connection to covering spaces. Then we will introduce the singular homology of a topological space and study it extensively. In the end, we will consider cohomology and homotopy groups and explore their relation to singular homology. We will also consider various applications of these invariants to topological and geometric problems.

Topology

Pure Mathematics
Elective
Mathematics
Concentration Module

Lecturer: Yuchen Bi
Language: in English

Time and place

Lecture: Di, Do, 12-14h, SR 226, Hermann-Herder-Str. 10, to be confirmed
Tutorial: 2 hours, date to be determined and announced in class

This course offers an introduction to differential geometry with a focus on the structure of smooth manifolds. Key topics include the construction and properties of vector fields, differential forms, and their applications. The course will also include an introduction to Riemannian metrics if time permits, though the treatment will remain at an introductory level.

Required: Analysis~I–III, Lineare Algebra~I and II \ Prior exposure to curves and surfaces (“Kurven und Flächen”) and topology is beneficial.

Pure Mathematics
Elective
Mathematics
Concentration Module

Lecturer: Stefan Kebekus
Assistant: Xier Ren
Language: in German

Time and place

Lecture: Di, Do, 8-10h, HS II, Albertstr. 23b
Tutorial: 2 hours, date to be determined and announced in class
Sit-in exam: date to be announced

Analysis I and II, Linear Algebra I

Pure Mathematics
Elective

Lecturer: Amador Martín Pizarro
Assistant: Charlotte Bartnick
Language: in English

Time and place

Lecture: Di, Do, 12-14h, SR 404, Ernst-Zermelo-Str. 1
Tutorial: 2 hours, date to be determined and announced in class

In this course the basics of geometric model theory will be discussed and concepts such as quantifier elimination and categoricity will be introduced. A theory has quantifier elimination if every formula is equivalent to a quantifier-free formula. For the theory of algebraically closed fields of fixed characteristic, this is equivalent to requiring that the projection of a Zariski-constructible set is again Zariski-constructible. A theory is called \(\aleph_1\)-categorical if all the models of cardinality \(\aleph_1\) are isomorphic. A typical example is the theory of non-trivial \(\mathbb Q\)-vector spaces. The goal of the course is to understand the theorems of Baldwin-Lachlan and of Morley to characterize \(\aleph_1\)-categorical theories.

necessary: Mathematical Logic \
useful: Algebra and Number Theory

Pure Mathematics
Elective
Mathematics
Concentration Module

Lecturer: Guofang Wang
Assistant: Florian Johne
Language: in German

Time and place

Lecture: Mo, Mi, 10-12h, HS II, Albertstr. 23b
Tutorial: 2 hours, date to be determined and announced in class

The aim of the calculus of variations is to minimise or maximise certain mathematically treatable quantities. More precisely, we consider \(\Omega \subset {\mathbb R}^n\) functionals or variation integrals of the form \[F (u) = \int_\Omega f(x,u (x ),Du (x))dx, \quad \hbox{ f\"ur } u : \Omega\to {\mathbb R}\] on \(\Omega \subset {\mathbb R}^n\).

Examples are arc length and area, as well as energies of fields in physics. The central question is the existence of minimisers. After a brief introduction to the functional analysis tools, we will first familiarise ourselves with some necessary and sufficient conditions for the existence of minimisers. We will see that compactness plays a very important role. We will then introduce some techniques that help us to get by without compactness in special cases: The so-called compensated compactness and the concentrated compactness.

necessary: Functional Analysis \
useful: PDE, numerical PDE

Pure Mathematics
Elective
Mathematics
Concentration Module

Lecturer: Abhishek Oswal
Assistant: Damian Sercombe
Language: in English

Lecture: Mo, 14-16h, SR 125, Ernst-Zermelo-Str. 1
Tutorial: 2 hours, date to be determined and announced in class

There is no information available yet.

There is no information available yet.

Pure Mathematics
Elective
Mathematics
Concentration Module

Lecturer: Maxwell Levine
Language: in English

Time and place

Lecture: Do, 14-16h, SR 226, Hermann-Herder-Str. 10
Tutorial: 2 hours, date to be determined and announced in class

Developments in artificial intelligence have boomed in recent years, holding the potential to reshape not just our daily routines but also society at large. Many bold claims have been made regarding the power and reach of AI. From a mathematical perspective, one is led to ask: What are its limitations? To what extent does our knowledge of reasoning systems in general apply to AI?

This course is intended to provide some applications of mathematical logic to the field of machine learning, a field within artificial intelligence. The goal of the course is to present a breadth of approachable examples.

The course will include a gentle introduction to machine learning in a somewhat abstract setting, including the notions of PAC learning and VC dimension. Connections to set theory and computability theory will be explored through statements in machine learning that are provably undecidable. We will also study some applications of model theory to machine learning.

The literature indicated in the announcement is representative but tentative. A continuously written PDF of course notes will be the main resource for students.

Background in basic mathematical logic is strongly recommended. Students should be familiar with the following notions: ordinals, cardinals, transfinite induction, the axioms of ZFC, the notion of a computable function, computable and computably enumerable sets (a.k.a. recursive and recursively enumerable sets), the notions of languages and theories and structures as understood in model theory, atomic diagrams, elementarity, and types. The concepts will be reviewed briefly in the lectures. Students are not expected to be familiar with the notion of forcing in set theory.

Pure Mathematics
Elective
Mathematics
Concentration Module

Lecture: Mo, 12-14h, SR 404, Ernst-Zermelo-Str. 1
Tutorial: 2 hours, date to be determined and announced in class

This course provides an introduction to analytical methods in mathematical physics, with a particular emphasis on many-body quantum mechanics. A central focus is the rigorous proof of the stability of matter for Coulomb systems, such as atoms and molecules. The key question - why macroscopic objects made of charged particles do not collapse under electromagnetic forces - remained unresolved in classical physics and lacked even a heuristic explanation in early quantum theory. Remarkably, the proof of stability of matter marked the first time that mathematics offered a definitive answer to a fundamental physical and stands as one of the early triumphs of quantum mechanics.

Content:

• Mathematical background: \(L^p\) and Sobolev spaces; Fourier transform; \\
  
• Introduction to quantum mechanics and prototypical examples; \\
  
• Many-body quantum mechanics; \\
  
• Hamilton operator and its properties; Lieb-Thierring inequalities, electrostatic inequalities, Coulomb energy; \\
  
• Proof of Stability of Matter.

Analysis III and Linear Algebra are required. \
No prior knowledge of physics is assumed; all relevant physical concepts will be introduced from scratch.

Pure Mathematics
Elective
Mathematics
Concentration Module

Lecturer: Mikhail Tëmkin
Language: in English

Time and place

Lecture: Mo, 10-12h, SR 404, Ernst-Zermelo-Str. 1
Tutorial: 2 hours, date to be determined and announced in class

Real-world data is often given as a finite set of points in ℝⁿ, called a point cloud. Topological data analysis aims to extract features of a point cloud algorithmically. At its core, it is a pipeline of tools from pure mathematics. These tools are of fundamental theoretical importance, and many have practical applications of their own (which the course will briefly discuss). The tools span geometry (convex sets, Delaunay triangulation), topology (simplicial and chain complexes, homology), and algebra (quivers). The course provides a thorough introduction to them and culminates by assembling them into persistent homology, the main object of study in topological data analysis. Although targeted at students in the “Mathematics in Data and Technology” program, it may also interest pure mathematicians because of the close interplay between the two areas.

Linear Algebra

Pure Mathematics
Elective
Mathematics
Concentration Module