Module 24-B-TG Topology and Geometry

Non-official translation of the module descriptions. Only the German version is legally binding.

Students master the basic concepts of set-theoretical topology, i.e. they are able to use these concepts in a technically correct way and thus gain a connection to spatial visualisation for many initially abstract and vague problems. They can use their extended spatial visualisation skills to carry out mathematical proofs independently. They can deal with various geometric objects of central importance in a technically appropriate manner and thus have the basic knowledge and skills required in in-depth courses on algebraic geometry, algebraic topology, differential geometry, global analysis, functional analysis, algebra, number theory and mathematical physics. They are confident in applying the methods of geometry and topology and can successfully transfer them to new problems in geometry and topology.

In the tutorials, students demonstrate the acquisition of skills in the basic techniques of mathematical work in the field of topology and geometry, the ability to apply the methods and carry out mathematical proofs under supervision as well as presentation and communication skills and perseverance as basic mathematical skills in through the study requirements. Further understanding of the contexts and concepts, independent proofs and confidence in applying the methods to new problems are demonstrated in the final exam.

1. Allgemeine Topologische Eigenschaften: Metrische und topologische Räume, stetige Abbildungen, Vergleich von Topologien (gröber, feiner), Kompaktheit, Trennungsaxiome, Zusammenhang, Satz von Tychonov, Produkttopologie, Summen- und Quotiententopologie.
(Optional: Funktionenräume, Sätze von Urysohn und Tietze, Zerlegung der 1, Kategorien und universelle Eigenschaften)

2. Mannigfaltigkeiten, Differenzierbarkeit, Beispiele: Projektive Räume, Grassmann'sche.
(Optional: Tangentialbündel, Lie Gruppen und Homogene Räume, Vektor- und Faserbündel, Garben)

3. Überlagerungen, Hochhebungseigenschaft, Homotopien, Fundamentalgruppe, Klassifikation von Überlagerungen, Seifert-van Kampen.
(Optional: Details der Galois-Korrepondenz, i.e. Galois-Überlagerungen, Automorphismen vs Normalisatoren, u.s.w., Orientierungen von Mannigfaltigkeiten, Graphen und freie Gruppen, Browersche Fixpunktsatz und Fundamentalsatz der Algebra)

Kompetenzen der fachlichen Basis in Analysis und Linearer Algebra (24-B-MG1, 24-B-MG2) sowie je nach gewählter Vorlesung weitere Kompetenzen

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