Dr. Claudio Paganini

This lecture is a continuation of the differential geometry I course held in WiSe 25/26.
The main focus of the course is Lorentzian geometry, with particular emphasis on structures and phenomena relevant to general relativity. Starting from Lorentzian metrics and geodesics, the course develops the theory of causal structures, including causality conditions, and global hyperbolicity. The geometric formulation of the Einstein field equations is introduced, and their implications are explored. Accordingly, the Lorentzian part of the course culminates in the study of singularity theorems. An excursion into black holes will be added if time permits. 
A secondary part of the course is devoted to symplectic geometry. We will introduce the basic definitions and study some obstructions to the existence of symplectic structures. We aim to understand the constraints symplectic geometry imposes on the underlying manifold. 

In this course we will treat the basics of differential geometry following the script of Prof. Christian Bär.

Lectures (Lecturer: Dr. Claudio Paganini): 

Monday: 16:00-17:30  Room M102

Tuesday: 12:30-14:00 Room M101

Exercises (TA: Marco van den Beld Serrano): 

In this course we experiment with a new form of exercise sessions. 

As usual there will be weekly exercise sheets. These will be solved in teams of two to three. However, instead of the usual exercise class, each team will be assigned a slot of ~30-45 minutes with the TA where one of the team has to explain the (partial) solutions you obtained. 

The timeslots will all be on Thursday between 10:00-14:00. (10-12 in PHY 5.1.01, 12-14 in M 201)

The assignment of the timeslots takes place on Monday October 13. in the first lecture.

Exam qualification requirement: satisfactory explanation of 10 solutions throughout the semester. 

Exam: 30min oral exam. 

Der Kurs führt die Grundbegriffe der Linearen Algebra ein und wir üben den Umgang mit diesen Objekten an vielen Beispielen. Der Fokus ist dabei hauptsächlich auf den Anwendungen, wie sie danach z.B. in der Analysis 2 gebraucht werden. 

Der Kurs besteht aus vier Vorlesungen a 90 Minuten, jeweils am 

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Dienstag 9. April, 10:00-11:30, im H31 

Vektorräume

• Definition
• linear abhängige/unabhängige Vektoren
• Span
• Basis
• Dimension
• Unterräume

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Mittwoch 10. April, 10:00-11:30, im H31

Vektorraum Homomorphismen

• Homomorphismus (Isomorphismus, Endomorphismus, Automorphismus)
  
• lineare Abbildungen
• Bild und Kern

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Donnerstag 11. April, 10:00-11:30, im H31

Matrizen

• Matrix Produkt
• Ähnliche Matrizen
• Spur
• Determinante
• Eigenwerte

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Freitag 12. April, 10:00-11:30, im H31

Abbildungen nach K

• Dualraum
• Skalarprodukt
• Norm
• Orthonormal Basis

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Es wird jeden Tag ein Übungsblatt geben. Diese Übungen besprechen wir jeweils 

Dienstag 9. April, 16:00-17:00, im H31

Mittwoch 10. April, 16:00-17:00, im H31

Donnerstag 11. April, 16:00-17:00, im H31

Freitag 12. April, 16:00-17:00, im H31