Lehrveranstaltungen

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Aktuelle Lehrveranstaltungen

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Preliminary Program

This lecture aims at students in the programs International Master in Mathematics, in Mathematics, and Mathematical Physics. It may also be interesting for students in the master program Physics.

If there are students of the International Master in Mathematics then the lecture will be offered in English. Otherwise the lecture will be in German. The lecture notes will be in English anyway. Please get in contact with me early enough then we can discuss and arrange the details.

This master course is a first introduction to the topics of differential geometry. We will discuss differentiable manifolds as geometric objects in an intrinsic approach. A particular emphasize will be put on the global calculus on manifolds, showing how the coordinate-based calculations can be minimized as far as possible. After manifolds, we discuss vector bundles as the next important ingredient in differential geometry. Integration on manifolds will be presented in two ways, using an orientation and without orientation. We will see the most important cohomology theories attached to manifolds. If time permits, we will also give a short introduction to Lie groups.

The lecture will have a second part in the following summer term: this will be most probably an introduction to geometric mechanics, symplectic and Poisson manifolds. Beside that, this lecture serves as the starting point of various other courses in the above master program: there will be seminars and RiGs where are a good understanding of differential geometry as provided in this course is mandatory. For students of mathematical physics, this lecture might be (depending on the lecturer) also a starting point for the Analysis und Geometrie klassischer Systeme. Of course, it can also serve as background to lectures in general relativity.

Differentiable manifolds
Vector bundles and their sections
Calculus on manifolds
Integration and cohomology
Lie-Groups, Lie algebras, and their actions

Lehrveranstaltungen

Prerequisites

We expect good knowledge from the bachelor courses in analysis and linear algebra. In particular, we will need some aspects of multilinear algebra and tensor products (which will be briefly recalled if necessary). The bachelor course Geometrische Analysis can be seen as a motivation: there one considers submanifolds of the euclidean space, now we treat manifolds intrinsically. However, this course will not be required. Finally, some basic knowledge in point-set topology will be useful: we will recall the relevant information if necessary.

Literature

The list of textbooks is rather long, there are many good texts in differential geometry. Many of the references should be seen as background information. At the beginning of the lecture we will point out some more details to particular texts.

Dates

Lecture: Most probably Wednesday 10-12 SE 30 and Thursday 10-12 SE 31.
Tutorial: Upon negotiations, not yet fixed.

Lehrveranstaltungen

Links

On WueCampus, there will be a course for this lecture. As soon as this is enabled, it will replace this homepage.

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Programme

This Research in Groups aims at master students in the programs International Master in Mathematics, Mathematics, and Mathematical Physics. It may also be interesting for students in the master program Physics.

If there are international students then the course will be offered in English. Otherwise the lecture will be in German. The student talks and the proceedings can be in a language of your choice (within reasonable bounds, as usual). The lecture notes will be in English anyway. Please get in contact with me early enough then we can discuss and arrange the details.

The course can be seen as a follow-up of the last semester course on Analysis and Geometry of Classical Systems. We will discuss the construction of gauge theories based on the usage of principal fiber bundles and associated bundles. Beside the construction of several relevant models of gauge theories we will see other applications of principal fiber bundles like characteristic classes and resulting invariants. In the WueCampus course you can find a more detailed preliminary program.

The second component will be a seminar by the students on more particular topics. I expect the participants to write a small proceeding-like summary of their seminar talks. Details on the topics can be found in the WueCampus course.

Many of the topics can also be shared by two or more students. The precise content of the talks and the corresponding literature will be explained individually.

Prerequisites

This RiG is a continuation of the last semester lecture. It will be necessary to have some background in differential geometry, in particular concerning principal fiber bundles, connections and their curvature. Background on the physics of gauge theories might help to understand the motivation but is not strictly necessary.

If you are in doubt, please contact me directly and we will find a solution. We will explain the necessary things either in the lecture part or directly.

Literature

A general list of references can be found in the WueCampus course. This will be discussed in the first meeting. More references will be given individually.

Dates

If there are collisions with other lectures or seminars, please contact me early: maybe one can still shift things around a bit.

Lecture: Friday 10-12 SE 31.
Seminar: The seminar will take place at one or two days at the end of the semester. The precise date will be announced.

Lehrveranstaltungen

Links

On WueCampus there will be a homepage for this RiG. The WueCampus homepage will replace this site as soon as it is activated.

Zukünftige Lehrveranstaltungen

Vorlesung mit Übungen
Dozent: Prof. Dr. Stefan Waldmann

Inhalt

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