This course (in English, as the first part of a three semester sequence) will be an introduction to the theory of Convex Polytopes, a core field of Discrete Geometry. The lectures will

• introduce basic facts about polytopes, with an emphasis on the methods that yield the results (Fourier-Motzkin eliminiation, Schlegel diagrams, shellability, Gale diagrams, and oriented matroids
• discuss important example and elegant constructions (cyclic and neighborly polytopes, zonotopes, Minkowski sums, permuathedra and associahedra, fiber polytopes, and the Lawrence construction)
• show the excitement of current work in the field.

The course will be based on Günter M. Ziegler's 1995 text book "Lectures on Polytopes". This book has quickly become a standard text book for this field - but it also has triggered a lot of research, many of the open problems from 1995 have been solved by now, so the book does not any more cover "the excitement of current work in the field" - however, the course will try to provide this. In parallel to the course we will revise and update the book, towards a 2016 second edition of "Lectures on Polytopes". You are invited to get involved in this: By criticizing the old text, suggesting improvements and additions, researching new references, contributing problems, etc.

Here is the list of chapters of the book, and thus (roughly) the course contents:

1. Introduction and Examples
2. Polytopes, Polyhedra, and Cones
3. Faces of Polytopes
4. Graphs of Polytopes
5. Steinitz' Theorem for 3-Polytopes
6. Schlegel Diagrams for 4-Polytopes
7. Duality, Gale Diagrams, and Applications
8. Fans, Arrangements, Zonotopes, and Tilings
9. Shellability and the Upper Bound Theorem
10. Fiber Polytopes, and Beyond

References:

• Branko Grünbaum: "Convex Polytopes" (Original edition: Interscience 1967; 2nd edition, Graduate Texts in Math. 221, Springer 2003) 
• Günter M. Ziegler "Lectures in Polytopes" (Graduate Texts in Math. 152, 1995; 7th revised printing 2007)