CGC Prague-Berlin DocCourse, Part II --- TU Berlin, Summer term 2004



Discrete Geometry
(Polytopes and More)

Prof. Günter M. Ziegler

Fakultät II
TU Berlin
Institut für
Mathematik

On Wednesday June 9 we have constructed a mind map that summarizes the content of the first half of the term:
Mind Map

Last session Wednesday July 14, 2004

This this the Homepage of the lecture "Kombinatorische Geometrie I: Diskrete Geometrie", summer/spring term 2004 at the TU Berlin, and the Prag-Berliner "DocCourse"-Program. The lecture will be continued in the fall by the lecture "Kombinatorische Geometrie II: Algorithmische Geometrie".
If you are looking for older content of this page, check here.

Contents

  1. Examples: 3-dimensional polytopes, products, cyclic polytopes, regular polytopes, hypersimplices, 0/1-polytopes, ...
  2. The representation theorem, polarity, face lattices and their combinatorics; the models of Eulerian lattices and strongly-regular cellular spheres
  3. 3-Dimensional polytopes, Steinitz' theorem, circle packings
  4. 4-Dimensional polytopes, Schlegel diagrams (and their pitfalls); fatness and complexity, examples
  5. f-Vectors and flag vectors; Shellability, the Dehn-Sommerville equations, the upper bound theorem, and the g-theorem

    (3 weeks break)

  6. Duality theory: Gale diagrams, non-rational polytopes; Rigidity and the lower bound theorem
  7. Hyperplane arrangements and point configurations
  8. Tilings, Delaunay triangulations, Voronoï diagrams, lattices; Complexity bounds: Incidence bounds, and the zone theorem
  9. High-dimensional aspects: Rounding, measure concentration, ...

Excercises:

The problem sets are handed out on Tuesday. Homework is to be handed in until Wednesday the following week.

20.04.2004 First Problem Set ps-file pdf-file polymake excercises
27.04.2004 Second Problem Set ps-file pdf-file polymake excercises
03.05.2004 Third Problem Set ps-file pdf-file polymake excercises
10.05.2004 Fourth Problem Set ps-file pdf-file polymake excercises
17.05.2004 Fifth Problem Set ps-file pdf-file polymake excercises
07.06.2004 Sixth Problem Set ps-file pdf-file
14.06.2004 Seventh Problem Set ps-file pdf-file
21.06.2004 8. Übungsblatt ps-file pdf-file
28.06.2004 9. Übungsblatt ps-file pdf-file

Schedule:

Lecture Tuesday 9:00 - 12:00 MA 313/314 Prof. Günter M. Ziegler
Übung Tuesday 14:15 - 15:45 MA 313/314 Dagmar Timmreck
Supervised Computer Experiments Tuesday 16:15 - 17:45 MA 313/314 Thilo Schröder and Niko Witte
Lecture Wednesday 9:00 - 12:00 MA 313/314 Prof. Günter M. Ziegler
Tutorium Wednesday 14:15 - 15:45 MA 313/314 Dagmar Timmreck
Work on Projects etc Wednesday 16:15 - 17:45 MA 313/314 Thilo Schröder and Niko Witte

First phase of lectures (5 weeks): April 20 to May 19.

Each DocCourse participant will have a project to work on after this. (Optional, but recommended, for TU Berlin students.)

Optional second phase of lectures (4 weeks) June 15 to July 7.

Requirements for scholarship holders

There are two kinds of scholarship holders participating in the course: The five predoc scholarship holders and several doctoral scholarship holders from the Graduate College Combinatorics, Geometry, and Computing (CGC).

Predoc scholarship holders: You are required to do the exercises that are handed out to you every week (this refers to both courses); furthermore, you are required to do one project (which may by from one course or the other).

CGC scholarship holders: If you want this course to be counted as one of your block courses then you are required to either complete the exercises handed out every week or to do a project. This refers to both courses separately.

For details (in particular: topics of possible projects), please contact Günter M. Ziegler or Volker Kaibel.

Contact:

Lecturer: Prof. Günter M. Ziegler MA 628 314-25730 ziegler@math.tu-berlin.de
TU Assistant: Dagmar Timmreck MA 621 314-25753 timmreck@math.tu-berlin.de
DocCourse Assistant: Nikolaus Witte MA 621 314-25753 witte@math.tu-berlin.de
DocCourse Assistant: Thilo Schröder MA 621 314-25753 thilosch@math.tu-berlin.de
Secretary: Elke Pose MA 627 314-23354 pose@math.tu-berlin.de


References:

Software:

Computation of examples in Polymake, visualization based on JavaView.