Winter Term 2013/2014 - BMS Advanced Course

Welcome to the website! The lectures on Discrete Geometry I are taught by Günter M. Ziegler. If you have any questions, please ask them during class or send an email!

Those of you who were not registered with Campus Management for DG I can now pick up their "Scheine" (course certificates) in Elke Pose's office at Arnimallee 2.

# Course Description

This is the first in a series of three courses on Discrete Geometry. We will get to know fascinating geometric structures such as configurations of points and lines, hyperplane arrangements, and in particular polytopes and polyhedra, and learn how to handle them using modern methods for computation and visualization and current analysis and proof techniques. A lot of this looks quite simple and concrete at first sight (and some of it is), but it also very quickly touches topics of current research.

For students with an interest in discrete mathematics and geometry, this is the starting point to specialize in discrete geometry. The topics addressed in the course supplement and deepen the understanding of discrete-geometric structures appearing in differential geometry, optimization, combinatorics, topology, and algebraic geometry. To follow the course, a solid background in linear algebra is necessary. Some knowledge of combinatorics and geometry is helpful.

- Polyhedra and polyhedral complexes
- Configurations of points, hyperplanes, subspaces
- Subdivisions and triangulations (including Delaunay and Voronoi)
- Examples and Problems

- Arrangements of points and lines: Sylvester-Gallai, Erdös-Szekeres,
- Szemeredi--Trotter
- Arrangements, zonotopes, zonotopal tilings, oriented matroids
- Examples and Problems (Challenge problem: simplicial line arrangements)

- Representations and the theorem of Minkowski-Weyl
- Polarity, simple/simplicial polytopes
- Shellability, face lattices, f-vectors, Euler- and Dehn-Sommerville
- Graphs, diameters, and the Hirsch (ex-)conjecture

- Regular polytopes, centrally symmetric polytopes
- Extremal polytopes, cyclic/neighborly polytopes, stacked polytopes
- Combinatorial optimization and 0/1-Polytope

- Linear programs, simplex algorithm, LP-duality

# Contact

Contact | Office Hours: | ||

Lecture | Prof. Günter M. Ziegler | ziegler(at)math.fu-berlin.de | TBA |

Tutorial | Marie Litz | m.litz(at)fu-berlin.de | TBA |

Tutorial | Albert Haase | a.haase(at)fu-berlin.de | TBA |

# Lectures

| ||

TUE | 10:15 - 11:45 | Arnimallee 6 Room 007/008 |

WED | 10:15 - 11:45 | Arnimallee 6 Room 007/008 |

## Lecture Notes Etc.

Bear in mind that these lecture notes are 'preliminary'. There are no guarantees. If you find errors, please email us.

# Tutorials and Problems

Tutorials | ||

WED | 12:30 - 14:00 | Arnimallee 2, Room 001 |

WED | 14:15 - 15:45 | Arnimallee 6, Room 031 |

In addition to the lectures there will be two weekly (identical) tutorials. Please choose one and attend regularly. In the tutorials, we will occasionally review topics from the lectures but mostly discuss examples and solutions to problems. You are encouraged to pitch in by presenting a solution every once in a while.

Every week each student is required to solve problems and hand them in. The sheets containing the problems will be uploaded on Wednesdays and solutions should be turned in before the second lecture in the following week. Please bring them with you to the lecture on Wednesday and hand them to Professor Ziegler by 10:15 AM. You will receive points for your solutions based on whether your solutions are correct and well-written.

Course requirements are the following: (1) You must score at least 50% of the total points of the problems assigned in each half of the semester. There will be 7 problem sheets in the first half of the semester. A sheet will have problems worth roughly 20 points. Note that it is ok to score less than 50% on a sheet as long as you reach 50% of the total points by the end of the first respective second half of the semester. (2) You must pass a written exam at the end of the semester which alone will determine the grade that you get for this course.

## Problem Sheets

- Problem Sheet 1
- Problem Sheet 2 (TeX File)
- Problem Sheet 3 (TeX File)
- Problem Sheet 4 (TeX File)
- Problem Sheet 5 (TeX File)
- Problem Sheet 6 (TeX File)
- Problem Sheet 7 (TeX File)
- Problem Sheet 8 (TeX File)
- Problem Sheet 9 (TeX File)
- Xmas Bonus Sheet (TeX File)
- Xmas Bonus Sheet (some Solutions) (TeX File)
- Problem Sheet 10 (TeX File)
- Problem Sheet 11 (TeX File, Bibliography)
- Problem Sheet 11 (Solutions for the first exercise) (TeX File)
- Problem Sheet 12 (TeX File)