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Hinweis

There will be no exercise sheet for the coming week. During the lecture on Tuesday, Feb 5, Marie and Albert will review important topics from the lecture so far and take a look at one or two problems. If questions arise as you study, you are welcome to email us or ask them during the lecture on Tuesday. Alternatively, Marie and Albert will offer a Q&A after the lecture on Tuesday. Good luck studying. We hope that you have had fun so far and we are confident you will do well on the exam!

If these times don't suit you, Marie will be at her office (basement of the Villa, room 007) Monday 12-6 and Tuesday from the end of Q&A until 6.

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The written exam will take place on Wednesday, February the 5th in the lecture room, Arnimallee 6 Room 007/008 and will last for 90 minutes. It will start at 10.15 AM and end at 11.45 AM. You will register for the exam simply by showing up on the 5th and giving us your name and student ID. Please arrive at 10 AM so you can register and be seated in time.

You are allowed to take a small one-sided handwritten cheat sheet (about the size of German A5) with you to the exam. Except for pen and paper, all other materials as well as phones and laptops etc. are not allowed. Professor Ziegler will give some details about the nature of the exam during the lecture on January 21.

If you decide not to take the exam or if you don't pass it (which hopefully won't happen) you may make an appointment after February 11 with Elke Pose for an oral exam with Professor Ziegler. Keep in mind that if you don't take part in the first exam, the oral exam will be your only chance. Oral exam dates will probably be at or right before the beginning of the next term. A note for FU Bachelor's students: you may take both exams and choose the better grade as stated in the your Prüfungsordnung. This does not apply to Master's or BMS students.

 

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Congratulations to all who took the written exam. You did very well! For those of you who may repeat it, slots for oral exams are available in the first week of the summer term, on April 16:

  • 9:00 - 11:15
  • 14:00 - 16:15

Oral exams will last for about 30 minutes and are held by Prof. Ziegler and a co-examiner. Please email Elke Pose to register. Keep in mind that your registration is binding. That means if you register and don't show up, we will count it as a failed attempt.

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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.

We will cover a selection of the following topics:
Basic structures in discrete geometry
  • Polyhedra and polyhedral complexes
  • Configurations of points, hyperplanes, subspaces
  • Subdivisions and triangulations (including Delaunay and Voronoi)
  • Examples and Problems
Combinatorial geometry / Geometric combinatorics
  • 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)
Polytope theory
  • 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
Examples, examples, examples
  • Regular polytopes, centrally symmetric polytopes
  • Extremal polytopes, cyclic/neighborly polytopes, stacked polytopes
  • Combinatorial optimization and 0/1-Polytope
Geometry of linear programming
  • Linear programs, simplex algorithm, LP-duality

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