TA: Sandro Roch

VL: | Monday | 10-12 | (MA 141) |

VL: | Tuesday | 14-16 | (EB 202) |

Üb: | Thursday | 14-16 | (EW 203) |

Exercises will be posted weekly. The solutions will be discussed during the exercise sessions. The exercise sessions will also complement the lectures with more examples and results.

This is a BMS Basic Course, which will thus be given in English.

Please register via email to the course.

- Definitions of polytopes
- Face lattices of polytopes
- Graphs of polytopes
- Linear programming and the simplex algorithm

- Joswig and Theobald: Polyhedral and Algebraic Methods in Computational Geometry, Springer 2013.
- Ziegler: Lectures on Polytopes, Springer 1995.

- Thursday, July 28,
- Friday, July 29,
- Monday, August 1, and
- Tuesday, August 2.

A second round of exams will take place at the end of September. More precise information will be added here and shared via email with registered students.

- Week 1: April 25 - 28

Lectures: Introduction to the course. Linear, affine and convex hulls and related defintions. V-polytopes and their faces.

Exercises: Exercise Sheet 1 and Carathéodory's Theorem [Chapter 1.6, Zie]. - Week 2: May 2 - 5

Lectures: Separating hyperplanes and separation theorems [Appendix B, JT], and the faces of a polytope [Chapter 3.1.1 and 3.1.2, JT]

Exercises: Exercise Sheet 2 - Week 3: May 9-12

Lectures: H-representation of polytopes and the representation theorem of polytopes [Chapter 3.1.3, JT]. Introduction to face lattice and Combinatorial type of polytopes.

Exercises: Exercise Sheet 3 and Radon's and Helly's Theorems. - Week 4: May 16-19

Lectures: The face lattice of a polytope [Chapter 3.2, JT]. Important example: simplicies.

Exercises: Exercise Sheet 4 - Week 5: May 23-26

Lectures: duality and polarity [Chapter 3.3, JT]

No exercise session on Thursday, May 26. - Week 6: May 30-June 2

Lectures: Bounds on entries of f-vectors and Euler's formula [Chapter 3.4, JT].

Exercises: Exercise Sheet 6 and cyclic polytopes [Theorem 0.6, Zie] - Week 7: June 6-9

No lecture on Monday, June 6.

Lecture: Cyclic polytopes and the upper-bound theorem [Chapter 3.5, JT].

Exercises: Exercise Sheet 7 - Week 8: June 13-16

Lectures: Introduction to linear programming, graphs of polytopes and comments on Hirsch's conjecture [Chapter 4.1, JT] and [Chapter 3.2, Zie].

Exercises: Exericse Sheet 8 - Week 9: June 20-23

No lectures and exercise session. - Week 10: June 27-30

Lectures: Duality of linear programs and the simplex algorithm [Chapters 4.2 and 4.3, JT]

No exercise session on Thursday, June 30. - Week 11: July 4-7

Lectures: Balinski's Theorem [Thm 3.14, Zie], Kalai's simple way to tell a simple polytope from its graph [Thm 3.12, Zie], bound on the diameter of 0/1 polytopes [Thm 3.11, Zie]

Exercises: Exercise Sheet 10 - Week 12: July 11-14

Lectures: Simplex algorithm and how to compute a start vertex [Chapters 4.3 and 4.4, JT], few comments on convex hull computations.

Exercises: Exercise Sheet 9 - Week 13: July 18-21

Lecture and exercise session: More topics in discrete geometry.