This is a BMS Advanced Course, which will thus be given in English (online). To participate registration via email is mandatory. In that email please give your name, your affiliation, and indicate whether you wish to receive credits (the latter is only possible for students of FU, HU and TU Berlin and BMS students).

VL: | Monday | 10-12 | video/riot |

Wednesday | 10-12 | video/riot | |

UE: | Tuesday | 12-124 | zoom |

Teaching assistant: Holger Eble

The course will be organized as follows: Each lecture will have a prerecorded and an interactive part. The videos (of about 45min each) will be available online, at the day of the lecture at midnight. Despite the fact that the videos will be kept and therefore can be watched any time, it is strongly recomended to watch the videos at the times suggested above. The interactive part of the lecture will begin at 11:00 (sharp) on Mondays and Wednesdays. For this we will use a riot/matrix chat server; details TBA.

The first lecture will be given live on zoom, on Monday, 2 Nov 2020, at 10:15.

Assuming a basic background in polytope theory, this course covers topics in polytopal combinatorics with a view towards applications to solving systems of polynomial equations. The prerequisites are a course Discrete Geometry I or equivalent: see the table of contents of last summer's lecture given by Mario Kummer.

Subject overview:

- graphs of polytopes: simple polytopes, Balinski's theorem
- lattice points and Ehrhart polynomials
- triangulations, regular subdivisions, mixed subdivisions and mixed volume
- secondary fans
- Theorems of Bernstein, Kushnirenko and Khovanskii
- a tiny bit of toric varieties and tropical geometry

Running BKK example:

- preliminary OSCAR ipynb [PDF]

- Beck and Robins: Computing the continuous discretely. UTM. Springer, 2007.
- Cox, Little, O'Shea: Ideals, varieties, and algorithms. Third edition. UTM. Springer, 2007.
- Cox, Little, O'Shea: Using algebraic geometry. Second edition. GTM, Springer, 2005.
- Dickenstein and Emiris (eds.): Solving polynomial equations, Springer 2005.
- De Loera, Rambau and Santos: Triangulations. Springer, 2010.
- Ewald: Combinatorial Convexity and Algebraic Geometry. Springer, 1996.
- Joswig and Theobald: Polyhedral and algebraic methods in computational geometry. Springer, 2013.
- Gelfand, Kapranov and Zelevinsky: Discriminants, resultants and multidimensional determinants. Reprint of the 1994 edition. BirkhĂ¤user, 2008
- Saito and Sturmfels: GrĂ¶bner deformations of hypergeometric differential equations. Springer, 2000
- Thomas: Lectures in geometric combinatorics. Student Mathematical Library, 33. IAS/Park City Mathematical Subseries. AMS, Providence, RI; Institute for Advanced Study (IAS), Princeton, NJ, 2006.
- Ziegler: Lectures on polytopes. GTM. Springer, 1995.