This is a block course held jointly by the Institute for Mathematics of the Polish Academy of Science (IMPAN) and Berlin Mathematical School (BMS), organized by Tadeusz Januszkiewicz and Michael Joswig.

The course will be held one week in Berlin (Nov 27 - Dec 1, 2017) and one week in Bedlewo (Mar 19 - Mar 23, 2018). There will be lectures and exercises during these two weeks. For the time in-between there will be additional project work.

The course is also open for students of all three Berlin universities, who can get credits equivalent to a regular course "Discrete Geometry III" of TU Berlin. There will be a first meeting on Tuesday, Nov 21, 10:00 in MA 621, where mathematical requirements, the organization of the projects etc will be discussed.

It will be assumed that the participants have some basic knowledge in the following subjects: cellular homology and cohomology, polytopes and linear programming, smooth manifolds, group actions.

Topics include the following:

- Topological Toric Varieties, Reflection Groups and Buildings, Symplectic Manifolds, Moment Map
- Lattice Polytopes, Algebraic Toric Varieties, Polytope Combinatorics and f-Vectors, Algorithms
- Abramenko & Brown: Buildings - theory and applications, Springer 2008
- Buchstaber & Panov: Torus actions and their applications in topology and combinatorics, AMS 2002
- Buchstaber & Panov: Toric topology, AMS 2015
- Cox, Little & Schenck: Toric varieties, AMS 2011
- Davis: The geometry and topology of Coxeter groups, Princeton 2008
- Dold: Lectures on algebraic topology, reprint of the 1972 edition, Springer 1995
- Humphreys: Reflection groups and Coxeter groups, Cambridge 1990
- Joswig & Theobald: Polyhedral and algebraic methods in computational geometry, Springer 2013
- Stanley: Combinatorics and commutative algebra, 2nd ed., Birkhäuser 1996
- Warner: Foundations of differentiable manifolds and Lie groups, corrected reprint of the 1971 edition, Springer 1983
- Ziegler: Lectures on polytopes, Springer 1995
- Baralić & Živaljević: Colorful versions of the Lebesgue, KKM, and Hex theorem, J. Combin. Theory Ser. A (2017)
- Davis & Januszkiewicz: Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. (1991)
- Joswig: Projectivities in simplicial complexes and colorings of simple polytopes, Math. Z. (2002)

Personal application (via email) is mandatory since there is a limit on the number of participants.

Monday, Nov 27 | 9:00-10:30 | Januszkiewicz | tba |

11:00-12:30 | Exercises | ||

14:30-16:00 | Joswig | Affine Toric Varieties | |

16:30-18:00 | Exercises | ||

Tuesday, Nov 28 | 9:00-10:30 | Joswig | Projective Toric Varieties |

11:00-12:30 | Exercises | ||

14:30-16:00 | Januszkiewicz | tba | |

16:30-18:00 | Exercises | ||

Wednesday, Nov 29 | 9:00-10:30 | Kastner | polymake Demo: Polytopes and Toric Varieties |

11:00-12:30 | Exercises | ||

Thursday, Nov 30 | 9:00-10:30 | Januszkiewicz | tba |

11:00-12:30 | Exercises | ||

14:30-16:00 | Joswig | Sparse Colorings of Simple Polytopes | |

16:30-18:00 | Exercises | ||

Friday, Dec 1 | 9:00-10:30 | Joswig | A Colorful Lebesgue Theorem |

11:00-12:30 | Exercises | ||

14:30-16:00 | Januszkiewicz | tba | |

16:30-18:00 | Exercises |