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 was a first meeting on Tuesday, Nov 21, 10:00 in MA 621, where mathematical requirements, the organization of the projects etc were 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.

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

Monday, Mar 19 | 14:30-16:00 | Joswig | h-vectors of polytopes and spheres |

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

Tuesday, Mar 20 | 9:00-10:30 | Januszkiewicz | Symplectic geometry, hamiltonian group actions, convexity theorems |

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

14:30-16:00 | Joswig | The g-theorem | |

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

Wednesday, Mar 21 | 9:00-10:30 | Januszkiewicz | Goresky-MacPherson-Kottwitz and other cohomology computations |

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

14:30-16:00 | Joswig | Computing face lattices and f-vectors | |

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

Thursday, Mar 22 | 9:00-10:30 | Januszkiewicz | Singular symplection structures and group actions: symplectic origami and related singularities |

11:00-12:30 | Exercises |

Monday, Nov 27 | 9:00-10:30 | Januszkiewicz | General strategy and simple examples |

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 | Right angled Coxeter groups | |

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 | Topological toric manifolds |

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

14:30-16:00 | Joswig | Even 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 | Right angled buildings | |

16:30-18:00 | Exercises |

- Exercise Sheet (to be continuously updated)
- Jupyter notebook for polymake demo [PDF]

- 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
- Fulton: Introduction to toric varieties, Princeton 1993.
- 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

- Bahri, Bendersky, Cohen & Gitler: The polyhedral product functor: a method of decomposition for moment-angle complexes, arrangements and related spaces. Adv. Math. 225 (2010), no. 3, 1634-1668.
- 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)
- Park, Park & Park: Graph cubeahedra and graph associahedra in toric topology, arXiv:1801.00296