## Polyhedral Methods in Algebraic Geometry

**Christian Haase**

class times: Tue 10:00-11:30, 12:30-14:00

**Coming up**: Prüfungskolloquium Monday, August 26, room A3.024

You pick a topic (see below for examples) which you will present during our conference. The presentation will be followed by a discussion/questions which are not restricted to the specific topic. I will then write down a grade. This grade can be improved by active participation in the discussion during the other talks.

Possible topics: proofs, proof strategies or techniques left out in class; worked example of a result mentioned in class; a result or an application related to but not treated in class, ...

-- Gröbner walk algorithm

-- Boolean rings and VLSI design

-- path signature varieties

-- toric phylogenetic models (group based models)

-- proof of Kapranov's Theorem

-- Bernstein's Theorem via homotopy continuation

-- algorithmic complexity of mixed volume and mixed cells

-- polyhedral intersection theory (toric or tropical)

-- matroids, Grassmannians and Bergman fans

-- space of trees and M_0,n

-- toric patches in geometric modeling

Time | Speaker | Topic |
---|---|---|

10h00 | Hannah | Gröbner walk |

10h40 | Yumeng | toric phylogenetic models |

11h20 | Marie | signed tropical numbers |

12h00 | -- Lunch -- | |

13h40 | Karin | toric Cartier divisors as polytopes |

14h20 | Xiangying | hyperplane arrangements, Bergman fan |

15h00 | Kemal | Grassmannians and the space of trees |

15h40 | -- Pause -- | |

16h00 | Lena | Okounkov bodies -- general and toric |

16h40 | Niklas | toric intersection theory |

17h30 | -- Kaltgetränke at Alter Krug -- |

**Past:**

04/09 | 0. Overview I. Algebraic geometry primer |

04/16 | I.2 Gröbner basics, Hilbert function, dimension, degree II. Polyhedral primer |

04/23 | II.2 Point lattices III. Toric Varieties |

04/30 | III.4 generation of toric ideals (Markov bases) |

05/07 | III.5 regular subdivisions and toric Gröbner bases |

05/14 | III.5 Gröbner fan, state polytope, secondary polytope |

05/21 | III.6.1 Polytopes with regular unimodular triangulation III.6.2 Hierarchical statistical models, Conditional independence III.6.3 Group based phylogenetic tree models |

05/28 | III.6.4 Path signature varieties IV. Tropical geometry |

06/04 | Problems for the algebraic geometry semester |

06/11 | IV.1 Tropical arithmetic |

06/18 | IV.3 Tropical varieties -- multiplicities and balancing |

06/25 | IV.4 Bernstein's Theorem and Seminar |

07/02 | IV.5 Grassmannians, matroids, trees III.7 Moment map of toric varieties |

07/09 | Another interesting seminar @HU involving toric geometry. Please attend! |

This course has (at least) two target audiences: on the one hand, this course is for students of algebraic geometry who are curious to see their field interact with polyhedral geometry, and to even see it applied to statistics, economics and optimization. On the other hand, this course is for students of discrete geometry, optimization or combinatorics, from numerical analysis, stochastics or other fields who want to get a glimpse of what algebraic geometry is about, and see examples of applications to their favorite objects.

The goal is to provide background and to get students up to speed so that they will be able to participate actively in a project group of the anticipated Thematic Einstein Semester Varieties, Polyhedra, Computation during the winter term 2019/20.