VL: Discrete Geometry II (Winter 2025/26)
This is a BMS Area 5 Core Course.
It continues Discrete Geometry I.
The exercise (UE) and tutorial (Tut) sessions will be given by Marcel Wack.
| VL: | Tue | 10-12 | MA 141 |
| Thu | 10-12 | MA 841 |
| UE: | tba | tba | tba |
| Tut: | tba | tba | tba |
Contents
This course will discuss more advanced topics in polyhedral geometry, with a view toward plane algebraic curves.
This will also include a first glimpse of tropical geometry.
Prerequisites: Discrete Geometry I (linear optimization, polytopes, convex hull algorithms).
While the bulk of the necessary algebra will be explained in the course, some knowledge of basic undergraduate algebra helps (e.g., it is useful to know that ideals in polynomial rings are finitely generated).
- Regular subdivisions of point configurations
- Tropical hypersurfaces
- Tropical plane curves
- Plane algebraic curves, over various fields
- Real plane algebraic curves
- Secondary cones and secondary fans
References
- De Loera, Rambau and Santos: Triangulations. Springer, 2010.
- Gelfand, Kapranov and Zelevinsky: Discriminants, resultants, and multidimensional determinants. Springer, 1994.
- Joswig: Essentials of tropical combinatorics. AMS, 2021.
- Joswig and Theobald: Polyhedral and algebraic methods in computational geometry. Springer, 2013.
- Lang: Undergraduate algebra, 3rd ed. Springer, 2004.
- Maclagan and Sturmfels: Introduction to tropical geometry. AMS, 2015.