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: Wed 14-16 H 3008
Tut: Thu 12-14 MA 841

In the exercise session (UE) Marcel will discuss examples and homework. In the tutorial session (Tut) the students work out examples, prove small results, work with the computer; with Marcel present to answer questions. Both, UE and Tut, start in the second week of the semester.

New exercise sheet each Wednesday; to be discussed on the subsequent Wednesday.

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).

  1. Regular subdivisions of point configurations
  2. Tropical hypersurfaces
  3. Tropical plane curves
  4. Plane algebraic curves, over various fields
  5. Real plane algebraic curves
  6. Secondary cones and secondary fans

References

  1. De Loera, Rambau and Santos: Triangulations. Springer, 2010.
  2. Gelfand, Kapranov and Zelevinsky: Discriminants, resultants, and multidimensional determinants. Springer, 1994.
  3. Joswig: Essentials of tropical combinatorics. AMS, 2021.
  4. Joswig and Theobald: Polyhedral and algebraic methods in computational geometry. Springer, 2013.
  5. Lang: Undergraduate algebra, 3rd ed. Springer, 2004.
  6. Maclagan and Sturmfels: Introduction to tropical geometry. AMS, 2015.
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Last modified: Wed Oct 01 09:52:57 UTC 2025 by joswig