Michael Joswig: VL Discrete Geometry II, WS25/26

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.

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 (polytopes [JT, Chap.3], linear optimization [JT, Chap.4], convex hull algorithms [JT, Chap.5]). 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).

Below, for each lecture, pointers to the relevant literature are given.

1. Regular subdivisions of point configurations

Tue, 14 Oct: basic definitions and examples [ETC, §1.2]
Thu, 16 Oct: placing triangulations [ETC, §1.2], [DLRS, §4.3.1]
Tue, 21 Oct: mother of all examples, secondary cones [DLRS, §5.2.1]
Thu, 24 Oct: beneath-and-beyond algorithm [BB]; fitness landscapes [EJLL19, EJLL23] [slides]

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

De Loera, Rambau and Santos: Triangulations. Springer, 2010. [DLRS]
Eble, Joswig, Lamberti and Ludington: Cluster partitions and fitness landscapes of the Drosophila fly microbiome. J. Math. Biol. 79.3 (2019) [EJLL19]
Eble, Joswig, Lamberti and Ludington: Master regulators of biological systems in higher dimensions. Proc. Natl. Acad. Sci. USA 120.51 (2023). [EJLL23]
Gelfand, Kapranov and Zelevinsky: Discriminants, resultants, and multidimensional determinants. Springer, 1994. [GKZ]
Joswig: Beneath-and-beyond revisited. In: Algebra, geometry, and software systems. Springer, 2003. [BB]
Joswig: Essentials of tropical combinatorics. AMS, 2021. [ETC] [addenda and errata]
Joswig and Theobald: Polyhedral and algebraic methods in computational geometry. Springer, 2013. [JT]
Lang: Undergraduate algebra, 3rd ed. Springer, 2004. [Lang]
Maclagan and Sturmfels: Introduction to tropical geometry. AMS, 2015. [MS]