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.

Entry for this course in TU's ISIS

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

Tue, 28 Oct: tropical polynomials [ETC, §1.1]
Thu, 30 Oct: duality between regular subdivisions and tropical hypersurfaces [ETC, §1.2]
Tue, 04 Nov: max vs min; homogeneity [ETC, §1.4]; integer linear programming [ETC, §2.4]
Thu, 06 Nov: constant coefficients [ETC, §1.5]

3. Tropical plane curves

Thu, 06 Nov: unimodular triangulations of lattice polygons; Pick's theorem [BR, §2.6]
Tue, 11 Nov: examples of tropical plane curves [BJMS15]
Thu, 13 Nov: metric planar graphs from tropical plane curves [ETC, §1.6]

4. Algebraic plane curves, over various fields

5. Real plane algebraic curves

6. Secondary fans and tropical moduli spaces

References

Beck and Robins: Computing the continuous discretely. Springer, 2015. [BR]
Brodsky, Morrison, Joswig and Sturmfels: Moduli of tropical plane curves. Res. Math. Sci. 2.4., 2015 [BMJS15]
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]
Gathmann: Plane algebraic curves, Class Notes RPTU Kaiserslautern 2023 [G]
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]

Software

Gawrilow, Joswig and the polymake team: polymake; see also the article polymake: a Framework for Analyzing Convex Polytopes
Richard Morris: ImplicitPlot and SingSurf; see also the article A Client-Server System for the Visualisation of Algebraic Surfaces on the Web
The OSCAR team: OSCAR; see also the book The Computer Algebra System OSCAR