## VL: Discrete Geometry III: Tropical Geometry (Summer 2024)

This is a BMS Advanced Course, which will thus be given in English.
It is accompanied by a seminar, for those who are interested.

VL: | Tuesday | 10-12 | MA 549 |

| Thursday | 10-12 | MA 650 |

### Contents

We will give an introduction to tropical geometry, assuming a solid background in polyhedral geometry (as taught in Discrete Geometry I+II).
Knowing bits and pieces about topology and algebraic geometry does not hurt.
Two topics will be discussed in greater detail: plane and abstract tropical curves, tropical linear spaces and their connection to matroid theory.
My goal is to make the connection to research topics of recent interest.
In particular, work of Chan, Galatius and Payne on the topology of moduli spaces of tropical curves, and of Adiprasito, Huh and Katz on combinatorial Hodge theory.

- Tropical hypersurfaces
- Tropicalization
- Moduli of tropical plane curves
- Stable tropical curves
- Matroids and tropical linear spaces
- The Heron-Rota-Welsh Conjecture

### References

- Adiprasito, Huh and Katz: Hodge theory for combinatorial geometries, Ann. Math. 188 (2018).
- Ardila and Klivans: The Bergman complex of a matroid and phylogenetic trees, J. Combin. Th., Series B (2006).
- Baker, Payne and Rabinoff: Nonarchimedean geometry, tropicalization, and metrics on curves, Algebr. Geom. 3 (2016).
- Brodsky, Morrison, Joswig and Sturmfels: Moduli of tropical plane curves. Res. Math. Sci. 2.4 (2015).
- Bunnett, Joswig and Pfeifle: Generalised cone complexes and tropical moduli in polymake, Proc. ISSAC 2023.
- Chan: Combinatorics of the tropical Torelli map Algebra Number Theory 6 (2012),
- Chan, Gelatius and Payne: Tropical curves, graph complexes, and top weight cohomology of M
_{g}, J. Amer. Math. Soc. 34 (2021).
- De Loera, Rambau and Santos: Triangulations, Springer 2010.
- Joswig: Essentials of tropical combinatorics. AMS 2021.
- Maclagan and Sturmfels: Introduction to tropical geometry. AMS 2015.