Tropical geometry is geometry over the tropical semiring, where multiplication is replaced by addition and addition is replaced by minimum. One can tropicalize algebraic varieties in this way to get polyhedral complexes that retain some information about the original varieties, and which can be combinatorially studied. The last few years have seen an effort to develop solid algebraic foundations for the field.
In this minicourse, I will start by introducing some of the basic tools and results in tropical geometry. I will also explain the central role that matroid theory plays in the field, particularly in the study of tropical linear spaces. With this, I will present the notion of tropical ideals -- introduced as a sensible class of objects for developing algebraic foundations -- and I will discuss work studying some of their main properties and their possible associated tropical varieties.
Some references:
- Diane Maclagan and Bernd Sturmfels, Introduction to tropical geometry, American Mathematical Society, 2021.
- Diane Maclagan and Felipe Rincón, Tropical schemes, tropical cycles, and valuated matroids, Journal of the European Mathematical Society, Vol. 22 (3), 2020.
- Diane Maclagan and Felipe Rincón, Tropical ideals, Compositio Mathematica, Vol. 154 (3), 2018.
-