I am currently thinking about Hurwitz numbers and how to define and maybe compute \(\mathbb{A}^1\)-versions of them.
Master's thesis - k-phase structures
In my Master's thesis, I generalized real phase structures, which are defined over the real numbers, to arbitrary fields. A k-phase structure
stores additional information to the valuation of a Puiseux-series, it collects the "signs" of the leading coefficients. It is closely related to
quadratic enrichments of tropical varieties. Roughly, they place the same information, the "signs" of the preimages under valuation,
on the variety or "between" the facets, resprectively. These satisfy a "balancing condition". You can find the PDF here.
There are a few new remarks and subsections compared to the text I handed in, the changes are noted in the beginning.
Bachelor's thesis - About the Tropicalization of the
Catalecticant and the Lüroth
Invariant
As the title suggests, in my Bachelor's thesis, I considered the Catalecticant and Lüroth invariant and studied how they behave
under tropicalization. In particular, I studied how their properties "invariant vanishes implies the polynomial has a property" behaved for their
tropicalization. The results were mostly negative. You can find my Bachelor's thesis here.
As this was my first contact with tropical geometry, this text also includes a short introduction into tropical geometry. Moreover, in the
appendix you can find a horribly inefficient MatLab code for plotting tropical curves. Over at coding, you can find
a way better and faster Python script for that.