The PI has accepted an offer for a position as a Junior Professor at Goethe University Frankfurt am Main. That is why he has decided to terminate his Marie-Curie Fellowship after only 6 months and many of the original goals have only been partially reached.
Let us give a quick summary of the progress so far:
Regarding (1):
- Significant progress has been made on the goal of giving a stack-theoretic treatment of the universal tropical Picard variety. The PI and his collaborators, Samouil Molcho (Hebrew University), Margarida Melo (Roma Tre University), Filippo Viviani (Roma Tre University), and Jonathan Wise (University of Colorado), are currently writing up their results and a preprint will be available shortly.
- Together with Yoav Len (Georgia Institute of Technology), the PI completed a project on the tropical and non-Archimedean geometry of Prym varieties. The foundational techniques developed in this project, allowed them to proof new upper bounds on the dimensions of Prym-Brill-Noether loci for double covers of special curves, expanding on the classical work of Welters and Bertram. A preprint, with the title “Skeletons of Prym varieties and Brill-Noether theory” is already available: arXiv:1902.09410.
Regarding (2):
- Before the start of his fellowship the PI has already completed a project on the realizability problem for tropical canonical divisors, in joint work with Martin Moeller and Annette Werner (both at Goethe University of Frankfurt). Since the PI has accepted a position at Goethe University, the PI has suspended his work on that aspect of the project until he is in Frankfurt.
- Together with Madeline Brandt (University of California, Berkeley), the PI completed a project that investigates the tropical and non-Archimedean geometry of symmetric powers of algebraic curves using the moduli space of effective divisors, the PI had constructed earlier in the project with Moeller and Werner. A preprint, titled “ Symmetric powers of algebraic and tropical curves: a non-Archimedean perspective”, is available: arXiv:1812.08740.
- Together with Dmitry Zakharov (Central Michigan University), the PI is currently completing the final write-up on the tropical geometry of the double ramification locus. A preprint will be available soon.
Regarding (3):
The original idea for Project (3), as described in the proposal, needed significant refinement. During the fellowship the PI found several collaborators to investigate these refined questions:
- Together with Yoav Len (Georgia Institute of Technology) and Dmitry Zakharov (Central Michigan University) the PI is currently working on a classification of finite abelian covers of tropical curves in terms of a stratified cohomology group. A preprint will be available soon.
- Together with Mattia Talpo (Imperial College), the PI has outlined an approach to the theory of specializing fundamental groups from algebraic to tropical curves. This appears to have unexpected applications to tropical Hurwitz theory and the construction of logarithmic compactifications of moduli spaces of curves with Teichmueller level structures. Work on a preprint has been started, but will still take some time until completion.
