From Riemann surfaces to tropical curves (and back again)

• Applicant: Professor Martin Ulirsch, Ph.D.
• Subject Area: Mathematics
• Term: since 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 456557832

Tropical geometry is the piecewise linear geometry of combinatorial degenerations and compactifications of algebraic and analytic varieties. The goal of this project is to significantly extend these methods in the case of Riemann surfaces both in its theoretical foundations and with a view towards applications in algebraic, arithmetic, and complex geometry. The central meta-problem of this project, which arises in many different incarnations, is the realizability problem. A particular focus will be on the following topics:- A tropical approach to Prym-Brill-Noether theory (a version of Brill-Noether theory that takes into account the natural symmetries of Riemann surfaces) and the tropical study of Prym varieties. - A theory of logarithmic linear series that interpolates between the classical theory of limit linear series and the their tropical counterpart.- The nascent connections between the geometry of flat surfaces and of tropical curves.In this project the perspective of non-Archimedean analytic geometry provides us with a manifold of techniques to study the process of tropicalization and the realizability problem. In addition to these techniques, the key new ingredient the PI is bringing into the field are the methods of logarithmic geometry in the sense of Fontaine-Kato-Illusie that allow us to consequently treat the geometry of tropical curves from a moduli-theoretic point of view.

• DFG Programme: Research Grants