TropicalModuli · Foundations and applications of tropical moduli theory

Foundations and applications of tropical moduli theory

One of the central ideas of modern geometry is to not only study a geometric object by itself but also to understand it as naturally sitting in a moduli space parametrizing objects of the same type together with their degenerations. In fact, just like a prism disperses light into a spectrum of colors, in many cases moduli spaces allow us to access many a priori hidden properties of the original object. Tropical geometry is the geometry of the combinatorial objects associated to degenerations and compactifications of algebraic (or analytic) varieties. This explains why, as in algebraic geometry, the tropical geometry of moduli spaces is one of the richest and most fundamental parts of this field, with many of the features of tropical geometry only being visible through the prism of moduli spaces. The experienced researcher (in following referred to as the PI) proposed to systematically develop the foundations of tropical moduli theory, using the new stack-theoretic methods developed by him and his collaborators R. Cavalieri, M. Chan, and J. Wise, and to investigate applications to other parts of mathematics using techniques from logarithmic geometry in the sense Fontaine-Kato-Illusie. During the fellowship the PI intended to focus on the following three types of moduli spaces (largely out of reach without the use of tropical stacks), and to explore applications to classical problems in arithmetic and algebraic geometry: (1) the universal Picard variety, with a view towards applications to theta characteristics and spin structures, Prym varieties, and Brill-Noether theory; (2) the moduli space of k-differentials, with a focus towards a solution of Eliashberg's problem (in the case k=0) and the study of compactifications of strata of abelian differentials (in the case k=1) and of quadratic differentials (in the case k=2); and (3) the moduli space of G-admissible covers using the theory of graphs of groups.

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