Tropical geometry and the moduli space of Prym varieties

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Algebraic geometry is concerned with geometric objects that arise as solutions to polynomial equations. Such objects, known as algebraic varieties, are at the heart of many real-world problems and have been studied since the dawn of maths. In recent decades, it was observed that significant mileage may be gained by stripping away some of the geometry and focusing instead on combinatorial aspects of those object. This pivotal shift in perspective is the basis for tropical geometry and has recently led to major breakthroughs in the geometry of curves, enumerative geometry, combinatorics, mathematical physics, and number theory. The current proposal is concerned with a family of important algebraic groups known as Prym varieties, which play a key role in rationality questions for threefolds, construction of compact hyper-Kähler manifolds , and the birational geometry of the moduli of abelian varieties. Despite numerous attempts by various authors, it is not yet known how to fully construct a compact space classifying them, and what the structure of such a space would be. However, by appealing to tropical tools such as the recently discovered tropical Prym variety, the time is ripe to study Prym varieties through a combination of algebraic, non-archimedean, logarithmic, and combinatorial techniques