Tropical geometry and the moduli space of Prym varieties

代数几何关注的是作为多项式方程的解而出现的几何对象。这种对象被称为代数变量，是许多现实世界问题的核心，自数学诞生以来就一直在研究。近几十年来，人们观察到，通过剥离一些几何形状，转而关注这些物体的组合方面，可能会获得显著的成果。这种观点的关键转变是热带几何的基础，最近在曲线几何、枚举几何、组合学、数学物理和数论方面取得了重大突破。目前的建议是关于一个家族的重要代数群被称为Prym变种，它在三倍的合理性问题，紧hyper-Kähler流形的构造，以及阿贝尔变种的模的两族几何中发挥关键作用。尽管许多作者进行了许多尝试，但如何完全构建一个对它们进行分类的紧化空间，以及这样一个空间的结构是什么，目前还不知道。然而，通过吸引热带工具，如最近发现的热带Prym品种，通过代数、非阿基米德、对数和组合技术的结合来研究Prym品种的时机已经成熟

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