Tropical geometry and the moduli space of Prym varieties

代数几何研究的是作为多项式方程的解而出现的几何对象。这类对象被称为代数族，是许多现实世界问题的核心，自数学诞生以来就一直被研究。近几十年来，人们观察到，通过剥离一些几何图形，转而专注于这些对象的组合方面，可以获得显著的里程数。这一关键的视角转变是热带几何的基础，最近在曲线几何、计数几何、组合数学、数学物理和数论方面取得了重大突破。目前的建议涉及一族重要的代数群，称为Prym簇，它们在三重合理性问题、紧致超Kähler流形的构造以及交换簇的模的二元几何中起着关键作用。尽管不同的作者进行了多次尝试，但人们还不知道如何完全构建一个对它们进行分类的紧凑空间，以及这样一个空间的结构会是什么。然而，借助于最近发现的热带Prym品种等热带工具，通过结合代数、非阿基米德、对数和组合技术来研究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