Tropical Hurwitz loci

计数几何是代数几何的一部分，它计算满足一定条件的几何对象，例如通过指定重数的给定点的平面曲线。这些问题通常很容易表达，但很难用基本的技术解决。一种通常成功的方法是将一个枚举几何问题转化为一个适当模空间上的交集问题。这就是为什么计数几何在推动代数几何的新发展以及与其他领域如辛几何或弦理论的联系方面取得了非常丰硕的成果。热带几何是一个最近迅速发展的领域，在这个领域中，代数几何对象退化为某些分段线性组合对象，称为热带簇。尽管退化了，许多代数几何不变量仍然延续到了热带世界。在这些情况下，热带几何是研究代数几何问题的有效计算工具。它还与其他领域有联系，如最优化或生物数学。热带几何学在研究列举问题方面特别成功。热带计数几何领域是由米哈尔金开创的，他的著名对应定理将平面曲线的经典数与其对应的热带曲线联系起来。这一建议建议从计数几何和热带几何的相互作用中研究问题，以期得到这两个领域的结果，并更深入地了解它们之间的联系。主要感兴趣的对象是称为(双重)Hurwitz轨迹的投影线的分支覆盖的轨迹。零维Hurwitz轨迹被经典地称为Hurwitz数。Cavalieri，Johnson和我已经引入了双Hurwitz数的热带类比。在固定分支数据的项中，双Hurwitz数具有有趣的分段多项式结构。热带方法有助于发现这种分段多项式结构的新特征。在这个建议中，我建议将图像扩展到某些分支覆盖的曲线的模空间中的高维循环。我打算研究经典的和热带的-Hurwitz轨迹，它们的分段多项式结构和跨越墙的公式。双轨方法-经典的和热带的-有两个目的：第一，我们可以使用这两种方法的全部机制来获得最大结果。其次，我希望通过对赫维茨轨迹理论的两个方面的粗略研究，更多地了解古典几何和热带几何之间的联系。此外，对Hurwitz轨迹及其与热带对应位置的联系的研究将推动热带模空间及其交集理论的研究，并有助于对双Hurwitz轨迹的交叉公式有更多的几何理解。

Enumerative geometry is a part of algebraic geometry in which one counts geometric objects satisfying certain conditions, e.g. plane curves passing through given points with assigned multiplicities. These questions are typically easy to formulate, but hard to solve with elementary techniques. An often succesful approach consists in translating an enumerative geometric question into an intersection problem on some appropriate moduli space. This is a reason why enumerative geometry has been very fruitful in pushing new developments in algebraic geometry and in making connections to other fields such as symplectic geometry or string theory.Tropical geometry is a recent and quickly growing field in which algebro-geometric objects are degenerated to certain piecewise linear combinatorial objects called tropical varieties. In spite of the degeneration, many algebro-geometric invariants carry over to the tropical world. In these cases tropical geometry is an effective computational tool to study problems in algebraic geometry. It also has connections to other fields such as optimization or biomathematics.Tropical geometry had particular success in the study of enumerative questions. The field of tropical enumerative geometry was pioneered by Mikhalkin with his celebrated Correspondence Theorem relating classical numbers of plane curves to their tropical counterparts.This proposal suggests to study questions at the interplay of enumerative geometry and tropical geometry, aiming at results in both fields as well as at a deeper understanding of their connections. The main object of interest are loci of ramified covers of the projective line called (double) Hurwitz loci. A 0-dimensional Hurwitz locus is classically known as Hurwitz number.Tropical analogues of double Hurwitz numbers have been introduced by Cavalieri, Johnson and myself.Double Hurwitz numbers have an interesting piecewise polynomial structure in the entries of the fixed ramification data. The tropical approach was helpful to discover new features of this piecewise polynomial structure.In this proposal, I suggest to extent the picture to higher-dimensional cycles in the moduli space of curves parametrizing certain ramified covers. I intend to study - classical and tropical - Hurwitz loci, their piecewise polynomial structure and wall-crossing formulas.The double-tracked approach - classical and tropical - serves two purposes: first, we can use the whole machinery of both methods to gain maximal outcome. Second, I hope to learn more about the connection between classical and tropical geometry by thouroughly investigating the two sides of the theory of Hurwitz loci. In addition, the study of Hurwitz loci and their connections to their tropical counterparts will advance the study of tropical moduli spaces and their intersection theory in general and might contribute to a more geometric understanding of the wall-crossing formulas for double Hurwitz loci.