Real Hurwitz numbers

Hurwitz数计算具有固定度和属且具有固定分支数据的黎曼球的分支覆盖。它们提供了各种数学领域之间的有趣联系，如几何、组合学和数学物理。在这个提议中，我们专注于赫维茨数的真实类似物。热带几何可以看作是一种退化技术，将凸几何对象与保留许多重要性质的代数变体联系起来。热带几何已成功地应用于枚举几何问题，特别是赫维茨理论问题和实枚举问题。在这个项目中，我们提出在现代赫尔维茨理论的背景下系统地研究真实的赫尔维茨数。主要的工具是热带几何。一方面，我们的目标是借助热带方法在实际枚举几何中获得新的结果，另一方面，我们也设想在热带几何领域取得进展，使这一工具在赫尔维茨理论中得到应用。

Hurwitz numbers count ramified covers of a Riemann sphere of a fixed degree and genus and with fixed ramification data. They provide interesting connections between various mathematical areas such as geometry, combinatorics and mathematical physics. In this proposal, we focus on real analogues of Hurwitz numbers. Tropical geometry can be viewed as a degeneration technique associating convex geometry objects to algebraic varieties that preserve many important properties. Tropical geometry has successfully been applied to problems in enumerative geometry, in particular to problems in Hurwitz theory and to real enumerative problems. In this project, we propose the systematic study of real Hurwitz numbers in the context of modern Hurwitz theory. The main tool will be tropical geometry. On the one hand, we aim at new results in real enumerative geometry with the aid of tropical methods, on the other hand, we also envision progress in the area of tropical geometry, sharpening this tool for its use in Hurwitz theory.