Real Hurwitz numbers

Hurwitz数计算具有固定次数和亏格且具有固定分支数据的黎曼球面的分枝覆盖。它们在几何学、组合学和数学物理学等不同的数学领域之间提供了有趣的联系。在这个提议中，我们关注的是赫维茨数的实类比。热带几何可以被视为一种退化技术，它将凸几何对象与保持许多重要性质的代数簇联系在一起。热带几何已成功地应用于计数几何中的问题，特别是Hurwitz理论中的问题和实际计数问题。在这个项目中，我们建议在现代Hurwitz理论的背景下对实Hurwitz数进行系统的研究。主要工具将是热带几何学。一方面，我们希望借助热带方法在实数列几何中得到新的结果，另一方面，我们也展望了热带几何领域的进展，使这一工具在Hurwitz理论中的应用变得更加尖锐。

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.