This survey grew out of notes accompanying a cycle of lectures at the workshop Modern Trends in Gromov–Witten Theory, in Hannover. The lectures are devoted to interactions between Hurwitz theory and Gromov–Witten theory, with a particular eye to the contributions made to the understanding of the Double Ramification Cycle, a cycle in the moduli space of curves that compactifies the double Hurwitz locus. We explore the algebro-combinatorial properties of single and double Hurwitz numbers, and the connections with intersection theoretic problems on appropriate moduli spaces. We survey several results by many groups of people on the subject, but, perhaps more importantly, collect a number of conjectures and problems which are still open.

中文翻译：

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赫维兹理论与双重分支周期

这项调查是在汉诺威的Gromov-Witten Theory的现代趋势研讨会上进行一系列讲座的基础上得出的。这些讲座专门讨论Hurwitz理论和Gromov–Witten理论之间的相互作用，特别着眼于对理解Double Ramification Cycle的贡献，Double Ramification Cycle是曲线的模量空间中的一个循环，它压缩了双重Hurwitz轨迹。我们探索了单个和双Hurwitz数的代数组合性质，以及在适当的模空间上与交点理论问题的联系。我们调查了许多人对此主题的一些结果，但也许更重要的是，收集了一些尚待解决的猜想和问题。