Hurwitz数是计数几何中的经典对象，它和曲线模空间的几何以及对称群的表示论密切相关。借助于对称函数，Hurwitz数的生成函数可以写成很紧凑的形式，并且可以由此证明它满足cut-and-join方程。Hurwitz数与Gromov-Witten理论紧密相关，cut-and-join方程也被用来证明许多与Gromov-Witten理论相关的定理。受到 orbifold Gromov-Witten理论的影响，人们很自然的考虑加入一个有限群 G 的作用来推广Hurwitz数。一个自然的推广就是G-Hurwitz 数，它的生成函数满足colored cut-and-join方程。在本项目中，我们准备研究G-Hurwitz 数的各种性质以及colored cut-and-join方程的可能的应用，并找出单G-Hurwitz 数的镜像曲线。

Hurwitz numbers are classical objects in enumerative geometry, which relate the geometry of Riemann surfaces to the representation theory of symmetric groups. The generating series of Hurwitz numbers can be written in a quite neat form using symmetric functions. In this form, one is able to prove that it satisfies the cut-and-join equation. Hurwitz numbers are closely related to Gromov-Witten invariants, and the cut-and-join equation is used to prove many theorems in Gromov-Witten theory. Inspired by the development of orbifold Gromov-Witten theory, one is naturally led to consider the problem of generalizing Hurwitz numbers to the orbifold setting. A natural generalization is the so called G-Hurwitz numbers, whose generating function satisfies the various colored cut-and-join equations. In this project, we plan to study the various properties of G-Hurwitz numbers and the possible applications of the colored cut-and-join equations, and find the mirror curve of single G-Hurwitz numbers.