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。Hurwitz数的生成函数满足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理论的影响，人们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。利用玻色费米对应，我们可以把G-Hurwitz 数的生成函数写成一个算子的真空期望值，从而证明它是2-Toda可积方程簇的一个tau函数。本项目中，我们准备研究colored cut-and-join方程的进一步应用以及G-Hurwitz 数的chamber结构和wall-crossing公式。

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 satisfies the cut-and-join equation. Hurwitz numbers are closely related to Gromov-Witten theory, and the cut-and-join equation is used to prove many theorems in Gromov-Witten theory. Under the influence of orbifold theory, people try to generalize Hurwitz numbers by adding a finite group. One such generalization is called G-Hurwitz numbers, whose generating function satisfies the colored cut-and-join equations. Using boson-fermion correspondence, we can write the generating function of G-Hurwitz numbers as the vacuum expectation value of certain operators, thus proving the it is a tau function of the 2-Toda hierarchy..In this project, we plan to study the further applications of the colored cut-and-join equations ang the chamber structure of G-Hurwitz numbers and the corresponding wall-crossing formulas.