Xianguo Geng，Zhengzhou University

We first consider the basic theory of algebraic curves, including the compact Riemann surface, Riemann-Hurwitz formula, Baker-Akhiezer function, three kinds of Abelian differentials, the construction of Riemann theta function and so on. Then, we consider the application of algebraic curve theory in integrable systems. We study the properties of algebraic curves induced by integrable systems, and calculate the factors of Baker-Akhiezer functions and meromorphic functions. An illustrative example of our approach is given, from which quasi-periodic solution of the corresponding integrable nonlinear evolution equations are obtained.