报告题目：On the number of zeros of Abelian Integrals
摘要：In this talk, we will introduce some new methods to estimate the lowest upper bound of the number of isolated zeros of Abelian integrals, which is called the weakened 16th Hilbert problem proposed by V. I. Arnold. Some algebraic criteria are obtained for the number of isolated zeros of Abelian integrals along energy level ovals of potential systems. As applications of our main results, we study three kinds of Abelian integrals along algebraic or non-algebraic level ovals, obtain the algebraic criteria on the Abelian integrals having Chebyshev property with accuracy one, simplify some known proof on the cyclicity of quadratic reversible centers, and give all the configurations of limit cycles from Poincar\'e bifurcation of two quadratic reversible systems with two centers, one of which has a non-algebraic first integral with logarithmic function. This talk is based on the joint works with Changjian Liu.