We will begin with billiards on rational polygon tables, and establish the link between them and Riemann surfaces with holomorphic differentials. Then we will explain the reason to introduce Teichmuller geodesic flows on the moduli space, open up huge possibility to use dynamical systems methods to create new algebraic geometry objects, and apply algebraic geometry techniques to study the original problem in dynamical systems. Under this philosophy, I will introduce Chen-Moller and Yu-Zuo's proof of the Kontsevich-Zorich conjecture.
Furthermore, we conjecture that the partial sum of Lyapunov spectrum (which measures the stability in dynamical systems) is larger than or equal to the partial sum of Harder-Narasimhan spectrum (which measures the stability in algebraic geometry) on Teichmuller curves. We will discuss the deep connections between them and the integral of the eigenvalues of the Hodge bundle curvature by using Atiyah-Bott, Forni and Moller's works. Finally, we will introduce the proof of this conjecture by Eskin, Kontsevich, Moller and Zorich recently.