We will begin from billiards on rational polygon tables, establish the link between them and Riemann surfaces with holomorphic differentials. After that we will explain why introducing Teichmuller geodesic flows on the moduli space, how it opens huge possibility to use dynamical systems methods to create new algebraic geometry objects, and to apply algebraic geometry techniques to study the original problem of 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 (measure the stability of dynamical systems) is bigger than (or equal) the partial sum of Harder-Narasimhan spectrum (measure the stability of algebraic geometry) on Teichmuller curves. We will discuss the deep connections between them and the integral of eigenvalues of the Hodge bundle curvature by using Atiyah-Bott, Forni and Moller's works.We will also give several interesting applications of this conjecture to dynamics of the Teichmuller geodesic flows(including an old conjecture of Kontsevich-Zorich on asymptotics behavior of the second Lyapunov exponent on hyperelliptic components).

Recently our conjecture has been proved by Eskin, Kontsevich, Moller and Zorich.