On $C^{2,\alpha}$ estimates for levels sets of Allen-Cahn equation

I will discuss recent new developments in De Giorgi Conjecture for Allen-Cahn equation. In dimensions up to 10, we shall establish the \$C^{2,\alpha}\$ estimates of level sets for stable solutions of Allen-Cahn. By applying reverse gluing method we show that the obstruction to \$C^{2,\alpha}\$ estimates is the existence of Toda System in one dimension less. Applications to classifications of finite Morse index solutions and some open problems will be discussed.