A general theme in geometry is the classification of algebraic/differential geometric structures which satisfy a positivity property. I will describe an "asymptotic" version of this theme based on joint work with Cheltsov, Martinez-Garcia, and Zhang. On the algebraic side, we introduce the class of asymptotically log Fano varieties and state a classification theorem in dimension 2, generalizing the classical efforts of the 19th century Italian school. The novelty here is the use of a convex optimization theorem that reduce the asymptotic positivity to determining intersection properties of high-dimensional convex bodies. On the differential side, I will give a conjectural picture for existence of singular Kahler-Einstein metrics, explain progress towards this conjecture, and relations to singular Kahler-Ricci solitons. Time permitting, I will also explain some conjectures and results about the "small angle limit" when the angle tends to zero, making ties to non-compact Calabi-Yau fibrations and steady Ricci solitons. The talk should be accessible to non-experts.