turkmath.org

The resolution of Calabi's Conjecture by S.-T. Yau in 1977 is considered to be one of the crowning achievements in mathematics in 20th century. Although the statement of the conjecture is very geometric, Yau's proof involves solving a non-linear second order elliptic PDE known as the complex Monge-Ampere equation. An immediate consequence of the conjecture is the existence of Kähler-Einstein metrics on compact Kähler manifolds with vanishing first Chern class (better known as Calabi-Yau Manifolds). In this expository talk, I will start with the basic definitions and facts from geometry to understand the statement of the conjecture, then I will show how to turn it into a PDE problem, and finally I will highlight the important steps in Yau's proof.