Łojasiewicz inequalities and their implications

Abstract: Uniqueness of limit structures is a central topic in many problems in geometry and PDEs. In geometric analysis, Łojasiewicz inequality has been widely used a robust tool to derive uniqueness. In this talk, I will give a brief introduction to Łojasiewicz inequalities and their applications, focusing on two problems on uniqueness of limit – for the tangent cone of an Einstein manifold, and for the tangent flow of mean curvature flow.