Optimal Coordinates for Geometric Problems

Tuesday, 21-November-2023 

Abstract

One of the most important defining properties of a manifold is the "locally Euclidean" property, i.e., the existence of local coordinates around any point. Since there are potentially many choices of these coordinates, an important part of differential geometry is defining relevant geometric quantities in a way that does not depend on the choice of coordinates. Nevertheless, not all coordinates are created equal; there are some that are much more useful for studying geometric quantities, depending on the context. In this talk, I will discuss some of the historically popular coordinate choices for Riemannian manifolds, as well as a relatively new coordinate system that has proven useful in solving some modern problems in geometry.