Length Bounds for Critical Points of the Energy Functional

The goal of quantitative geometry is to provide effective versions of known existence theorems for geometric objects. For example, following Serre's proof of the existence of infinitely many geodesics connecting any two points on a closed Riemannian manifold, one may attempt to prove a length bound for these geodesics. In this talk, we will provide a survey of current quantitative theorems concerning various types of geodesics and explore how such results can be proven. In particular, we discuss recent work on the existence of "short" geodesics that meet a given submanifold orthogonally, as well as recent work on the existence of "short" geodesic nets.