when trying to solve partial differential equations, a common practice is to enlarge the space of possible solutions to the class of non-differentiable functions, where it is easier to find “weak” solutions (i.e. potentially very irregular). as we are usually interested in “strong” solutions (i.e very regular), one is then confronted with the following problem: how do we upgrade the regularity? a fundamental tool in these situations is a monotonicity formula, an object that allows to study the infinitesimal behavior of solutions of pdes by reducing it to a classification problem. more concretely, a monotonicity formula is an identity implying that a certain quantity related to the problem at hand is monotone, or conserved. i will try to convey the gist of this idea that has found applications in many areas at the intersection of geometry and analysis, e.g. harmonic maps, minimal surfaces, free boundary problems, yang-mills connections to name just a few. i will try to maintain the level of analysis needed at a minimum, you only need to remember that the first derivative of a smooth function at an interior minimum is zero. i will explain the rest.