Abstract

Using a partial differential equation to ‘flow’ some geometric structure on your manifold towards a ‘more canonical’ one is a powerful technique in many parts of differential geometry. Unfortunately, solutions to these equations usually don’t exist for all time: ‘singularities’ can form, which can sometimes be ‘removed’ with some sort of ‘surgery’ operation — a local geometric modification to your manifold that allows the flow to continue. I’m definitely not an expert in this field but I’ll try to give a down-to-earth explanation of what the terms in quotation marks mean, illustrated with some examples I’ve managed to understand. I’ll finish with some wild speculation about how all this might apply to symplectic geometry (plus some joint work in progress).