Something I’ve been trying to get my head around lately is ‘Asymptotic safety’. This is a reasonably in depth idea in Quantum Field Theory (QFT) but, since I’m too lazy to sit down and work through a book on QFT, I’m trying to glean the basic ideas from papers and Wikipedia. Some of the things I’ve been reading are: this and this.
So, Asymptotic safety has something to do with a QFT reaching a non-trivial fixed point in the space of physical couplings as the theory undergoes scale transformations. Apparently this implies that we can non-perturbatively renormalize a QFT, which is, of course, a good thing. Clearly I need to keep reading about this before I can explain it to someone else, so hold on for a more in-depth/better explanation.