Right now I'm a fourth year PhD student in theoretical physics, working at the interface of quantum information and quantum gravity. Many of the subjects I end up learning for my research lack good introductory references. The physics subjects are often explained in research papers that were written decades ago in now-outdated notation and terminology; the math subjects are explained in textbooks for mathematicians that mostly lack physical intuition. For aspiring physicists like me, it can be helpful to have concepts that are well-understood by experts re-interpreted and re-explained in concise, pedagogical terms. While learning new math and physics subjects for my research, I often end up writing detailed "explainers" for myself that I think fit this niche. This blog will serve mostly as a repository for these explainers. I'll post explainers here as I write them in the hopes that they might be useful to other researchers trying to penetrate formidable subjects. At