For our Freudenthal Topology Seminar, I recently gave a talk about algebraicity of chromatic homotopy theory. It was an introduction to the papers by Pstrągowski, Patchkoria-Pstrągowski, and more recently Barkan. You can find the notes below. While the latter two papers discuss much more than ‘just’ the chromatic case, in my talk I decided to focus on the chromatic case only, as this is already a very interesting case, and to keep the length of the talk reasonable.
These notes are not by any means comprehensive or complete. Rather, I’m hoping that these notes can serve as a helpful starting point for studying the aforementioned papers.