This book chapter is published open access.
While the author is a professional mathematician, he is by no means an expert in the subject area of these notes. The goal of these notes is to share the author’s personal excitement about some results of James Maynard with mathematics enthusiasts of all ages, using maximally accessible, yet precise mathematical language. No attempt has been made to present an overview of the current state field, its history, or to place this narrative in any kind of broader scientific or social context. See the references in Section 11 for both professional surveys and popular science accounts that will certainly give the reader a broader and deeper understanding of the material.