Weekend research talk: Barry Mazur on Bridges between Geometry and Number Theory

Note: The previous post (on a possible direction for a competition essay) did not go out by email to everyone, so check it out if you're interested.

Now, in this very friendly and accessible talk by Barry Mazur, he talks about some of the themes we have touched on in recent RH Saga chapters (like the analogy between number fields and function fields), but also about things we haven't seen yet, like the idea that the integers should be viewed as a three-dimensional space, and the prime numbers as one-dimensional loops or knots embedded inside it. This is one of the key intuitions (but not the only one!) about the nature of $\mathbb{F}_1$-geometry.

The talk of Mazur is beautiful and ties together several different RH Saga themes. But if you also want a more technical and cutting-edge research talk on the same theme, you can check out the MPIM recording of Scholze's talk: What does Spec Z look like? This talk is not on YouTube as far as I can tell.