I’ve finally gotten around to posting slides from a three-week lecture series I gave at Google a few months ago called Differential Geometry: Definitions and Pictures. The goal of the talk was to introduce the audience (mostly Google employees working in machine learning) to the way mathematicians think about the subject; there are lots of pictures and intuition and no proofs. The target audience has had a class in multivariable calculus and probably linear algebra.
As always when reading slides without seeing the talk, some parts might not make sense out of context, but I hope this can still be helpful to someone in its current state.
I’ve posted a new article in the physics for mathematicians series. It gives an overview of classical electromagnetism from several perspectives, showing how the electromagnetic field strength can be thought of as the curvature of a connection on a principal bundle. It closes with a (very) brief discussion of classical Yang-Mills theory, which is a natural generalization of this idea and provides the template for almost all of the interactions in the Standard Model of particle physics.
This article is going up in a rougher state than some of the earlier ones. If you notice a problem or you just dislike the presentation of some or all of it, please shoot me an e-mail; I’d love to hear from you.
I’m still working hard on this series; I expect to keep pumping out articles in it until I get bored, which hasn’t happened yet! My current plan is to start work on a piece about statistical mechanics and thermodynamics next, but if anyone happens to be reading this and would like to see something else first, let me know!
I just reorganized the way this site is presented, and also it’s been a long time since I’ve updated anything here at all, so I thought, for the sake of the approximately twelve people who will ever read this, that I should explain what’s happening.
Most of the content here so far has consisted of long articles about math, and that’s probably going to continue to be true. But I’d also like to leave myself the option of also making shorter, blog-style posts that are less polished and permanent. So, from now on, the front page of this site is a blog and not a list of articles. I’ve made a separate list of articles that’s accessible from the navigation bar at the top, and the pages that used to house the articles I’ve already written have been retroactively converted into blog posts announcing where the article can be found.
As far as new articles go, right now I’m nearly done with a new article for the physics series about connections and, since I’ve recently left my job, I plan to also produce some more pieces aimed at a wider audience, but nothing more specific than that has been worked out yet.
I’ve posted a new article. This one is aimed at undergraduates with a strong understanding of linear algebra. More information at the link.