After a bit longer than I anticipated, I’ve posted a new article. This one is a followup to the previous article on generating functions, although you should be able to follow it just fine if you even just the definition of generating functions. It’s about an application of generating functions to the question of counting objects called ``integer partitions,’’ culminating in a proof of a beautiful result called the Pentagonal Number Theorem.
This article grew out of an activity that I ran last year for my students at New York Math Circle, and it’s one of my favorite arguments in all of combinatorics. Nothing in this presentation is especially original — you can find essentially the same proof on Wikipedia — but if you enjoyed the first generating function article and want to read more in the same style, I hope you enjoy this one as well.
I’ve posted a new article in the physics for mathematicians series. This one is on renormalization, and it forms the end of the main arc of the story I’ve been telling in the quantum field theory series. If you’ve been following those articles, I hope you find this one interesting, and I’d very much like to hear from you if you have any questions, suggestions, or corrections.
I’ve been engaged in this project in some form for around ten years, so it feels very nice to reach this milestone. There are still several quantum field theory stories that I’d like to learn and present here in the future, though. My current plan is for any future QFT articles to have more of a branching structure rather than building on each other in sequence like the existing ones have done. (Four articles in a row is already more than I originally wanted to end up with!) The first of these will probably be on the functional integral formulation of quantum field theory and its relationship to the Wilsonian perspective on renormalization.
The next article I’m working on for the site, though, is a followup to the generating functions piece from a couple months ago. As always, if you’re enjoying my content or you have any thoughts about what you’d like to see here, feel free to reach out by email. I’d love to hear from you.
No new article this time, but I thought it was a good moment for a brief update on what I’m working on.
As I mentioned in the previous post, I have a couple articles in the works that are designed to be a bit more accessible than a lot of what’s on this site; the next one of those will probably be a follow-up to the generating functions piece. I’m also very nearly done with the fourth article in the quantum field theory series, and that one will probably appear in the next couple weeks. There’s also now an RSS feed available, thanks to an email from a reader.
If you’ve been reading any of these articles and you have thoughts or corrections or just want to say hi, please don’t hesitate to reach out! If you or someone you know is interested in tutoring, I’d also love to hear from you.
Happy New Year, everyone.
I just posted a new article. This one is on generating functions, a fun and powerful technique in combinatorics, and it’s designed to be accessible to anyone who’s taken calculus and remembers at least a little bit about how Taylor series work. This will hopefully be the beginning of a series of two or three articles where we explore some fun applications of the concept.
This article is a part of an effort to put some content on here that’s a little more accessible than a lot of the articles I’ve written so far. (I’ve very much enjoyed putting together the physics series, and I plan to keep working on it, but the prerequisites for those articles can be quite steep!) There are a lot of topics I’ve explored with my students in the last few years that I think could be a nice fit here, and I plan to adapt more of them into articles for this website in the coming months.
As always, if you’re reading this and have a suggestion for a topic you’d like to see explored in this format, don’t hesitate to reach out!
There’s another new article in the physics for mathematicians series. This one picks up the story of quantum field theory right where the previous one left off. It explores how we can take the ideas we’ve developed so far and turn them into an algorithm for computing scattering amplitudes. In particular, this article finally introduces the famous Feynman diagrams and explains where they come from and how they fit into the story we’ve been telling across this series of articles.
In order to make this article flow better, I had to do something I don’t ordinarily like doing and make a small notational change to the previous article in the series. (In the older version, I didn’t distinguish between the vacuum state of a free theory and the vacuum state of an interacting theory, and wrote for both. In the newer version, the free vacuum is called and the interacting vacuum is called .) That change is now live as well. My hope is that if you’ve been following this series so far this isn’t too big a disruption, and I’ll try not to change the past out from under you again if I can help it!
My current plan for the quantum field theory series is that there will be one more article (on the topic of renormalization) in the “main” series, but that I might end up writing a few one-offs on some smaller topics in quantum field theory to go along with them. If this plan ends up being what happens, the picture is that the first four articles will serve as the trunk of the tree, and that the others will branch off from it and could potentially be read in any order.
It’s been very fun for me to finally arrive at the point in my study of this subject that I feel competent to put together these expositions! Quantum field theory is something I’ve been working toward mastering for a very long time now, and I’m happy to finally get the chance to share what I’ve managed to learn with others to whatever extent these articles can do that. As I always say, do let me know if there’s any topic you’d like to see covered in this style, whether it’s physics-related or not, and I’d always love to hear from any readers if you have comments, questions, corrections, or anything else you’d like to talk about.