I visited Mathcamp again this summer, and I’ve posted my notes for the class I taught there.
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!