Nicolas James Marks Ford

Another New Mathcamp Class

Sat, 12 Jul 2025 00:00:00 -0400

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

New Article: Generating Functions II - Partitions, Pentagons, and Power Series

Mon, 24 Feb 2025 00:00:00 -0500

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.

New Article: Quantum Field Theory IV - Renormalization

Sat, 18 Jan 2025 00:00:00 -0500

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.

Year-end Update

Tue, 31 Dec 2024 00:00:00 -0500

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.

New Article: Generating Functions I - How to Count with Power Series

Sat, 23 Nov 2024 00:00:00 -0500

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!

New Article: Quantum Field Theory III - Perturbation Theory and Feynman Diagrams

Wed, 03 Apr 2024 00:00:00 -0400

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 $\vert 0 \rangle$ for both. In the newer version, the free vacuum is called $\vert 0\rangle$ and the interacting vacuum is called $\vert \Omega\rangle$ .) 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.

New Article: The Lagrangian Approach to General Relativity

Sun, 11 Feb 2024 00:00:00 -0500

I’ve just finished a supplement to the general relativity article I posted a few weeks ago in the physics for mathematicians series. This one is about the Lagrangian approach to general relativity; while it’s a bit more abstract than the approach I took in the main article, I found when learning the theory that it offered a very useful perspective, so I wanted to share it. The article also contains an exploration of a topic that I got quite stuck on when learning this material myself: the relationship between the energy-momentum tensor as it emerges from Einstein’s theory and the concept of energy-momentum we already have from non-gravitational physics.

I’m still working on other articles for this series. The next one might be the one on Lagrangians in classical field theory I mentioned in the last update, and it might also be the third entry in the quantum field theory sequence, which I currently expect will end up being around four to six articles long. As always, if there’s a topic you’d like to see covered in this style, please reach out and let me know!

New Article: General Relativity

Wed, 17 Jan 2024 00:00:00 -0500

After almost a year I’ve finally finished another new article in the physics for mathematicians series. This one is a quick introduction to general relativity, especially aimed at people who are already at least a little familiar with concepts like connections and curvature in the context of Riemannian geometry. This one was very fun for me to write, so I hope that if you’re interested in the topic you enjoy it too!

I do intend to keep working on this series, although probably still at this very slow pace. I’m currently thinking that the next one will be an overview of how to use Lagrangians in classical field theory. As always, though, if there’s a topic you’d be interested to see covered in this style, feel free to reach out and let me know.

More New Mathcamp Notes

Sat, 05 Aug 2023 00:00:00 -0400

I visited Mathcamp for just one week this summer, and I’ve just posted notes for the class I taught there. If you’re one of the students who was in that class and you happen to be reading this, thanks again for a wonderful week!

New Article: Quantum Field Theory II - Interacting Fields and Scattering

Sat, 25 Feb 2023 00:00:00 -0500

I’ve just posted another new article in the physics for mathematicians series. This one introduces interacting quantum field theories, including a discussion of how to talk about particles in an interacting theory, moving from there to a discussion of particle scattering and a result called the LSZ formula, which we’ll use in the subsequent installment to do computations with the famous Feynman diagrams.

More than any other topic I’ve covered in this series, I am very aware that this one is reaching an even smaller audience than usual. It’s a topic I care a lot about, and I’m going to keep working on it for that reason, but I think for the next physics article I’d like to aim for a topic with a bit more reach. (By the way, if you’re enjoying the QFT articles and I haven’t heard from you, you should reach out!) Right now I’m thinking that that means general relativity, which has the advantage of not requiring a lot of physics background to appreciate, but I am of course open to suggestions.

New Article: The Riemann Zeta Function and the Distribution of Primes

Thu, 23 Feb 2023 00:00:00 -0500

I’ve posted a new article, this time outside the physics series. As the name suggests, it’s an overview of the relationship between the Riemann zeta function and the distribution of the primes, and my hope is that it’s readable to anyone who knows enough complex analysis to have seen the Residue Theorem but who might not know anything at all about analytic number theory. This article grew out of a series of lessons I put together for one of my tutoring students, and I really enjoyed learning the material well enough to write it. (By the way, I also currently have some openings for new students! Reach out by email if you’re interested.)

I am also finishing the final edits on a second quantum field theory post, which should be up pretty soon.

New Article: Quantum Field Theory I - Free Fields

Mon, 17 Oct 2022 00:00:00 -0400

After a very long delay and lots of hand-wringing, I’ve posted a new article in the physics for mathematicians series about free fields in quantum field theory. More than any other article in this series I am posting this one without being completely satisfied with the quality of the exposition, but it’s been long enough that I thought it was better to stop delaying and just get it out there, flaws and all. If anyone happens to be reading this who has any suggestions for how it could be improved, I’m very happy to listen!

I’ve been very bad at guessing how long this series will take to write, but my current plan is that the next thing I post will be a direct continuation of the QFT story. As always, let me know if you have any ideas for something you’d like to see.

New Mathcamp Notes

Mon, 29 Aug 2022 00:00:00 -0400

This year was the first in a long time that I’ve taught at Mathcamp full time, and I’ve just posted notes for two of the classes I taught there this summer.

If you’re one of the very small number of people who are still interested in the quantum field theory article I’ve been promising for years, I have good news: I have actually, finally written most of the first post, and I anticipate having it done soon! More news on that when it’s closer to done.

Bean and Nothingness Released, and Other News

Thu, 30 Dec 2021 00:00:00 -0500

I’m pretty sure my audience here is small and mostly already knows this, but just in case I’m wrong: the video game I’ve been working on since 2012 has finally been released! Right now it’s only available for Windows (Mac and Linux versions are likely coming pretty soon) and you can get it right here on Steam. This has been a labor of love for the four of us who’ve been working on it and we’re pretty proud of what we’ve managed to put together. If you like puzzle games, we think you’ll really like this one.

You can also follow us on Twitter and/or join our Discord server.

In other news, I’m still tutoring, and I’m going to be returning to Mathcamp this summer as academic co-coordinator. It’s been a long time since I’ve published any expository articles on this site, and I’m not sure how much longer it’s going to be. I’m still planning to do a couple on quantum field theory, but it’s been much harder to find the time to work on it than I thought. If you’re still following that series, all I can say is hang tight!

Tutoring Services Available

Tue, 19 Jan 2021 00:00:00 -0500

A bit over a year ago, a friend of mine connected me to his cousin who was interested in learning some of the math he never had a chance to study in college. That meeting inspired me to start working as a tutor for adults interested in proof-based mathematics, which I have been doing ever since. I have room for a couple more students, so I’ve just put up some more information about the work I do. If you think you might be interested in working with me, don’t hesitate to reach out! You can e-mail me at njmford at gmail dot com or click the envelope icon on the top of this page.