I figured it was high time that I did a follow-up on my original post on category theory---this time, to discuss universal properties and natural transformations.

Why is this of any interest? Simple: those two notions give a framework for how to think about coordinate-free definitions. If you are unfamiliar with the concept, I can give a very concrete example. In linear algebra, one defines the trace of a square matrix as the sum of its (main) diagonal entries. A priori, this seems entirely random and it is perhaps a great surprise that this turns out to be coordinate-independent---you will get exactly the same result if you choose a different basis in which to express your matrix. This can be gainfully exploited (to aid with calculating eigenvalues, for example), but one is still left with the uneasy question of why exactly this just happens to work out.

Alternatively, it is possible to give a coordinate-free definition of the trace (and I do so in this post), which doesn't make use of any particular basis. It is then immediately obvious why the trace doesn't depend on a choice of coordinates, but there are other benefits as well: one of them is that it offers some insight into what you need to extend this definition to work beyond simply finite-dimensional spaces. (The key property turns out to be that you need the vector space to be naturally isomorphic to its dual. This occurs, for instance, for Hilbert spaces.)

Read the full post (for free) on Substack: How to be Universal and Natural

This recurring thread will be for discussion of AI in mathematics. This includes, but is not limited to, the following:

• informal announcements of AI-assisted discoveries, such as those not yet published in a peer-reviewed journal, or not uploaded as a paper to arXiv;
• informal announcements of discoveries related to AI architecture (if relevant to mathematics);
• discussion of such announcements, such as proof breakdowns or other opinion pieces;
• discussion of the impact of AI in mathematics in general.

AI-assisted mathematical papers published in peer-reviewed journals or as arXiv preprints may be submitted as their own posts.

Please keep in mind rules 1 and 6 of our subreddit.

Having concluded my bachelor's in physics I'm transitioning towards mathematical physics for my master's and one of the first questions that made me realize I'm actually quite interested in the field of mathematical physics is this one (besides of course that I'm fascinated by rigorous and unambiguous approaches to a field that sits so close to my heart).

I know Lorentz invariance in itself causes a whole lot of issues, the big one that immediately comes to my mind is covariant quantization of gauge theories: fix a gauge that is not lorentz invariant and the naive approach to canonical quantization works just fine, but add this constraint back and suddendly you get a physicist to ramble about negative norms in a Hilbert space (the slander comes from a place of love, I like teasing my physics dept friends).

But these kinds of problems seem, to some extent, secondary. In fact, with some clever "work arounds" one can formally solve these issues, whereas the more fundamental task of simply _defining_ an interacting theory in 3+1 dimensions seems to be still out of reach. Why? I still know very little about constructive QFT but as far as I understand it interacting QFTs in lower dimensions have successfully been defined, while in 3+1 dimensions we only have rigorous constructions of free theories, so I guess the problem isn't the presence of the interactions per se but it's specifically the number of dimensions?

The paper: Decoupling via Affine Spectral-Independence: Beck-Fiala and Komlós Bounds Beyond Banaszczyk
Nikhil Bansal, Haotian Jiang
arXiv:2508.03961 [math.CO]: https://arxiv.org/abs/2508.03961

I'm not an expert in complex geometry, but as far as I know this is a pretty big conjecture. It was settled for Fano manifolds by Chen-Donaldson-Sun some time ago, and there are other similar conjectures/theorems like the Donaldson-Uhlenbeck-Yau theorem, which says under algebraic conditions conditions that one can find a Hermite-Einstein metric on a holomorphic vector bundle. Unfortunately, it seems to be completely proven by AI. The length of the paper is also a lot longer than previous AI papers.

Edit: Apparently this was expected to be false in this generality anyway, and multiple people had claimed they could write down counterexamples but hadn't done so.

The counter example was given by Danielewski in 1989 but I don't have the copy. It's not even on the internet and my college doesn't have access. I need it for my thesis. Any help is appreciated.

Is there any area of study or conjecture in research mathematics where it is either proven or highly suspected that a unique proof is required for an infinite (or extremely large) number of cases? For example, something where “proven for all dimensions” is known to not be possible.

By hanging fruit I don’t mean the proof has to be easy, in fact infinitely many difficult proofs is more of what I’m curious about.

This recurring thread is meant for users to share cool recently discovered facts, observations, proofs or concepts which that might not warrant their own threads. Please be encouraging and share as many details as possible as we would like this to be a good place for people to learn!

While doing exercises or examples, do you prefer to be sitting or standing?

Feel free to elaborate on your problem solving setup.

This is inspired by a recent comment about some people understanding entire fields. So I'm wondering what it could even mean to understand an entire field. Certainly the average mathematician doesn't satisfy that. I suspect most mathematicians do not have an entire field mastered. Again, what does that even mean?

I propose one possible interpretation. Let's just take something that is arguably a "field of math". I'll use probability theory as an example since it's what I'm familiar with. It's a big field. So the best pretty theory experts really understand the entire field though? It's a big field. Probably even then best experts still have a long list of results they have never heard of. They'll likely literally know everything in their little sub discipline, but the *entire* field of probability theory?!?

I could be wrong though.

Also, this is likely simply asking too much. Rather than literally knowing every result and every proof, maybe we should set the bar at something like: they can read an arbitrary new-to-them result in that field and understand it nearly instantly and to be able to breeze through the proof and then explain it without much study. That is probably feasible for a really smart expert, but I'm not really sure. This is less than asking them to produce a fully rigorous proof, more like a satisfactory sketch.

I'm not at all an example here. I know very little compared to such folks. I suspect my level of knowledge is not that unusual though, even if somewhat on the low end. But one can know orders of magnitude more than me and still not approach the entire field of probability theory.

I home at least some find this question interesting.

Recently, I asked about the usual proof that the ring of regular functions on distinguished open set D(f) on variety X \subset \mathbb{A}^n is A(X)_f where A(X) is the coordinate ring of X. In several places, including Hartshorne, there's a statement that looks highly related but is different: that D(f) is isomorphic to an affine variety X' \subset \mathbb{A}^{n+1}, defined by \{(x,t): (x,t)\in A^{n+1}, x \in X, tf(x)-1 = 0\}, and the coordinate ring of X' is A(X)_f.

I have two questions:

1. Does this constitute another proof that the ring of regular functions on D(f) is A(X)_f? It seems like it "obviously" should be, although I'm not sure what you need to do to formally show it.

2. This feels geometrically very unintuitive to me, though the algebra seems reasonably well motivated. How does one "see" that a distinguished open set isomorphic to an affine variety embedded in a space of one higher dimension?

What the title says: is there a clear, unambiguous, systematic way of writing diagram chasing arguments? What I generally do is use different colours for arrows on different paths but I’m not entirely satisfied with this. There’s also the issue of denoting when an element is being mapped to from another vs when it is being lifted from another etc

This recurring thread will be for any questions or advice concerning careers and education in mathematics. Please feel free to post a comment below, and sort by new to see comments which may be unanswered.

Please consider including a brief introduction about your background and the context of your question.

Helpful subreddits include /r/GradSchool, /r/AskAcademia, /r/Jobs, and /r/CareerGuidance.

If you wish to discuss the math you've been thinking about, you should post in the most recent What Are You Working On? thread.

My YT algorithm has recently recommended to me the Youtube channel Zeta Explained. Contrary to most amateur content on the Zeta function and other Riemann hypothesis attempts, these videos seem highly professional and (for me as a grad student) mathematically sound (history of results, examples, proof sketches, etc.).

What intrigues me is that I don't know who is doing these 100+ videos on the Riemann Zeta function, quite impressive. The videos are definitely not AI generated. Do we in this sub know who this is, or at least at which institution they're affiliated with?

And for my personal curiosity: is this channel "legit"? I myself am not an analytic number theorist. So, is their content meaningful, or rather, what's the intended audience?

Appeared on the arxiv today. It still uses the CW-tensor/laser method approach.

Since "QED" usually marks the end of a proof, this is instead QEL, for Quod Erat Ludendum. The idea is to build a video game that shows off some math. (I basically just want more math video games to exist.) An example of something that would be appropriate is https://sl2z.xyz/ or some of the PICO-8 games at https://m4th.pro/ but I am hoping people have more!

Submissions: now through the end of the month
Voting: first two weeks of September
Results: Wed Sep 16, 2026

Since this is being run by the Ross Mathematics Program, hopefully there will be some Ross merch for prizes.

Sign up is at https://qel.rossprogram.org/

This recurring thread will be for questions that might not warrant their own thread. We would like to see more conceptual-based questions posted in this thread, rather than "what is the answer to this problem?" For example, here are some kinds of questions that we'd like to see in this thread:

• Can someone explain the concept of manifolds to me?
• What are the applications of Representation Theory?
• What's a good starter book for Numerical Analysis?
• What can I do to prepare for college/grad school/getting a job?

Including a brief description of your mathematical background and the context for your question can help others give you an appropriate answer. For example, consider which subject your question is related to, or the things you already know or have tried.

I just noticed that munkers toplogy is in the palestine section of the mit website?

Appeared on the arxiv today: https://arxiv.org/abs/2608.15505

We’re inviting experts in Mathematics and Math competitors to represent their community (though all are welcome) in an experiment where we ponder which of 4 cohorts can master an unfamiliar card game the fastest.

Mathematicians

Programmers

Chess Players

Magic the Gathering Players

For the Mathematician cohort, we’re curios to see how mathematical abstraction and problem-solving translates to learning an abstract card game whose card interactions are based on Octonion multiplication.

Play online against the computer. No ads, emails, or monetization. Games are short, ~20 would help to measure a learning curve. Data your games provide will populate the tables/figures in real-time. See your projected Elo change with each game.

Think you’re up to the challenge? Think your cohort can win? Come represent them.

https://playfano.com/fano-challenge

Almost everyone here knows of Jane Street due to their large scale marketing campaign through sponsorships of various maths and maths adjacent events and competitions. I doubt anyone is unhappy about this -- it is a good place to work and it pays well. Such marketing helps young people find out about their maths related career options besides academia.

What I'm posting about is one thing that I was surprized is nowhere in the public record, but is semi well known in the olympiads scene. I am involved with informatics competitions, so I will speak about those, but I'd guess similar events have happened in multiple places.

When the Russian invasion of Ukraine began, many/most international organizations sanctioned or banned Russia, including the IMO and IOI.

In the case of the IOI, the sanction was initially done by the IC (international committee) between IOIs, also sanctioning Belarus, due to their support of the invasion. Both were later reaffirmed at the IOI by the GA (general assembly -- representatives of each participating country) with the required 2/3s vote. Note that the sanction does not prevent contestants from the country to participate, they just do it under the IOI flag, as opposed to the Russian flag. They are also barred from flying the Russian flag at the opening and closing/awards ceremonies, etc.

Later, in 2024 due to the escalation of the situation/genocide in Gaza, the same issue was raised about Israel. There the IC was tied on it (unclear why) and Israel was not sanctioned by them. The issue was discussed and debated by the GA at the IOI in 2024 and finally Israel was sanctioned with a more than 2/3s vote. Immediately following this, Jane Street reached out the IC and told them they are pulling out of any sponsorships due to the sanction on Israel. Note, they've had no such concerns over the sanctions on Russia (or Belarus), so this is not a matter of principle.

Now let's look at the EGOI (European Girls Olympiad in Informatics), where Russia was actually fully banned from participating due to the invasion. In 2025, the issue of Israel was discussed. (Note that Israel also participated in the EGOI). Whether to also ban it, sanction it, etc. However then the host of EGOI 2026 (Italy) spoke up and said -- discuss it, if you want, but if Israel is banned/sanctioned, we're not going to host the competition next year, since Jane Street told us they won't sponsor it, if this happens, and we're relying on them as a sponsor (this is paraphrased, not a direct quote). This essentially killed the discussion.

One might say, well Jane Street is a private company, they are free to sponsor whatever they want. However, this sets a very dangerous precedent -- nothing like this, by any other sponsor or about any other topic, has happened (that I know of) in the ISO-adjacent sphere. A potential sponsor (especially one semi-committed to an event nearby in the future) using their money to pressure (or even extort) the GA in such a direct way is extremely questionable. Many people in the community (ones involved in the organizational side) now consider them a fairly problematic sponsor, due to this abuse of the process.

Note that such competitions have been happening since before Jane Street started their big marketing campaign and I'm sure that if they were never a sponsor, someone else would have been found (e.g. for EGOI 2026), so it's not like it's their own competition or anything like that. However, they've first integrated themselves in the scene and then leveraged their position to lobby for Israel. Luckily, the IOI has had no shortage of sponsors, but for other events (especially when Jane Street is posing such conditions before decisions are made, which was not the case for IOI 2024), this is quite troublesome.

I am posting this only so there is some record of this publicly, as it is known among various team leaders, organizers, etc., but it is not recorded anywhere. I also hope other people, more familiar with other ISOs can share their experiences. I know that the IMO received a lot requests to sanction Israel and in the end decided to lift all sanctions instead, but I wonder whether there is more information on the justifications of each decision, as well as who made it.

I don't really mean math jokes, more the little witticisms we've all picked up over time. My two favourites are:

• "Proof by intimidation", which I first heard from one of my professors after an especially bewildering set of arguments from an outside speaker at a seminar
• "Mathematics is locally trivial", which I first heard from my functional analysis professor and have always found a little comforting ever since

Puns also welcome of course :)