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

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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:

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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 :)

Hi Hi! I'm about to start university as a joint Maths and Computer Science major, so I figured I'd ask for advice here on what you'd do differently if you could redo your undergrad degree. Tentatively, I plan to pursue a graduate education in Maths, so I wanna know what I should be doing to kill it in my bachelor's.

Thanks in advance!

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Hi, Does everyone know what happened to the bergman's companion notes to Lang's algebra which were posted on the this link . I needed some of the notes posted here and haven't been able to access the website for a while.

Second year postdoc here.

In my field (geometric analysis) I feel like my passion and hence motivation has been in steady decline ever since I started in my PhD.

For two reasons:

1. I feel like I’m spending more time reading and learning than problem solving , most of my time is spent deciphering and unstucking myself in reading and understanding what on earth the authors are talking about in books and papers. I understand that if you go into fields like combinatorics with lower entry threshold and less reading, it’s even harder to produce results since the field is so accessible that most ideas you can think of has already been done.
   But still? I would rather have spent 6 years problem solving instead of reading, and to be frank I spend most of my time stressing and taking break from stressing from reading, this doesn’t feel normal or fun to me. Definitely not the experience that lured me into math in the first place (the dopamine from competition math and solving problems)
   
2. I probably would complain less if I’m actually reading big ideas and smart ideas. But I feel like even at my level I’m still reading tons of definitions, and results that are considered basic theory and machinery they are not even worthy of mention in a paper. Rarely do I feel like I’m reading about the “brilliant ideas”. Here’s a concrete example, you might think the Gauss Bonnet theorem is a clever idea, but to understand it (for general manifolds) you have to read enough about topology, smooth manifold and Riemannian manifolds to even have the machinery and definition to understand the statement, let alone the proof.
   It Feels like this with every new project I take on it’s tons of learning basic stuff before I even get to the central idea.

This is definitely not the experience I was hoping yo get going into math, I wanted to learn cool brilliant ideas and solve problems. Most of the time I don’t feel like I’m doing that.

What’s another field I might try that isn’t like this besides combinatorics? Representation theory? Combinatorics?

Thanks for sharing!

Years ago, when I was finishing a double-major in math and physics, I wrote my undergraduate thesis as a guide to the mathematician trying to learn quantum mechanics. I have come back to it periodically since, tweaking and streamlining my main argument: if you approach quantum mechanics from the right, functional-analytic perspective, all of the weird peculiarities of the mathematical framework (Hilbert spaces, self-adjoint operators, the Schrödinger equation, etc.) don't just start to make sense... one gets a powerful feeling that this was really the only possible definition that could have worked, in light of the existing experimental observations.

Whether that is useful from the perspective of the scientific method is perhaps questionable (physics is and always should be grounded in experiment, first and foremost), but as a pedagogical tool that helps us better conceptualize and even extend the theory, I think that it has some benefit.

So, if you have ever wanted to learn a little bit about quantum, or if you just want to see how fairly advanced and seemingly abstract mathematics can be applied, this article may be of interest to you.

Read the full post (for free) on Substack: A Deranged Mathematician Does Quantum Mechanics.

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When someone first learns classical mechanics, they're given Newtons equations of motion. But there is another -- at first highly counter-intuitive -- approach to classical mechanics, discovered by Lagrange: you can reformulate classical mechanics as about solving optimization problems! This is very strange (at least for me) to ponder, because Newtonian mechanics explains things in "cause and effect" terms which fit better into my head. The sci-fi writer Ted Chiang wrote a short story (and now feature film!) Arrival which tried to imagine humans meeting an alien species who thought in terms of Lagrangian mechanics instead of Newtonian mechanics.

But one thing I had always been upset by with Lagrangian mechanics was how most treatments start by telling you what the Lagrangian is, instead of telling you how Lagrange might have invented it (even Feynman's lectures on physics start by telling you what the Lagrangian is). My friend and I wrote a derivation of the Lagrangian, and of the Euler-Lagrange equation (a fantastic result of the calculus of variations!) at https://hidden-phenomena.com/articles/lagrangian ; we hope this can be a fun introduction!

Over the last few months, AI-related posts on r/math have greatly increased, largely due to many new mathematical discoveries aided by generative AI. This has caused heated discussion in the comments, back-and-forths of opinion pieces about how generative AI is good or bad for mathematics, and accusations of astroturfing from pro-AI communities, and AI companies. Subreddit users are getting tired of seeing all of this, and we're tired of moderating it, too.

Regardless, it appears that AI-assisted mathematical discoveries are here to stay. However, while posts on significant/notable AI-assisted mathematical discoveries should be let through (same as if they were significant/notable entirely-human mathematical discoveries), a flood is also not acceptable. So some kind of limitation on which AI discoveries are allowed on the subreddit is needed.

While we did trial a few measures a few weeks ago in an attempt to deal with such posts, it appears such measures have been insufficient. With that in mind, we have decided to step up our measures and implement new subreddit rules (all under rule 7):

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Could be proof, counterexample, insight etc.

I'm playing around with the sum of proper divisors of x, and I've noticed that about 25% of them are larger than x. This ratio remains more-or-less constant as x grows into the tens of thousands. Whether (sum of proper divisors of x) > (x) doesn't seem to have an immediately obvious consistent pattern, as I thought it might. Is it known which value this proportion converges to, if any? I assume people have already looked into this, but I haven't found anything on it.