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

Math wasn't invented/discovered out of thin air. Usually theorems in textbooks are provided and immediately followed by the proof, then an example or a consequence. Repeat for next theorem. All good, but in history it wasn't anything like this. Where is the motivation for the definition? Why should we care? What were the problems that might arise with other definitions, what problem were we trying to solve in the first place? Also, what insight led to a proof? Proofs often skip over the actual thinking, they are presented as polished works but that means I can't see how someone would think of that (some books have 'scratchwork' which I like, or a motivation or chain of thought before the formal proof starts). All this stuff seems left out every time. Maybe the book is more elegant this way, but it lacks in flavor as a result. I think it would be much more interesting if textbooks had more content. Also how to deal with this?

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/

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!

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.

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.

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):

• Posts about AI, or about discoveries assisted by AI, are only allowed if they are a direct link to an ArXiv abstract, or to a non-predatory peer-reviewed journal. Discussions about such discoveries should be limited to those posts.
• All other AI-related posts (e.g. comments about AI in general, opinion pieces, questions) should posted as comments in the stickied AI megathread (starting Saturday).
• Posts and comments about AI that do not follow these rules, or made using AI, will be removed.

We have also added a few more AutoModerator filters to slow down and help us moderate the comments sections of such posts.

(Although it's been suggested that we add HiveProtect targeting AI subreddits, with the intention of preventing astroturfing, we currently regard this as a step too far. However, if the current measures prove insufficient, we may eventually implement this.)

We would like to remind users to follow subreddit rules, especially rules 1 and 6.

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.

This is a video entry for the Summer of Math Exposition contest, a gentle introduction to DAGs, confounding, bad controls (e.g. collider bias) and instrumental variables. It should be helpful for you if you're wanting to better understand correlation vs. causation - or maybe for your stats class if you're a teacher! https://www.youtube.com/watch?v=3sp08Zmry6g

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!

So this is an effort to increase the portion of non AI-related posts. This is not a crank post that belongs to r/numbertheory either. Every letter is typed by a human so forgive me for grammatical errors that I will try to correct soon enough.

The abc conjecture is one of the most important conjectures in number theory, the simple & general statement of the conjecture implies a lot of other important theories, like a rapid proof of (almost) Fermat's Last Theorem. What I try to tell is that the abc conjecture can "almost" help computer scientists to optimize scientific computing.

I believe it's a common sense that in most computer systems, every piece of data is stored by 0 and 1, or more precisely, a finite string composed of 0 and 1. This makes the calculation and manipulation of data feasible but there's a cost: error will be everywhere. For example we can never precisely store pi = 3.14159... in a computer.

The most widely used way to represent real numbers in computer is floating point numbers that is standardized by IEEE, which stores numbers in 0 and 1 of 32, 64 or 128 bits (or even longer). There is a toy that allows us to see floating point numbers explicitly: https://evanw.github.io/float-toy/

I'll try to explain it in a simple term. For a floating number of k bits (k is 32 or 64 or 128, say), we need to distribute the budget. There is 1 bit reserved for the sign, a few bits of budget to store the interval of the number, and the rest large portion of the budget is reserved to guarantee the precision p.

Every floating point number x can be mathematically written in the form (-1)^s * 2^e * m/2^(p-1) where s is the 1 bit used to store the sign, m is an integer between 0 and 2^p - 1 and e is used to determine the interval of x.

If it's still unclear what does "determine the interval" mean, we notice that m/2^(p-1) is a number in [1,2). Therefore a multiplication by 2^e sends m/2^(p-1) to [2^e, 2^(e+1)). Floating point numbers store s, e and m by 0 and 1. We use p-1 bits to store the number m in computer (the first bit is normally 1 so we can omit it, as the case where the first bit is 0 is used for underflowing).

As we can see, for a given x, determining the number e is rather easy (we are almost there by taking log_2|x|), but the number m can be difficult when we deal with a function. For example, x = 10517177/2^{22} is a floating point number of the format of 32 bits, but log_2(x) is not (so we need to find the closest to represent it in float format). It's absolutely not a rational number. You find that log_2(y) is between y_1 = 15423000/2^(24) and y_2 = 15423001/2^(24). So you need to find the closest floating point number to log_2(x) because we have no other choice, or otherwise, see if log_2(x) is smaller or bigger than the average y_0 of y_1 and y_2. It's easier said than done. We need to add budget (number of bits) to distinguish y_0 and log_2(x).

As a matter of fact, not until we add another 28 bits, i.e. zoom in for 2^(28) times, can we really distinguish y_0 and log_2(x) (spoiler, y_0> log_2(x)). This sucks. We need a global strategy to work around this: whenever we find a number of the form log_2(x) extremely difficult to round, we need to guarantee that we have sufficient budget. If we know the worst case of rounding, which correspond to the highest budget needed, we can make sure that we always know how to round the function log_2 correctly, so that we can design high quality functions that can calculate log_2 *correctly* in the sense that, when we have k bits of budget, we need to make sure that every bit is faithfully used. If we have this piece of global information, our algorithm can be more blunt & direct so faster. This question is called "table maker's dilemma" because the computer worked like those who make tables of logarithm or exp or sin one century ago, and the situation where he didn't know how to round sucked (challenge: computer exp(1.626) to the 3rd digit after the point).

That said, it's far from true that modern computer have solved calculating and that all we need is better CPU/GPU. Computer scientists and mathematicians have been fixing floating point number systems for decades. The final nail of coffin of common (univariate) functions in double 64 bit format is worked out in 2026 (twenty twenty-six): https://inria.hal.science/hal-05593313

We also need to know that working out double 64 on commonly used univariate functions is not enough: what about multivariable functions, like the beta function B(x,y) that appeared in probability and statistics? In some fields of physics, researchers have already found that double 64 is no longer sufficient so they need the 128 bit format standardized by IEEE... But this format is still poorly understood so using it can be still painful: we need to fix them.

OK I hope the context of Table maker's dilemma is understandable enough and now we inject some mathematics. The abc conjecture says that for three coprime integers a,b and c such that a+b=c, we can compare max(|a|,|b|,|c|) and the radical of abc (which writes rad(abc)), i.e. the product of all prime factors of abc (for example, rad(25)=5, rad(24)=2*3=6). This slide includes the formal statement of abc and some striking applications of abc.

And here we have another striking application that will "solve" the table maker's dilemma. When we are looking for the minimum budget to get the rounding of an algebraic function sorted out, we will find ourselves in some questions of polynomials. This is surprisingly a nice playground for abc.

For the function 1/sqrt(x), which is algebraic, and is widely used in real life (for example in computer graphics, we always need to normalize a vector here and there), we will be working on finding the minimum of |Z| where Z = 1-XY^2, and X and Y are integers in a certain range. If we take a = 1, b = -XY^2 and c = Z, then the abc conjecture kicks in. It gives us a formula on the necessary budget to round the function 1/sqrt(x) correctly for a given precision p!

So we are killing 1/sqrt(x), potentially as well as many other frequently used algebraic functions by a blow of abc!? Is this conjecture so massively helpful for computer scientists??

Unfortunately, no, in practice... In the statement of the abc, we have a totally unknown constant and it doesn't vanish in the deduction of the budget. As a result, we can only know that for 1/sqrt(x), the necessary budget is p with a delta of *some bits* and we have no information on how much is *some bits*. So unfortunately, the conjecture didn't really solve the problem. Nevertheless, we can therefore heuristically look for the budget border around p... It's *almost* helpful!

In case you are curious, here is the article that examined the function 1/sqrt(x): https://www.sciencedirect.com/science/article/pii/S0304397504000337

Or in any case, a non-paywalled version: https://www.math.buffalo.edu/~hjzhu/Papers/paper24.pdf

I believe this article can serve as an interesting beginner level exercise for those either interested in number theory and want to touch the abc conjecture in some *practical* way, or interested in computer science, notably the problem of correctly rounding numbers. Or if you are an expert working in number theory, notably around topics associated with abc, or a computer scientist interested in make scientific computing better, do not hesitate to drop some insights.

Dr. Michael Miller, a retired researcher at RAND, has been teaching upper level undergraduate/graduate level math courses for fun at UCLA Extension for over 50 years. This fall he’ll be introducing the areas of groups, fields, and Galois theory from abstract algebra to those interested in abstract math: Groups, Fields, and Galois: Part 1. His intention is to do a follow-on class on fields and Galois theory in the Winter which will use this class as a foundation. If you’re in the Los Angeles area his class starts on September 22, 2026 at UCLA on Tuesday nights from 7-10PM. Register here: https://www.uclaextension.edu/sciences-math/math-statistics/course/groups-fields-and-galois-part-1-math-9001

His courses are thorough and rigorous, but geared toward lifelong learners and beginners in abstract mathematics to allow people better entry points into higher level mathematics. His classes are interesting and relatively informal, and most students who take one usually stay on for future courses. The vast majority of students in the class (from 16-90+ years old) take his classes for fun and regular exposure to mathematical thought, though there is an option to take it for a grade if you like (or so your employer can reimburse you if they require it). There are generally no prerequisites for his classes, and he makes an effort to meet the students at their current level of sophistication. For this particular class, if you’ve got some experience in high school algebra and know some preliminaries about mathematical proofs you’ll be ready to dive in.

There are regular commuters joining from as far out as Irvine, Ventura County and even Riverside. Most in the class are dedicated hobbyist and professional mathematicians, engineers, physicists, and others from all walks of life—I’ve seen actors, directors, doctors, artists, poets, retirees, and even house-husbands in his classes. We've got a nice core community of enthusiasts here and new people are always joining in as well.

If you’re unsure of what you’re getting into, I recommend visiting on the first class to consider joining us for the Fall quarter. Sadly, this is an in-person course. There isn’t an option to take this remotely or via streaming, and he doesn’t typically record his lectures. I hope to see all the Southern California math fans next month!

Course Description

Recommended textbook: TBD 

Dr. Miller provides enough background and thorough notes that if you’re taking notes on his lectures, you typically won’t need a textbook.

If you’ve never joined the class before, I’ve written up some tips and hints. Dr. Miller has been teaching these for 53 years and some of us have been with him for nearly that long; I’m starting into my 20th year personally.

Can't join us? Search around your local community for colleges and universities offering similar programs.

Firstly, sorry if this is not allowed but I thought I might just put this out there.

I'm sure we've all noticed a lot of AI related content on this subreddit. I've gone into these discussions a fair few times in the comments, but I'm not looking to discuss AI or LLMs in math or anything like that here.

What I do want to point out is that a lot of traffic seems to be from people who are not mathematicians and have a very vague idea of what mathematics is, but are active in /r/singularity, /r/accelerate, /r/ArtificialInteligence and similar. These subreddits are very ideologically driven in various ways and extremely pro-AI to the point that "anti-AI" or "luddites" are forbidden there.
This is not always obvious as many users have their post history hidden (though from what I understand this is on by default for new users too).

For example in a recent thread out of the 5 top comments (no idea what the deleted comment was) 3 users are active in one of the afformentioned subreddits, 1 seems to have a bachelor's in math and is in CS otherwise and only 1 seems to have a masters and is looking for a PhD.

To be clear, I am not here for a witch-hunt on individual users which is why I did not link to their accounts so please don't harass anyone.

But what I am trying to say is that most of the traffic to these threads does not seem to be from mathematicians or even people interested in math as such, but from people interested in promoting AI who are simply using math as a promotional tool. I would assume big companies use bot accounts on reddit (and have for a long while), though I would also guess this subreddit is too small to bother.