r/math

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.

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 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.

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

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?

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.

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/

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.

https://preview.redd.it/kdii8z5qc6kh1.png?width=418&format=png&auto=webp&s=97fd825df0f7a3a195434915fec57eef8d9f7b26

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

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 :)

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

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!

A consensus is emerging among respected mathematicians that there is a decent chance AI will exceed humans in both brute force verification and complex, creative problem solving at the highest levels. Few frontier theorems will be proven by humans alone, perhaps, in a matter of years.

More controversial, AI may also outpace humans in shaping the correct definitions, building theories, and making connections between disparate areas of mathematics, oft considered the peak of human creativity in mathematics. New areas of mathematics may be created without much human guidance.

Perhaps mathematicians become as helpful to AI as toddlers are to mathematicians. One cannot confidently rule out this scenario - Terrence Tao may find himself completely useless in building a rich, beautiful body of new mathematics.

In such an extreme scenario, humans would still matter in mathematics!

Lockhart's Mathematician's Lament argues for the intrinsic beauty of mathematics as being of primary importance. Humans, as knowledgeable appreciators of beauty, thus play an important a role as spectators and enthusiastic amateurs in mathematics, even if they cannot be world-renowned "competitors" in theorem-proving and theory-building. This mirrors the situation in chess, where the vast, vast majority of human chess players and appreciators will never contribute to the frontier of advanced lines, and arguably even the most skilled like Magnus Carlsen rely on AI to develop their strategies, and would be crushed by such AI in competition. Being completely uncompetitive does not make chess playing and appreciation valueless.

> Amateur: from French amateur "one who loves, lover"

But there is more beyond this. Mathematics allows you to understand things that are otherwise impossible to understand. Some of these are important for fairness and justice: Arrow's impossibility theorem, statistical bias, observer relatively and other tricky concepts around coordinate systems (map != territory), locally-trivial globally-nontrivial (global obstructions), forgetful maps to extract the essential structure and remove irrelevant details, limits of computation, etc.

Understanding such mathematical concepts allows you to make moral judgements in ways that would be impossible otherwise. Some super-smart machine might tell you Arrow's theorem is true, but internalizing it yourself gives you the rich understanding of fairness in democracy necessary to consciously shape it. As with humans surpassing the capabilities of their own eyes with optical then radio telescopes, we are not impoverished by using tools that allow us to extend our reach into things we can never directly perceive or understand.

It can be frightening because the life's work of someone of the previous generation can be reproduced and surpassed flippantly. Gauss himself spent a significant amount of time manually factoring prime numbers by hand, a tedious exercise upon which his conjecture on the distribution of primes (the prime number theorem) was based. Gauss died before his conjecture was proven. His notebooks full of rote calculations could be reproduced today in a fraction of a second so short you could not perceive it. Anyone today repeating an endeavor like Gauss by hand would be thought a fool, just as an astronomer who refuses to use a telescope.

That doesn't make the pursuit of understanding pointless. As the limitations of our use of AI will stem from limitations of our own minds, it will still be profoundly rewarding to practice mathematics. Indeed, we may spend less time performing rote exercises and miring in false conjectures. Already the body of mathematics is too large for any single person to understand. One can pessimistically reduce mathematics to mechanics, or optimistically find meaning in your particular path through the mathematical version of the library of babel. Because we shape our minds, our society, and our world with mathematics, we will always matter as sentient beings who reify mathematics by subjecting ourselves to reason, and better ourselves because of it.

This recurring thread will be for general discussion on whatever math-related topics you have been or will be working on this week. This can be anything, including:

* math-related arts and crafts,
* what you've been learning in class,
* books/papers you're reading,
* preparing for a conference,
* giving a talk.

All types and levels of mathematics are welcomed!

If you are asking for advice on choosing classes or career prospects, please go to the most recent Career & Education Questions thread.

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!