There is a big hype over LLMs solving problems in some areas of mathematics.

Now, I'm not here to ask whether this will "replace mathematicians" (whatever that means) or not. That's already being done and a post like that is probably being typed as I'm typing this.

What I'm interested in, assuming that LLMs become better at solving problems than humans, how will we train new PhD students?

This seems like something which could drastically lower the general proficiency of mathematicians. Will a mathematician become just a person who checks LLM outputs to see whether they are correct?

Could someone familiar with algebraic geometry give some insights about this?

AI, for the last year, has been excellent at proving lemmas. I'm beginning to wonder, to what extent hirers should begin to be suspicious as to whether a student/postdoc with good preprints and publications from a minor university - or who had been an unproductive before - is actually talented vs just being early adopter who paid OpenAI $200 in 2025? There are already some young people in my field that I'm a bit suspicious of, as the quality of their publications has no correspondence whatsovever to the ideas they generate during real time discussion. This was historically an issue with collaborations, but we could at least get letters from coauthors there, but now do we need to be similarly suspicious of single author publications?

Is this just eventually going to mean that the only postdocs being hired in 2028 are from the top universities? I believe this is the already the norm in other subjects like English.

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.

In these examples, the numbers seem to melt away. There is a physical tangible conundrum / toy / puzzle that needs abstract mathematics to solve. It's permutation adjacent. Somewhere in discrete maths. What else is there? I know about all the above. Any other cool things? I want a rabbit hole to jump down. I'm not a mathematician, I'm too poor, and I'm also kinda stupid (I blame it on being poor). But I think these are really cool. it's just neat ig.

Math-heads seems to be mainly interested in geometric calculations, calculus, 4d shapes, topology, and logic puzzles. I AM NOT SAYING THIS TOPICS ARE LAME, I simpllyyyy am currently interested in those mentioned above. Any other tricks for puzzles and rabbit holes to jump down?

Thanks in advance for any help. Gracias! Merci!!!!

Say I have constructive proof and I want to write a code on Sagemath. I don't know much about coding so I take help from AI. So while publishing the paper what should I do? I'm confused should I add the code or not

If anyone knows how to paint a doomy gloomy future it's /r/math, so here's an open ended question for all of you.

No one can predict exactly to which extent the rise of LLMs will change how the workday of a working mathematician. The only thing that is certain is that the future is rather uncertain. The doomiest comments on this subreddit also show that this uncertainty is particularly tough on people who are early on in their studies, and who would probably like to know to which extent it is useful (in the sense of putting bread on the table as opposed to, say, having fun) to pour a lot of energy into learning how to proof stuff.

Now, for even younger people – kids in elementary school or high school say – the future is even more uncertain.

So my question is: has the rise of LLMs changed how you would advise a younger kid to navigate the future? Chances are that it will always be useful to have a good baseline understanding of maths and logic, but would you be less inclined to advise a skilled high school student to pursue a theoretical career? Or would you have more reservations about enabling a particularly gifted elementary school student, so as to not accidentally set them up for a bumpy course?

Or on a more practical level, have you changed your guidance on which skills are the more useful to acquire given that, for example, software development in particular has have its nature completely changed?

I know that this is adjacent to the “Career & Education” thread, but I'm more interested in the high-level picture than in personal advise.

Normally I would be extremely skeptical, but the result is checkable by simple computation. Remarkable!

The two-dimensional case remains open, however.

Team:
People's Republic of China 1 232
United States of America 2 207
Russia 3 196
Singapore 4 169
Viet Nam 5 168
Republic of Korea 6 167
India 7 166
Democratic People's Republic of Korea 8 163
United Kingdom 8 163
Japan 10 159
Brazil 11 155
Romania 12 152
Canada 13 151
Iran 14 147
France 15 146
Hungary 16 144
Türkiye 16 144
Israel 18 143
Bulgaria 19 142

Individual, gold medal perfect:
Leyan Deng People's Republic of China 1 G 42
Che Liu People's Republic of China 1 G 42
Bolun Zhang People's Republic of China 1 G 42
Hyeonjun Lee Republic of Korea 1 G 42
Alex Chui United Kingdom 1 G 42
Liam Reddy United States of America 1 G 42
Alexander Wang United States of America 1 G 42

https://www.imo2026.com/Results/Individual_Results.htm

https://www.imo2026.com/Results/Team_Results.htm

Not yet on the official website: https://www.imo-official.org/editions/2026/

I know just enough geometric measure theory to pretend to know what I’m talking about, so if all of this post is nonsense feel free to disregard.

I’ve been on a bee binge recently (likely also drinking far too much mead) as I distract myself from how inadequate I feel trying to do math and stumbled upon the honeycomb theorem.

Read the proof for 2 dimensions and understood slightly more than nothing so then observing the machinery I naively assumed this actually had a lot of applications in industry if you could generalize it.

As far as I could see there have been proposed structures as solutions for 3 dimensions but those aren’t proven and when we get higher than that we know basically nothing.

What interests me is that there is apparently a non trivial link between this and vector quantization.

Is this a real active area of research or am I way off base? The machinery here seems above my pay grade even if the required mathematical maturity may not be

ex:

Alice: I shrug big at number theory, it just doesn't spark anything for me.

Bob: It's a handy medium for learning proof techniques. Also cryptography has neat stuff going on in both applied and pure settings. One thing in particular...

Looking for some inspiration for a math themed party for my son's 4th birthday. My son's been really into numbers/math/geometry/time for a while, so I knew I'd have to do this at some point. I like doing a different theme for each birthday because it makes it more fun and easier to plan out decorations.

I'm looking for mostly food ideas and entertainment ideas, since that's where I'm struggling with the most. For decorations I was thinking of printing out a bunch of different equations that equal 4. Like 3 + 1, √16, 8/2, 2026 - 2022, etc. and getting some number balloons.

The cake will be a working clock cake, since he also loves clocks. And I've been trying to think of ways to include fractions in the food. Like having cupcakes or mini pizzas or something with a slice taken out and putting the fraction on it. I don't know. Maybe I'll get a pie and write the digits of pi on it (he gets excited about that kind of stuff). Any other ideas would be really helpful.

I was originally thinking of hiring a science experiments entertainer and I know he'd love that, but the recommended age for most of the ones in my area is 5+. Not sure if I should abandon the math theme for the entertainment and just get something fun like a bubble performer.

For party favours, I was thinking of mini cutesy calculators or those animal shaped measuring tapes. Would kids like that? I know my son's into that stuff, so it's hard for me to gauge what other kids might like. I still want it to be a fun party for them. There'll likely only be 2-3 other kids there his age.

Last week, I wrote a post about the motivation for functional analysis---this is currently my #1 most upvoted post on Reddit, so I figured I should do a follow-up. (The poll at the end of the post told the same story.) Thankfully, I already had something in mind: what is a Hilbert space, and what is it used for?

A surprisingly common, but erroneous answer is that it comes from quantum mechanics. It is true that Hilbert spaces entered into the physics literature via quantum mechanics, and that this connection bolstered their development. But Hilbert spaces came first, and you can already see their utility just from Fourier series, which is entirely classical. We'll see how it helps answer some of the problems we left unsolved in the previous post.

Read the full post (for free) on Substack: WTF is a Hilbert Space?

This post got some attention over at r/soccer, but I figured the folks here might appreciate the math somewhat more.

There had been a question as to how FIFA picked the match-ups involving the qualifying third-place teams at this year's World Cup. FIFA provided a giant 495-row lookup table depending on which teams qualified, but it seemed like nobody had figured out how they came up with this table.

It turns out that that FIFA used maximum-weight perfect matchings: they had a secret weight vector on the match-ups, and they picked the perfect matching with highest total weight. Showing that this is not a coincidence - that is, that most potential choices of match-ups do *not* admit such a secret weight vector - is a fun exercise in high-dimensional geometry. I definitely didn't expect the first time I'd use Schläfli's inequality to be in soccer analysis!

Take a look at the link above.

Currently training in masters for applied mathematics and I'm looking for reading material where difficult math problems have emerged from thinking about social justice and the solutions have helped communities in one way or another.

Papers I'm finding in this domain tend to be public policy which is not quite what I'm looking for.

So I am interested in stochastic matrices P of size N×N such that for any initial distribution x, as k goes to infinity P^k x goes to (1/N, 1/N, ..., 1/N), the uniform distribution.

I am curious what general properties such matrices have. For example, I have the feeling that such matrices must be symmetric, but I have no clue how to go about proving or disproving this.

Any suggestions on how to get started and what to read and such when studying a problem like this?

Sorry for the weird title. I wasn't sure how to describe it.

Basically, I'm looking for ideas on how to make things like addition, subtraction, division more visual. Something similar to how a clock is very visual.

I've noticed that my son is able to do mental math very easily whenever time is involved but sometimes struggles if it's just plain numbers. For example, if it's 9:28 and we're waiting for someone to come at 10:00, he can instantly tell me there are 32 minutes left. He can also instantly convert minutes into seconds (like 3 minutes is 180 seconds). But if I ask him what's 9 + 5, he'll sometimes struggle with that and need to use his fingers or a number line. My theory is that clocks are very visual, at least more so than a number line or doing addition with blocks. I'm wondering if there are other things I can use to make basic arithmetic more visual.

He's still quite young, so none of this technically matters but he loves math and is a self-learner. He learned to read a clock pretty young and has a good grasp on double digit addition, his times tables, fractions, and percentages. Most of his play is all very typical and we still focus on pretend play and socialization, but I just figured it doesn't hurt to help him bridge any gaps he's missing while he's playing with numbers, since his understanding of it is all over the place.

The efforts over two years of a group of authors led by Kato reached the conclusion that Mochizuki’s IUT-based proof of abc is unformalizable, but they reserve judgement since Mochizuki has recently evolved on certain points. Kato is posting about this on x here

Here's the report and an interesting quote from the final section of the report

Of course, it should be noted here that there are also several points in common between our analysis and that of Scholze-Stix. Perhaps the most important common point is that both reports point out a problem in the “process of deriving Corollary 3.12 from Theorem 3.11,” and that this issue relates to the “identification of copies of the real number line R.” However, to elaborate further on the former point, although Scholze-Stix went on to argue that “the suggested proof has [a problem] so severe that, in [their] opinion, minor modifications will not rescue the proof strategy,” we are not making any claims regarding the possibility or difficulty of remedying. Furthermore, regarding the question of whether a proof of the “abc Conjecture” exists, while many LANA members hold the view that “the original paper does not contain at least a formalizable proof,” the members were unable to reach complete consensus on this point.

Ngl, this looks somewhat bleak for IUTT, to put it mildly

In school, people are often taught the quadratic formula, but almost never told that there is a formula to solve cubic and quartic equations (like x^3 + 4x + 2 = 0 or x^4 - 5x^3 + 6x^2 - 7x + 8 = 0).

This is for a good reason: the cubic and quartic formulas are considerably more complicated than the quadratic one! It took people a long, long time to discover them.

However, the French mathematician Galois had a wonderful idea that allows people to re-discover the cubic and quartic formulas much more efficiently: one could use symmetries of numbers to derive the formula. When people discuss Galois theory, they usually use it talk about a negative result: Galois theory proves there is no formula to solve quintics. But this positive result uses the same basic ideas, and has the benefit of giving you a cubic formula at the end!

At https://hidden-phenomena.com/articles/quartic , my friend and I wrote a blog post to explain how to use the symmetries of numbers, a la Galois, to derive the quartic formula. Next week, we'll explain how to solve the cubic formula.

This order might seem funny to you, but actually it follows history: the Italian mathematician Ferrari discovered how to solve quartics in terms of cubics, and then later his teacher Cardano found (by asking Tartaglia...) how to solve cubics! So, Ferrari knew how to solve quartics in terms of cubics before he knew how to solve cubics.

-----

For experts, here I will say a little about the modern Galois theory way of describing this solution, but if you don't know Galois theory, please read the blog post https://hidden-phenomena.com/articles/quartic instead, as it is entirely elementary!

Anyway, here it goes. There is a surjective group homomorphism S_4 -> S_3 (coming from the fact that 4 = 2+2 in three ways), with kernel Z/2 \oplus Z/2. In particular, Z/2 + Z/2 is a normal subgroup of S_4, and so Galois theory tells us that if L/k is any S_4-extension, say with k characteristic 0, then there is an intermediate field F so that F/k is Galois with Galois group S_3, and L/F is Galois with Galois group Z/2 + Z/2. In particular, L is obtained by adjoining two square roots to F; the cubic formula will tell us how to build F from k with cube roots and square roots, so that L can be built out of k from cube roots and square roots, and hence we can find a quartic formula using cube roots and square roots!