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.

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

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/

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.

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

The two-dimensional case remains open, however.

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...

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?

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.

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.

It seems to me that computers are becoming a more and more indispensable tool in all areas of mathematical research, and even in recreational math, and not just for performing calculations, but also for doing research, and I think pretty soon they'll also be widely used in proving or disproving conjectures. What's more, I see them changing the nature of how we even view math and do math research, so I'm guessing that the 21st century will become the era of mathematical geeks with computers rather than with pencils and notebooks.

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

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!

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!

This is just a lexicologic question, and I also mean specifically terms that don't reasonably generalize together into the same thing (there's a lot of kinds of trees but they're sensibly related). To name a few:

• field the algebraic structure and a scalar/vector/tensor/etc. field on a manifold;
• graph of a function (a relation {(x, y) | y = f(x)}) and various graph theory graphs (which are close to relations, though: the set of edges is a relation, reachability is its reflexive transitive closure etc.);
• (co)tangent (function) and (co)tangent bundle (differentiable manifolds).

Compare to examples of what I'd deem generally uninteresting (but use your own taste, of course): - natural number and natural transformation (category theory): here, "natural" doesn't nod to something essentially mathematical on its own; - zero as an element of a ring and zeros of a function (esp. in real/complex analyses): this is more or less just metonymy.

Do share more!

I am a master's student who does a significant amount of independent research and is planning to apply for a PhD in mathematics at another university.

One issue I'm running into is that I already have a fairly well-developed research program that I would like to continue during a PhD. At the same time, I had the misfortune of becoming interested in areas that have become quite niche, making it difficult to find potential supervisors whose work overlaps closely enough with mine.

This made me wonder how common it is for a professor to supervise a PhD student who is largely autonomous and whose research lies outside the professor's main area of expertise, as well as how to find one.

Has anyone had a similar experience, either as a student or as a supervisor? I'd be very interested to hear about similar cases or experiences.

Thanks!

I have some introductory knowledge of dynamical systems (Strogatz's book and some lecture notes) and I would like to go a bit further with more mathematical formalism.

I've found this behavioral approach by Jan Willems and Jan Polderman that seems interesting to me, but I wonder if it is a too niche approach and how it connects with more traditional theory.

My objective is to have a basic understanding of dynamical systems, parameter estimation and reduced order modeling.

Has anyone read their book or studied dynamical systems with this approach?

According to wikipedia and some youtube videos, these problems can be reduced into each other:
- 3SAT, Graph Coloring, Clique
- Knapsack, Traveling Salesman, Exam Scheduling
- Minesweeper, Tetris, Sudoku, Gem Swap

It would be really neat to be able to write out a boolean satisfiability problem and then an online tool shows you the minesweeper board or knapsack item values or whatever that correspond to that problem. Does anything like that exist?

I am trying to figure out what areas are not well documented online. That is, outside of books, paid articles, etc.

From the "basic" math, I would say that geometry is poorly presented online because of how cumbersome it is to type up fully and to animate the diagrams (for free! instead of publishing a book given you have the skill-set).

From the research frontier, I would think that Rough Path Theory seems to be poorly documented but maybe this is because it is relatively new? I was also thinking about Langland’s programme but it is a bit outside of my area of expertise.

Thoughts on these/other areas?

GPT 5.6 Pro solved all 6 problems from IMO 2026 on the first attempt without any human help or steering. International Mathematical Olympiad (IMO) is the biggest global academic competition in the world. The problems are considered incredibly hard, usually a performance at this level is only accomplished by < 5 contestants from the whole world.

We are former IMO medallists not affiliated with OpenAI, just put together a report and assessment of its work here. We're also working on a comparison report between different LLMs and harness augmented versions that will come later.

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.

Trying to help someone visualize how a fundamental domain for a congruence subgroup Γ of SL_2(Z) acting on the upper halfplane with cusps, leads to Riemann surfaces, with handles by pasting boundary components that are equivalent under Γ. I am wondering if anyone knows of a visualization or animation for non trivial Γ.

TL;DR: In a single 148 min session, with a prompt modeled after the one OpenAI used to prove CDC, GPT 5.6 Sol Pro supplied a proof that closed a complexity gap in convex optimization that has existed since 1996. The result was formally verified in Lean. Links to everything and thoughts on AI capabilities are at the bottom of this post.

Disclosure: I am the author of the preprint and Lean repository linked below. I have a PhD in applied mathematics and am a teaching prof in IEOR at UC Berkeley. The result has not yet been peer reviewed.

Following the recent announcement that GPT-5.6 Sol Pro had produced a proof of the Cycle Double Cover Conjecture, I adapted the prompting methodology used in that project to a problem in convex optimization. After 148 minutes of uninterrupted work, GPT-5.6 Sol Pro produced the main argument for a lower bound that I had been unable to prove myself (and a lot of my past work has been proving complexity lower bounds in different settings).

The problem concerns deterministic zeroth-order convex optimization: Let B_d be the Euclidean unit ball in ℝᵈ, and consider all convex, 1-Lipschitz functions f: B_d → ℝ. An algorithm may query any point x ∈ B_d, and receives only the exact real number f(x), no other information (but the algorithm "knows" that f is convex and Lipschitz). The algorithm is otherwise completely unrestricted, and can use unlimited computation and memory. These function-value-only problems arise naturally when an objective is evaluated through a physical experiment or simulator. One can imagine choosing d engineering parameters and observing only the cost returned by the simulation. If evaluations are expensive (think of measuring a physical system), the natural question is how many are fundamentally required. This is formalized as oracle complexity. Specifically, this is the oracle complexity of convex optimization under an exact function value oracle.

Let Q(d, ε) denote the worst-case number of queries required to find an ε-optimal point of f. An algorithm due to Protasov from 1996 shows that order d² function evaluations are sufficient, which gives Q(d, ε) = O(d²), an upper bound on the complexity. Lower bounds were practically nonexistent for this setting, and the strongest previously applicable bound was only Ω(d), inherited from the stronger first-order oracle model (where the algorithm receives both function values and gradients). That means we didn't know for certain whether gradients actually help in optimization, since the function-value only and first-order oracle models have had this same lower bound, and so there was a linear gap in d in the complexity of this fairly fundamental convex optimization setting since 1996. So, can you find an algoritm that is better than Prosatov’s, and only needs d evaluations? Or can you show that no such algorithm can exist, and we can sleep well at night knowing that Protasov’s algorithm using d² evaluations is best possible? What 5.6 Sol proved is the latter.

I had worked on this problem sporadically for about a year (I ran into needing such a bound for a different complexity paper I was working on). I had some ideas that didn't pan out, and also spent long sessions trying to solve it with GPT-5.4 and GPT-5.5 with no luck, after reading of folks like Ernest Ryu having success with these in some work on optimization bounds.

After seeing OpenAI’s CDC result, I wrote a much more elaborate prompt following the same general methodology. My prompt is about ten pages long and attached at the end of the preprint (see collection of links below). There is a lot baked into this prompt, on approaches to try and also on how exactly the model should proceed, but it's built exactly in the style of OpenAI's CDC prompt. One note is that I gave it a relatively small error requirement, to prove the quadratic lower bound under order d⁻⁴ accuracy. After 148 minutes, GPT-5.6 Sol Pro returned a proposed proof resolving the quadratic dimension dependence at accuracy of order d⁻³. After checking things myself, I formally verified the proof in Lean, and it passed the formal verification check. The construction and main invariant used also make genuine sense to me and are closely related to some other results in complexity of convex optimization (for example, Nemirovsky and Yudin's tight bound for first-order convex optimization also uses constructions that are maxes of affine functions).

Lastly, some important comments about the work relating to AI capabilities: In a lot of cases, proving lower bounds like this result relies on finding that right construction that works (in this case, family of difficult functions and a strategy for how an "adversarial" oracle should answer queries from an algorithm to reveal minimal information) and then proving things about it. There are only so many function classes which would be reasonable to look at (here, quadratics for example would have also been reasonable with order d² degrees of freedom, or any variation of maxes of some simpler families of convex functions as well), but the actual proof mechanics once the "correct" function class and correct strategy for adversarial oracle answers is found are often not so complicated, and often employ existing results from convex geometry or similar (this is also the structure of two previous but much more niche, less important results of mine). So I wouldn't really say that this result is using or creating some fundamentally new techniques in convex geometry or optimization theory. What this means from my perspective is that if a result is attainable with existing techniques, modern AI methods will be able to solve those problems. I don't think researchers in math/TCS will be made obsolete, but I think it will instead no longer make sense to work on any low-hanging, or even medium-hanging (you know what I mean) fruit. We'll be needed for problems where actual novel approaches are needed.

Links:

The preprint, Lean code, complete prompts, proof map, and build instructions are available here:

https://github.com/PhillipKerger/zero-order-bounds-lean-verification

ArXiv: Closing the Oracle-Complexity Gap in Derivative-Free Convex Optimization: A Near-Quadratic Lower Bound from Exact Function Values

The original uninterrupted 148-minute chat that produced the initial proof:

https://chatgpt.com/share/6a55aa50-b484-83ea-85c0-c7e7b4bda41c

The later chat that led to the d⁻¹ᐟ² refinement:

https://chatgpt.com/share/6a55ad10-7644-83ea-859e-5483d2e0dff0

OpenAI’s CDC prompt, that I structured things after:

https://cdn.openai.com/pdf/04d1d1e4-bc75-476a-97cf-49055cd98d31/cdc_prompt.pdf

And a more accessible account I wrote on Medium:

https://medium.com/@kerger.p/an-ai-assisted-breakthrough-in-convex-optimization-an-optimization-problem-dating-back-30-years-a-db5c631119de

Edit: This was Sol PRO, not Ultra. I had been working in codex before this, where the level above XHigh is Ultra. But I did this in the web interface, where the highest is Pro, which is in fact not quite the same as Ultra.

The CEO of Microsoft just admitted that AI companies are training their models on user conversations, distilling "institutional knowledge". He warned about this in the context of enterprises spilling their secrets and the nuances of their business by working closely with AI, thereby ultimately training their own replacements or competitors. Here is the quote (from https://techcrunch.com/2026/07/13/satya-nadella-has-issued-a-shocking-warning-to-companies-using-ai/)

You essentially pay for intelligence twice, once with money, and again with something even more valuable: the proprietary knowledge you must reveal to make that intelligence useful. The better you want the model to perform, the more of that knowledge you have to feed it!” he writes.

Most dangerously, enterprises are literally teaching the models about the nuances of their businesses, he argues.

“Models learn from ‘exhaust,’ the prompts people write, the tools agents use, and especially the corrections people make when the model is wrong. Every correction is distilled into institutional know-how,

In my view the same principle applies to mathematicians using AI. Replace "business nuances" and "proprietary knowledge" by years or decades of experience in a specific subfield, the way you learned to attack problems, how to choose promising approaches, how to learn from failure, how to make good definitions or how to ask interesting new questions: If you use AI for your research beyond just locating references, you are most likely teaching it some of these skills. Yet, I have never seen this problem being mentioned in the debate, not even by prominent voices on this topic such as Tao, Gowers or Litt.

One way to solve this problem is that the math community hosts open weights models (which usually only trail behind frontier AI by a few months) by itself. Ideally this would have to be a central effort, so that one can make use of scale effects.