In this thread I want to share some thoughts about the FrontierMath benchmark, on which, according to OpenAI, some frontier models are scoring ~20%. This is benchmark consisting of difficult math problems with numerical answers. What does it measure, and what doesn't it measure? I'll try to organize my thoughts around this problem. To be clear I don't intend any of this as criticism of the benchmark or of Epoch AI, which I think is doing fantastic work. Please understand anything you'd read as criticism as aimed at the problem's author, namely me.

Had a sort of a funny experience with OpenAI’s Deep Research tool, which I wanted to share since I think it reveals some of the tool’s strengths and weaknesses. .@srivatsamath recently suggested to me that (as a result of the Internet democratizing access to advanced math), there’s been an increase in important math research done by young people. I was curious if this is true.

Some very brief first impressions from my attempts to use OpenAI's new Deep Research project to do mathematics. I'm very grateful to the person at OpenAI who gave me access. Some short caveats: I'm just trying to evaluate the product as it currently stands, not the (obviously very rapid) pace of progress. This thread is not an attempt to forecast anything. And of course it's possible I am not using it in an ideal way.

Some brief impressions from playing a bit with o3-mini-high (the new reasoning model released by OpenAI today) for mathematical uses. First of all, it’s clearly a significant improvement over o1. It immediately solved (non-rigorously) some arithmetic geometry problems with numerical answers that I posed to it, which no other models have been able to solve. I consider these problems pretty tricky.

I want to explain in down-to-earth terms what this paper is about, since it ultimately boils down to what I think are some really concrete and fundamental questions. 1/n

https://twitter.com/mathAGb/status/1882657278710665216

If you ask a middle-schooler, they might tell you that mathematics is about "solving equations." What does this mean in practice for modern mathematicians, though? Often one abstractly proves a solution with some nice property exists, studies the set of solutions, etc. 2/n

I've recently been talking a bit about how difficult it is to carefully check even well-written mathematics. I want to try to explain something about this by telling the story of some errors in the literature that (in part) led to the two papers below. 1/n
These papers began with an attempt to prove an open conjecture in surface topology—the Putman-Wieland conjecture. At some point in mid-2021 Aaron and I were pretty convinced we had a proof, and began writing up the details. 2/n

A couple more brief thoughts on o3’s (incredible) performance on FrontierMath. First of all, everything I’m saying is based on the 5 publicly shared problems. I don’t know what the rest of the benchmark looks like, but based on these it’s clearly hard.

The correct answer is “Bob.” Congrats to the 10% who got it right — those few brave dreamers.

https://x.com/s1lknoose/status/1769177202816434655?s=46&t=41cpGRZavPniB0OfiIMsDA

Here are the numbers from 10 to 100 in alphabetical order:

18, 80, 88, 85, 84, 89, 81, 87, 86, 83, 82, 11, 15, 50, 58, 55, 54, 59, 51, 57, 56, 53, 52, 40, 48, 45, 44, 49, 41, 47, 46, 43, 42, 14, 19, 90, 98, 95, 94, 99, 91, 97, 96, 93, 92, 17, 70, 78, 75, 74, 79, 71, 77, 76, 73, 72, 16, 60, 68, 65, 64, 69, 61, 67, 66, 63, 62, 10, 13, 30, 38, 35, 34, 39, 31, 37, 36, 33, 32, 12, 20, 28, 25, 24, 29, 21, 27, 26, 23, 22.

OK, let me explain precisely what this means and why it's true. Let c be a complex number, and let R be the subring of the complex numbers generated by c. The differential operator

d=(d/dz-c/z)

acts on R[z, z^{-1}].

1/n

https://twitter.com/littmath/status/1729996751568859627

How does it act? Well,

(d/dz-c/z)z^n=nz^{n-1}-cz^{n-1}
=(n-c)z^{n-1}.

2/n

ChatGPT “proves” the cube root of 27 is irrational, then computes it to be 3, then admits it was wrong about its irrationality, and then finally, when asked to find its error, claims it was right all along. Undisputed king of BS.





o no

A math thread about some of the questions I've been thinking about lately. The goal is to answer some concrete questions about 2x2 matrices. Our starting point is the equation below--but it will take us on a journey through algebraic geometry, classical analysis, and more. 1/n Everything here is joint work with Aaron Landesman and Josh Lam -- if you're an expert, you can read the paper here:

2/narxiv.org/abs/2308.01376

One of my pet peeves is when a mathematician will claim some other area of math overcomplicates things, or is broadly uninteresting—most math (IMO) is pretty cool, and I think it’s good for the soul to approach other areas with a spirit of curiosity, rather than superiority. Of course it’s fun to post a screenshot of some complicated text and tweet “math is the language of the universe lol” or something—I’ve done this myself—but I think it’s best done in with a bit of self-mockery, rather than playing into some imagined rivalries.

In this thread I will try to explain what a short exact sequence is, with an interested 8th grader as the target audience. All I'll assume is some familiarity with systems of equations, which is typically a middle school topic. 1/n Most of you think this is impossible (see the poll below)! But I think it can be done, admittedly with some imprecision.

https://twitter.com/littmath/status/1648315697791942658

One of the most misleading things about learning mathematics is that you are seeing the *output* of a great deal of trial, error, sweat, and tears, presented as if it’s obvious, or at best a product of cleverness. I often try to explain how one might have discovered the proof I’m presenting, but even so one can’t show all the blind alleys one would go down in actual research.

Continued studying games in my Intro to Proofs class today, and one of my students beat me in a game! Couldn’t be more proud :).

https://twitter.com/littmath/status/1577799621920915456

(i’m not owned, i’m not owned…)

A thread on some of the math I've been thinking about lately. I think it's really fun: it involves connections between low-dimensional topology, algebraic geometry, classical analysis and ODEs, dynamics, representation theory, number theory, and more. 1/n This thread is meant for people with some background in math -- maybe partway through a math major, plus or minus a bit. If you're a professional, you can read the intro to the relevant paper here: arxiv.org/abs/2205.15352
2/n

Thinking about this beautiful thread again. If I understand correctly the proof works (essentially) by arguing that certain primes are not square in the 2-adic numbers; this cannot work for primes which are 1 mod 8, as they are 2-adic squares. And 17 is the first such prime!

https://twitter.com/andresecaicedo1/status/1302697810446483456

This suggests that in some sense the proof sketched in the thread does generalize to arbitrary primes p; one just has to choose the right auxiliary prime such that p is not a square in Q_l. And there are infinitely many such, by Chebotarev.

@YuriSulyma @sarah_zrf @SC_Griffith I don’t know a good de Rham-Witt formalism; things are much more subtle in positive/mixed charactetistic, and there are multiple inequivalent notions of connection (e.g. flat vector bundles are no longer the same as O-coherent D-modules)… @YuriSulyma @sarah_zrf @SC_Griffith I recommend Chapter 9 in Grothendieck’s “Dix Exposés” for one nice PoV on this. Basically the point is that (in char 0) a connection gives a rule for extending sections along nilpotent thickenings (that is, Taylor expansion), and you can formulate this as a sheaf on the site…

Different images generated by an AI art app (wombo.art) with the prompt “encyclopedia of alien botany” Some more:

i call it "silicon valley" yo i solved the trolley problem, it was easy