I used ChatGPT to solve an open problem in convex optimization. *Part I* (1/N)

I crossed an interesting threshold yesterday, which I think many other mathematicians have been crossing recently as well. In the middle of trying to prove a result, I identified a statement that looked true and that would, if true, be useful to me. 1/3

New from @ArminPCM : how Snorkel builds reinforcement learning (RL) environments that train and evaluate agents in realistic, enterprise-grade settings.

Another AI-assisted proof. This time in optimal transport / convex analysis by @AdilSlm This report details the interactions with the LLM and is a very informative case study on how to work together with ChatGPT.

I don’t know what the endgame will look like. Whether an LLM will one day autonomously prove a major unsolved math problem remains to be seen, but I am certain that LLMs will greatly accelerate progress in mathematics. (4/4)

This will also happen in mathematics. One by one, different fields will begin to pick the low-hanging fruits of the LLM-assisted research era. Occasionally, there will be contributions that feel distinctly non-LLM, and those will be celebrated as flashes of human genius. (3/4)

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In Chess, Magnus Carlsen said: “There was a period … where you could very clearly see which players have been using these [AI] and which players didn't. [We] got into it … got an advantage over basically everybody … it just made us understand the game a lot better.” (2/4)

I firmly believe we are at a watershed moment in the history of mathematics. In the coming years, using LLMs for math research will become mainstream, and so will Lean formalization, made easier by LLMs. (1/4)

For reference, here are the links to the thread series: Part I https://t.co/6HDAr6y8Z9 Part I(b) https://t.co/GygToZ3EJa Part II https://t.co/v9qi1P9e80 Part III https://t.co/iWx5fCL6u9 (2/2) https://t.co/6HDAr6y8Z9

Preprint on using ChatGPT to resolve a 42-year-old open problem (point convergence of Nesterov’s accelerated gradient method) is out. Mathematical results are complete, though still need to expand the discussion of historical context & prior work. (1/2) https://t.co/Dmd9huMjXS

Thoughtful commentary. 100% agreed.

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(1/4) An example from Ernest highlights an essential point. a) GPT or AI can undoubtedly accelerate scientific discovery — people already know this fact But more importantly, b) domain knowledge still matters. You must know what the right research question is

@RaduIoanBot This concludes my three-part Twitter series. Now that we have the mathematical results sufficient for a publication, I will post our work on arXiv on Monday. After a week or two of polishing the writing and gathering feedback, I’ll submit it for peer review. 14/N, N=14

@RaduIoanBot Again, ChatGPT is now at the level of solving some math research questions, but you do need an expert guiding it. I strongly encourage fellow mathematicians to try incorporating AI assistance into their workflow. It takes some getting used to, but it can be worth it. 13/N

@RaduIoanBot Overall, this entire journey took just a week, less than 30 hours of my time. ChatGPT’s assistance provided a significant speedup, and without it, I would most likely have given up after three days of slow progress. (As I did in the past.) 12/N

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@RaduIoanBot Conclusion: Starting from the key idea of the first tweet, we extended the convergence result to several related settings and resolved the main 42-year-old open problem, with ChatGPT doing most of the heavy lifting along the way. 11/N

@RaduIoanBot They also argue that the FISTA method [Beck, Teboulle 2009] exhibits point convergence. Check out their work when it goes public! (As far as I know, [Bot ̧ Fadili, Nguyen 2025] did not use AI tools.) 10/N

@RaduIoanBot The arXiv preprint of [Bot ̧ Fadili, Nguyen 2025] will go public in a few days. They also prove weak convergence in the infinite-dimensional Hilbert space setting. (ChatGPT will now be happy.) https://t.co/JJqkuqKcd3 9/N

In particular, Radu Ioan Boţ, Jalal Fadili, Dang-Khoa Nguyen reached out with a preprint of their own, capitalizing on the ideas in the continuous-time proof to also establish point convergence of Nesterov 1983 and more! @RaduIoanBot 8/N

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After the initial tweet went out, several colleagues in the optimization community reached out with excitement and amusement. It was a lot of fun reconnecting with old friends. And I now have some interesting conversational material for the next optimization conference. 7/N

In 2016, Kim and Fessler introduced a variant called OGM, which improves upon NAG by a constant factor of 2. We also show that OGM exhibits point convergence. 6/N

The proof, cleaned up and typed up by me, resolves the 42-year-old open problem: 5/N