Fascinating discussion including showing how LLMs are helping push the state of the art:
> I still did not see how to prove this inequality, but I decided to try my luck giving it to ChatGPT Pro, which recognized it as an {L^1} approximation problem and gave me a duality-based proof (based ultimately on the Fourier expansion of the square wave). With some further discussion, I was able to adapt this proof to functions of global exponential type (replacing the Fourier manipulations with contour shifting arguments, in the spirit of the Paley-Wiener theorem), which roughly speaking gave me half of what I needed to establish (2).
> As a side note, this latter argument was provided to me by ChatGPT, as I was not previously aware of the Nevanlinna two-constant theorem.
This mirrors my experience with these tools for math. Great for local problems and chatting through issues. Still can’t do the whole thing in one shot but getting there.
The proof proceeds by a modification of the Duffin–Schaeffer argument, together with the two-constant theorem of Nevanlinna (and some standard estimates of harmonic measures on rectangles) to deal with the effect of the localization. (As a side note, this latter argument was provided to me by ChatGPT, as I was not previously aware of the Nevanlinna two-constant theorem.)
Which I find funny in a way: I'm sure that almost no one here understands the article, grasps the significance of this problem in mathematics, or can meaningfully comment on the difficulty of solving it. But we'll still have opinions because the article mentions a popular tool some of us like, some of us dislike, and some are ambivalent about.
It would be surreal if a carpentry forum was regularly abuzz about mountaineering because climbers use a hammer-shaped tool.
Nevanlinna theory isn’t that obscure (in the sense of mathematics, I suppose) but it is very difficult (for me, probably less so for Tao) when working to have the whole of 21st century analysis in your head at once and see what could be applied where. I can see how an LLM would be quicker than a human at recognizing a context where a theorem from an apparently unrelated subfield could be applied.
From a few older posts, I estimate that there are at least 10 mathematicians here, some doing math research and some doing other stuff. This is bleeding edge math, so probably only 100 persons in the word are working in something close enough to understand this now. [I guess I can understand it if I take a month [1] to study this and drop everything else.] I worked in harmonic analysis [2], but this looks more related to maximals that is a topic that I tried to avoid. I'm not sure about the areas of the other mathematicians here.
Anyway, there are a lot of topics in the discussion in HN and many times someone can give some insight. Sometimes it's about compilers and there are a few users that know about that. Sometimes it's about rockets and there are a few users that know about that. Sometimes it's about steel alloys and there are a few users that know about that. ... Not everyone here is a programmer.
I'm using very little AI, but my wife has been using it a lot. We both agree that it's at the level of a Gold medal in the IMO, that means that it can kick our ass hard. Anyway, sometimes the AI hallucinates and sometimes makes stupid error, so it's necessary to verify the output.
[1] It looks short, so perhaps a week is enough, but I feel I'm being too optimistic. Let's say a month just to be safe.
[2] Protip: If you see a circle, take the Fourier Transform and cross your fingers. You can't believe how many problems it solves. If it's useful, add me as a co-author.
We've always been bikeshedders. For example, back in Slashdot days, some company would decide to migrate something from Windows to Linux. Immediately the debate became whether they should have gone with Debian or SuSE instead of Red Hat.
Fascinating discussion including showing how LLMs are helping push the state of the art:
> I still did not see how to prove this inequality, but I decided to try my luck giving it to ChatGPT Pro, which recognized it as an {L^1} approximation problem and gave me a duality-based proof (based ultimately on the Fourier expansion of the square wave). With some further discussion, I was able to adapt this proof to functions of global exponential type (replacing the Fourier manipulations with contour shifting arguments, in the spirit of the Paley-Wiener theorem), which roughly speaking gave me half of what I needed to establish (2).
> As a side note, this latter argument was provided to me by ChatGPT, as I was not previously aware of the Nevanlinna two-constant theorem.
This mirrors my experience with these tools for math. Great for local problems and chatting through issues. Still can’t do the whole thing in one shot but getting there.
Presumably one reason this is of interest here:
Which I find funny in a way: I'm sure that almost no one here understands the article, grasps the significance of this problem in mathematics, or can meaningfully comment on the difficulty of solving it. But we'll still have opinions because the article mentions a popular tool some of us like, some of us dislike, and some are ambivalent about.
It would be surreal if a carpentry forum was regularly abuzz about mountaineering because climbers use a hammer-shaped tool.
Nevanlinna theory isn’t that obscure (in the sense of mathematics, I suppose) but it is very difficult (for me, probably less so for Tao) when working to have the whole of 21st century analysis in your head at once and see what could be applied where. I can see how an LLM would be quicker than a human at recognizing a context where a theorem from an apparently unrelated subfield could be applied.
From a few older posts, I estimate that there are at least 10 mathematicians here, some doing math research and some doing other stuff. This is bleeding edge math, so probably only 100 persons in the word are working in something close enough to understand this now. [I guess I can understand it if I take a month [1] to study this and drop everything else.] I worked in harmonic analysis [2], but this looks more related to maximals that is a topic that I tried to avoid. I'm not sure about the areas of the other mathematicians here.
Anyway, there are a lot of topics in the discussion in HN and many times someone can give some insight. Sometimes it's about compilers and there are a few users that know about that. Sometimes it's about rockets and there are a few users that know about that. Sometimes it's about steel alloys and there are a few users that know about that. ... Not everyone here is a programmer.
I'm using very little AI, but my wife has been using it a lot. We both agree that it's at the level of a Gold medal in the IMO, that means that it can kick our ass hard. Anyway, sometimes the AI hallucinates and sometimes makes stupid error, so it's necessary to verify the output.
[1] It looks short, so perhaps a week is enough, but I feel I'm being too optimistic. Let's say a month just to be safe.
[2] Protip: If you see a circle, take the Fourier Transform and cross your fingers. You can't believe how many problems it solves. If it's useful, add me as a co-author.
It is deeply hilarious to watch software engineers become, at long last, bikeshedders.
What do you mean, "at long last"?
We've always been bikeshedders. For example, back in Slashdot days, some company would decide to migrate something from Windows to Linux. Immediately the debate became whether they should have gone with Debian or SuSE instead of Red Hat.
Think of this as an algebraic jewelers loupe. Zoom in on local structure and push aside the global. It’s fascinating.