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Fields Medals 2026

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Re: Fields Medals 2026

#51

Post by GuB-42 »

siva7 wrote: post #20

"harmonic analysis and geometric measure theory, including applications of multiscale and decoupling techniques to the local smoothing conjecture for the planar wave equation, and major advances in Fourier restriction, Falconer distance sets, Furstenberg sets in the plane, and the Kakeya problem in three dimensions." I'm not sure there is another profession in the world where it's impossible to explain to a layman on…

Besides the usual jargon, what I find interesting is that the tradition of always naming things after their discoverers: Fourier, Falconer, Furstenberg, Kakeya. 4 names in one sentence.

Other fields do it, but it is almost systematic in math, and arguably, it makes things even harder to understand as people names are not descriptive.

Re: Fields Medals 2026

#52

Post by yorwba »

leonvoss wrote: post #49
orochimaaru wrote: post #44

Earlier quoted context omitted.

Not true. Novel mathematical methods precede their application by at least a decade and widespread use by about a century. Calculus was invented in 1670, it was about 1680-1700 till it started actually being used in astronomy. The uptake was probably faster because at that time a lot of mathematicians were Astronomers as well. There is a lot of mathematics created but we don’t yet know how to use it. My hope is that…

Quantitative scientists are also mathematicians. I'm not attacking mathematics, just probably-useless subfields. Is there any good quantitative evidence that actually estimates what percent of math work today will be useful? Because to me it seems like <1% and I feel like we could easily make that number a lot higher. Particularly I want to see massive improvements in quantitative social science; physics already gets…

What open problems in quantitative social science do you have in mind where better math could achieve massive improvements? I'd expect the bottleneck to be data availability nearly always.

Re: Fields Medals 2026

#54

Post by CogDisco »

GuB-42 wrote: post #51
siva7 wrote: post #20

"harmonic analysis and geometric measure theory, including applications of multiscale and decoupling techniques to the local smoothing conjecture for the planar wave equation, and major advances in Fourier restriction, Falconer distance sets, Furstenberg sets in the plane, and the Kakeya problem in three dimensions." I'm not sure there is another profession in the world where it's impossible to explain to a layman on…

Besides the usual jargon, what I find interesting is that the tradition of always naming things after their discoverers: Fourier, Falconer, Furstenberg, Kakeya. 4 names in one sentence. Other fields do it, but it is almost systematic in math, and arguably, it makes things even harder to understand as people names are not descriptive.

But then there's the opposite in maths where there's an overload on "plain" terminology. "Normal" means a billion different things. Same as "regular" or "simple".

Omitting the human history of a field does not automatically make it easier.

Re: Fields Medals 2026

#55

Post by applfanboysbgon »

falcor84 wrote: post #41
applfanboysbgon wrote: post #23

Earlier quoted context omitted.

"Looks like" being the operative keyword there. Do you feel like you're having a conversation with a peer when you prompt an LLM in the topic you're an expert of? For the love of God, I'd hope not. The whole point is that, even though these things are really good at generating what looks like human output, they are still just regular software algorithms.

> Do you feel like you're having a conversation with a peer when you prompt an LLM in the topic you're an expert of? For the love of God, I'd hope not. Yes, I do feel that. Make of it what you will.

The only thing I can think is ... how?

If I were to anthromorphize my experience with frontier models, it would be as a mentally challenged child with complete memorization of an encyclopedia and thesaurus. It has the ability to rapidly experiment and potentially succeed at tasks through trial-and-error, but not without constantly corralling it in the correct direction because it would stick a fork in an outlet if unattended for five minutes.

Tao's chat certainly doesn't give me a vibe of talking with a peer. Do you much often have conversations with colleagues where you write one sentence and then get five pages dumped on you, repeating ad infinitum? LLMs can be useful for rubber ducking, and sometimes the plausibly-related word-soup it generates so quickly will help your thinking along faster, but that's not the same thing as a genuine conversation. And it mostly looked like Tao was using it as an advanced calculator, firing off his own ideas for it to quickly do calculations on. I don't know why we need to anthromorphize these tools just because they generate sentences.

Re: Fields Medals 2026

#56

Post by orochimaaru »

leonvoss wrote: post #49
orochimaaru wrote: post #44

Earlier quoted context omitted.

Not true. Novel mathematical methods precede their application by at least a decade and widespread use by about a century. Calculus was invented in 1670, it was about 1680-1700 till it started actually being used in astronomy. The uptake was probably faster because at that time a lot of mathematicians were Astronomers as well. There is a lot of mathematics created but we don’t yet know how to use it. My hope is that…

Quantitative scientists are also mathematicians. I'm not attacking mathematics, just probably-useless subfields. Is there any good quantitative evidence that actually estimates what percent of math work today will be useful? Because to me it seems like <1% and I feel like we could easily make that number a lot higher. Particularly I want to see massive improvements in quantitative social science; physics already gets…

I think in social sciences it maybe more of an inability/reluctance of the domain group to use the mathematics as opposed to the mathematics being absent.

Take category theory for example. The initial mathematics appeared in 1942. The application to social sciences started in about 1970 and I’m not sure of the level of uptake at the current time but a quick AI search says applications have accelerated in the past decade (needs verification).

Re: Fields Medals 2026

#57

Post by lhd1 »

yewenjie wrote: post #38
hackandthink wrote: post #34

Scary stuff from one of the winners: "A Taxonomy of Omnicidal Futures Involving Artificial Intelligence" (Jacob Tsimerman, Andrew Critch) https://arxiv.org/pdf/2507.09369

Does anyone know if Tsimerman talked about AI extinction risk at other places? Critch has worked long-term in the field (MIRI, CHAI) but I was surprised to see this colab.

they are friends according to https://www.ams.org/journals/notices/202607/noti3372/noti337...

Re: Fields Medals 2026

#58

Post by nine_k »

GuB-42 wrote: post #51
siva7 wrote: post #20

"harmonic analysis and geometric measure theory, including applications of multiscale and decoupling techniques to the local smoothing conjecture for the planar wave equation, and major advances in Fourier restriction, Falconer distance sets, Furstenberg sets in the plane, and the Kakeya problem in three dimensions." I'm not sure there is another profession in the world where it's impossible to explain to a layman on…

Besides the usual jargon, what I find interesting is that the tradition of always naming things after their discoverers: Fourier, Falconer, Furstenberg, Kakeya. 4 names in one sentence. Other fields do it, but it is almost systematic in math, and arguably, it makes things even harder to understand as people names are not descriptive.

There are no way to name most problems both compactly and descriptively. The inability to attach an acceptably descriptive moniker to a problem leads to calling the problems (and solutions, theorems, etc) after the author.

Re: Fields Medals 2026

#59

Post by anvuong »

falcor84 wrote: post #19
anvuong wrote: post #14

Earlier quoted context omitted.

It's like saying: - winners in the 30s were the last time we have pure human to win (before computer) - winners in the 70s were the last time we have pure human to win (before internet) - winners in the 90s were the last time we have pure human to win (before search engine) Why can't we treat LLMs as just another tool like computers, search engines, computing libraries? Why do people keep trying to anthropomorphizing…

But it's not the same thing. I went through this conversation between Terry Tao and ChatGPT about the Jacobian Conjecture counterexample [0] and it looks a lot more like a conversation between peers than him using a tool. [0] https://news.ycombinator.com/item?id=49010345

Well, how many humans do you think are able to prompt this to the LLM?

"The homogeneity in x is an intertwining between a dilation (x,r,u) to (lambda x, r,u) and a dilation (P,Q,R) to (lambda^-2 P, lambda^-1 Q, lambda R) which seems to collapse the 3d jacobian to a sort of twisted 2d jacobian. Is there a general theory of such twisted jacobians and do you have any sense why those particular dilation weights were used?"

"I can see why the five-dimensional Jacobian has a nice monomial form in rho. Why does this make the three-dimensional Jacobian after restricting to c_2 = rho = 1 and eliminating the delta, eps variables also a monomial (now in x)? Is there some block-diagonal structure or something in the 5D Hessian that allows for a nice reduction? I would have expected some sort of Schur's complement type operation to appear."

I, and probably most people on here, won't be able to get the LLM to write such a detailed conversation, because we are not experts in this field. They are tools.

Re: Fields Medals 2026

#60

Post by coffeeaddict1 »

orochimaaru wrote: post #44
leonvoss wrote: post #31

Earlier quoted context omitted.

So overhyped. Yet they have no power but some prestige among nerds. The reason why it is bad is that the money/status is very limited relative to the amount of smart people. I would rather praise developments in quantitative sciences.

Not true. Novel mathematical methods precede their application by at least a decade and widespread use by about a century. Calculus was invented in 1670, it was about 1680-1700 till it started actually being used in astronomy. The uptake was probably faster because at that time a lot of mathematicians were Astronomers as well. There is a lot of mathematics created but we don’t yet know how to use it. My hope is that…

> Novel mathematical methods precede their application by at least a decade and widespread use by about a century.

This isn't really a good argument. The assumption here is that the "applications" were possible because of the math itself, but it leaves out the possibility if the math didn't exist somehow it will be discovered/invented because the applications demand so.

> There is a lot of mathematics created but we don’t yet know how to use it.

The vast majority of mathematical work is complete useless. Only a small percentage finds use in the real world (even if you consider the maths from centuries back).

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