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作为经常做数学研究的人，你常常会有一个想法，然后你会想，
Often as a practicing mathematician, you have an idea, and then you kind of think
这也许行得通。
it might work.
然后你试上几个小时、几个星期，最后在某一点上，你就放弃了。
Then you try for a few hours, a few weeks, and at some point, you give up.
但对于GPT来说，好吧。
Whereas for GPT, like, okay.
有人告诉我这么做。
A human told me to do this.
那就这么做吧。
Like, let's just do this.
所以这就是为什么我们正处于这样一个复兴时期，
And so that's why we're sort of in this renaissance of, like,
成果触手可及。
reachable results.
这是这个问题最棒的地方，真的没人知道答案，因为模型只是在疯狂地猜。
This is the best part about this problem, which is really nobody had any idea because the model just guessing in some
以一种不可思议的方式。
insane way.
到目前为止，它似乎没有上限，但它还没有那个背景知识。
It doesn't seem like there's a limit so far, but it doesn't have that context yet.
如果世界能更快地发现数学，那就太好了。
It'd be nice for the world if we'll find mathematics on a lot faster.
数学问题的难度上限是很高的。
The ceiling for difficulty of a math problem is pretty high.
即使人工智能在数学方面继续以指数级的速度进步，它也可能永远无法解决像P对NP这样的问题。
Even if AI continues getting exponentially better at math, plausible will never solve something like p versus n.
数学界理想中会如何接受这个呢？
What's the ideal way that this is being taken up by the math community?
可能现在大多数人都觉得，好吧。
Probably most at this point are like, okay.
AI显然在做一些不简单的事情。
AI is obviously doing some nontrivial stuff.
所以AI不仅仅是在数学基准测试上表现更好。
So AI isn't just getting better at math benchmarks.
它开始在一些困扰人类几十年的数学问题上取得进展。
It's beginning to make progress on mathematical problems that have resisted humans for decades.
在本期节目中，a16z基础设施合伙人Lisha Li与OpenAI的数学家Mehtaab Sawhney和Mark Sellke坐下来，了解实际正在发生的变化。
In this episode, a16z infra partner Lisha Li sits down with OpenAI mathematicians Mehtaab Sawhney and Mark Sellke to understand what's actually changing.
他们讨论了球堆积、编码理论和群论的最新成果，并解释了为什么这些进展不能简单归结为蛮力计算。
They walk through recent results in sphere packing, coding theory and group theory, and explain why these advances can't be reduced to brute force.
模型尝试不同的方法，放弃死胡同，跨领域连接想法，在某些情况下，产生的推理读起来非常像人类数学家的笔记。
The models try different approaches, abandon dead ends, connect ideas across fields, and in some cases, produce reasoning that reads surprisingly like the notes of a human mathematician.
他们还探讨了一个更大的问题。
They also tackled the bigger question.
当证明一个结果不再成为瓶颈时，数学会变成什么样？
What happens to mathematics when proving a result becomes less of a bottleneck?
从数学品味和人类判断力，到理解大量新成果的涌现，丽莎、梅塔布和马克一起探讨了AI如何不仅改变哪些问题能被解决，还改变了数学实践本身的意义。
From mathematical taste and human judgment to understanding an explosion of new results, Lisha, Mehtaab, and Mark explore how AI could change not just what problems get solved, but what it means to practice mathematics.
好的，感谢两位的到来。
Well, thank you guys for coming.
这真的很令人兴奋，因为我觉得数学在AI的推动下发展得太快了。
This is really exciting because I think math has been moving so fast with AI.
我特别想和两位既是实践数学家又在OpenAI工作的朋友聊聊这些成果。
I'd just love to get to both practicing mathematicians and who work at OpenAI to chat on some of these results.
今天我们请到了马克·塞尔克和梅塔布·索尼。
So we have with us Mark Sellke and Mehtaab Sawhney.
我们其实有联系，因为Yufei曾经是你的导师。
We're connected actually because Yufei was actually your adviser.
而且你们两位在数学上的钻研比我深得多，我退出数学领域已经十多年了。
And so both of you guys have worked much more deeply in math since I have quit many, many like, over a decade ago.
所以能听到你们对OpenAI如何应对这些挑战的想法，以及你们对数学在AI飞速助力下的未来走向的看法，真的非常激动人心。
So this is very exciting to kind of hear your download of your thoughts on how OpenAI has been sort of approaching this and also just, like, where you think math is going with the incredibly rapid advance of how AI has been helping.
我们可以从一些非常基础的问题开始。
We can start off with some very basic questions.
你们在能分享的范围内，都在做些什么？
What do you do to the extent that you can, of course, share?
你们是怎么从实践数学家变成在OpenAI工作的？
And how did you come from being a practicing mathematician to working at OpenAI?
对。
Yeah.
我觉得我们俩大概都是去年模型在数学上真正开始突飞猛进的时候，变得特别兴奋的。
I mean, I guess we both broadly got excited last year when the model started to really take off in math.
对。
Yeah.
所以我加入得比Mehtaab稍微早一点。
So I joined a little bit before Mehtaab.
我基本上就是去年夏天看到IMO金牌的时候，决定加入的。
I saw the IMO gold medal last summer, basically.
我当时想，这太厉害了。
I thought, this is amazing.
我想看看他们到底是怎么做到的。
I want to see what the heck they did.
对。
Yeah.
让我去瞧瞧。
Let me go see.
然后呢，到了秋天，Mark给了我一个GPT-5的账号，我就开始玩这些模型，很快就确信，玩起来真的特别兴奋。
And then, yeah, I guess in the fall, Mark gave me a GPT five account, and then I started playing with the models and very quickly became convinced that, yeah, it was extremely exciting to play with them.
是的。
And yeah.
你们俩之前就在合作，
You you two were collaborating
在那之前。
before then.
对。
Yeah.
是的。
We yeah.
我们认识有一段时间了。
We've known each other for a while.
对。
Yeah.
对。
Yeah.
我们其实还合作写过一篇论文。
We had one paper we actually wrote jointly.
对。
Yeah.
对。
Yeah.
所以GPD五就是你的转换点。
So g p d five was your conversion.
对。
Yeah.
那神奇之处是什么？你向它提了什么问题？
What was the magic that sort of what question do you throw at it?
什么过程？对。
What process Yeah.
所以我觉得，实际上，是的。
So I think actually yeah.
所以我觉得这件事是怎么开始的，至少对我来说，起点是这样的：有一系列问题，叫做保罗·厄尔多斯，他是一位非常著名的数学家。
So I think how this started was, at least for me, the starting moment was something like there's a collection of problems called so Paul Erdrich is a very famous mathematician.
他提出了一堆问题，现在这些问题都被收集在一个网站上。
He posed a bunch of problems, and so they've now all been collected on this site.
而我主要研究组合数学，这些问题中有很多都是最重要的。
And so I specifically worked in combinatorics, a lot of these questions are among the most important.
所以翻翻这个网站总是很有趣。
So it's always fun to flick through this light.
但有一件事经常让我特别沮丧，就是我看到一个问题，上面标着“未解决”，但我其实不确定它是不是真的还没解决，因为文献往往很难搜索。
But one thing that often happened to me that was extremely frustrating was I would look at a question, see that it's marked as open, and then not actually know if it's correct, not actually know if it was still unsolved because the literature is often quite hard to search.
对。
Yeah.
有一次，我把它输入到GPD Five里，五分钟后它就找到了一个参考文献。
And one instance, just plugged it into g p d five, and five minutes later, it found a reference.
那一次，其实我几个朋友已经开始思考网站上的那个问题了。
And this was a case where a few of my friends had actually started thinking about the problem on the site.
我当时在和他们聊。
I was talking with them.
而且，我们花了好几个小时。
And, I mean, we had spent a few hours.
当时还不清楚这个问题是否在可解决的范围内。
It wasn't clear if the problem was within reach.
那感觉真的很好。
It was just very nice.
好的。
Okay.
有人告诉我，是的，这个问题是可以解决的。
To be told, yes, this is in reach.
这是方法。
Here's how you do it.
然后，GPT-5告诉了我这些。
And yeah, GPT-five told me this.
然后我把这件事告诉了马克，然后，是的。
And then I told Mark about this and this yeah.
这有点，嗯，怎么说呢。
This is sort of yeah.
对我来说，这真是个令人惊讶的时刻。
This for me was quite a surprising moment.
对。
Yeah.
然后我们进一步研究了一下，发现了大约10个类似的案例。
And then we looked more into it and we found 10 more cases sort of like this.
当时，我觉得在建立联系方面做得更好，就像你说的，比如搜索之前是否有类似的结果或相关的东西，这对人来说确实很难，但对机器来说可能更容易。
At the time, I feel like being better at maybe making connections between, as you're saying, like, the search for whether there has been a result or a related thing earlier is just kind of humanely hard, but maybe better for machine.
但我想，随着过去一年的进展，令人印象深刻的成就已经远远超出了这个范围。
But I imagine as the progress has happened in the last year, what has been impressive has kind of reached beyond that.
也许通过更抽象地讨论，或者通过最近宣布的一个问题来讨论会更自然，比如Astra，让我了解一下，最近的进展不仅仅是搜索更多领域、建立这些领域之间的联系，而是实际上更深入、更数学化的推理，类似于一位在职数学家的工作。
And maybe through talking about it more abstractly or if it's more natural to talk about it through one of the problems that has been recently announced through, you know, Astra, and kind of enlighten me as to, how the recent progress has been a lot more than just searching through more areas, making these connections between the field and perhaps just actually deeper, more mathematical reasoning that's similar to a working mathematician.
对。
Yeah.
我的意思是，我认为这种熟悉一切的搜索能力仍然是相对的优势，对。
I mean, I think this, like, search point of being familiar with everything is still definitely like a relative strength that Yeah.
也许这影响了AI现在正在解决的问题类型。
Maybe informs, like, the types of problems that AI is solving now.
我觉得还有一些其他的相对优势和劣势。
I think there are some other relative strengths and weaknesses.
另一个比较明显的优势是，一旦有了想法，它执行起来非常在行。
Another relative strength that's pretty noticeable is just like it's very good at executing on some idea once it has it.
每当你有个想法，通常需要把各种细节都理顺。
Whenever you have an idea, there's usually some amount of getting everything lined up.
比如，epsilon 是否小于 delta 这类问题。
Is epsilon, like, smaller than delta, this kind of thing.
你得把所有细节都弄对。
You have to get everything correct.
而且，对于人类来说，很容易在这些细节中迷失，而我觉得 AI 总能完美地处理这类论证。
And, like, for a human, it's easy to get lost in these kinds of details, and the AIs just kind of always nail these kinds of arguments, I find.
对。
Yeah.
我觉得你们对这方面会更了解一些。
I feel like you guys will know more detail on this.
但就拿单位距离问题来说，那里的方法确实有OpenAI的贡献，但最初的想法可能还是Erdős提出的。
But, like, for the unit distance problem, it was just like, the approach there was definitely contributions from OpenAI, but, like, the approach perhaps was suggested even originally by Erdush.
而真正完成推理过程本身，是一个非常非常了不起的壮举。
And then it's just that the actual reasoning was a very, very, like, momentous feat.
所以对人类来说，你会想，我时间有限。
And so for human, you're like, well, I only have a limited amount of time.
如果走了这么多步之后，还是不清楚。
And if after so many steps, it is still not clear.
我的意思是，也许你像安德鲁·怀尔斯那样，真的花十年时间独自做点什么，但风险收益比是否划算，还真不好说。
I mean, maybe you're like Andrew Wiles and you actually spent ten years alone and do something, but, like, it's not clear that the risk reward is not good enough.
但对GPT来说，好吧。
Whereas for GPT, okay.
我会说，有人告诉我
I'll like, a human told me
去做这个。
to do this.
那我们就直接做吧。
Let's just do this.
所以这就是为什么我们现在处在一个成果触手可及的复兴时期。
And so that's why we're sort of in this renaissance of, like, reachable results.
这么说你能理解吗？
Does that track?
那你觉得在ASTER的结果中，主要优势是不是就在这些方面，还是说这里有什么额外的成分或者魔法？
And did you feel like with the ASTER results, is that sort of, like, where the strengths have been primarily or there's an extra ingredient or magic here?
我觉得单位距离的例子挺能说明问题的，可能就在于那个具体的构造。
I feel I think the unit distance example is about it's quite telling in the sense of maybe the exact construction.
你可以把它做得跟人们之前尝试过的非常相似。
You can make it look very similar to what people had tried before.
但我认为，我
But I think, I
意思是，作为一个实践中的数学家，你常常会有
mean, often as a practicing mathematician, you have
有个想法，然后你大概觉得它可能行得通。
an idea and then you kind of think it might work.
然后你试几个小时、几天、几周，最后在某一点上放弃了。
Then you try for a few hours, a few days, a few weeks, and at some point you give up.
然后一个不算少见的情况是，一两年后你发现别人把你认为行不通的想法做出来了。
And then a not so uncommon experience is that you find out a year or two later that somebody else got the idea to work that you thought that didn't work.
所以有时候，让一个想法真正落地，就已经是战斗的大半了。
So somehow getting an idea to work can even be a large portion of the battle.
而且我觉得，尤其是在单位距离猜想这个例子里，有太多极其琐碎的细节。
And I think, especially in the case of the unit distance conjecture, there's just a lot of extraordinarily finicky details.
很多时候做数学，你其实是在跟问题赌博。
And very often when you're doing mathematics, you're kind of gambling against the problem.
你会想，也许该试试这个方法，但看起来真的不太可能，而且不值得花时间。
You're like, maybe I should try this approach, but it seems really unlikely and just not worth my time.
而这个模型，我觉得在好几个例子里，通过结合它已有的知识和良好的品味，做出了正确的数学。
And the model, I think in several of these cases, by combining what it knew and sort of having good taste, kind of made the correct math.
你可以从我们发布的思维链摘要中看到这一点。
And you can kind of see this in the summarized chain of thought we released.
你可以大致看一下。
You can sort of look at it.
它像数学家一样推理，因为它知道一些非常正确的关键点，所以能做出正确的决策，最终剪枝搜索树。
It's reasoning like a mathematician, and because it knows a few very correct bits, it it makes the right decisions and eventually able to prune the search tree.
它并不是真的什么都试。
It's not really trying everything.
它尝试了很多不同的东西。
It tries a lot of different things.
它非常执着。
It's extremely dogged.
我是说，它不可能尝试每一个想法。
I mean, it can't try every idea.
它只能尝试有限的一组想法，而且它能利用自己的知识加上良好的数学判断，找到正确的路径继续前进。
It has to try a limited set of ideas, and it's able to kind of use its knowledge plus good mathematical judgment and and find the right path to go along.
所以，对我来说，这之所以特别，是因为这个问题很多人都思考过。
So, I mean, that for me was like because this was a problem which a lot of people have thought about.
而且，这个想法并不算陌生，这很可能说明很多人都试过，或者至少几位非常严肃的数学家尝试过。
And, the fact that the idea is not so foreign probably indicates that a lot of people had tried it, or at least a few very serious mathematicians had tried.
我觉得这正是它看起来特别有意思的地方。
And I think that's what made it really interesting to see.
我觉得看到这些证明时，我还有一种感受，就是如果我自己有个想法并试图去实现，可能我对如何让事情成功会有一个错误的计划。
I think something else that I feel when I see these proofs is if I have an idea and I'm trying to execute, it might be that I have some wrong plan for how to get things to work.
作为人类，如果你走了一条错误的路，走了一段时间，可能很难重新调整你的大脑，重新开始尝试另一条路。
And as a human, if you have some wrong path you go down for a while, it can be hard to rewire your brain to start over and try a different path.
最初的想法在你的大脑里，和那些最终没成功的事情是联系在一起的。
The initial idea is kind of linked in your brain with these other things that ended up not working.
这有点像你的上下文窗口有点被污染了，你不能从上周克隆一个自己，然后说，别这样做。
It's sort of like your context window is, like, a little polluted, and you can't just make another clone of yourself from last week and say, don't do this.
试试别的办法。
Try something else.
往另一个方向培养你的直觉。
Build your intuition in another direction.
对。
Mhmm.
但用AI来做这件事就很容易。
But it's very easy to do this with an AI.
所以我觉得这也是另一个原因，一旦你有了一个好的大致方向，把细节做对突然就不再是那么大的障碍了。
So I think this is another reason that it's like getting the details right once you have some good general direction is much less of a barrier all of a sudden.
你说用AI来做更容易，是不是意味着它其实并没有受到人类干预的引导？
And when you say it's much easier to do with AI, it's like it's not actually being directed with human interference too.
就像你说的，在推理轨迹中。
It's as you're saying in the reasoning traces.
它就是在做这些选择。
It's like making these choices.
也许它会回溯，但它不会被思考问题时的上下文所干扰，就像通过这些机制一样。
Maybe it backtracks, but then it's able to not be distracted by maybe the context in which it's thinking about the problem, like, via these machinery.
那你会看到它也会回头吗，还是说它只是做出了好的选择？
And it's like, do you see it kind of go back as well, or is it just making good choices?
就是说，它是碰巧选对了，还是真的像数学家那样在推理，好吧。
Like, is it a lucky sample, or is it actually reasoning like a mathematician where, okay.
它在这条路上走得不好，但它会回头，然后不让那影响后面的判断？
It doesn't do well in this path, but it goes back, but then it doesn't let that pollute?
不会。
No.
我的意思是，它确实会犯错，然后它会回头再想想。
I mean, it definitely makes mistakes, and then it goes back and thinks about it.
我觉得它某种程度上非常精于计算，非常正确。
I think it's somehow very calculating, very correct.
我是说，对。
I mean Yeah.
作为人类数学家，你在做这些决定时并不总是完美的。
As human mathematicians, you're not always perfect at making these decisions.
而且第一次某个方法行不通的时候，你会自动降低对这个方法成功可能性的预期，然后你会反复试几次。
And the first time something doesn't work, you automatically kind of downgrade how likely this approach is to work, you keep doing this a few times.
这个模型似乎比人类更能更新对路径成功可能性的判断，至少在我们看到的几个解决方案里是这样，是的。
The model somehow is much better able to like it seems for several of the solutions we've seen, it somehow seem it seems much better able to update, like, how likely the path is to work, like Yeah.
相比之下，人类在拒绝一条路径时可能没那么灵活。
Versus rejecting a path versus a human doing it.
但我觉得即使不是这样，你可以重新开始一个新的模型会话，你知道，这种情况总是会存在的。
But I think even if it weren't, the fact that you could just start another model session over into, like, you know, it's always going to be the case that that
明白了。
Got it.
明白了。
Got it.
这里。
Here.
所以从某种意义上说，它还是在利用你可以并行运行多个智能体来处理这个问题的优势。
So in some sense, it is, like, still leveraging the fact that you could, like, run kind of parallel, you know, agents on the problem.
对。
Yeah.
但如果它是在回溯的话，那确实让它看起来更像一个人类数学家。
But if it were kind of backtracking, then does make it seem much more like a human, you know, mathematician.
而且它可能确实也在做类似的事情，因为很明显，我们得犯错误才能对解空间产生直觉，知道哪些路径是不可能的。
And perhaps it is kind of doing some of that stuff too because, like, obviously, like, we have to, like, make mistakes in in in order to, like, even gain intuition for, like, that solution space is, like, not, you know, not in the the set of paths that it could be in.
我觉得这种事情在人类身上也会发生，比如你卡在某个方法上，你可能会把你的大致想法告诉另一个人，然后他们回来帮你找到让这个方法奏效的办法。
I mean, I think this kind of thing happens with humans too where, like, if you get stuck on some approach, you might tell another human your kind of general idea, and then they'll come back and, like, figure out how to get it to work.
而且，你知道，就是，对。
And, you know, it's just Yeah.
和人类做这件事需要更多时间。
It takes more time to do this with humans.
我在想，也许这涉及到你能讨论的程度，当然，不谈训练配方之类的，但有趣的是，如果你只是从数学论文中学习，那是一个非常差的训练集，对数学来说，因为也许数学教科书甚至是更纯粹的例证。
I wonder I mean, know, maybe this gets to the extent that you can actually talk about sort of like obviously, don't talk about the training recipes or whatever, but it's it's interesting that if you're just studying, for instance, from math papers, it's like a very poor training set, like a priority for math because I mean, maybe math textbooks are even a pure example of this.
它实际上很难重建事物背后的动机，你知道，就像，不是说，也许有些人喜欢，但别从那里学实分析。
It's, like, really bad at actually reconstructing the motivation for why things were, know, you it's like, don't I mean, maybe some people like it, but don't learn real analysis from.
它已经非常干净利落了。
It's just like it's very clean already and crisp.
我觉得这不太好，因为它没有展现出我们当初为了定义某些概念而经历的挣扎。
And I think that that's bad because it doesn't show the struggle that made us formulate definitions in a certain way.
比如说，为什么我们需要用这种超级抽象的方式来定义实数呢？
Like, why do we even need to have real numbers be defined in this, like, super abstract way, etcetera?
所以，我觉得论文也是这样，除非你是为了教别人成为数学家，否则大多数人写论文时并不会带着这种教育的目的。
And so, you know, I think paper is also I mean, unless you're most people don't write papers with the the context of I need to educate somebody to do to be a mathematician.
所以，实际上学习数学的课程体系，并不一定体现在我们作为数学家的很多成果中。
And so, like, the actual maybe curriculum of, like, learning math is not inherent in, like, a lot of our artifacts as mathematicians.
所以，也许换个问法，就是推理追踪器生成的东西，是不是真的像这样，好吧。
So maybe another way to ask this question is if the reasoning tracers are actually producing things that it's like, okay.
这其实更接近数学思维。
This is actually more close to mathematical thought.
那这是怎么产生的呢？
Like, how does that arise?
我的意思是，是的。
I mean, I yeah.
我觉得OpenAI一直是推理模型的先驱，也是以这种方式教AI推理的先行者。
I guess OpenAI has been, like, the pioneer of reasoning models and teaching AI to reason in this way.
所以我们正在尽一切可能的方向，努力教模型在不同领域更好地、更长时间地推理。
So we're doing a lot of work at of all possible directions on teaching models to reason better and for longer in all kinds of different domains.
我觉得我们是在训练通用的推理模型，而且我们描述的很多行为，比如回溯或者重新开始，这些其实并不是数学特有的。
I mean, I think we're training general purpose reasoning models and kind of one a lot of these behaviors that we're describing mathematically, backtracking or kind of starting again, I mean, these are these are not really specific to mathematic.
虽然我们在数学例子中特别看到了这些行为，但它们其实是通用的推理工具。
I mean, we're seeing them specifically in mathematics in these examples, but kind of they're general purpose tools for reasoning.
我觉得如果你在推理上花功夫，最终应该会看到这些模式。
And I think if you work hard at reasoning, you should see these patterns eventually.
对。
Mhmm.
所以这只是自然涌现出来的，因为我觉得这就是为什么OpenAI的方法如此……它不依赖于自动形式化来引导推理。
So it's just emergent because it's I mean, I do think that's why the OpenAI approach was so I mean, it's like it doesn't rely on, you know, doing auto formalization in order to, like, guide the reasoning.
我觉得这显然更像我们人类的方式。
I think that's, like, obviously more like us.
但问题在于，如果你只是在一个像数学证明的语料库上训练，比如自动形式化并转换成Lean语言，你未必能获得那种如何思考的投影。
But it's just, like, so not obvious that if you're just, like, training on, say, a corpus of, like, math proofs, auto formalized and lean, that you get get the sort of, like, projection of, like, how to think well.
换句话说，也许我们可以用代码来思考这个问题。
Like, put another way, like, with maybe maybe if we think about it with code.
代码之所以是很好的训练语料，是因为它是少数几个具有如此大上下文的数据库之一。
Like, code is such a good corpus to train on because it's one of the few datasets that has such large context.
你可能会在书籍中看到类似的情况，但它们的结构联系没那么紧密。
You just like mean, maybe you see this kind of with books, but they're less structurally interconnected.
我觉得那里的结构确实更少。
There's just, like, less structure there, I think.
可以说，平均而言，书籍中的结构比代码中的要少。
It's safe to say, like, on average in a in a book compared to, like, a piece of code.
所以对于数学论文，我觉得我们编码模型做得不好的地方，正是数据集里没有的东西，也就是那种语义。
And so, like, with math papers, I feel like but maybe what what we're so bad at with coding models is stuff that that dataset doesn't contain, which is, like, kind of the semantics.
语法是有了，但还缺少一点更高层次的语义，就是那些产出背后的东西。
Like, the syntax is there, but there's a little bit of, like, the higher level semantics of what produced.
比如，为什么非得这么写？
Like, why do I have to write it this way?
我不是想扯太多哲学层面的东西，但真的很有意思，它居然能做出这么好的数学，这完全是涌现出来的。
It's not I'm kind of getting too too much of the philosophical, but it is just, like, really interesting how it's still emergent that it's doing good mathematics.
如果你们想更详细地聊一些具体问题，我们可能会深入探讨，但不仅仅是按预期那样，比如暴力破解那种。
And we'll probably get into this in more detail if you guys, you know, wanted to talk in more detail about some of the problems, which is just like it's it's not just doing, like, the expected, like, we'll push the brute force thing.
你显然对某些推理过程印象深刻，而这些并不是从我们想象中那种简单的训练集里就能轻易得到的。
Like, You clearly are impressed with some of the reasoning traces, and it's just not obvious that's gleaned from what we would imagine would be the easy training set here.
对。
Yeah.
确实。
Absolutely.
我觉得，正是因为这个，我们才觉得有必要发布这些结果的总结性思维链。
I mean, I think this kind of thing is one reason we decided it was important to release these summarized chains of thought for these kinds of results.
因为如果你从没见过这些，只看到一堆证明出来，你根本不知道那是什么意思。
Because if you've never seen these and you just see all these proofs coming out, you're not sure what it means.
比如，模型是不是在用某种疯狂的方式瞎猜？
Like like, is the model just guessing in some insane way?
它是不是在用一种完全陌生的方式思考？到底是怎么回事？
Like, is it thinking in some totally foreign like, what what's going on?
但实际上，它的推理方式惊人地像人类专家。
But but actually, it it's it's reasoning kind of shockingly like an expert human would.
对。
Yeah.
对。
Yeah.
对。
Yeah.
这真的很像在读同事的笔记。
It's it's very much like reading a colleague's notes.
我是说，从某种程度上讲，它有点更乱。
I mean, it's a little more disorganized in some way.
他们有点像，特别是如果你和合作者工作得很近，有时候你会看到他们直接在邮件里把想法倒出来。
They kind of like, especially if you work close enough with the collaborator, sometimes you'll just see them like spill out their thoughts in an email to you.
感觉就像读了很多这样的邮件连在一起。
It kinda it it feels like reading a lot of those chained together.
所以，是的，头几次真的挺让人惊讶的。
So it's, yeah, it's it's very it's quite surprising the first few times.
你们俩是不是特别参与了选择要发布的这些问题，就是Astra最近提交的那10道题？
Were you two sort of very involved in choosing the problems to release in this, like, last 10 problem set that Astra has applied to?
你最喜欢哪一道？
Which was your favorite?
是的。
Yeah.
我们肯定参与了。
We are definitely involved.
你想从哪个开始？对。
Do you wanna start on Yeah.
我是说，也许从打包开始？
I mean packing maybe?
对。
Yeah.
我觉得也是。
I guess yeah.
所以，我个人在这些问题里最喜欢的是下面这个。
So I guess my personal favorite among these problems is the following.
这个问题非常简单，就是关于你能多高效地放一个面包。
It's it's extremely simple question, which is just like it's just about how efficiently can you put a bun.
我画的圆不太好，而且大小也不一样。
My circles are not very good, and they're not all the same size.
但是
But
但我们假设它们是均匀的。
But we're assuming they are.
对。
Yeah.
所以问题就是，你能把一堆球放得多密——你有一堆半径为一的球，在d维空间里。
So the question is just, like, how dense can you place a bunch of so you have a bunch of spheres you have a bunch of spheres of radius one and d dimensions.
所以问题就是，它们能堆得多密？
So the question is, how densely can they pack?
然后，对，在二维情况下，就像——d等于一的时候，这个问题没什么意思。
And so, yeah, so so in two dimensions, it's kind of like the so so d equals one, this is not an interesting question.
基本上就是一条实数线，然后，对，你可以把它切分。
Kind of it's just the real line, and, yeah, you can cut it up.
一维的球就是一个单位线段，对吧，你可以覆盖所有东西。
Sphere in dimension one is just a unit segment, okay, you can cover everything.
所以当d等于二的时候，就是那种大家都喜欢的图景。
So in d equals two, it's kind of the picture that you know, that everybody loves.
就是一堆球体，它们排列成类似六边形的格子结构。
It's just like it's just a bunch of spheres which sort of form, like, a hexagonal lattice.
对。
Mhmm.
希望我画得够好，但我可以画个六边形。
Hopefully, I've drawn it well enough, but I can draw the hexagon.
这有点暴露我对这个问题的不了解。
Kinda betraying my naivete on this problem.
这个是不是很明显？
Is that, like, obvious?
是不是有一个非常优雅的证明，说明它……
Is it, like, a very elegant proof that it's
不是规则的格子？
It's not regular lattice?
对。
Yeah.
这能行并不是那么显而易见的。
It's not so obvious that this should work.
我记得好像是六十年代才被证明的。
It was only proven in the sixties, I think.
虽然有个简短的论证，但也不是那么容易。
There's a short argument, but it's not it's not so easy.
对。
Mhmm.
是的。
Yeah.
那直觉在哪里呢？
Where's the intuition?
就是说，这个论证的核心机制大概是什么样的？
Like, what is kind of like the machinery of the argument?
我的意思是，它有点像……
Mean, it kind of like
我的意思是，它看起来确实应该能行。
I mean, it definitely looks like it should work.
所以我才……
That's why I'm
是的。
Yes.
我觉得，但是所以……
I think But so
确实。
did.
对。
Yeah.
我觉得这个问题最妙的地方就在于，真的没人知道该怎么解决它。
I think this is the best part about this problem, which is really nobody had any idea about the problem.
对。
Yeah.
所以，是的。
So yeah.
所以，说实话，我最好的例子就是蜜蜂就是这么做的。
So, I mean, honestly, the best situation that I have for this is that, like, bees do this.
如果有更高效的方法，蜜蜂可能就会用别的方式筑蜂巢了。
And if there was a more efficient way, then probably bees would have honeycombs some other way.
因为进化是高效的。
Because evolution is efficient.
对。
Yeah.
我觉得除此之外，我也没有更好的论据了。
I think beyond that, like, I don't have a great argument.
而且，我们对这个问题的了解之少，从这一点就能看出来：好，d等于三。
I mean, and I think how little we know is demonstrated by the fact so, okay, d equals three.
答案就像是在杂货店里怎么堆橙子一样。
The the answer is just like it's what it's how you pack, like, oranges in a grocery store.
对。
Mhmm.
这个结论是海尔斯在2000年代才证明出来的。
And this was only this was proved by Hales sometime in the two thousands.
而且我们还没有一个简短的证明。
And, like and we don't have a short proof of this.
我觉得最短的证明也有几百页吧。
Like, I think the shortest proof is, like, a few 100 pages.
这个证明主要涉及哪些领域？
What area does it, like, draw from in order
主要是大量的线性规划论证，还有非常精细的几何。
to So it's a lot of linear programming arguments, and it's very delicate, like, geometry.
其实挺难看的。
It's it's quite ugly, actually.
是的。
Yes.
这真是个出了名的难看证明。
Like This is like a famously ugly argument.
哦，不会吧。
Oh, no.
然后最著名的两个结果是维度等于8和24的情况。
And then the two most famous results are d equals eight and twenty four.
8和24。
Eight and twenty four.
那肯定是一些奇怪的子空间吧。
It must be some, like, weird subspace
对。
Yes.
是的。
Yeah.
没错。
Exactly.
所以这个是在……像是某种发光的东西上做的。
So this was done in It's like glowing something.
对。
Yeah.
所以这个是在2017年做的。
So this was done in 2017.
不过稍微好看一点。
Slightly prettier, though.
是的。
Yeah.
所以，之所以在这两个特别的维度下能成立，是因为这个叫做格点堆积。
So so the reason it works out in these two very special dimensions is that so this is called a lattice packing.
所以它看起来非常规整。
So it's, like, kind of very regular.
嗯，明白。
Mhmm.
结果在这两个维度里，有两种非常特别的晶格。
And it turns out in these two dimensions, there are two very special lattices.
它们叫E8晶格和利奇晶格，非常漂亮，而且密度高得不同寻常。
They're called the e eight and Leach lattice, and they're very nice, and they're, like, unusually dense.
它们是从数学的其他领域来的，结构特别优美，而且事实证明它们就是最优的结构。
Like, kind of they're just very, very pretty structures coming from other areas of math, and it turns out that they're the optimal structures.
但它们还是规则的，对吧？
But they're still, like, regular
对。
Yeah.
它们非常规则。
They're very regular.
但除此之外，我们就不知道其他精确的维度了。
But, I mean, beyond this, so we don't know any more exact dimensions.
我们知道这五个维度，其他的就都不清楚了。
We know these five dimensions, we kind of don't know anything else.
而且，我的意思是，为了说明我们知道的有多有限，关于这个有两点非常令人惊讶。
And, I mean, like, to give an indication of how little we know so there are two very surprising things about this.
你可以定义delta d为d维空间中最密的球堆积。
So you can define, like, delta d to be, like, the densest sphere packing in d d dimensions.
所以有一个简单的下界，大概是2的负d次方。
So there's kind of an easy lower bound of, like, two to the minus d.
基本就是这样。
Basic yeah.
嗯，这个嘛。
This is Mhmm.
这个证明起来不是很难。
This is not so hard to show.
基本上，任何不能再放进另一个球里的堆积都必须达到这个密度。
Basically, any packing where you can't put it in another sphere has to have this density.
所以，好吧，这个下界不算太小，而且我们知道它必须按指数衰减。
So, okay, it's not ridiculously small, and we know that it has to decay exponentially.
所以它必须衰减，就像它增长为1减c，至少对于某些高密度来说是这样。
So it has to decay like, it grows like one minus c for some at least for some hot density.
所以在高维空间里，你只能覆盖一个极其微小的部分。
So in large dimensions, you can only cover like a vanishingly small portion.
但除此之外，我们几乎一无所知。
But we know like basically nothing else.
所以从
And so from
这仅仅是因为高维球体，对，
it's just because like the high dimensional sphere, like Yeah.
就是它们占据空间的那种情况。
Thing where they occupies.
就像，对，
It just like yeah.
体积的行为很奇怪。
The volume behavior is weird.
对。
Yeah.
所以基本上，我的意思是，基本上，他们
So basically, I mean, basically, they
不想互相挨着。
don't want to touch next to each other.
我不确定有没有特别简短的方法能看出它是指数级小的，但已知它是指数级小的。
I I don't know if I don't think there's a particularly short way to see that that it's, like, exponentially small, but it's known to be exponentially small.
而且很长一段时间里，最好的结果就是那个奇怪的数字，比如二的负零点五九九次方。
And for a long, long time, the best found was something like this funny number, like two to the minus point five nine nine d.
这是两位数学家在上世纪七十年代证明的。
And this was proved by two mathematicians in the seventies.
科普蒂扬斯基。
Koptyansky.
K。
K.
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这个数字真奇怪。
That's a weird number.
它是怎么来的？
Where is that?
是算出来的吗？还是组合数学里的？
Spit out of It's not It's like combinatorial?
还是说……
Or It's it's It's
是某个特别复杂的优化问题的答案。
the answer to some extremely ugly optimization problem.
其实背后有一个挺巧妙的策略。
There's, like, there's, like, a nice underlying strategy.
好的。
Okay.
但是……
But
那挺好的。
that's nice.
是的。
Yeah.
它的音乐性体现在哪里？
What's the music of it?
对。
Yeah.
对。
Yeah.
关于这个，我最后再说一点。
I'll I'll say one last thing about this.
好。
Yeah.
所以，是的，我的意思是，这是两位俄罗斯数学家，在七十年代。
So, yeah, I mean, were these two Russian mathematicians in the seventies.
对。
Yeah.
其实他们的论文特别难找。
It's actually very hard to find their paper.
就一页纸，大概两页那么长。
Like, one page it's, like, two pages long.
明白。
Mhmm.
对。
Yeah.
他们没写太多细节，因为那篇论文很聪明。
They don't write very many details because paper was smart.
但是
But
对。
Yeah.
所以两个负的d，就像那个方格子，就像那个笨的
And so Two negative d's, there's just like the square lattice, like the dumb
一个
one
或者，所以，是的。
or So so yeah.
其实没那么容易，所以这个论证是这样的。
It's actually not so easy to so the argument for this is as follows.
基本上，想象你有一组球，我告诉你，你可以构造一组球，使得你无法再放下另一个球。
Basically, imagine that you have a set of spheres, and I tell you can construct a set of sphere so that you can't put down another sphere.
因为如果你能放下额外的球，就继续放下去。
Because if you could put down an extra sphere, just keep putting it down.
所以你有一组球，使得没有其他球可以放下。
So you have a set of sphere so that there's no other sphere which you can put down.
那个球？
That sphere?
这几乎就像是一个
It's almost like a
随便取一个这样的堆积。
Just take any such packing.
好的。
Okay.
对。
Yeah.
对。
Yeah.
我断言，这至少得覆盖2的负d次方这个比例。
I claim that this has to cover at least two to the minus d fraction.
原因是，如果你把每个球都放大两倍，那它们就得覆盖空间里的每一个点。
The reason is that if you blew up each of these spheres by a factor of two, then they have to cover every point in space.
明白。
Mhmm.
原因是，否则的话，如果有任何空的地方，你就可以在那里放一个球。
And the reason is, otherwise, you could put down if there was any empty space, you could put down a sphere there.
对。
Yeah.
所以我想，如果你用通常的那种排列方式，其实会有更多的地方可以斜着放东西。
So I guess if you take the usual lettuce, there are actually, like, more places you can put things kind of diagonally.
对。
Yeah.
好。
Okay.
对。
Yeah.
对。
Yeah.
所以那其实不是，那是个更差的界限。
So that's actually not it's like a worse bound.
对。
Yeah.
你可以一直往里放东西。
You can just keep plopping things in.
对。
Yeah.
所以这，这其实还行。
This is So this is, like, more okay.
好。
Good.
对。
Yeah.
这让我想到一个特别有意思的事实，就是如果你在高维空间里取一个立方体，然后在每个点上都放一个球。
This is related to this, like, really funny fact where you, like, put a sphere on every point on if you take a cube in high dimensions, you put a sphere on every point.
而且这些球会小到几乎看不见。
And it's, like, vanishingly small.
它小得几乎可以忽略不计。
It's vanishingly small.
它小到可以在中间再放一个球。
It's so small that you can put another sphere in the middle.
对。
Yeah.
对。
Yeah.
刚好能放进去。
It fits.
高维球体的行为真的很奇怪。
High dimensional sphere behavior is weird.
是的。
Yeah.
非常奇怪。
It's very weird.
所以，最棒的一点是，这个模型展示了以下内容。
So so the great part is that so the model shows the following.
我来写两样东西。
So I'll write two things.
这个叫Astra，我想这么称呼它比较合适。
So this is Astra, I guess, probably the right way to refer to this.
对。
Mhmm.
好，我要写一个叫LP界的东西。
So, okay, I'm gonna write something called the LP bound.
我马上解释一下，它表明这个值比这个非常好的数字要小。
I'll explain this in a second, and it shows that it's smaller than this very nice number.
你想说等于吗？
Do you wanna say equals?
是的。
Yeah.
其实，是等于。
It's equals, actually.
D 的 2π 次方，加上小 o 1 的 d 次方。
D to the two pi plus little o one to the d.
如果你能算出这个数，它大概大约是 2 的负 0.6 次方。
And if you you can work out what this number is, it's like it's like roughly something like two to the minus point six.
差不多吧。
Close about.
对。
Yeah.
挺让人惊讶的。
It's surprising.
但这个，
But this one,
你知道，你会觉得，哦，也许这里有什么更漂亮的结构浮现出来了。
you know, you're like, oh, maybe there's some, like, nicer kind of, like, structure there that that that fell out.
而且MO的也还行。
And it's MO's alright.
这大概就是二的负零点六零一……次方，差不多这样。
This is, like, roughly something like two to the minus point six zero one dot dot dot d.
对。
Mhmm.
如果那是数值的话。
If that's the numerics.
我以为那是六零。好。
I thought it was six zero Great.
是的。
Yeah.
这显示的是我的。
This shows mine.
对。
Yeah.
好，那行。
So okay.
所以有几件事要讲。
So there are a couple of things.
首先，这个d的LP是什么？
So first, what is this LP of d?
其实Zascha的工作是建立在一些早期工作之上的。
So via Zascha's work actually builds on some earlier work.
结果发现，有一种方法可以通过所谓的线性规划界来攻克球堆积问题。
It turns out that there's a way to attack sphere packing via what's called a linear programming bound.
LP就是线性规划的缩写。
So LP just stands for linear programming.
Cohen和Elkies提出了一个基于线性规划的球堆积方法。
So Cohen and Elkies gave an approach for sphere packing based on linear programming.
这就像在一个凸集上做线性优化问题，但都是无限维的。
So it's like a it's like a linear optimization problem over a convex set, but it's all kind of infinite dimensionals
这里。
here.
嗯。
Mhmm.
基本上，这个问题的核心就是，你要试着理解接下来的内容。你要做的是，构造一个函数 f。
And basically, what this reduces down to is you try to understand the follow so what you try to show is you basically, you construct a function f.
这是在 d 维空间里的，它映射到实数集，并且具有以下性质。
So this is in d dimensions, and it's mapping to r, and it has the following properties.
首先，f(x)。
So first, f of x.
这是一个在 d 维空间中的函数。
So this is a function in d dimensions.
当 x 的大小大于 1 时，它总是小于零。
So it's always less than zero if, like, the size of x is bigger than one.
其次，x 的傅里叶变换总是非负的。
And you second have that the Fourier transform of x, this is always nonnegative.
所以这其实就是一个线性规划问题，因为傅里叶变换是线性算子。
So this is just this is a linear program because the Fourier transform is a linear operator.
还有，夜赛。
And night race.
所以你只是取一个满足条件的任意函数 f？
So you're taking just some arbitrary f that satisfies property?
对。
Yeah.
所以你可以取任何满足这些性质的函数。
So you can take any f that satisfies these properties.
明白。
Mhmm.
他们证明的是，δ_d 被傅里叶变换在零点的值除以傅里叶变换在零点的值乘以半径二分之一球的体积的比值所界定。
And what they prove is that delta d is bounded by the ratio of the Fourier transform at zero to its to the the four the Fourier transform at zero f of zero to the to over the Fourier transform at zero times the volume of the ball of radius one half in data matches.
所以，好的。
And so okay.
这个证明其实不算短。
This this proof is not so short.
对于神秘数学家来说，可能半页就证明完了，但稍微有点技巧性。
For mysterious mathematicians, like, it's a half a paragraph to prove it, but it's a little bit tricky.
关键是，这其实是一个松弛问题。
And the point is so it turns out so this is a relaxation problem.
不能保证取最优解就能给delta d一个很好的界。
There's no guarantee that taking the optimal left will give you a good bound on delta d.
但关键在于，这也是她2020年或2022年获得菲尔兹奖的重要原因之一，就是她在8维和24维构造了一个函数，使得这个上界恰好匹配那两条非常特殊的线。
But so what did, and this was sort of the key I mean, a large part of the reason she won a fuels metal in in 2020 was that or in 2022 was that she constructed a function in eight and twenty four dimensions such that this upper bound matches exactly these two very special lines.
这算是自然界的奇迹，既能构造出这样的函数，又能给出最优的界。
And these are kind of miracles of nature that both you can construct this function and that it gives you the optimal bound.
但你可以看到，这是一个非常非常自然的问题。
But you can just this is a very, very natural problem.
就是说，一个函数只有两个非常简单的性质，你只是想了解它在高维d的情况下表现如何。
I mean, it's a function with a very two very simple properties, and you just want to understand how this behaves in for large dimensions d.
那一直是个很大的谜团。
And that was a big mystery.
科恩和其他几个人发表了一篇数值计算的论文，他们只是根据数值结果猜测答案是这样的，但完全不知道为什么。
There was a numerics paper by Cohen and several others, which conjecture that just based on doing numerics, that this was the answer, but they had no idea why this would be the answer.
而这个模型表明，实际上在高维情况下，线性规划界限具有非常好的渐近行为。
And what the model shows is that actually the linear programming bound in large dimensions has this extremely nice asymptotic behavior.
这个证明也解释了这种界限是从哪里来的。
And the proof kind of explains where this is tunneled from.
而且因为你完全理解了这种线性规划界限，这实际上给出了对delta d更好的界限。
And because you understand this LP bound perfectly, this actually just gives a better bound on delta d.
事实证明，这个旧的界限可以在这个框架下重新解释。
It turns out that this old bound can be kind of reinterpreted in this framework.
而这个模型的作用是展示你通过这个框架能获得的最佳界限。
And what the model does is it shows you the best possible bound you can get by this framework.
所以这个模型把两者联系起来了。
So the model made the connection.
对。
And Yeah.
那大概是什么样的？
What is the sort of like?
我们，嗯，对。
What we yeah.
那个模型在那里做了什么？
What did the model do there?
我觉得模型给出了一个函数 f，首先它构造了一个函数 f，这个函数给出了这个界限，然后它证明了没有任何函数 f 能做得更好。
I think so the model gives a function f, which so first, it constructs a function f, which gives you this bound, and then it shows that there's no function f which does any better.
所以这是一个不等式，非常强。
So it's inequality, which is quite strong.
我们现在在高维情况下对这个问题的理解已经很透彻了，这相当了不起。
We now understand this problem in high dimensions very well, and that's pretty remarkable.
而模型只是被要求去分析这个高维线性规划。
And the model was just kind of told, analyze this linear program in high dimensions.
去玩吧。
Go have fun.
明白了。
Got it.
为了说明它是怎么知道的，我觉得这个猜想基本上就是靠数值计算得出来的。
To give an indication of how this would know it, I think this conjecture was based basically only by doing numerics.
非常聪明的数值计算，但终究是数值计算。
Extremely clever numerics, but numerics.
所以，你得想办法弄清楚为什么这是正确的目标，而它确实做到了。
So, yeah, you have to kind of figure out why this is the right thing to aim for and it does.
这真的很了不起。
And that was pretty remarkable.
是啊，其实我读研究生的时候，有段时间大概花了六个月想这个问题，结果一点进展都没有。
Yeah, I mean, I had actually thought about this problem for about six months at some point when I was a graduate student and yeah, I remember just making absolutely zero progress on it.
所以后来有人解释清楚为什么是对的，感觉特别好。
So it was very nice to be like, explained why it was true.
所以那是一次很愉快的经历。
So that was a pleasant experience.
我觉得总的来说，这也是那种很多人尝试过的问题，但它的解法特别了不起，尤其是对于LP不能做得比这更好的证明，其实相当简短。
I think also in general, was one of these solutions which I knew several people had tried the problem, it's pretty remarkable because the model solution, especially for this being that the LP can't do better than this, was quite short.
就几页复分析的推导，但恰好就是最对路的方法。
It's a few pages of complex analysis, but it's kind of exactly the right approach.
一旦你看到了，简直难以置信。
Once you see it, it's like unbelievable.
就像，为什么之前没人想到这么做呢？
Like, why why hasn't somebody done this before?
就像，数学有很多种好法，但我觉得其中一种就是，你一看就感叹，天哪。
It was it was like, there are many types of good mathematics, but I think one of them is just like, you see it and you're like, oh, man.
我怎么没想到这个？
Why didn't I think of this?
而且真的很有意思。
And it was really fun.
其实我大概知道自己为什么没想到，但看到这个解法还是很开心，也很有趣。
And I mean, I sort of knew why I didn't think of it, but it was quite nice to see it, and it was fun to see.
对。
Yeah.
这就是我特别喜欢它的原因。
That's why I like this a lot.
对。
Yeah.
所以这是Astra解决的10个问题中的第一个。
So so this is the first of the 10 problems that Astra solved.
明白。
Mhmm.
但第二个问题其实和它密切相关。
But the second is actually closely related.
明白。
Mhmm.
所以这是球体堆积问题。
So this was sphere packing.
第二个是球面码和二进制码。
The second one is spherical and binary codes.
对。
Mhmm.
那什么是画一个码呢？
So what's you draw a code.
你画了一个堆积。
You you drew a packing.
这会是同样的图。
It's it's gonna be the same picture.
对吧？
Right?
好的。
Okay.
当然。
Sure.
当然。
Sure.
不然的话，我们脑子里就会一直想着那个六边形堆积的图。
Otherwise, we're gonna have his picture of hexagonal packing in our minds.
对。
Yeah.
球形码其实就是球堆积，只不过是在另一个球面上。
A spherical coat is literally just a sphere packing, but on another sphere.
对。
Yeah.
所以，我的意思是，球形码基本上就是在另一个球面上的球堆积。
So, I mean, a a spherical coat is basically just a sphere packing on the surface of another sphere.
所以没错。
So yeah.
对。
Yeah.
我接受这个球体。
I'll accept the sphere.
对。
Yeah.
好。
Okay.
所以和之前一样的画面，只不过你是在一个曲面上。
So same same picture as before except you're kind of on a curved surface.
好。
Okay.
好。
Okay.
好。
Okay.
那为什么叫代码呢？
So why is it called a code?
嗯，我想可能是因为二进制代码，这其实也是类似的东西，只不过现在是在立方体上。
Well, you can I I guess the the reason is because of binary codes, which is again the same sort of thing, but now it's on a cube?
好的。
Okay.
对。
Yeah.
行。
Fine.
我来画一个立方体的图，然后在上面画一个最简单的代码。
Let me me draw a picture of a cube and some like simplest possible code on it.
所以这其实讲的是纠错码。
So like when sending so this is really about error correcting codes.
什么是纠错码？
What are error correcting codes?
所以就像是我给你发送一串比特。
So it's like I send you some string of bits.
对吧？
Right?
然后我可能担心我发送的一些比特会被弄错。
And maybe I'm worried that some of the bits I send you get corrupted.
对吧？
Right?
所以可能只是因为系统里的一些错误，这个比特就变了。
So maybe just because of some errors in my system, this one gets changed.
我们想要一种通信协议，让你能纠正这些小的错误，恢复出我想传达的信息。
And we want some communication protocol so that you can decode this small amount of error and recover what I was trying to tell you.
普通的英语其实也有这种特性。
And normal English language kind of has this sort of property.
如果我打错几个字，你还是能明白我在说什么。
If I make a few typos, you're gonna be able to understand what I'm saying.
但如果我们的通信方案太脆弱，那就行不通了。
But if we have some really brittle communication scheme, it's not going to work.
所以编码就是解决这个问题的方法。
So codes are kind of the the way you you solve this.
从数学上讲，就是说，你知道，像这样的二进制字符串是什么？
And mathematically, it just means, like, you know, what's a binary string like this?
固定长度的话，就像超立方体上的一个点。
So the fixed length, it's like a point on some hypercube.
对。
Mhmm.
我们想要一个允许的码字字典，这些码字彼此之间要有一定的距离。
And we want a dictionary of allowable code words that are, like, separated from each other.
所以在这种情况下，如果我不想让任何两个相邻，我就会选这四个顶点，比如那些数字加起来是偶数的顶点。
So, like, in this case, if I don't want any two to be adjacent, I would kind of take these four vertices, kind of the, like, even ones if you sum up the the digits.
对吧？
Right?
然后，好吧。
And and, like, okay.
我想也是。
I guess okay.
在这种情况下，我想如果出了错，你没法知道是哪个错，但至少你能知道有错。
In in this case, I guess if I if I have an error, you can't tell which one it's from, but at least you can tell it's like not at least you can tell there was an error.
哦，我明白了。
Oh, I see.
明白了。
I see.
对。
Yeah.
对。
Yeah.
因为那有点稀疏，对吧。
Because it's like kind of sparse in the yeah.
就是说，它不是那种相邻的，所以当汉明距离是……
It's like it's not shoe adjacent so that like is this like when the Hamming distance is
不是，对。
not Yeah.
对。
Yeah.
没错。
Right.
没错。
Right.
你想要，对。
You want yeah.
所以你想要任意两个不同的……之间的汉明距离都很大。
So you want like a large Hamming distance between any distinct
好的。
Okay.
在字典里。
In dictionary.
而且，是的，我想如果你取两个相对的角，那么如果我有单个比特错误，我总能恢复出它来自哪个点。
And, yeah, I guess if you take two opposite corners, then if I have a single bit error, I can always recover which point it was coming from.
所以它肯定是最接近的那个。
So one that it's definitely closest to.
对。
Yeah.
所以在这两种情况下都有同样的问题，在高维设置中，你能达到什么样的速率？
So there's the same question in both of these cases, in a very high dimensional setting, what kind of rate can you get?
对于二进制码来说，这确实是一个非常实际的问题。
And for binary codes, it's really an extremely practical question.
这有点像，如果我给你发送一个n位字符串，有1%的错误率，我的消息需要变长多少才能容忍这么多错误？
It's sort of like if I send you an n bit string and there's 1% error rate, how much longer does my message have to become to tolerate that amount of errors?
这是通信中一些基本的信息论极限。
This is some fundamental information theoretic limit of communication.
你可以看到，球形的情况确实很像球体填充。
You can see certainly the spherical case, it looks very much like sphere packing.
比如说，如果你把这些小球做得非常小，那么大球的曲率就不太重要了，看起来就像在无限空间里填充球体一样。
For example, if you make all these little spheres really small, then the curvature of the big sphere is kind of not going to matter so much, and it it looks like just packing spheres in full space.
对。
Mhmm.
事实上，是的，这些问题最终被证明是非常相关的。
And in fact, yeah, these these problems turned out to be very related.
所以对于这些问题，来自这些KL作者的有类似的界限。
So for these problems, there were similar bounds coming from these KL authors.
球体有某种结果，立方体也有某种结果，但本质上都是同一类东西。
There's something for the sphere and something for the cube, but it's all kind of the same stuff.
而我们的模型在这些情况下也找到了更好的界限。
And our models found better bounds for these cases as well.
实际上，如果你把它们写出来，技术看起来相当不同。
Techniques look pretty different actually, if you write them out.
所以这个全空间线性规划的分析，其实只用到了复分析。
So this full space analysis of this linear programming was using just complex analysis.
但是对于这些情况，我们用的方法是表示论。
But if you the method for these cases, we're using representation theory.
球体和立方体都有很多对称性。
Both the sphere and the cube have a lot of symmetry.
基本上，证明的思路就是充分利用这种对称性。
And basically, the idea of the proof was to really leverage this symmetry.
之前已有的方法里也用到了一些对称性，但真正的改进在于，我们更深入地运用表示论，让代数对称性以更复杂的方式发挥作用。
There's some amount of this in the previous existing method, and really the improvement is to lean into the representation theory really hard and kind of make the algebraic symmetry enter in a more sophisticated way.
然后，从表示论的公式出发，如果你在球面码的情况下取小球极限，就能恢复部分结果，也能得到这个值。
And then it turns out that from the representation theory formulas, if you kind of take this, like, like, small sphere limit in the spherical code case, you you recover, like, part of this result and you recover this this value.
所以，这个结果并不是一个特例。
So, like, this result isn't a special case.
你只是从码的角度得到了一个方向的界，但两者之间其实有非常紧密的联系。
You kind of only went one direction of the bound from from looking at it from the code's point of view, but, like, there's, like, a very close connection.
好的。
Okay.
对。
Yeah.
是的。
Yeah.
因为你们让这个并行运行。
Because you guys let this, like, run-in parallel.
所以它是在某种程度上自行发现的，因为你们并没有在喂它数据。
So it's, kind of discovering or because you're not sort of, like, feeding it.
所以实际上，这是唯一一个涉及交互的情况。
So so the so actually, this was the one case where there was some interactivity involved.
有意思。
Interesting.
所以除了这一对之外，其他所有情况，我们只是遇到了一些问题。
So for for all of so except for this pair, it was just, you know, we had some problems.
我们把数据喂进去，然后模型就给出了一些解决方案。
We we fed them in and we, you know, the the model the model came back with some solutions.
对。
Mhmm.
这里发生的事其实挺有意思的。
What happened here is actually pretty interesting.
所以我们先让它改进这些代码的边界。
So so we first asked it to improve the bounds for the codes.
对。
Mhmm.
它给出的改进方案用了一些表示理论。
And it came back with an improvement that, like, used some amount of representation theory.
然后我们就问它，嘿，你能再进一步吗？
And then we kind of asked it, Hey, can you push this further?
结果会怎样？
What happens?
然后它给出了一个更复杂的表示论结果。
And then it came back with some much more sophisticated representation theory.
结果发现，通过把这种方法推到极致，你就能得到那个关于全空间球堆积的猜想值。
And it turned out that you got this conjectured value for full space sphere packing out of that method by pushing it as far as it can go.
所以我们接着直接分析那个天空，试图把整个图景补全。
So then we kind of asked to directly analyze the sky and try to complete the picture.
好的。
Okay.
对。
Yeah.
好的。
Okay.
所以这种关系并不是巧合。
So so the relationship, like, isn't a coincidence.
对。
Yeah.
对。
Yeah.
有意思的是，你刚才说的第一个提示，可能听起来特别基础。
It's, like, interesting when you're sort of saying the first prompt, which is, you know, maybe so basic.
我们就像是在问，你能不能再往前推一步？
We're just like, can you push this further?
这确实需要数学家们做出一些判断。
It does require some judgment from mathematicians.
但你可以想象，随着模型规模的扩大，最终可能就不需要这么做了。
And but, like, eventually, you would imagine by scaling the models, you don't need to do that.
或者另一种观点是，这个“框架”其实很重要，它本身就是整个装置的一部分。
Or there's another view that the harness actually does matter, and this is kind of part of the harness apparatus.
你们在和Astra合作的过程中，特别是面对不同代的模型，你们怎么看这个问题？你们需要输入多少，或者说这个框架到底有多重要？
Do you guys have any views on that with your working with Astra, especially generations of models and how how much you have to kind of input or how much the harness matters versus not?
我的意思是，是的。
I mean, I yeah.
我觉得确实有一些这样的小怪癖，基本上就是取决于你让模型做什么。
I I guess there have been some, like, funny quirks like this that just come from, like, exactly what you asked the model to do, basically.
比如，在这个案例中，模型最初被要求做的是把代码的界限改进一个指数因子。
Like, in in this case, what the model was asked to do originally for codes was to improve the bounds by some exponential factor.
所以它真的体现在这个前导常数上。
So it it really shows up in this leading constant up here.
对。
Mhmm.
它改进了界限，但没有试图推得太远。
And it it improved the bounds and it it didn't try to push things too much further.
有时候你会看到它这么做，但有时候它就不管了。
Sometimes you see it do, but sometimes it just doesn't bother.
但没错，再问它一次，它就会更进一步。
But yeah, just ask it again and it goes further.
所以这不是能力问题。
So wasn't like a capabilities issue.
当时就是不想做而已。
Just didn't feel like it at the time.
你觉得那算是判断力吗？
Do you call that judgment?
因为，怎么说呢，确实有那种情况。
The because, What like, there there's a yeah.
我的意思是，他们认为
What do you I mean, they think
模型通常都是非常任务导向的。
models tend to be pretty task oriented.
如果你告诉它去做一个任务，它完成了下一个，就会很开心。
If you if you if you tell it to do a task, accomplishes the next, but it's pretty happy.
所以，是的，任务导向性，但我们是期望那个水平能提升上去吗？不是说它们会变得不那么擅长任务导向。
So, yeah, the the task orientedness, it's like but do we expect that level to kind of ascend up to it's not that they will be less good at being task oriented.
只是说它们会提升到那种，好吧，的水平。
It's just like they'll ascend to the level of like, okay.
不是。
No.
我们往这个方向走吧。
Let's let's go in this direction.
你们也会有判断力的，因为你们也有判断力。
You'll have the judgment too because you guys have the judgment too.
比如说，好吧。
Like, okay.
这看起来挺有前景的。
This is pretty promising.
看起来你用了很多表示论。
Looks like you're using a lot of representation theory.
目前好像没有限制，但它还没有那个上下文。
It doesn't seem like there's a limit so far, but it doesn't have that context yet.
但我想说的是，这个可能很难去外推。
But, like, I guess what I'm trying to say is, like, this one is hard to maybe to extrapolate.
但从之前的几代来看，你可能需要给它更多的提示，更多的引导工作，但最终可能还是得减少提示。
But from, like, previous generations when you had to give it more maybe prompting more of that harness work, but eventually, probably had to give it less.
所以这大概能给你一些信心，说明确实存在这种快速上升的趋势。
So it probably gives you some confidence that there's this, like, really fast ascension.
那你是不是也看到了，嗯，对。
And and do you see yeah.
比如，在哪里——
Like, where
就是，解决一个更难的数学问题，其实就像要解决很多个稍微没那么难的小问题。
is Somehow somehow solving a harder math problem is, like, you have to solve many smaller, like, somewhat less hard math problems.
而且，数学问题越来越难，这本身就说明模型能够在单个连续单元里承担越来越多的任务。
And and the fact that the math problems are getting harder is kind of an indication that you're the model is able to take on more and more work in, like, a single continuous unit.
我觉得这看起来确实很有前景。
And I I think that's that's the thing that looks very promising somehow.
就像这些解决方案，不是说你有了一个想法就万事大吉了。
Like, any of these solutions, it's not like one idea, then you're kind of home free.
你需要几个部分，让它们相互配合、相互沟通。
You need several you need several pieces to kind of interact and talk to each other.
模型嘛，我的意思是，它不会一下子把所有想法都想出来。
The models, I mean, the the model doesn't come up with all the ideas at once.
对吧？
Right?
它不会瞬间把所有东西都提取出来。
It doesn't pull everything out in in an instance.
所以，它需要看到这个部分和另一个部分怎么相互作用，这本身就是解决一个问题，或者说是在拼凑很多问题。
So kind of the fact that it needs to sort of see how this piece interacts with another piece, that's kind of like solving a problem in itself or piecing together many problems in itself.
可能就是这样，当你告诉它‘再往前推一步’的时候，这个指令的难度和它正在解决的每一个小问题的难度是同一个量级的。
It could just be that, okay, when you're telling it, okay, push this even further, That was of the same order of, like, magnitude as, like, all the smaller things it's solving as well in in each way.
所以你不觉得那只是一个特别的方向。
And so you don't think that's just kind of like a privileged direction.
它就像是，嘿。
It's just sort of like, hey.
我们再给它一点帮助吧。
Let's give it, like, one one more help.
或者你真的觉得，我想问的更大的问题是，模型有没有一种好的品味？
Or you actually think that there's guess I what I'm trying to get at a bigger question is, like, is there a good sense of, like, you know, taste?
因为，比如说，当人们谈论模型在研究方面做得怎么样的时候。
Because, like, when people talk about, for instance, how well the models are getting at, like, doing research, for instance.
这就是我们，
And that's what we we,
你知道，想要一点
you know, want a little bit
递归自我改进。
of RSI.
而且，关于这种改进有一些令人惊讶的地方，然后还有，哦，你知道，也许现在
And and sort of, like, there's surprising things about how that improves, and then there's, like, the oh, you know, maybe right now
它仍然处于一个，像，
it's at a level of still, like,
一个初级研究员。
a junior researcher.
就是，其实不太会问对的问题。
It's, like, not really asking, like, the right problems.
所以我就是想了解一下，你看到每一代模型进步的过程中，这个能力是怎么提升的。
And so I'm just trying to get, like, a maybe a sense of, like, where you're seeing that progress through the model advancements each generation.
我的意思是，品味到底是什么？
I mean, what is taste even?
对。
Yeah.
我觉得我对品味的看法比较实用主义。
I I think somehow I tend to be pretty utilitarian in my view of taste.
就是说，如果你能通过更好的判断更快地解决问题，那我觉得这就是我对品味最好的衡量标准。
And, like, if you're able to solve problems faster by making better judgments, like, I think that's, like, the best, like, general proxy I have for taste.
而且，能解决更难的问题，基本上就意味着品味更好。
And somehow, the fact that solving harder problems means it has kind of by definition means it has better taste.
我觉得确实有这些，是的。
I think there there are these no yeah.
我觉得有时候，因为它们是以任务为导向的，你确实会偶尔看到这些迹象，就像，哦，它显然取得了突破。
I think occasion occasion because they are task oriented, you do occasionally get these these symptoms of like, oh, it clearly has made a breakthrough.
它有点明白自己取得了突破，然后它并没有完全推到极限，因为那不是你要求的。
It kind of understands it's made a breakthrough, and then it doesn't kind of push all the way to the limit because that's not what you asked.
但看起来确实是这样。
But that seems yeah.
但比起我们目前看到的进展状态，这似乎还是次要的。
That seems seems rather minor compared to, like, the set the the state of progress we've seen so far.
嗯。
Mhmm.
嗯。
Mhmm.
好的。
Okay.
对。
Yeah.
不。
No.
我觉得这挺清楚的。
I think that's a pretty clear.
我觉得吧，如果你同时想完成一个具体的长期任务，又想在过程中展现品味，那确实容易搞混。
Like, I think it's it's like maybe you're liable to get confused if you're trying to, like, do a concrete long horizon task and show taste kind of at the same time.
但你知道，如果有一个模型负责品味，另一个模型作为监督AI的下属，专门去长时间解决难题，那就不一样了。
But, like, you know, if if you if you have one model that's responsible for taste and one model that's responsible for going out and working for a long time at solving a hard problem kind of as the underling of the of the the supervising AI.
我觉得目前这样应该没问题。
I feel like that's kind of going to be fine currently.
哦，有意思。
Oh, interesting.
因为那等于说这两件事是分开的，或者说就算不完全分开，至少不应该互相干扰对方的上下文。这其实是个挺强的说法，我是说，可能比我想的还要强一些。
Because that is like saying that they these two things are somewhat sep or if not separate, at least they shouldn't kind of pollute each other's context, which is a little bit I mean, it could be potentially like a strong stronger statement than I guess yeah.
不是。
No.
只是觉得挺有意思的，因为可能就像你说的，咱们就采取功利主义的答案吧。
It's just kind of interesting because it's it might just be, like, to your point, it's you know, let's take the utilitarian answer.
它在解决越来越难的问题。
It's solving harder and harder problems.
它做的远不止是暴力破解那么简单。
It's doing a lot more than just, like, you know, brute forcing something.
它在做选择。
It's making choices.
就像是在一个巨大的可能路径空间里进行剪枝，最终得到一个既可行又极其狭窄的路径。
It's like pruning, you know, a vastly large space of possible pads into something that's, like, really you know, it's both tractable, but then ends up being like it's a diminishingly small path within that space.
但是，为什么需要一个单独的模型，或者一个单独的生成版本，或者一个不同版本的模型来负责品味呢？也许这太抽象了。
But, like, having like, why would would be, like, a separate model, a separate generation or something that's a different version of the model that would contribute to taste, or maybe that's totally like, it's too abstract.
完全说不通。
Doesn't make any sense.
你懂吗？
You know?
我们其实应该让真正的问题变成——是什么让我们得到更好的智能版本，是模型外面的框架，还是模型本身？
We should just let the actual this might like, the related question be like, you know, what is what is the the thing that gets us to a better version of intelligence, the harness in the model, or is it just the model?
而且我们看到，至少在应用AI或创业公司里，这就像一场持续的拉锯战：你需要框架，但框架对新模型的适应很差，因为有时候一个非常简单的框架反而是最能发挥模型原始能力的方式。
And it's like, we we see this in, you know, at least in applied AI or, you know, startups where it's like it's a continual battle of, like, you need the harness, but then the harness adapts very poorly to a new model because sometimes, like, a very, very minimal harness is still the best way to expose to the raw power of the model.
但现在我们也有这些训练机制，要求框架必须跟模型一起训练——部分原因是为了让东西更专有，让别人更难使用。
But then now we also have these, like, training regimes where we require the harness to be, you know, trained with I mean, part part of this is to keep things more proprietary and harder to harder for other people to use it.
但我觉得部分原因可能真的是为了对你在意的推理过程有更多控制。
But I think partially, it's it's maybe actually that helps have more control on, like, the reasoning traces you care about.
这说起来有点绕，但其实就是，对。
It's a long rambling way of saying it's like, yeah.
我其实真的——看到模型在数学上进步这么多，还有像“品味”这种抽象又难描述的东西，真的很有意思。
I I don't actually like, I I this is so interesting to see how the models have gotten better at math and maybe something that's, like, very abstract and hard to describe, like, taste.
这是一种厘清到底什么才是真正必要的方法。
It's a way to tease out, like, what is actually necessary here.
我觉得我唯一有点非传统的想法是，就是当你做任何任务的时候，有时候你会钻牛角尖，然后拼命工作，这时候有个朋友在旁边看一眼，说你在干嘛呢？
I think my only, like, nontraditional thought here is that, like, when you're working I mean, just when you're doing any task, occasionally, you get pigeonholed and you, like, work really hard and just having a friend look over your shoulder and be like, what are you doing?
然后只要退一步，花十秒钟想想，这通常非常有用。
And then just having that one bit of step back for ten seconds, this is often very useful.
对。
Yeah.
我觉得没有理由认为人类和模型会有那么大的不同。
I mean, I see no reason why humans would be so different than models somehow.
是的。
Yep.
是的。
Yep.
模型和人类不会有那么大的不同。
Models would be so different than humans.
几个人一起合作通常比一个人单干更有力量。
Having having a few humans working together is often more powerful than just having one.
对。
Yeah.
而且在这种协作的过程中，你其实是在人为地创造这种动态，但它跟真实的动态非常相似。
And it's like in this kind of collaborative thing, you actually you kind of, yeah, artificially created it, but it's very similar and dynamic.
但我觉得，品味很大程度上也在于你能判断哪些问题适合用你想到的方法去解决。
But I I think a lot of taste is also like having a sense of what problems you or like some method you have in mind are going to be good at solving.
明白。
Mhmm.
当然，这里面确实有一些纯粹的审美因素。
Like, it's I mean, certainly there's some amount of like absolute aesthetic point.
对吧？
Right?
但还有一点，就是你要有一种直觉，知道该去追求什么，因为你能在这方面取得进展。
But there there's also just like, having a nose for what you might want to pursue because you'll be able to make progress.
而且我觉得，作为擅长完成任务的一个副产品，你自然会培养出这种能力。
And I think for that, there's you would expect that as a side product of being good at completing tasks, you would you would get there sort of.
对吧？
Right?
你们告诉我，我们还要不要做一节关于sofic群的内容，因为我觉得，你们决定吧。
Let me know if we still wanna do, like, a section on sofic groups because I think, you know, up to you guys.
那绝对非常有意思。
It's definitely super interesting.
所以，第一个问题可能是：什么是群？
So maybe the the first question is what is a group?
我们来回顾一下。
Let's remind ourselves.
群就是一组元素，带有一个乘法运算。
So a group is a set of elements with some multiplication operation.
基本上，这就是数学家思考对称性的方式。
And basically, this is how mathematicians think about symmetry.
所以，基本上，如果g和h是群里的元素，那么g h就是群里另一个定义明确的元素。
So so you're like basically, like, if g and h are elements of your group, then g h has some is some other well defined element of your group.
而且还有结合律，还有逆元。
And you have like associativity and you have an inverse.
所以对于每个g，都有一个逆元，而且群里还有一些特定的元素，算是单位元。
So for every g, there's some inverse, and there's some, like, specific elements in the group that is is kind of the identity.
好的。
Okay.
所以这有点像对操作进行组合的一种抽象。
So it's some like abstraction of composing operations.
这些可以是数字。
So these could be like numbers.
也可以是矩阵乘法。
They could be like multiplying matrices.
也可以是旋转某个东西，这是矩阵乘法的一个特例。
They could be like rotating something, which is a special case of multiplying matrices.
而一个群是SOFIC的，
And a group is SOFIC,
如果
if
嗯，其实有精确的定义，但粗略来说，它意味着……我应该说，群可以是有限的，也可以是无限的。
well, there's some precise definition, but roughly it means it So I should say groups that can be finite or infinite.
比如一个正方形，它的所有旋转就构成一个群，里面有四个元素。
So if you have a square, all the rotations of it form a group with four elements.
如果是圆，那它的旋转构成的群就有不可数个元素。
If you have a circle, then the rotations form a group with uncountably many elements.
所以SOFIC群要么是有限的，要么是可数的。
So SOFIC groups are either finite or countable.
你可以把它们想象成可数无穷的。
You should think of them as being countably infinite.
也就是说，它们的元素个数和整数一样多。
So there's like the same number of elements as like the integers.
而SOFIC的意思是，在某种意义上，它可以被有限群来近似。
And it's SOFIC, if in some sense it can be approximated by finite groups.
所以我们之前不知道是否存在非SOFIC群。
So we didn't know if there was a non SOFIC group.
所以，Aster证明的结果就是存在一个非SOFIC群。
So so the the the result that Aster proved is simply that there exists a non-sofic group.
对。
Yeah.
而且，在进入那个证明之前，我觉得数学里很多项目都是这样的，就是，好吧。
And without like I mean, we can you know, before going to to that proof, it is like you know, it's like I feel like a lot of the programs in math is like, okay.
我们是如此有限的生物。
We are such finite creatures.
让我们看看我们的有限近似能做得怎么样。
Let's see how well our finite approximations are, you know, do.
在这种情况下，尤其是对于可数的情况，你可能会把它和Aldis Leon的东西联系起来。
And in this case, especially for the countable case, it's like maybe you'll you'll be relating it to, like, the Aldis Leon's thing.
它就像帮助锚定这个图景，看起来像是，如果那是真的，那会是个不错的结果，但事实并非如此。
It's just like it helps kind of anchor the picture of, like, it seems like such a I mean, it's a nice result if it were true, but it it's not.
而且这看起来几乎挺合理的。
And it seems almost like reasonable.
所以，是的。
And so yeah.
我其实没去，我很想听听它是怎么找到反例的解释。
I I actually didn't didn't go I would love to hear the explanation of, like, how it found a counterexample.
对。
Yeah.
我的意思是，我觉得，你知道，人们原本希望不存在非SOFIC群，所以每个群都有这种近似性质，这有点像人们期待奇迹发生。
I mean, I I would say that, like, you know, the the hope that there was no non-sofic group, so every group has this kind of approximation, like, maybe this is sort of like people hoping that there's a miracle.
因为事实证明，这样的群有很多好的性质，因为你可以对有限群进行某些证明，然后按照SOFIC定义所允许的方式去近似它们，从而得到结果。
Because it turns out that groups like this have a lot of nice properties because you can run certain proofs for finite groups and then kind of approximate them in whatever way the definition of being SOFIC lets you approximate them and get the results.
所以，这里有一个概念叫surjunctive群。
So so, like, there's this notion of being a surjunctive group.
所以有一个事实是，任何SOFIC群也是surjunctive的。
So there's there's some fact that any group which is sofic is also a surjunctive.
Surjunctive是群上动力系统的一种性质。
Surjunctive is some property of like dynamical systems on the group.
我想最初的问题是，是否每个群都是surjunctive的。
And I guess the original question was whether every group is surjunctive.
这是Gottschalk在七十年代提出的一个问题。
This is some question of Gottschalk from the seventies.
这种先证明有限群的情况，然后进行近似的方法，正是促使人们思考是否存在非sofic群的原因。
And this fact that follows this pattern of prove it for finite groups and then do this approximation is what motivated the question about if there's a non-sofic group.
对。
Yeah.
也许我可以简单说说Aldous Lions猜想，然后……
Maybe I'll say a little bit about this Aldous Lions conjecture and
好的。
Yeah.
当然可以。
Sure.
对。
Yeah.
对。
Yeah.
我之前确实听说过一些，因为概率论里有一个相关的更强猜想，是Aldous和Lyons推广的。
So I guess I had heard of this a little bit beforehand because there's a related stronger conjecture in probability that was made popular by Aldous and Lyons.
这个猜想大致是说，任何具有某种良好性质（叫单模性）的无限图，都可以用大的有限图来近似。
This conjecture roughly what it says is like any infinite graph with some nice property called unimodularity.
随机单模随机图可以被大的有限图近似。
Random in modular random graph can be approximated by large finite graphs.
所以，要解释这些概念想表达什么，而不陷入技术细节，最好的办法就是说说它们在整数上的含义。
So maybe the the way to, explain what these kinds of things are trying to say without getting into technical weeds is to say what they mean about the integers.
那么，我怎么把整数画成图呢？
So so how would I draw the integers as a graph?
这个图叫做Healy图。
So this is called the Healy graph.
就是把最近的邻居连起来。
You're just gonna connect nearest neighbors.
对吧？
Okay?
所以有一种标准的方式，让这个图代表整数。
So there's some kind of canonical way in which this is like the graph that represents the integers.
明白。
Mhmm.
对吧？
Okay?
而且某种意义上，你可以用有限图来逼近这个。
And there's some sense in which you can approximate this by finite graphs.
为什么？
Why?
嗯，如果你看模n的整数，你会得到类似的图，只不过是一个大圆圈，而不是一条无限长的线。
Well, if you look at integers mod n, then you kind of get the same picture, but you have a big circle instead of an infinite line.
关键是，你看这里的任何一个点和那里的任何一个点，在附近看起来都是一样的。
And the point is if you look at any point here and any point here, in nearby, things look the same.
你得走得很远很远，才能看到这种整体的几何结构，发现它是一个圆而不是一条直线。
You have to go, like, very far away to kind of see this global geometric structure that you have a circle and not a line.
对。
Mhmm.
实际上，整数和模 n 的整数都是群，只要通过加法，或者模 n 加法就行。
And in fact, the integers and integers mod n are both groups just by, like, adding numbers or adding numbers mod n.
对。
Mhmm.
所以这些模 n 的整数就像是完整整数的 SOFIC 近似。
So these integers mod n are like SOFIC approximations for the it full integers.
所以这种近似就是为什么整数是 SOFIC 群的原因。
So this approximation is kind of why the integers are a SOFIC group.
对。
Mhmm.
所以，这个“每个群都是Sophic”的论断，其实是“你可以用群来做这种近似”这个事实的一种推广。
So so the the Sophistic the the statement that every group is Sophic is sort of a generalization of the fact that you can do this approximation with groups.
而那个“所有线猜想”是一个更广泛的猜想，大意是说，任何网络你都能做这种近似。
And this all those lines conjecture is kind of a broader conjecture that like any network, you can do this.
而且它大概不需要那么多代数结构。
And it like, you don't require as much algebraic structure roughly.
所以它是一个更广泛的猜想。
So it's it's kind of a broader conjecture.
所以这个猜想在大概两年前被推翻了，那真是一项了不起的工作。
So so this conjecture was disproved earlier, like two years ago, and it was kind of a really tour de force work.
它有250页，还建立在另外200页的工作之上。
It was two fifty pages building on another 200 pages.
它用了量子复杂性理论。
It uses quantum complexity theory.
所以它真的搭建了一座非常复杂的桥梁，我觉得没多少人能理解这个。
So it really builds this very complicated bridge, I think not very many people could understand this.
所以既然这是一个更强的猜想，那么推翻它比推翻所有群都是Sothic的这个命题要弱一些。
So since this is a stronger conjecture, the disproof is weaker than disproving this statement that all groups are Sothic.
但事实证明，直接证明存在一个非Sothic群要简短得多，也容易得多，相比那个推翻Aldous Lyons猜想的惊人工作。
But it turns out that the direct proof that all that there's a non Sophic group was much shorter and easier than this this really amazing disproof of the Aldous Lyons conjecture.
大概就15页吧。
It's like 15 pages maybe.
它没有用到那些和量子复杂性理论非常复杂的联系。
It doesn't have any of this very complicated connection with quantum complexity.
它基本上就停留在群论的领域里。
It just kind of stays in group theory land.
我的意思是，它确实用到了其他数学家的一些重要现有成果，比如Kuhn和Kuhn和Tom的，但整体上是一个非常合理、正常的证明。
I mean, uses some important existing results by other mathematicians like Kuhn and Kuhn and Tom, but it's it's like a it's like a very reasonable normal kind of proof.
对。
Yeah.
而且，可能说得有点明显，但Sothic群命题和那个之间的联系就是，你取Cayley图，那就是他们用来证明Aldous Lyons猜想的东西。
And and kind of spelling maybe the obvious, but, like, the the connection between the Sofic group statement is just you take the Cayley graph, and that's the one that is like, what they use or for the eldest Leon.
对。
Yeah.
对。
Yeah.
所以这就是为什么它有点像，你知道的，对。
And so that's why it's like a, you know, a Yeah.
所以呢，
So of
没错。
Right.
所以，对，基本上就是这样。
So, yeah, basically, happens is yeah.
所以对于一个群，你可以直接取一个凯莱图。
So for a group, you can take exactly a Cayley graph.
你取一些生成群的元素，然后把相邻的元素连起来。
So you you take some, like, elements that generate the group and you kind of connect elements that are adjacent.
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