But the wording of the result makes it sound like we don't know what the lowest possible complexity bound might be. So, prior to this result did we think there couldn't be a lower possible bound? Or did the arithmetic circuit community think there were lower possible bounds but didn't see it as a high value target for experts to tackle (maybe a problem that was instead regularly given to students to study).
For example, despite our best efforts, the state of the art lower bounds on time complexity of algorithms for solving 3SAT is O(n). In contrast, our best algorithms for the task run in time roughly O(2^n). That’s an exponential gap. This is despite decades of trying to find lower bounds.
The sofic groups question was the outstanding question about sofic groups. Almost everyone thought that non-sofic groups existed, and there were plausible candidates, but proving a group was non-sofic was out of reach. Now that we know how to do it once, we can probably do it a lot more.
The Connes rigidity conjecture I think people thought was false, but it was a provocative claim to make. The significance of conjectures is frequently not that the answer to the question is "yes", but that we don't know how to answer the question. And now, apparently, we do.
a colleague was telling me that the base idea for proving that something is not sofic already appeared in the literature around 2019 or so (this is the "expanders graphs" that are mentioned in OpenAI s paper. no one had managed to find a concrete example though. this doesn't make the result less impressive in any case.
The general consensus of developers is that AI can only do the work of a strong 'junior'. Yet as soon as we are presented with pure mathematical results, people seem incredibly ready to accept that AI can do more than what a strong student could achieve.
If it works better here than for programming, then I would guess it's because you can give it a very precise prompt, so you either solve the problem or you don't. If you read the prompts people have shared for problems like this, then the instructions are basically "Solve this problem. Don't give up early. Don't solve a similar problem."
Your worry.... is because they used the word advanced? For marketing? The word is used very appropriately here. There were PhD's who spent a big part of their career tackling these problems.
I'm trying to understand if these specific problems were the kinds of problems that would have justified an expert investing weeks or months to solve. Or if they were the kinds of problems that would normally have been given to students to investigate.
Mundane incremental research is cobbled from existing citations that already appear nearby in the record.
Basically, innovative research is a measure of bridging thought and domains that were previously not bridged. It's quite concrete as a measure in the citation record.
So we can know pretty conclusively.
Puja Ohlhaver gave a talk on this[1], and ran some experiments (that I had the pleasure to support on)
Also on HN front page today: AI's debt binge can't last, hidden borrowing reaches $1.65T (fortune.com)