I think Tim Chow does a great job of explaining why questions like this are off-topic: they are primarily opinion based, they are a subjective evaluation of the quality of someone's research, and they flout the text on the "don't ask" part of the help page, which instructs us all to "avoid asking subjective questions where every answer is equally valid: 'What’s your favorite ______?'"
Despite being clearly off-topic on MathOverflow, it seems a large segment of our user base enjoys such questions, and argues that we should periodically make exceptions to our policies, e.g., around the holidays. There are several problems with periodic exceptions:
- Future users are getting mixed signals: one buried in the help page that new users probably won't see, and one represented by a highly upvoted question (that experienced users know is off-topic but allowed anyway).
- Exceptions are not applied uniformly, and can appear to be the whims of moderators.
- Big list questions that garner many answers linger on the main page, pushing other questions down and making it hard for research mathematicians to get actual help on their research, if they have the misfortune of asking during a time when such a question is active.
As Tim pointed out, a question regarding the "most important results of the year" was asked on December 22, was quickly closed, and then prompted this meta thread on the same day. Then, on December 27, a different user asked essentially the same question, and got (as of this writing) 55 upvotes and 6 downvotes. Talk about a whim! I wonder how the person who asked the Dec 22 question feels. The user who asked the Dec 27 question, Bogdan Grechuk, said in the comments that he didn't check meta or the previous closed question, and just asked because he enjoys this kind of topic (and does not seem to care if the question is appropriate or not). A classic example of (1) above.
When relatively new users ask questions that are off-topic, experienced users often write comments or include text in answers that helps the user (and future readers) realize why their question is off-topic, sometimes while also answering. Like, if a question asks for career advice, an answer might begin "Well, this question is really more appropriate for academia.SE, but while it's here on MO, I'll share my own experience..." We see the same thing all the time for questions that are more appropriate for MSE but still get a quick answer on MO.
In that spirit, I answered the Dec 27 question, including the following text:
I believe this question should be closed, and that we should not have a holiday tradition of asking "what are the biggest results of the past year" because such a question is clearly "primarily opinion based" and is not about research mathematics. Voting patterns will show only which subfields of math are over-represented on mathoverflow, and that's a poor proxy for importance of the result mentioned.
I also gave the OP a link to the Quanta magazine article that lists the breakthrough results of 2023, and I mentioned a result in my area (that the telescope conjecture is false) that didn't make the Quanta list. Despite the entire thread being "primarily opinion based", various users panned my answer for "editorializing" or "muddying the waters." Eventually, a moderator, Stefan Kohl (who I respect a lot, by the way) edited my answer to take away the part that commented on why the question is off-topic for MO, pointing out that such discussion belongs on meta instead of main. Again, this feels like we're making exceptions for one thing, but not for another. As of this writing, my answer has 21 upvotes and 19 downvotes, and serves as an example of why "primarily opinion based" questions are not appropriate for mathoverflow. Anyway, I rolled back Stefan's edit because I'm not ok with having an answer of mine to a clearly off-topic question without a disclaimer, contributing to the problem where future users will see the highly upvoted question and think they should ask the same thing for 2024. If my answer remains, I think it's important that it contain a disclaimer that the question is not appropriate. If the moderators feel strongly that such a disclaimer should not appear in an answer, then they can delete my answer entirely.
I do think the community should settle once and for all the controversy of whether or not these kinds of questions are allowed, to avoid this recurring problem. The situation feels analogous to populist political movements, so I think we need to hear from the moderators, who were elected to think about what's best long-term. I support closing the question for each of the previous years, so that users will know they are off-topic and will not ask again in future years. Or, if the moderators think these questions should be allowed, then having them asked on a predetermined date each year by some special account, with text that makes it clear that this is a holiday exception to the rules. Otherwise, I predict we'll see this kind of question asked every year, and it'll be a race among new users who want a highly upvoted question, to ask the question before anyone else does.
EDIT: This debate seems to elicit strong feelings in people. After posting this answer here on meta, a user kept vandalizing my original post on the "breakthrough results" question, so much so that I had to ask the moderators to stop that user. So, for the moment, that answer is locked to prevent tampering, but I strongly suspect that this group of angry users who want to silence opposition to their position will go back and vandalize my answer more when the lock expires. I thought I'd put an update here on meta to explain the situation. Also, I wanted to add a few examples of other places where someone commented in an answer about the appropriateness of the question:
https://mathoverflow.net/a/126065/11540
https://mathoverflow.net/a/372901/11540
https://mathoverflow.net/a/312328/11540
https://mathoverflow.net/a/454778/11540
https://mathoverflow.net/a/56229/11540
I am sure there are many more examples, but I don't want to spend tons of time cataloging them all.
EDIT 2: In the end, the moderators decided to delete my answer, then repost just the part about the Telescope Conjecture. So, it seems the decision is to ignore the rules on the help page when it comes to asking a clearly off-topic question, but then strictly enforce an unwritten rule that says not to comment on the appropriateness of a question when answering it. I suppose that if the moderators want to avoid a double-standard, they should also edit the five answers I linked above that commented on the appropriateness of their respective questions. The deletion of my answer also removed the following text, which is yet another reason why this "breakthrough" question should have never been asked, since it already has a whole article of Quanta devoted to answering the exact same thing:
Furthermore, there is already a yearly article "The Biggest Discoveries of the year" in Quanta Magazine. Here is the one for 2023. It already mentions the Ramsey theory result that is currently the top answer here. It also mentions results in combinatorics, number theory, aperiodic tiling, and physics (namely, the math governing boundaries of black holes).
Overall, I am displeased with how this was handled. I think it contributes to the problem of lack of clarity in what is considered on vs off-topic, and to the perception that the moderators act according to whims rather than according to the written rules of the site.
EDIT 3: Here is a great example of a MathOverflow answer that consists of explaining why the question was a bad question. Like my answer, it received both upvotes and downvotes (as of this writing, 10 upvotes and 8 downvotes). Unlike my answer, it was not subjected to harassment or snarky comments, and was not deleted by the moderators. Oh, well. I guess it's not news to anyone that MO doesn't maintain consistency regarding the rules, or else the "breakthrough" question would have been closed as violating the rules of the FAQ that I linked to above.