I mentioned Math Overflow a while back (also see the general backgrounder on math overflow on this blog). At the time, I hadn’t joined Math Overflow or participated in it. I joined a week ago (January 6) and my user profile is here. Below are some of my observations.
Surprising similarity of questions with questions I’ve asked in the past
Looking over the group theory questions, I found that a number of questions I had asked — and taken a long time to get answers to — had been asked on Math Overflow, and had been answered within a few days. The answers given weren’t comprehensive; I added some more information based on my past investigations into the topics, but it was still remarkable that these questions, most of which aren’t well known to most people in the subject, were answered so quickly.

The question Are the inner automorphisms the only ones that extend to every overgroup? is a question that I first asked more than five years ago, when still an undergraduate. I struggled with the question and asked a number of group theorists, none of whom were aware of any pasy work on these problems. I later managed to solve the problem for finite groups, and then my adviser discovered, from Avinoam Mann, that the problem was tackled in papers by Paul Schupp (1987) and Martin Pettet (1990), along with the many generalizations that I had come up with (some variants of the problem seem to remain unsolved, and I am working on them). You can see my notes on the problem here and you can also see my blog post about the discovery here.

The question When is Aut G abelian? was a question that I had been idly curious about at one point in time. I couldn’t find the answer at the time I raised the question, but stumbled across the papers a few months later by chance (all well before Math Overflow). It’s interesting that the question was so quickly disposed of on Math Overflow. See also my notes on such groups here.

The question How can we formalize the naturality of certain characteristic subgroups is a more philosophical question with no real concrete answers, which I’ve considered for a long time too.

The question Balancing problem in combinatorics that I had, based on a generalization of an Olympiad question I had seen, turns out to be part of something called rainbow Ramsey theory, as the answer suggests.
Two things stand out: (i) all these questions are questions whose answers are not wellknown (the people I asked didn’t know the answers offhand) but are questions that many people do ask (ii) On Math Overflow, they were dealt with quickly.
I think the situation is similar in many other established areas of mathematics — the answers are out there, but they are not wellknown, probably because these problems are not “important” in the sense of being parts of bigger results. But they are questions that may naturally occur as minor curiosities to people studying the subject. These curiosities may either go unanswered or may get answered — but the answers do not spread to the level of becoming folk knowledge or standard textbook knowledge, because they aren’t foundational for any (as yet discovered) bigger results.
Math Overflow now provides a place to store such questions and their answers — thus, next time you have one of these questions, a bit of searching on Math Overflow would yield the question and the answers provided and stored for posterity. Apart from the questions I had thought of, consider this one that somebody thought up and turned out to have been considered in multiple papers: When is A isomorphic to A^3?.
The situation is probably different for areas where new, cuttingedge questions are being asked — i.e., areas where the questions are charting “new” territory and are helping build the understanding of participants. Some people told me that this is likely to be the situation with areas such as topological quantum field theory or higher category theory.
Skills needed and developed
So what skills are needed to participate in Math Overflow, and what skills get developed? In order to answer questions quickly, a combination of good background knowledge of mathematics and the ability to search ArXiV, Mathscinet, JSTOR, Google Scholar, and other resources seems necessary. In order to ask questions, it seems that a combination of a lot of background knowledge and a natural curiosity — of the bent needed for research, is what is needed.
Pros and cons of posting on Math Overflow
The obvious pro seems to be that a lot of people read your question. The sheer number of readers compensates for the fact that most people, even experts in the area, may not immediately know the answer. Because of the larger number of people, it is likely that at least a few will either have come across something similar or will be able to locate something similar or will be able to solve the question because he/she gets curious about it. It was pretty exhilerating when a minor question that I didn’t feel equipped or curious enough to struggle with was answered by somebody who came up with a construction in a few hours (see collection of subsets closed under union and intersection), or when a question I had about elementary equivalence of symmetric groups was answered within a few days (see elementary equivalence of infinitary symmetric groups).
The potential con could be that people may be tempted to ask a question on Math Overflow without thinking too much about it. This probably does happen but I don’t think it is a major problem. First, the general quality of participants is quite high, so even if people ask questions without thinking a lot about them, chances are there is something interesting and nontrivial about the question — because if there weren’t, people of the profile contributing would have been able to solve it even without a lot of thought. Further, even if a question is a good exercise for a person specializing in that subject — so he/she should struggle with it rather than ask others, it may be good for a specialist in another subject to simply ask.
The voting (up/down) system and the system of closing questions that are either duplicates or not suitable for Math Overflow for other reasons (such as undergraduate homework problem), combined with a reputation system for users linked to the votes they receive, seems to be a good way of maintaining the quality of the questions.
A revision of some of my earlier thoughts
In a blog post almost a year ago titled On new modes of mathematical collaboration, I had expressed some concern regarding the potential conflict between the community and activity that is needed to have frequently updated, regularly visited content, and the idea of a steady, timeindependent knowledge base that could be used as a reference. mathematics blogs, with regular postings and comments, and polymath projects, which involve collaborative mathematical problemsolving, are examples of the former. Mathematics references such as Mathworld, Planetmath, the mathematics parts of Wikipedia, The Springer Encyclopedia of mathematics, ncatlab, Tricki, and the Subject Wikis (my idea) are some examples of the latter, to varying degrees and in different ways. The former generate more activity, the latter seem to have greater potential for timeless content.
Math Overflow has the features of both, which is what makes it interesting. It has a lot of activity — new questions posted daily, new answers posted to questions, and so on. The activity, combined with the points system for reputation, can be addictive. But at the same time, a good system of classification and search, along with a wide participatory net, makes it a useful reference. I’m inclined to think of its reference value as greater than what I thought of at first, largely because of the significant overlap in questions that different people have, as I anecdotally describe above.
Math Overflow stores both mathematical data — questions and their answers, as well as metadata — what kind of questions do people like to ask, what kind of answers are considered good enough for openended questions, how quickly people arrive at answers, what kind of questions are popular, how do mathematicians think about problems, etc. This metadata has its own value, and the reason Math Overflow is able to successfully collate such metadata is because it has managed to attract highquality participants as well as get a number of participants who are very regular contributors. Both the data and the metadata could be useful to future researchers, teachers, and people designing resources whether of the community participation type or the reference type.
On the other hand…
On the other hand, Math Overflow is not the answer to all problems, particularly the ones for which it was not built. Currently, answers to questions on Math Overflow are broadly of the following three types (or mixtures thereof): (i) an actual solution or outline of a solution, when it is either a short solution arrived at by the person posting or an otherwise wellknown short answer (ii) a link to one or more pages in a short online reference (such as an entry on Wikipedia or any of the other reference resources mentioned above) (iii) a link or reference to papers that address that or similar questions.
For some questions, the links go to blog posts or other Math Overflow discussions, which can be thought of as somewhere in between (ii) and (iii).
With (i), the answer is clearly and directly presented, and the potential downside is that that short answer may not provide a general framework or context of related results and terminology. With (ii), a little hunting and reconstruction may need to be done to answer the question as originally posed, but the reference resource (if a nice one) gives a lot of related useful information. (iii) alone (i.e., without being supplemented by (i) or (ii)) is, in some sense, the “worst”, in that reading a paper (particularly if it is an original research paper in a journal) may take a lot of investment for a simple question.
In my ideal world, the answer would be either (i) + (ii), or (ii) (with one of the links in (ii) directly answering the question), plus (iii) for additional reference and indepth reading. But there is a general paucity of the kind of indepth material in online reference resources that would make the answer to typical Math Overflow questions by adequately dealt with by pointing to online references. So, I do think that an improvement in online reference resources can complement Math Overflow by providing more linkable and quickly readable material in answer to the kinds of questions asked.