What do moderators do? Is it just a matter of dealing with posts flagged for moderator attention, or are there other things too?
Sometimes we have to write to some one who is misusing the site in some way, either with good intentions (e.g. over-eager editing of old posts), through ignorance (particularly new users), or simply because they are misbehaving. Very often there's first a discussion on the moderator mailing list. Sometimes we also suspend a user, and we have some (semi-secret, I guess) rules of thumb for escalating suspensions for repeat offenders, although happily this hadn't been necessary so much recently.
In a handful of instances we've had to ask a 'senior' mathematician to change their behavior, or suspend them, or otherwise deal with unacceptable behavior. This is a little scary sometimes (one might worry about them subsequently holding a grudge which might affect job or grant applications later, say), but in practice it seems to be fine. We've always managed to find a sensible and mutually acceptable solution, and to our knowledge(!) haven't made enemies.
Dealing with flags for moderator attention is the main day-to-day obligation. Currently (October 2013) we do pretty well at this, averaging under 3 hours to process a flag. For various reasons, there are actually more flags for moderator attention now than there was before the transition to 2.0.
The software provides nice tools for going through flags, and in particular it's usually fine if a given moderator decides a particular flag is too complicated to deal with immediately. Occasionally a flag will 'escalate' to the point of someone started a conversation on the private moderators' mailing list.
Sometimes we find that someone is flagging inappropriately, and we need to write to them explaining our objection.
The moderators are also responsible for the long term administration of the site. We don't really know what will be required in the future, but below I give some examples of things we've done previously. (I should also note that individual moderators have contributed in all sorts of different ways to each of the points below; some have been handled by a single person, some have been joint efforts. I'm making no attempt to sign credit or blame here!)
- We solicited community input on the migration to 2.0, and negotiated the possible frameworks.
- We wrote, with the help of a lawyer, the SE folks, and the community via meta, the legal agreement governing the migration to the Stack Exchange network.
- We arranged for MathOverflow to incorporate (and, indeed, we've been meaning for a while to explain all the technical details of this!) so that we would have a durable legal entity able to sign this agreement, and to handle finances.
- We applied successfully for a Sloan grant, to cover the legal expenses of incorporating and vetting the migration agreement, and also to support work on integrating MathOverflow with the mathematical literature.
We try to spend enough time on meta to ensure that questions requiring our input get answered properly. Often this requires looking up some statistics from the moderator tools, helping write a Data Explorer or API query, looking at a deleted post, or merely remembering some ancient thread on tea. Sometimes it requires input from the SE folks, which we can request.