The Math Stack Exchange chat room has had some heated arguments/suspensions in the past and I was wondering if there were any plans for the newly opened Math Overflow chat room to have its own moderators to provide the required special attention to prevent similar problem situations?
The site http://mathoverflow.net is not a new site; it exists since more than three and a half years (end of september it will be four). Only it joined the SE-network recently. Then like now it has moderators (the same as before). Who they are can be seen on the already provided page (in analogy with other sites) https://mathoverflow.net/users?tab=moderators
Some of these moderators were elected some others not, as they started the site "from scratch" and there was not yet anybody to elect them. In the old version of MO, Anton Geraschenko, the main founder of the site, was not only moderator but also administrator.
Please, do not think of this site as some site in "beta" or something along these lines.
Moderators of Math Overflow are also moderators in chat. That said ...
Math Overflow can certainly have an election like any other site, and the same procedure would apply as on any other site.
Once a year, we check in with the moderation teams to ask how they're doing and if they feel they need any additional help - and we generally just go with whatever the existing team says. If they indicate that they need two more moderators, and that seems sane, then we schedule an election for them.
Once in a while we may prod more frequently if we notice that flags are taking an exceptional amount of time to clear - but we'll usually have heard cries for help when this is consistently the case.
If and when the existing Math Overflow moderation team reaches out to us to ask for more help, we'll gladly schedule an election for them using the same STV system that Stack Overflow and the rest of the sites use. We like leaving this up to the moderation teams themselves, because they generally have the best feel for when one might be needed.
Regarding chat - Yes, other moderators on the network have moderation abilities in chat. When Math Overflow transitioned to the Stack Exchange 2.0 platform, some of our other community moderators from other academic sites joined chat to make themselves available to anyone that had questions or needed guidance. This was a very nice thing for them to do, they're genuinely helpful people and wanted to help make everyone feel welcome.
Still, your existing mod team are the ones that are in charge of making sure that the site runs smoothly.