In my opinion, the best way to advertise the chat (and specific chat rooms) is using them. If a user comes to chat and sees the chatrooms associated with math.SE, they might have a peek in some of them. It is more likely that they enter a room where there was some recent activity. If they find out that chat can be useful and/or entertaining, maybe they will come back.
You may also notice that occasionally some rooms are displayed in the side-bar. You probably saw something like this while browsing math.SE:
I do not know how exactly are the room shown there selected, but based on past experience, usually some of the rooms which recently had some messages are displayed there. So keeping some activity in the room is important if you want to advertise it in this way.
So if you think that some chat room could use more activity and that it would be good to make more users aware of this room, you can do various things. Let us take - as an example - analysis chatroom. (According to the room description, the intended topics are calculus, real analysis, functional analysis, etc. Which covers rather broad area.)
- If you have interesting link or you have noticed interested question, which is related to analysis, you can post it there. (Maybe it will start some discussion. Even if not, at least some users might learn about something interesting from your post.)
- If you are discussing something on the main and the discussion grows too long for comments, you might invite the other user to chat. If the question is related to analysis, you can continue the discussion in this room. (In this way, other users interested in analysis might notice the discussion - especially if they follow this chat room.)
- The same thing can be said about discussion in the main chatroom. This chat room is often quite busy and it happens sometimes that two or three discussions are going on at the same time. If you have a longer discussion with somebody and it is difficult to follow because there are other things going on, you might decide to continue the discussion in a separate chat room. Again, if this is related to analysis, you can go to the room devoted to this area.
- Even if some topic related to analysis is discussed in other room, you might still mention it in this room by posting a link to some message from other room. Or you can link to a bookmarked conversation in another room. Or you can add link to a room imported from comments, where something of interest is discussed. (Again, if we assume that there are some users who favorited analysis chat room or who follow this room in some other way, they might notice some discussion which they might have missed otherwise.)
- If you see that somebody asked a question in this chat room and you are able to answer it or at least give some useful comment, you should do so. (If somebody came to chat in order to ask about some specific problem, their experience from using chat is much better if they get some feedback or if at least somebody talks to them.) Or even if you do not know answer but have some reasonable advice, you may say so. (For example, reply like this can be useful: "I am not familiar with this topic myself, but I think that searching on the main in the questions tagged baire-category might be a reasonable thing to do." It is possible that the user already tried this. But it is possible that they have learned something new from this reply - maybe the user was unaware that search can be restricted by tags, maybe they
were unaware that there exists a tag closely related to their topic. What I want to illustrate with this example is that even a user which is not familiar with the problem in hand, but has some experience with Math.SE, can give some useful advice.)
I will add a few more things, which do not address directly your questions, but are somehow related. (If there is a need for separate thread for this issue, I can move this part of the answer there. But I am not sure whether this is important enough to deserve a separate post on meta.)
Nowadays if you look at the rooms associated with this site you usually see several chat rooms. (It is worth noting that the list changes a lot if you view frozen and deleted
There are some rooms which have purpose related to some type of site maintenance or "bureaucracy" (Mod's office, blog, tagging, closing and reopening questions, unanswered question...) I guess that it is more-or-less agreed that these room can be useful and it is worth keeping them alive. But you will also see several rooms which are dedicated to some specific area of mathematics. These are the room which I want to speak about. (And since I saw this comment posted around the same time as this meta questions, I guess that your question is intended to be mainly about this type of chat rooms.) Some of these rooms had some reasonable activity in the past, even if they are relatively inactive nowadays. Some of them contain only occasional isolated messages. (Personally, I consider total number of messages and also number of bookmarked conversations and starred posts reasonable indicators showing how active a particular room used to be. If there are some starred posts and bookmarked conversations, it shows that some information given in that room was interesting enough for somebody to mark it for the future reference.)
I have to admit that I am the culprit responsible (at least partially) for their existence - I have started some of them. Some of them were started by other users, but I helped keeping them alive. Normally a room is frozen if there is no activity for 14 days. But posting just anything once in a fortnight can keep the room alive, although kind of artificially. Since I have been criticized for these actions by some users, or at least told that it is probably not the ideal way to use chat, I would like to defend such practice a bit and explain why it might be occasionally useful or at least why it is not harmful to the site. (In my opinion. Or at least I'd guess that this is not a bannable offense.)
Of course, it would be much better if instead of posting just messages which keep rooms alive in this artificial way I would try to create some real activity in those rooms. (Following the advice I have given in the first part of my answer.) But, well, not everything can be ideal. I can at least promise that I will try to do that.
The disadvantage of keeping rooms which are not really needed alive is clear - they take space in the list of rooms. But I think that bumping a room once in a fortnight is not really that bad.
What could be possible advantages? (I have previously posted some brief arguments here, I
will try to elaborate here a bit more.)
- Having such specialized rooms can help with better organization of chat. For example it may help searching. (Although you can search for messages in all rooms, but restricting to some special room can help you narrow search results.) It might help user notice interesting topics. In the main chatroom there is a lot of activity every day. But if somebody is interested in some specific area, it will be easier for them to notice that there was some discussion related to this area if it is posted in a room with less activity. (For this reason I would consider these specialized room as useful even if no discussion was going on there, but some users would post there links to discussions in other rooms related to the particular area of interest.)
- This can save a bit of work for moderators - since they are the people which we bother with such requests if we want to unfreeze a room. (Admittedly, this does not happen often, but at least occasionally there are such requests.)
- Even if we decide to let most of these specialized rooms freeze, I think it could be reasonable to have them at least for the areas which have the most questions here. (Judging by the number of tagged questions, this seem to be calculus and linear algebra.)
- Related to the previous point - beginners' questions are, as far as I can say, the type of questions which most often get downvoted/close/deleted. If some of such questions are asked in chat (and, ideally, also answered or at least somebody can push the asker into the right direction), this might remove some noise from the main site. And even if the question is not answered in chat, the asker can get some suggestions in chat and which may result in better quality of the question, if they decide to post on the main site after all. (And even if the only thing the user asking in chat gets is some advice how to search and they find out that the question would be a duplicate, that still might help reduce the number of such questions.) Of course, users need at least 20 reputation points to talk in chat, many users posting "newb" questions cannot use chat. On the other hand, some of the users might have sufficient rep from another SE site.
- Such rooms might help create a community of users which are interested in the topic. For example, it is very probable that each academic year we will have influx of new users who are currently studying calculus (or linear algebra, or multivariable calculus, or some other area). Having some possibility to communicate with other users with similar interests might be useful.
- One interesting example is commutative algebra. Now we have a room
for commutative algebra here. But previously there used to be a room with a related topic here. And in particular this room was very active for some period. Maybe it is just a manifestation of my obsessive-compulsive side, but having single room would somehow "feels" better. There is no doubt that there will be many more users studying this subject. So it is not that unlikely that even if the current room is frozen, a new room for commutative algebra might be created again.
I should say that I definitely do not plan on do this indefinitely with every room. There are some rooms which I helped keeping alive for some time, but when it was apparent that no activity is going on, I simply let them die. Like graph theory or functional analysis
or finite group theory or analytic number
theory.) But I will certainly admit that the point at which I let go of some chat room is much later that others might decide. (And it depends on many factors. For example whether the rooms used to be rather active
in the past. Or whether the topic is particularly close to me.)