A few days ago, the chat went into a deep measure theoretic discussion and a separate room was spawned to keep the main chatroom for the general chat.
So we came up with the idea of having a separate room, in which an invited speaker could give a lecture and later on answer questions. Of course this is less likely to go wonderfully smooth without in-chat $\LaTeX$ support, but there are workarounds.
The ideal is to have a weekly session. Of course this may be too much so have a lecture once every two weeks, much like a colloqium or a seminar.
I have created a separate chatroom which is a "gallery" chatroom, it means that it is read only for everyone and the speaker will be granted a voice. After the lecture will be finished we will work out a method for asking questions (perhaps in a different room) so it will not clutter the chat log.
What I would like to ask the good folks of math.SE is to come up with ways that we can make this even better.
Suggestions about how to get $\LaTeX$ work on the chat (as Jeff remarked that they do not plan on adding such support to chat) are most welcomed.
Along side a list of "What you'd like to hear". Remember that this is not a 1-1 technical hour, and should probably be similar to what you'd hear in the seminar at your local department.