Does anyone know a good IRC infrastructure that allows for quickly entering and displaying TeX formulas?

There's a plugin for Pidgin called pidgin-latex which handles just that.

ETA: If people start using this plugin (or, more generally, if we use TeX/LaTeX in any capacity for this study group), it might occasionally be helpful to use the detexify handwritten symbol recognizer - for when you want to use a symbol and can't quite remember the command that produces it.