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math.stackexchange.com has a chat capability, maybe do it there.
MO is not a social networking site for mathematicians. What you suggest would be much better done with a separate tool or just "by hand" (emailing people, starting a group blog, etc.).
@Qiaochu: In my opinion MO clearly is a social networking site for mathematicians...just not in the way that is sought after in this question. Anyway, I agree that the site should not (and currently cannot) be bent to this purpose.
No Farmville so far, though!
Mariano, that depends on what you want to grow and how you like to view it. I'd say MathOverflow is a good contender for the right audience.
To use existing features to help mohamedun, edit mohamedun's user page to put a list of requested study groups, have mohamedun drop oh so very occasionally an appropriate comment briefly mentioning the existence of said list, and see if the search features allow one to search the user pages by interest. I know a search user feature is present on meta.mathoverflow.
Added 2011.06.13: Peter Krautzberger has shown us a way to search User pages, and Dr. Geraschenko has added it to the MathOverflow Tips and Tricks page near the bottom. I am suggesting a format in another thread to make certain searches more effective. I invite discussion and expansion of or changes to the suggested format.
Gerhard "Don't Need No Stinkin' Farmville" Paseman, 2011.06.07
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