Twitter has chosen the NYSE to be the platform for the upcoming IPO. They have also submitted a tentative date of November 15th, though of course that could change.
The Social Media giant will use the ticker symbol TWTR
I would guess the government would have to get back to work several weeks before twitter would want to start the process. The market could be very volatile shortly after our government decides to get back to work, so look for twitter to delay the ipo if the government shutdown continues.