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This is an old revision of the document!
After your Prelims you should already know what area of research you want to work in. Usually that revelation comes after taking a class you really enjoyed.
The standard procedure then is to approach the instructor of the class (or other faculty member working in the same area) and ask for a reading course on a topic that you're both interested in.
After a quarter or two of reading courses you should be able to tell if the faculty member fits your research interests and work ethic.
Together with your to-be advisor, you pick an open problem you want to work on, and that problem becomes the basis of your Qualifying Exam. That's why you must pick an advisor at least 6 months before your Qual, meaning you start taking reading courses a year before the Qual.
Here is what the graduate council has to say about the advisee/advisor relationship: Mentoring Guidelines
To see how to prepare for the Qual, go to The Qualifying Exam tab.
The timeline described here can be too late for some people. To find another version of the timeline, go to Wish I'd Known... tab.