Before the end of a student's 11th quarter in residence (Winter quarter of the 4th year), they must pass an oral exam. This is not quite what it sounds like; the "exam" is really more accurately described as a research proposal. It is your job to prepare an organized and thoughtful presentation of exactly what you plan to do for your thesis. The audience will include your doctoral committee, which consists of your thesis advisor, two other faculty members from the math department, and two faculty members from a department other than math (one of these two must be a tenured UCSD faculty member). Of course, other people are certainly allowed to come watch and cheer you on.
Seung is a student of Nolan Wallach, and advanced to candidacy at the end of his third year. For him, advancing provided a chance to organize his thoughts and make sure he had a clear understanding of his goals.
He recommends students advance as early as possible. Preparing for your advancement can take most of your time for a quarter, sogetting it out of the way allows you to focus on your research sooner.
That is, if your advancement talk is about topic X, but later you decide you are more interested in topic Y, you can write your thesis on topic Y. The main purpose of the advancement talk is for your committee to be able to see that you are able to elucidate a statement of a math problem, and outline some kind of rudimentary plan of attack.
It's certainly advantageous not to change research topics - for the simple reason that you will save time by not having to start over - but you should not feel that you are locked in once you advance to candidacy.
After finding a research topic, make sure you have a clear understanding of the problem you want to solve. More specifically, how does your problem fit in the landscape of other research in your field? Are there other problems (solved or unsolved) that are similar in nature to yours? What consequences will solving your problem have? You should also make sure that your advisor believes that your problem is tractable.
We've all seen incredibly simply stated math problems that take the world's best mathematicians decades (or even centuries) to solve. While no one can ever be completely certain that any given problem is tractable, your advisor will likely have a good idea of whether he or she thinks you have a reasonable chance of finishing a given problem in a reasonable amount of time.
Keep in mind that the talks are only about an hour long. The people on your committee may not have ever even heard of your particular research topic. Remember that two of your five committee members must be from departments other than math. If you assume too much background, they will be left completely in the dark. Of course, you will probably have too little time to give complete and careful definitions of everything related to your research. Striking the right balance will take some thought.
As you might have guessed, there is also a fair amount of paperwork that goes along with advancing. It’s important to get all of the paperwork completed correctly and submitted because it’s the paperwork that documents the fact that you actually did your talk and advanced. Here are the departmental and OGS requirements to keep in mind:
1) First, you must have passed the department’s one foreign language exam.
2) Committee Members: Committee members are usually chosen with the help of your dissertation advisor. The committee must consist of no less than five (5) members, three from our department and two from outside the department (one of whom must be tenured). Please note that asking faculty members to serve on your committee is clearly your responsibility, with the assistance of your faculty advisor. No less than three weeks prior to your Oral Qualifying Examination email your list of members to the Graduate Advisor for submission to OGSR for approval (please provide everyone’s full name, academic title and affiliated department). Please designate who is the Chair of your committee and if you have a Co-Chair.
3) It is your responsibility to make arrangements with each committee member for the date and time of your examination. Room reservations should be made at the Front Desk. Send an email to email@example.com with the date, time and location of your oral examination so that the information can be posted on our bulletin boards. It is your responsibility to remind the committee members close to the time of your exam.
4) A few days before your examination, go to the Graduate Advisor and pick up the “Report of the Qualifying Examination and Advancement to Candidacy for the Degree of Doctor of Philosophy” form (which requires the signatures of all your committee members), and the “Degree and Diploma Application.” After your examination return these forms to the Graduate Advisor, who will check for errors, obtain the Graduate Vice Chair’s signature, and then make copies for departmental records.
5) Before you take these forms to OGSR, you must first stop by the Cashier’s Office and pay a filing fee. They will validate your “Report of the Qualifying Examination and Advancement to Candidacy for the Degree of Doctor of Philosophy” form, and you can then take it, as well as the Degree and Diploma application, to OGSR for processing.
If you have any questions or concerns about the above step, the Graduate Advisor can assist you.
While you are not required to advance until the end of your 11th quarter, it is not a bad idea to advance earlier if you are ready. After you have advanced, you are eligible to teach your own course as an Associate Instructor. This is not only good experience, but can be very helpful when putting together a teaching portfolio for job applications.