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Below is a also a list of suggestions from students and faculty. Suggestions may differ between the Applied and Pure Exam but the timeline should still be the same. Also, if you want more advice beyond this department, check out the Grad Studies Qualifying Exam suggestions.
Some other useful resources:
For applied math students, this means finding a problem and knowing background for this problem. You will be evaluated on how well you can explain the problem, how well you know what directions you will take, what other people have done, and how fluent you are on the background material.
For pure math students, this means finding a research specialty and knowing what prerequisites it has. You will be evaluated on how well you know the prerequisite material. (This advice is for the post-presentation section of the quals. Presentation advice in the Preparation section.)
To get an idea of how ambitious (or not) the exam content needs to be, browse through the quals syllabi binder in the grad computer lab and consult your advisor.
Qual Proposal: Draft a qual topic proposal. Previous proposals are available in a binder in the grad lab. Circulate the draft among your committee, get their approval, then forward it to firstname.lastname@example.org for approval in the proper channels. Send a finalized copy to all members of your committee to have available to peruse prior to the exam.
Don't be overambitious with your list. There's plenty of post-qual time to learn more of what you need to write your thesis.
Don't put anything on your syllabus unless you are *sure* that you will know the material very very well by 2 months before the exam. (We mean it.)
You will need four professors if in pure math (five professors if in applied math), and one of them to serve as your committee chair. Try to choose professors who already have a good impression of you. Also, talk with your adviser to find out if he/she plan on being on your committee. (Your adviser can't be chair.) Some professors have strong opinions about being on their students' committees (some will, some won't).
For Mathematics majors, one member of your committee must be external to the Department, and the Chair must be a member of the Department.
For Applied Mathematics majors, one member of your committee must be external to GGAM, and the Chair must be a GGAM member.
Later you will need three members for your dissertation committee; these three don't have to be on your qual exam committee but it can be convenient if the groups overlap so keep this in mind when choosing.
Exam Scheduling (this can be the most stressful part): When everyone is satisfied with the proposal, arrange a qual date that everyone can make. Reserve a room for the qual through email@example.com, reserve 3 hours.
One of the hardest part of the exams is getting a committee of five very busy professors to not only agree on a meeting time but also to actually all show up at the same time for your exam.
Make sure you pay attention to the following deadlines. The above are good recommendations for timing your exam, but these deadlines are established by Graduate Studies and cannot be moved (easily):
Talk or email former students of your advisor to get their advice and ask what questions came up. Talk to other students who have shared the same committee members to find out what questions they were asked. Talk to your committee members to find out their expectations -- they may or may not tell you point blank what they intend to ask you and what they will not.
Study hard enough in advance that you can give yourself time to relax and watch a movie or have a glass of wine the night beforehand. You probably wouldn't be taking the exam when you are if your professors didn't feel you were ready.
Review statements of theorems and definitions, understand the main ideas of the proofs. By the date of the exam, you should know these statements cold.
For GGAM students: know your problem, inside and out. Know the rigorous mathematics as well as intuitive explanations of all the specifics of your problem, including the variables, expected outcomes, etc.
Plan your talk to be 40 minutes without questions.
If you have a computer presentation: Let the computer staff know the date and time of your exam so that they can be "on call" to help with anything that breaks (knock on wood).
DO NOT EDIT YOUR TALK THE NIGHT BEFORE. (Trust us.)
Try to practice/prepare enough so that under usual conditions, you could deliver the talk without notes.
Talk to your advisor and your committee about whether or not they allow you to bring notes with you for your talk. (This goes for any sort of talk, powerpoint or chalk.)
With respect to giving a talk during the exam (and this goes for any talk): Work on building on large pile of coherently connected knowledge and not just many small piles of knowledge. Not only will your talk be easier to follow, but you will also be implicitly demonstrating your ability to plan and perform research.
Do NOT bring up any lingo/jargon that you can't explain.
If you choose to give an expository talk, here's the advice I was given for how to choose a topic: pick something that you can finish with some detail in about 40 minutes. Your examiner's questions will add about twenty minutes. Expect many interruptions.
Don't pick something too famous; every professor will have a pet version of it, and you'll invite more questions than you want. On the other hand, don't pick something too obscure; the examiners might have trouble judging the weight of the theorem.
A couple of professors said they were impressed by a particular research talk, because it conveyed to them that the student knew where they were going. But if you don't have a research result that can be explained in some detail in 40 minutes, or you don't have a research result, it's okay if your qual talk isn't on research. I gave an expository talk on an theorem related my field, and I passed. (All I'm saying is: if you have some spiffy research result that you can present in under an hour, by all means, do so! But the world won't end if you don't.)
Don't forget to backup your presentation!
Practice your talk in front of advanced students who will understand your talk.
If you advisor has a lab group (or many students), practice your talk in front of them after your mock exam.
Take a mock qual: arrange for senior students to give you a practice exam. Whether or not you've taken an oral math exam or not, it's good to practice thinking extemporaneously at the blackboard in front of examiners. Take the mock qual soon after you've studied a bit. You may discover that you want to revise the way you've been studying -- for some, an oral exam is a very different experience than a written exam, and studying techniques may need to reflect this difference.
... that you exist and that they exist. Professors can be absent-minded. Remind your committee of the date, time, location 1 month, 1 week, and 1 day prior. It may not hurt to send out an email the day of.
Try to relax the day before and day of the qual. You've done all you can. Now show'em what you've got.
You will be asked to leave the room so the committee can come to agreement on the format, then you will give a 40 minute presentation (either expository or research-related), which will run up to an hour an a half by interruptions.Then you will be asked to leave the room, and return for more questions about your qual topics. Then you will be asked to leave again and you will finally hear your answer after the last pause.
Most exams take under three hours, and over two, though there are exceptions. Don't forget that when you are sent out of the room, it's so the examiners can take a break too. If you hear them laughing, they may just be sharing the latest spherical cow joke. It may have nothing to do with you at all.
Try to present yourself with confidence during the test, even when you are unsure of an answer. Your examiners can be swayed by your attitude.
One should tell oneself that not knowing an answer to a question is not an automatic fail and instead concentrate on getting some future questions right and improvise by discussing related topics that you know better.
Also, if you're not completely sure of an answer, but have some vague idea of how to start, don't just remain silent and try to think through the whole solution before answering. It's better to state your initial idea and usually the committee members will nudge you on from that point. They often only wan't to see if you can start out in the right direction.
When discussing work that you've done make sure to discuss "your results" to create ownership of your results.
Bring to the exam:
Now that you've passed your works isn't done. You must scrape together both the right mixture of signatures as well as the fee ($90 as of Spring 2017) to officially file for candidacy once you've finally passed the exam. (It personally took me two months to get this done!).