Qualifying Review

Students often stress a lot about the QR Exams. Here we gather some advice from students who have made it through this process. For details on the Qualifying Review Process, please see the official math department webpage. It is important to note that the Qualifying Review is not just a sequence of exams: each student is carefully discussed by the Doctoral Committee to determine readiness to pass the Qualifying Review, including their academic record at Michigan (courses, grades, comments of professors) as well as plans with a tentative advisor. The process is not just an all-or-nothing exam score above a certain number. 

That being said, we acknowledge that students do stress about the QR exams, and every math PhD student must pass exams in three subjects (although it is acceptable also to  “course out” of one of the subjects). The Math Department QR Exam page.  has  outlines of each exam’s syllabus,  past QR exams and some solutions. (Beware though, some solutions might have some errors! Write to us if you think you found one!)

Structure of the QR

Qualifying Review examinations (QRs) are offered in the following four areas – Algebra, Topology, Analysis, and Applied Analysis. Exams are offered before fall term (late August/Early September), before Winter Term (early January) and after winter term (early May). There is no penalty for trying and failing: students are encouraged to try an exam as soon as they think they might be ready.


Students must pass one of these exams by January of their second year. We advise and expect students to try one in May of their first year, after spending that year preparing for the QR exam in that subject by taking the corresponding “Alpha Courses;” if needed, student can make a plan to prepare through the summer and/or fall. Of course, students who have studied the material at the graduate level already are encouraged to take the exams even earlier: your score on the QR is a good data point for choosing classes. It is usually a big mistake to skip the alpha courses without having passed the QR exam in the corresponding subjects.


All three exams should be passed by January of the third year.  You can also `course out’ of one of the QR exams: instead of taking a qual, you can take two courses  specific courses in that area and earn at least B’s. Please read the more detailed rules on the math department website.  


The applied analysis qual is often cancelled, due to no one registering for it.   


General Advice

Take the alpha courses.

The alpha classes (Math 593, 594, 591, 592, 596, 597) are designed to prepare you for the quals. You might think that are not related to your field of interest, but the content of these courses is something every mathematician should be proficient in.  For those with more applied interests, you can substitute Math 556, 572 for 592 and 594.


Start taking the QRs early, and try again and again.

Start early and don’t be afraid of taking a QR exam even if you are not fully prepared. Sometimes the quals are easy and you might get lucky, and there are no penalties for failing a given qual. 


Don’t be afraid of coursing out.

If you only have one QR remaining and have not been able to clear it despite multiple attempts, consider coursing out. 


Focus on one Exam at a time

Try to get at least one QR done at a time. You can study for multiple quals at a time, but study more for one particular area, that you feel the most confident about. 


Form study groups

If you plan to study on your own, you might often not feel motivated enough to do it. In such cases, it is good to have form a study group to keep you focused. In addition, your study partners might know how to solve some problems that you don’t. You can easily find people studying for a particular exam by either asking around or sending an email to the math graduate student list. Additionally, the AWM sometimes has  QR study sessions before the exams.


Take timed practice exams

Many people perform very differently under time pressure. Try to solve a past QR exam in three hours. This will simulate the time pressure that you will face during an exam and will make you better prepared for it. 


Study Resources

  • The alpha courses are designed to prepare students for the QR exams, in addition to being a great overview of the particular areas by masters in the field (and potential dissertation advisors).  It is not a good idea to skip these courses unless you first pass the QR exam. 
  • Before taking the exam in a particular subject,  you can sit in the appropriate alpha courses without registering to review, and/or as seek out course notes and problem sets from previous alpha classes. You can make a habit to work with the first year students on their problem sets, even if you are not in the course.
  • Past QR Exams are probably the best resources to study for QRs for those who have already taken the courses.
  • Mel Hochster made extensive study materials for the Algebra QR exam, including practice problems and solutions, over a period of time in the early 2000’s:
  1. Review Materials from Fall 2002
  2. Review Materials from Fall 2003
  3. Also, scroll down Mel’s website:  toward the bottom are Old Algebra QR Exams with solutions from January 2004, May 2004, September 2004, and January 2005.


Advice from students

Sanal Shivaprasad

I took all the quals as soon as I came in. In retrospect, this was probably a bad idea. I was just settling in to a new place, and the quals were right after a tiring week of the teaching orientation. I remember that period as being one of the most stressful times at the university. 

I would recommend you try to take as many of the quals as you can when you come in (maybe 1 or 2), while not stressing too much about it. And, don’t expect to get any studying done in the week right the quals before, due to the teaching orientation.


Rachel Webb

For me, the best way to study for a test is to take old versions of the test. I start doing old tests from the beginning of my study, beginning with the oldest one available. At first it may take me several weeks to get through a test, as I need to go reread relevant parts of the textbook and maybe work several easier homework problems before I can solve the test problems. But solving the tests slowly gets faster, and by the end of my study I can solve a test in one or two days.


Yuxin Wang

I would definitely recommend studying for the quals through past exams. I personally went through around 4-5 past exams in Analysis and Algebra, and made sure that I understood each problem (some older Analysis exams do not have solutions, so I asked them on Stack Exchange). Taking notes on some common tricks is also useful — for instance, the density argument in Real Analysis. 

Another minor aspect is that for some Analysis/PDE students, the “Applied Analysis” exam might be worth considering. My experience is that, having learnt numerical differential equations, the Applied Analysis exam was a lot easier to me than the Algebraic Topology exam. The skills tested turned out more useful as well.