Previous qualifying exam survivors have identified these invaluable strategies to help you prepare for the exams. Please note subject to change without notice.
(i) Access old exams at the Kleene Mathematics Library, both online and hard copies. The previous two semesters' exams are also available on 2nd floor VV by the elevator. Solutions are available for past Algebra and Logic Quals. These can give you a good idea of the types of questions you’re likely to see, help you review the content, and get a feel for the structure of the exams. There are multiple copies of the exam books in the library. Be sure to check through them carefully, since some of them are missing an exam or two (or their solutions).
(ii) Form a study group with other students who are preparing for the same exam. By working with other students on old exams, homework problems, and general concepts, you will strengthen your understanding and preparation for the exam.
(iii) Talk with graduate students who have passed the exam in that area. They can provide you with valuable hints and insights, and might even offer to answer questions you have as you study.
(iv) Ask faculty questions. While there are fewer people around the department in the summer and over winter break, there are professors around, and you may find many of them helpful as you’re studying. Don’t hesitate to ask! You have nothing to lose.
In addition, here are some specific tips for each of the exams. While some of these “facts” are subject to change, they do reflect patterns in the exams over the past few years, so use them as guidelines.
Algebra The exam is generally five questions, with at least one problem in each of groups, rings, linear algebra, and Galois theory (the fifth question is a combination of these four). Groups, rings, and galois theory are covered in the first year courses (741-742), but linear algebra is not. If you need more work in linear algebra, consider taking Math 542, which will help fill in the necessary background.
Analysis The exam is based on 721 (real analysis) and your choice of 722 (complex) or 725 (functional analysis). There are usually nine problems, three based on introductory analysis / advanced calculus courses (such as 521/522), three on a first real analysis course (such as 721) and three on (your choice of) 722 (complex analysis) and 725 (second course in real analysis). You will be asked to solve six problems.
Applied Mathematics The number of questions varies from exam to exam, but you usually will have some choice of questions to answer--be sure to read the directions carefully. A good background in complex analysis can be a big help. If your background in this area is weak, you will need to do some extra reading, or consider taking Math 623.
Computational Mathematics The exam is based on Math/CS 714 and 715. There are usually five to six problems. Previous exams could be of help in knowing the type of problems being tested.
Logic The elementary section of the exam is based on 770 (foundations) and the second section is based on one of 771 (set theory), 773 (recursion theory), and 776 (model theory). Each section contains three questions, of which you must choose two. The exam is generally quite consistent, but as with all quals, varies slightly depending on who taught the first-year courses.
Geometry / Topology The exam usually contains nine questions, in three groups of three: basic topology, algebraic topology, and differential topology. You will be asked to do all three questions from the basic topology group either two questions from the algebraic topology group or two questions from the differential topology group. It is common for exam questions to come from the previous semesters' homework problems, as well as recent exams.