(i) Past exams are available and 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. With the exception of Logic, old exams may be accessed at the Astronomy, Math and Physics (AMP) Library. The Logic exam (with solutions) may be accessed at this page. Solutions for past Algebra exams are available here.

(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) 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!

## Exam Tips

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 Mathematic****s** 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. The exam is generally quite consistent, but as with all quals, varies slightly depending on who taught the first-year courses.

**Geometry / Topology** The exams contains six questions, in two groups of three: You will be asked to do two out of three questions from the algebraic topology 1 group (751) and either two questions from the algebraic topology 2 (752) group or two questions from the differential topology (761) group depending on which version of the exam you are taking. It is common for exam questions to be related to the previous semesters’ homework problems, as well as recent exams.