Qualifying Exam

Qualifying exams are the first major hurdle for me as a first-year math graduate student. Here at UC Santa Barbara, we’re required to pass exams in three different areas in two years: to stay in the Ph.D. program, you need to get at least two PhD passes and one Master pass. The exams are offered twice a year, giving us four total opportunities to get them done. I was able to pass all of mine in my first year, and in this post, I’d like to share my experience and some tips on how I prepared.

I chose to tackle Algebra, Analysis, and Topology. The topics for each exam are drawn from the standard undergraduate mathematics curriculum. For that reason, it’s highly recommended that you prepare during the summer before starting your Ph.D. program. This allows you to take the exams while the material from your undergraduate courses is still fresh in your mind!

UCSB offers a repository of past qualifying exams to help students prepare. If you’re an incoming student, you’ll receive access to this resource soon. You’ll also get access to a repository of solutions to past qualifying exams, maintained by fellow graduate students.

For Algebra and Topology, the exam material closely matches what is covered in the first-year graduate courses for each subject. Reviewing lecture notes, textbooks, and homework from those classes is especially helpful for preparation. For Analysis, Professor Craig has created a qual syllabus that outlines the main concepts and theorems, along with their corresponding textbook sections. Each week during the Spring quarter, I completed a problem set to prepare for the Analysis and Topology exams, which were scheduled for the last week of the term. I also recommend trying to solve each problem set within 4 hours, as this matches the actual test duration.

The PDF files below are the qual prep books I made for analysis and topology.