The information in this post may vary for different schools.

What are the PhD qualifying exams? Depending on the school, within the first 1-2 years of grad school, PhD candidates are expected to take typically 2-3 written exams in core areas (such as topology, algebra, analysis) and demonstrate proficiency. At UConn, these exams are about 6-7 problems and last 4 hours. Some schools call these exams quals and some call them prelims (for preliminary exams), but they are the same thing.

At UConn, we are required to PhD pass 3 prelims (our prelims are graded by a PhD pass, Masters pass, or fail). The prelim options are algebra, geometry and topology, complex analysis, and real analysis. I took (and passed) the algebra, geometry and topology, and complex analysis prelims. There is an overwhelming amount of material that is covered or that can be covered on these exams, so it is important to have a thorough understanding of the material. In this post, I will share how I studied for these exams and some tips/strategies that I used, and hopefully these will help you with your studying. We want to study hard, but effectively!

Coming into grad school, I was not prepared for the level of difficulty of the courses. I quickly realized that there were many gaps in my background and that I needed to catch up. While trying to catch up, I realized that I needed to learn how to study more effectively. Everyone is different and has their own style of studying. It’s important to figure out what works best for you and how you learn the most. Studying for prelims really starts when you are taking the prelim course. At UConn, the course instructor is the one who writes the prelim that year, so it’s extra important to study the homeworks, midterms, and finals for those classes. Also, definitely take advantage of office hours if you are stuck on a problem or having a hard time understanding something!

The biggest asset to studying for prelims is taking/studying past exams. Typically every school will post their past exams online, along with an exam syllabus, so you can view them and use them for studying. For example, here is a link to UConn’s past prelims. There are a few different ways to go about using these:

1. View the prelim syllabus. The first thing I did when I began studying was look through the prelim syllabus and make sure I understand everything that is defined/mentioned. I went through my notes and the course textbooks and wrote down important definitions and theorems. It’s good to review definitions and theorems that you know, but it’s more important to focus on the ones you don’t know. It’s also important to review the proofs to some of the important theorems and examples that are mentioned in class and the textbook.
2. Look through past exams and make a note of what kind of problems there are. This is not always easy to do, but sometimes you can sense a pattern with the kinds of problems that appear. For example, on the complex analysis prelim, there is typically always a contour integral problem, a conformal mapping problem, a problem that uses Rouche’s theorem, etc. So when studying for that prelim, it’s super important that you make sure you are able to do several of these kinds of problems. Of course it varies based on who is writing the exam, and this should not be the only way you study, but it definitely is a helpful starting point. This method helped me study for the topology and geometry and complex analysis prelims, where the problem types are more predictable.
3. Do as many recent prelims as you can. Since the exams change over time, it might not be as beneficial to study the oldest exams. It’s usually best to stick with the exams from at least the past 5 years or so. Personally, I did not sit down and take many timed 4 hour practice exams because I did not think that would be helpful for me. Instead I would sit down and do as much of an exam as I could in 2 hours and then go over it. If the 2 hours were up and I could still keep working on the exam, then I would, but if I was pretty much stumped after the 2 hours, then I would stop. It’s okay to not know how to do every problem when you first start studying.
4. Review all of the problems you solve and learn how to do the ones you don’t know. The most helpful thing to me was my friend, Surath. After I solved as many problems as I could on a prelim, I would send him a copy of my solutions and he would grade them for me. Not only would he let me know if my solutions and proofs were correct, but he would also give me tips on how to solve the problems that I didn’t know how to do. It is super important to know if you are doing these problems correctly, so I highly advise you have someone look over your solutions. This person can be someone who has already passed the exam, a professor, or someone who knows a lot about the subject. It’s also important that you learn how to do the problems that you’re stuck on so that you have a better understanding of the material and can solve more problems.
5. Learn some proofs. There are some theorems that are commonly asked about on certain prelims. For example, on the algebra prelim it has been asked quite a few times to “let $$F$$ be a field, prove that $$F[x]$$ is a Euclidean domain”. Often times they also ask for the statement of the Sylow theorems. It’s good to know which theorems you are expected to know the proofs of and to make sure you understand them and are able to at least sketch them.

Aside from all of that, make sure you are taking breaks and eating and mentally relaxing as well!