Qualifying Exam

I took (and passed!) my qualifying exam in October 2021. My topics were Algebraic Number Theory (major), Probability Theory (major), and Complex Analysis (minor). You can view my official syllabus and a full transcript from the exam.


Preparing for the Qualifying Exam can be a stressful and scary time. In talking to my advisor and other graduate students and thinking about my own process I collected a lot of good (in my opinion) advice on how to prepare for quals and compiled these thoughts into a short article of advice in the hopes that they may be useful for future students preparing for their quals. My preparation process is by no means the ‘right’ one, as everyone’s background and topics are different. However in presenting my process I hope it will help others get started or feel more confident in their own quals preparations.

The document contains some general advice for the exam itself, a list of types of quals questions to prepare for, details about how I studied for my quals (again, not the only way), and a detailed list of resources you can use for planning and studying for your qual. My advice is far from exhaustive and I highly recommend talking to your advisor and older grad students to learn from their experiences.

Sample Study Materials

Below are the study materials that I created in preparation for quals. Most of the value in these study materials is in creating them yourself. For this reason I highly recommend that if you are studying for quals you create your own study guides. This will also allow you to tailor them to your own topics and understanding, but more importantly is a far more effective study method than reading someone else’s materials.

How NOT to use these materials (for quals preparation)

How TO to use these materials (for quals preparation)

Typed Study Guides For each topic I created a TeXed study guide which had two sections: ‘memorization’ and ‘questions’. ‘Memorization’ contained terms, theorems, examples that I might want to memorize for the exam. ‘Questions’ is a list of a potential quals questions with solutions. For both, I marked the most important terms/questions in pink and created a table of contents of just the terms/questions.

Handwritten Review Notes During the later stages of reviewing I wrote out hand written review guides for each subject, organized by topic. This is a more condensed and selective version of the material, with an emphasis on how the material fits together.

Python Quizzing Scripts Finally, to practice going through questions and common theorems/proofs I wanted to create a script that would randomly generate a few questions. I wrote one in python for each subject, with questions stored by the chapter/topic and optional flags for focusing on particular potions of the material. To use these, download them and run python [topic]_practice_Qs.py in the terminal. You can also modify the questions or use the template for another subject entirely. [If you are worried about downloading python files directly, I have included txt versions as well which you can copy/modify as python files yourself]