Congratulations to those students who have been accepted to a PhD program in the mathematical sciences starting in the fall! You are about to start an unforgettable part of your life. What you will soon realize is that the first year of graduate school is a time of important transitions in the way you study, the way you think about mathematics, the way you think about yourself and the way you think of your professors. Below I offer some suggestions of what you can do this summer in order to be better prepared for the transition to grad school.
Review what you learned in your Real Analysis course: Regardless of the area of mathematics you end up choosing for your dissertation, Real Analysis will play a key role. This is why the great majority of PhD programs will require that you pass an Analysis exam. Don’t just rely on the fact that you took an Analysis course as an undergrad. Review this material during the summer before you take a graduate Analysis course.
Understand the requirements of your program: Most PhD programs have requirements that include written exams (Qualifying exams or Preliminary exams), an oral exam, a specific number of course credits, and a dissertation. Your program has a list of such requirements and the times by when they must be satisfied. For example, the written exams must be passed typically during your second year, sometimes sooner. Make sure you know exactly the requirements of your program and when each must be satisfied. Verify your understanding with the graduate coordinator or chair of the department.
Reflect on how you study best for learning (not for an exam): We all have a routine of how we study in college. Some of us like to leave things to the last minute and cram the night before an exam. Others are more methodical. One of the most important transitions to grad school is the fact that you will not be studying just to pass your classes; you will be studying to learn mathematics. In graduate school you cannot afford to forget what you learn because the material builds on previous material and the eventual goal is to reach the edge of what is known and use what you have learned to create new mathematics to push that boundary. So it is good to reflect on the best way for you to study to learn. The goal is to make all new material part of your mathematical understanding, not just a temporary accumulation of knowledge.
Be ready to change your study habits: This is related to the previous point. Most likely you will have to change the way you prepare for class and for exams. You will likely have to get used to the idea of collaborating with other students since the amount of homework and studying is significant. What I mean by “collaborating” is that you work together to learn together, not to split the work and do only your part. Perhaps you can consider reviewing every night the class discussions of that day while they are fresh in your mind or some other habit that works for you.
The most significant transition that you will make in graduate school is to go from being the student who is good at solving problems you are asked to solve to becoming the researcher who understands what the next problem that needs to be solved is. This requires understanding the bigger picture of where the problems fit. It helps to get comfortable talking to your professors about math. As a professor I am constantly looking for graduate students that are making this transition.
Once you start your graduate program you will have to learn to manage your time by constantly evaluating all the work you need to get done and adjusting your other activities appropriately. You will have to skip some weekend trips with friends at times. This is especially important if you also work as a Teaching Assistant or have other responsibilities. During your first year of grad school your main goal is to prepare to pass the written exams.