Research with Graduate Students

How and Why I Decided to Involve Students in My Research

When I was a PhD student at the University of British Columbia, a more senior student in our department, Kyle Hambrook, heard that I was working on a problem in $p$-adic harmonic analysis. He approached me with a problem he wanted to consider—adapting a construction from the Euclidean setting to the $p$-adic setting. He mentioned in passing that he thought that PhD students in our department did not do enough collaboration and he was hoping that we would be able to do this project together. Now, half a decade later, we are still collaborating and working on problems together, and some of my best work has been joint with Kyle. I don’t know if I could ever pay Kyle back for the trust he placed in me as an inexperienced PhD student, but as the saying goes, if you can’t pay it back, pay it forward. Thus my piece of advice to young researchers: always look for opportunities to involve graduate students in your research.

One of the most striking things I noticed as a postdoc was how lonely it can get. I had an excellent postdoctoral supervisor and a very welcoming research group and department, but even so I found it difficult to see myself as part of the department. I was a newcomer who had no understanding of the culture of the department I had suddenly become a part of, and knew that in a few short years I would have to leave again. As a postdoc, I found it difficult to get involved with the research group outside of interacting with my supervisor even though the department as a whole could not have been more welcoming. My postdoctoral years were exciting from a mathematical perspective, but knowing that I would be leaving soon made it difficult to integrate into the department.

One observation I made as a postdoc was that the graduate students at the University of Edinburgh learned very different background material from the graduate students at the University of British Columbia. In my second year at Edinburgh, I taught a course on additive combinatorics—an area that essentially all of the harmonic analysis students at UBC had encountered, but that was more-or-less unknown to the graduate students at Edinburgh. As a postdoc, you should try to get a feel for what the graduate students in your department know. Often it will be very different from what you learned as a PhD student.

While at Edinburgh, I had an idea for a new research project that I could tackle. I decided that, in order to become more involved in the department, I would send out an email asking if any PhD students were interested in working with me. I was happy when PhD student Reuben Wheeler expressed interest. This is something that might surprise newer researchers—many graduate students will jump at the chance to be a part of a new project.

Unfortunately, the project that I had proposed did not work out—the idea I had come up with to tackle the problem had a flaw, and at the time I felt that I had wasted Reuben’s time. However, I later came to realize that even an unsuccessful side project can be a good experience for a PhD student. Many PhD students have their first few projects hand-picked by their supervisors, and learning when and how to fail at a project is a valuable skill.

Later, I proposed a second project to Reuben Wheeler, as well as two number theorists at the University of York—Sanju Velani and Evgeniy Zorin. Sanju and Evgeniy encouraged Reuben and me to write up the project ourselves and it ended up being Reuben’s second publication. We are currently writing a follow-up project and this has become a very fruitful direction of research.

Why You Should Involve Graduate Students in Your Work

Giving PhD students a chance to work on a project with you can be a valuable opportunity for them to gain new research perspectives. Further, involving graduate students in your work can be a huge benefit to you as well.

If you are intending to pursue a research career in mathematics, you are at some point going to be supervising graduate students of your own. Bringing a graduate student onto a research project is a good way to build your skills as a supervisor. Everyone has their own quirks in terms of how they work on mathematics projects—for example, some people do their best work in the morning, and others do their best work in the evening. Some people think most clearly in the office, and some people think most clearly when out on an errand. Everyone is different, and when you start to supervise graduate students of your own, you are going to need to know how to supervise students who will not work in the way that is the most comfortable for you. It is important to practice this skill early in a relatively low-stakes setting so that you are ready to supervise students when you are hired as a professor.

Starting a joint project with a graduate student is a great way to expand your perspective on mathematics. If you work on a project with an experienced mathematician, they are likely familiar with a broad range of literature and will likely have read many of the same papers as you and many other subject matter experts. However, a graduate student will likely have read only a few papers, which may shape their perspective on your research area in a unique way. It can be advantageous to work with someone who has only read a few papers because it is likely that their perspective will be vastly different from someone who has been working in the field for a long time. Often I have found that working with a graduate student on a project gives me a broader perspective on problems that I thought I knew a lot about.

During your postdoctoral years (and perhaps even as early as the end of your PhD), you should be thinking about potential research projects for master’s and PhD students. I was actually surprised at how many problems generated by my research would be appropriate for someone early in their career. For example, it is often possible to sharpen an existing result by obtaining better estimates on an error term. Often, a theorem that works in one setting can be straightforwardly adapted to another setting. These kinds of simple but nontrivial arguments are perfect problems for future graduate students. Taking the time to work on problems with graduate students will help train you to think of good problems for them.

How to Involve Graduate Students in Your Work

No matter what stage you are in your career, you have an opportunity to involve graduate students in your research. If you are currently a graduate student, you can always work on a joint project with another graduate student in your department. Other students—especially more junior students—will usually jump at the opportunity to work with you.

Later in your career, if you want to work with a graduate student in your department, you can simply send an email and see if anyone is interested.

Another interesting idea is to talk to graduate students at other institutions. This is something you should be doing at every conference. When you attend a conference, it is tempting to try to talk only to people you already know, or experts who you are interested in working with. However, it is often rewarding to talk to graduate students about their work as well. Often, a graduate student at another institution will be working on an interesting problem you’ve never heard of. This is an excellent opportunity for the student as well—most graduate students do not have the opportunity to coauthor a paper with someone at another institution.

A great way to meet graduate students at conferences is through problem sessions. If there is a problem session at a conference, try to present a problem that will be within the reach of a graduate student. You might be surprised at how many graduate students are willing to work on problems with you. If you attend a conference that does not have a problem session on the schedule, you can always suggest holding one. Even an informal problem session will probably be very popular, especially with younger researchers.

Lightning Round—Some Miscellaneous Advice

It can be hard, but try to keep a regular work schedule. Think about what conditions lead to getting the most work done and try to do most of your work in a favorable environment. Never neglect your teaching duties—teaching well takes a lot of planning. Do not use class time to discuss homework problems. Try to maintain a positive relationship with your colleagues, if possible. Do not neglect your health. Make sure you take the time to exercise. Do not dwell on your mistakes or failures. Do not be intimidated by other researchers. Do not underestimate the impact of your research on the community. Do not assume you know everything about your research area. Do not assume you will have nothing to bring to a new research area. Do not be afraid to supervise a graduate student working on a problem that is slightly outside of your area of expertise. Do not be afraid to ask the senior professors in your department for advice. Always be grateful for the help and kindness you received from others, and try to provide the same help to people who might need it. Respect the wisdom of more experienced mathematicians, but remain humble when working with students or junior mathematicians. Before you start your first permanent position, make sure you know when your grant proposal deadlines are. Work on problems that will be of interest to other researchers in your area, but be wary of chasing fads. If you are going to work on a popular problem, try coming at it from a different angle from the rest of the experts. Your job will be stressful, but never forget the joy of teaching or doing research.