The title of this article is a question I usually get asked. Admittedly, it’s a hard question to answer briefly since a sentence like, “I take graduate-level courses to gain a general background in the subject and then conduct original research in mathematics in order to defend a thesis,” leads to many follow-up questions. This article will serve as a more in-depth response to share my experience of being a Ph.D. student in mathematics.
In the U.S., Ph.D. programs typically aim to satisfy the requirements of both a Master’s degree (taking a certain number of graduate courses in a specific field) and a Doctorate of Philosophy (preparing and defending a thesis that comprises research done by the student in a specific field). The graduate coursework helps build a breadth of knowledge for the Ph.D. student to have sufficient background in order to make meaningful contributions to their respective research community.
I’m just finishing up my first year of Ph.D. studies, which focuses more on taking courses. In the upcoming fall semester, I’ll be taking my “qualifying exams,” which test whether I’ve truly obtained the mathematical knowledge from the graduate courses I’ve taken. Most Ph.D. programs have some form of qualifying exam, which varies greatly in how they are administered – mine will be a series of written exams, while other departments hold oral exams or a combination of the two.
If I pass the qualifying exams, I am considered “qualified” to pursue a Ph.D. thesis. At this point, I will work with my Ph.D. advisor to form a doctoral committee of other faculty members or industry representatives who will guide me in my thesis work and provide suggestions or feedback as I conduct research.
The research in question is probably the most difficult to explain since it can very quickly involve a lot of jargon or abstract concepts. I work in an area of applied mathematics that seeks to design computer algorithms that can efficiently find approximate solutions to problems involving partial differential equations (I’ll break that down more in a bit!). There are many mathematical journals dedicated to this kind of research, so once I have publishable results, I can submit them to those journals and use those results to make up part of my thesis.
What exactly are the “approximate solutions” to these problems, though? I’ll give a few examples. Let’s say we are trying to figure out how the concentration of a dye changes over time as we drop it into a flowing river. From experiments, scientists have figured out that this problem is well-modeled by a partial differential equation called the advection-diffusion equation. If we have a computer algorithm that can solve this equation given the properties of the river and the initial concentration of the dye, then we have an approximate answer for how that concentration evolves over time.
If you’ve taken a differential equations class, you may remember finding solutions to several equations just by writing out a series of steps. These solutions exactly solve the problem without any approximations – so why do we need computers at all? Well, the problems of interest to applied mathematicians—complex mathematical models involving several parameters and tricky equations—rarely have exact solutions that can be found “analytically” (with just pencil and paper). My thesis will likely involve finding a set of tricky equations that show up a lot in applied settings and looking for computational methods to still obtain approximate solutions.
As I reach the end of my word count, I realize that there’s still a lot I didn’t mention, specifically about my day-to-day schedule, which can be hard to describe other than “doing research” or “working on coursework.” Also, many Ph.D. students are teaching assistants, which introduces more responsibilities. But I hope this gives a clearer overview of my general life as a Ph.D. student. In the future, I hope to provide some more specifics as I get further into my research projects and other endeavors (publishing, attending conferences, thesis work, applying for internships and postdoctoral positions, etc.). Stay tuned!
