In my day job, I am a graduate student in the MIT Math Department, an experience I’ve reflected on before in What I Wish I Knew When I Got to Graduate School and Why I Didn’t Do Research In Your Area. Both of those posts focused on graduate school as a whole, naturally inflected by my own experience but not primarily discussing aspects unique to my department. In this post, I’d like to focus on the particulars of going to graduate school in math, centered on five of my own previous misconceptions of it.
The hardest part of math research is solving a problem.
In my previous research experiences as an undergraduate student, the research plan was fairly clear from the beginning. Often we were given a paper with some open questions at the end, and encouraged to try to tackle them. It wasn’t ever as easy or predictable as a problem set or test, and often we could only make partial progress or tackle a similar but slightly different problem.
When I got to graduate school, my prospective advisor told me that he had lots of open problems, too many, in fact. I was initially elated — internally thinking something like “Let me at them!” But as I talked with him and other older graduate students, they talked about the need to have “good taste” in deciding which problems to try to tackle. The world of combinatorics was teeming with open problems — the true value to the field was not in solving them one by one, but in finding solutions which could teach the field something new and fundamental to many different problems.
As I got further in graduate school and chose to go in a different direction, the problems dried up. Or, really, I should say, the good problems dried up. I learned to see flaws with almost every problem I came across, and even major flaws in entire areas of research. In almost every case, I realized that there either wasn’t a good reason to hope there might be a solution, or there wasn’t a good reason to care whether a solution existed at all.
In the end, I learned that the hardest part of research is not finding the solution, it’s finding a good problem.
Pure math is harder than applied math.
Applied math is my current field, but I wouldn’t have expected that coming out of college. At Caltech, applied math (or technically, “Applied and Computational Mathematics” or ACM) was not a particularly popular major. Most of the applied math majors that I met tended to be washed-out pure math majors. It seemed to me that those that could, did pure math, and those that couldn’t, did applied math.
At MIT, I found a much healthier dynamic, with both departments together under the same umbrella. At the same time, as I began to realize my own desire, and even drive, for my work to have more than a blind hope of usefulness, I struggled with whether I was taking “the easy way out.” Instead of having to learn algebraic geometry, I could just take cross-listed computer science classes.
Little did I know, though, applied math is actually harder. This is where that first misconception comes in. The problems that arise in applied math could very well be easier to solve than the problems that arise in pure math, problem for problem. But finding a good applied problem is harder than finding a good pure problem. In addition to the problem being tractable by mathematical means, applied mathematicians need to find a problem that actually matters to some application downstream.
This is why there is always a temptation among applied mathematicians and computer scientists to quietly drop the need for applications and try to solve the simplest, most beautiful problem we can find. From P = NP to string theory, there are plenty of sexy research areas that are basically pure math with a thin semblance of a connection to the real world.
Pure math is the same sort of work as applied math.
As I slowly became more and more of an applied mathematician, I actually didn’t give much thought to what it meant. I imagined that I’d just do the same sort of things that I had been doing in pure math, but on math problems with applications.
This is probably my biggest regret. One aspect sticks out in particular: Collaboration. In my previous pure math experiences, I had taken the approach of drilling deep, studying my problem to an extent that no one had studied it before. This worked out great when there were new features to discover and all I needed to do was uncover them.
In applied work, though, it’s rarely enough to just discover things. If you find something useful, you have to then move to implement it in practice. In all but exceptional cases, this often means that you need to be a part of a team. Some of the most effective applied mathematicians and computer scientists that I know maintain active lines of communication with biologists, neuroscientists, chemists, or machine learning practitioners. To be an effective applied mathematician, it isn’t just nice to be talking to others outside your department; it’s necessary.
In the end, I sorely wish I had realized this earlier. I spent years working on a problem that seemed to me as a mathematician to have applications, and which other researchers at a similar point in the stream as myself praised as important. In the end, though, I wasn’t actually talking to anyone downstream to see what they needed, and perhaps there aren’t that many people who would use our ideas downstream in the first place.
You learn, and then you work.
Unlike other departments, in math, you aren’t expected to start producing research in your first year. Some excellent students do, but the norm is that you spend your first year or more taking classes and starting to talk to professors to figure out who you want to do research with. At MIT, there is also a requirement to take at least three classes each semester before you pass your qualifying exam, which often takes place sometime in your second year.
I came into graduate school still loving learning all sorts of things; I had just gotten a double major in chemistry “just for fun.” After I passed my quals, though, I decided that I needed to break that habit in order to actually start making my own research progress. I decided to narrow my focus, stop taking classes, and attend fewer seminars in far-flung areas (unless they featured someone I knew or at least a free meal). I stopped caring as much about papers that weren’t directly relevant or maybe one step away from my research.
This was a mistake. I had forgotten that graduate school is still school, and research is simply another form of learning. All of the professors that I’ve gotten to interact with have been very literate in areas far beyond their own. And if I don’t plan to stay in academia, the point of going to graduate school is to learn, beginning to end.
In my most recent project, then, I’ve started trying to at least learn and build some experience with the programming language R. This process has been frustrating at times, but it does bring back happy memories of when I first learned Mathematica during that summer of doing math research after my freshman year of undergrad. Even if what I’m trying to do doesn’t end up working, I’ll have come out with some hopefully relevant experience to my future work.
Paper-writing is about persuasive communication.
On a bit of a different note, I like to write, which you probably know if you read this blog. 🙂 But if the reviews I’ve gotten so far are representative, that experience only weakly translates to an academic context. The writing style I use, and like to read, is not as welcome there.
Naturally, that makes sense to some extent. I wouldn’t write an entire paper like I write a blog post. But in my experience, even at points in the paper where persuasion is needed (like when I justify my new model), even what I thought were my most illuminating analogies just plain rubbed a reviewer or two the wrong way. And with the way reviews are compiled, pretty much any red flags will doom a paper. No wonder academic writing is so dry!
Fortunately, writing is only one of the modes of communication that researchers employ. Giving talks provides an excellent way to break the formality, persuade, and interact with an audience, and you can also post your slides online. Unfortunately, to give a talk at a conference, you often need to submit a paper first, that is then judged on its merits as a paper, not as a talk. Fortunately, if you’re in the loop, there are smaller-scale workshops and mini-conferences that soften that requirement and probably provide the most real value out of any of these modes.
On a side note, this is all part of why I am skeptical of various efforts to encourage students at places like Caltech and MIT to write more. Blogging is very different from humanities essay writing, which is very different from academic paper writing. If alumni who went to graduate school wish they were better at paper writing, getting them to write essays in humanities class wouldn’t actually improve that.
I don’t know how widespread these misconceptions are. Your experience may vary, but I hope it’s at least helpful to articulate where my own impressions of my own program were mistaken.
Thanks for this! As an aspiring mathematician myself, this is really helpful to see!
These are very good points
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