We had our Open House in the MIT Math department last week, that time of the year when a bunch of the prospective grad students come to see the building, talk to the professors they think they want to work with, get a vague sense of what their social lives would look like in Boston, and gossip among themselves about how much better they expect it would be than at other places like Princeton.
I remember when I was in their shoes, filled with both the hope of possibility and an insatiable desire to prove myself among new colleagues. I’ve learned a lot more about grad school and the research process in the five years since then, and there’s plenty more I wish I knew. With that in mind, I’d like to write an imaginary letter (e-mail?) to my self five years ago, as a college senior deciding where to go for grad school. Here it is:
Hey 2012-Sam! This is 2017-Sam, writing something you’ll never see because time doesn’t work like that. Oh well.
You got into all the grad schools you applied to — good job! It seems that your good work in the couple of math REUs you did over the summers worked out. But now is not the time to relax and rest on those laurels.
First, let me settle the big question on your mind. Yes, you should go to MIT; you won’t regret it. Socially, it’ll be everything you expect and more. That really friendly guy you met in Stata during your visit weekend — he will grow into a very good friend over the years, as will many others in the Graduate Christian Fellowship you got a glimpse of at the lunch he invited you to. You’ll have even more of a chance to serve there and in your church than you even did the Christian fellowships and your church at Caltech. It’ll be rejuvenating for your heart of service that right now feels a bit exhausted. Oh, and I’m sure you’re curious: you will indeed find a wife there, as you told others at Caltech back in the fall that you hoped to. Be patient, though; you’ll know it when she actually agrees to date you without you having to talk her into it first.
MIT will also be a great option for your research, but not for the reasons that you’re placing hope in right now. You actually haven’t heard of most of the professors that will be important to your journey, and many aren’t even at MIT yet! I know you’re putting a lot of weight on the professor that you think you want to work with, but you should really be looking at the department as a whole at this point. After all, well, I probably shouldn’t be telling you this, but he’ll be leaving for Stanford in a few years…
More importantly, though, why do you think you want to research combinatorics? Sure, you seem to be decent at it, the bit you’ve tried, but that’s not enough. Why does the world need more people studying combinatorics? You recently applied for the NSF and NDSEG fellowships — use those experiences to guide yourself answering this question. Why should taxpayers pay you to study this stuff? I know you didn’t even try to offer direct applications of your research in your Broader Impacts statement, aiming to talk more about how you like to help other people along the way in your spare time. But what about those hours of research you’ll be doing every day? How is that going to help people? Be specific.
Actually, why do you want to go to math grad school at all? Think about that rather than giving some pat answer. Did you really truly enjoy those hours of listening to Owl City while staring at blank sheets of scratch paper in Avery conference room last summer, and want to do it for a full five years? It’ll get marginally better when you get yourself a set of white boards and markers, but in the end, most of the time will just be you and the math.
Remember how at some point in your first research project you couldn’t really explain your methods to your grad student mentor anymore and had to just write that massive paper alone? Research in grad school will be lonely too if you keep up that style of trying to dive deeper into a single question than anyone else ever has. You need to learn to work with others without leaving them behind.
At the same time, when making these sorts of decisions, you rely too much on what other people say and don’t do enough of your own in-depth research. You picked the schools you applied to based on the list that one professor gave you when you asked him where you could study extremal combinatorics. It’s great to get advice, but on some level, you also have to immerse yourself in the world of that field or professor. You don’t know what research in an area is like until you’ve read at least dozen papers in enough depth to explain main gist of what they did and how they did it.
To give this a spin, try following the arXiv. Add some RSS feeds like Math.CO to your Google Reader, and expand from there. (Alas, Google Reader’s also going away in a couple years, but you’ll easily find an adequate replacement.) Imagine yourself writing papers like that, and trying to find good problems to investigate based on them. Now imagine trying to justify those lines of work to an advisor, or worse, to your Devil’s Advocate self.
Because math grad school is actually rather different from REUs in one particular way: In math grad school, you mostly
get to have to decide what you want to work on. Remember that moment part of the way through your Duluth REU when another student tried to venture in a completely new direction he had picked on his own? Remember how you thought that was crazy (yet cool) but couldn’t imagine yourself coming up with a decent problem like that from scratch? That’s what grad school forces you to do. To be clear, you have time until that happens — you won’t really be expected to start research until you’ve taken quals midway through your second year, and you’ll have a bit of guidance from your advisor — but you should use that time to start trying to chart a course for yourself rather than playing addictive games.
(On that note, if you ever hear of a game called Threes, stay far away. You don’t want to look back at your summer a year and a half from now and realize that you averaged 20 hours a week playing that game.)
Right now you know so little, so little. You’ve taken plenty of classes at Caltech, but you don’t really understand what the research world is like. Beyond reading individual papers, to be successful as an academic, you generally have to find a program of ongoing research to join, like finding a train to ride. Only certain trains are running in certain places. Right now, you think that you want to ride the growth of extremal combinatorics, but you don’t really know anything about that field. What are the major questions they’re trying to answer? What are the tools they’re trying to leverage? And who even cares? Your process of applying to grad school, writing out what you want to do in various ways, has prematurely solidified your opinions, skipping over the step of actually learning what research in those fields would be like.
Grad school is going to be hard, no matter how smart you are. It’s not just a matter of solving individual problems like you imagine right now. You have to develop your own vision, your own hopes for what can be done that has never been done before, and steadily recalibrate those hopes to converge to or at least intersect with what is actually possible. The best grad students come in knowing what problem they want to work on and even have an idea of how they want to tackle it. You’ll meet some of those types at MIT, and they’ll graduate in four years and go on to get tenure-track faculty jobs. That won’t be the route you end up taking, and that’s okay. But you need to eventually get to that point in some field, and it’ll be more up to you to come up with than you expect.
Maybe academia isn’t the greatest fit for you after all. You’ve gotten much more excited about making a spreadsheet to coordinate rides to church or charting your connected component of the local graph of roommate connections among your friends than you’ve been about your research, especially before your major breakthroughs both summers. Keep coming up with random ideas and pursuing random projects like those to completion. Don’t worry about not having the programming prerequisites; online resources are excellent and you’ll pick up what you need to on the way just like how you taught yourself Mathematica in Duluth. It’s a lot easier to motivate your education with problems that you hope to solve than the other way around. You’ll find that motivation beyond “it’s the problem I was given” or “it uses the skills I have” will be absolutely necessary to your success in grad school.
In all, your time at MIT will be rewarding in many ways. It’s a dream opportunity to study pretty much whatever you want, especially in a math department like MIT’s. This wide breadth of options can be intimidating, but don’t let it go to waste. If you know what sort of challenges to expect and meet them head on, I’m sure you’ll go much further than I did. If only.