[The first in a sequence of retrospective essays on my five years in math graduate school.]
My favorite analogy for graduate school is the gravity turn: the maneuver a rocket performs to get from the launch pad to orbit. I like to imagine a first-year graduate student as a Falcon X rocket, newly-constructed and tasked with delivering a six-ton payload into low Earth orbit.
Picture this: you begin graduate school, fresh as a rocket arriving at Cape Canaveral and bubbling with excitement for your maiden voyage. Your PhD adviser, on the other hand, is the Hubble Space Telescope. Let's call her Dr. Hubble (not to be confused with the astronomer of the same name). Dr. Hubble is ostensibly the ideal guide for your first orbit insertion. After all, she is famously good at staying in orbit - she’s been up there since 1990.
But problems quickly arise as you probe Dr. Hubble for advice on how to approach the launch. Namely:
- She left Earth more than thirty years ago, and space technology has since been completely revolutionized.
- She states all advice at an extremely high level with birds-eye-view detachment, observing, as she is, from a vantage point a thousand miles overhead.
- Most fatally, the Hubble Space Telescope vessel does not include the lower-stage rockets that brought her into space. In fact, she doesn’t include large engines of any kind. Her thirty years of experience free-falling in orbit will do you very little good until you break out of the stratosphere.
The problem is even worse than this, however. It is not that Dr. Hubble, despite her best intentions, gives outdated advice. It is not even that Dr. Hubble cannot consciously articulate all the illegible skills she’s reflexively performing to stay in orbit. The problem is that even if you could perfectly imitate what Dr. Hubble is doing right now, you would likely still crash and burn.
What I didn’t understand going into graduate school is that academic mathematicians are often working in a state akin to the free-fall of orbit. The Hubble Space Telescope remains in orbit around Earth because it travels horizontally so quickly that, even as it’s continuously accelerating towards the Earth, it continually misses. The laws of physics have arranged it so that it is not possible - barring deliberate sabotage - for her to fall back into a sub-orbital trajectory.
Similarly, a successful research professor is embedded in an intricate system that, as surely as Newton’s laws, keeps her in a state of steadily producing new research. Many of her ground-breaking papers are not one-off productions - they produce sequels, variants, and interdisciplinary applications year after year. She has cultivated dozens of long-time collaborators of the highest level who freely share ideas and research directions, and has the reputation to find more at will. She attends conferences every other month that keep her updated on the leading edge of the field. Every year her research group grows, as if by clockwork, adding a couple graduate students and postdocs to whom she can delegate projects with only the gentlest supervision. As a result, the careers of many other people depend on Dr. Hubble to continue producing research at a steady rate. Every incentive is aligned for objects in motion to stay in motion, and it would take deliberate sabotage to bring Dr. Hubble out of her successful research trajectory.
This is not to say that academic researchers all start cruising in free-fall after they leave graduate school or make tenure. It is perfectly normal for a spaceship that reaches orbit to proceed onto its next adventure after some rest, continuing on to visit another planet or leave the solar system altogether. The best researchers I know are similarly courageous, taking on more responsibilities and pushing past their comfort zones time and time again. I’m merely remarking that once one reaches a certain horizontal velocity in space, it is actively hard to fall back down from the sky.
Contrast this to the sorry state of Dr. Hubble’s new graduate student stranded on the launchpad under the blistering Florida sun. He has no prior publications producing continuous dividends, no access to brilliant and dependable collaborators, no knowledge or intuition about what problems are within reach, no students to farm ideas out to, and no reputation to trade off for any of the above. Above all, nobody else really depends on him, so his motivation to succeed is mainly shallow self-interest. This is particularly hard on him, as there are many things he would do in a heartbeat for someone else that he can’t work up the energy to do for himself. The singular advantage he has over his adviser is youth - a finite amount of extra fuel that he must burn quickly and judiciously like a first-stage booster rocket in order to reach her altitude.
There is a paradox inherent to orbit insertion: rockets launch straight up, while orbit is all horizontal. For some diabolical reason, a spaceship must spend its initial phase accelerating in a direction completely perpendicular to its desired velocity. That reason is called the atmosphere: in order to avoid continuously paying the toll of air resistance, a rocket spends a period of time flying straight up. But any additional vertical motion past the upper atmosphere is wasted motion, so at some point (and sooner is better than later), the rocket starts turning smoothly towards the horizon and accelerating towards orbit. Thus is birthed the smooth quasi-hyperbolic curve known as the gravity turn, the ideal orbit insertion trajectory.
How is graduate school like a gravity turn? For one, it is an enormous error in a gravity turn to try to directly imitate the velocity vector of a ship in space while still at sea level. Regardless of its power, a rocket launched horizontally will quickly nose-dive into the Atlantic. Similarly, a student can rarely succeed in graduate school by solely imitating the activities of established researchers. The student must engage instead in certain activities, such as studying fundamental background material and actively networking, that are mostly orthogonal to a research professor’s day-to-day.
For another, it is an equally enormous error to dip your nose cone towards the horizon too late, and spend too much fuel accelerating vertically. Once you break the atmosphere, all excess vertical velocity is wasted motion. At some point during graduate school, the student must transition away from activities that only grant temporary altitude. Becoming knowledgeable gets you to a great place to start doing research at a higher level. But spending too much time studying without attempting original research renders you a mere encyclopedia. Taking classes, networking, applying for fellowships, and going to student summer schools all follow the same principle - there is an appropriate amount to do, past which they increasingly approach wasted motion as far as getting into orbit is concerned. (Of course, if you enjoy any given activity intrinsically, then by all means continue to do it as much as you want.)
An additional consideration is that, while the gravity turn is the most technically fuel-efficient method of orbit insertion, not everyone who arrived in orbit took this most efficient path. In every department there are superstar students who were outfitted with nuclear reactors in place of conventional rocketry, and these folks get to space by pointing their nose cones in any old direction and blasting off. If you’re such a person, just blast off; calculating the optimum gravity turn curve might be the real wasted motion. Also, many of your professors will likely have fallen in this rarefied category in their own graduate school experience, so their advice on efficient gravity turns will be entirely theoretical in nature.
It is worth remarking though, that even a nuclear rocket might learn something useful from practicing the gravity turn maneuver. Just because you have an easy time leaving Earth’s atmosphere and have no need of finesse, doesn’t mean your travels won’t land you on Venus someday. And breaching that monstrous atmosphere will take every ounce of efficiency you can muster.
A natural question remains: if many graduate school activities only count for temporary vertical altitude, what constitutes horizontal motion that is useful for permanently entering orbit? Examples include:
- Producing good research, as every nice paper you write continues to pay dividends year after year.
- Becoming an attractive collaborator, partly by acquiring enough reputation that people are willing to work with you, and partly by being productive and pleasant enough that they stick around.
- Learning to support the research of others, as much of your potential impact lies not in personal contribution, but in the network effects accumulated from being a positive community member.
This last skill begins at the very start of graduate school, where the biggest immediate impact you can likely have is facilitating your adviser’s and other collaborators’ research.
I will close by reminding the reader that the gravity turn maneuver is not a truth delivered from up high that holds for all time across all circumstances, but an engineered solution to an inelegant and ever-varying practical problem. Launching from a moon base, for example, does not require a gravity turn at all because the moon has no atmosphere to fight against. There, you could comfortably reach orbit by blasting off almost horizontally from the lip of a crater. Only you know exactly where you’re launching from and the thrust-to-weight ratio of your vessel. Adjust your gravity turn accordingly.
I hope it is a comforting thought that free-fall is possible: that one day through all the striving of graduate school you may reach a position where the system propels you forward in your research and all you have to do is sit back and relax. I hope that on that day you continue to strive anyway.