Be water, MIT friend
I just finished my third year of graduate school in the Mathematics Department. I work in Fourier analysis, a part of math that studies qualitative and quantitative questions about the overlapping patterns of waves. Before I was thinking about Fourier analysis, when I was a new graduate student, I was considering the question of what kind of work I wanted to do in graduate school. I think many new graduate students spend a good amount of time considering what they want to do in graduate school, as well as an important related question—how to choose an advisor or a lab.
Part of what made choosing an advisor hard for me was that I felt like to make the right decision, I had to have a strong understanding of the work that the professors in the Department were doing so I could plan out how I would acquire the necessary background to begin doing research. Publications and research pages are often highly technical documents, and I couldn’t see the flow that connected where I was as a college senior to where I would be as a graduate student doing research in such disciplines. It turned out I needed something else to help me decide.
For the Math Department at MIT, it is okay if this question isn’t resolved before the end of the first year of graduate school. We have the privilege and time to learn more about math and mull this question over in a vegetative state after a day of classes, or on walks through the brisk Cambridge air with a friend. It reminds me of a question I had about the waves on the Charles River on my morning commute across the Harvard Bridge from Boston to MIT.
To tell you more about this question, let’s take a walk over the Charles River. As we stride smoot-by-smoot across the ample concrete path of the Harvard bridge, connecting MIT and Kendall Square to downtown Boston, we look down and our vision is saturated with this intricate texture of waves:
The most fundamental property of waves, and the first thing that comes to my mind when I see this patch of water, is the phenomenon of superposition. That is, waves add on top of each other, and at each point on the Charles the pattern we see is the sum of the waves overlapping at that point. We can pick out quite a few eye-catching crests in this snapshot, and it’s fun to try and do the mental arithmetic of superposition. Lifting our eyes, we look downriver and see this part of the Charles just beneath the distant cityscape:
A swath of smooth water tightly hugs a stripe of choppy water
It’s a distinctive pattern, and it’s perhaps surprising that superposition might explain this pattern too. Which brings me to my question about this picture: if superposition is the most fundamental aspect of the overlapping waves, does that mean the choppy waves actually extend into the smooth part of the water, and there are a bunch of corresponding waves in the smooth part of the water perfectly canceling out the choppy waves, resulting in the overall pattern? That explanation doesn’t feel satisfying to me, and I’d wager there is something else other than the raw data of the individual waves and their superposition to consider. Something I originally thought was fundamental—superposition—probably isn’t the answer to how we should understand this pattern of waves. Likewise, I wanted to pick my advisor on the basis of my knowledge of the area they do research in, but that couldn’t be the key to how I made my decision. After all, acquiring that kind of knowledge is part of what graduate school is for! To answer my own question about this picture, I think the key to this wave pattern on the Charles has more to do with the way water moves because of external forces, as in the wake of a boat, than it does with superposition.
When making a big decision that feels like it will make waves in our lives years after we make it, I think it is helpful to acknowledge our own flexibility and ability to adapt. Bruce Lee has a nice quote about the way water moves that I think applies to all graduate students in our discussion:
“Be water, my friend.
Empty your mind.
Be formless, shapeless, like water.
You put water into a cup, it becomes the cup.
You put water into a bottle, it becomes the bottle.
You put it into a teapot, it becomes the teapot.
Now water can flow, or it can crash.
Be water, my friend.”
To tie this back to picking an advisor, meeting the people behind the webpages made all the difference for me. Trying to decide what I wanted to work on without this key input was like considering the pattern of waves on the Charles without thinking about the way water moves. There are so many waves, and likewise there are so many professors and “data points” about their research, and these are important pieces of the puzzle, but knowing these “data points” was not the deciding factor I wanted them to be.
Unlike for the waves on the Charles, there may be no explanation as to how to choose advisors or labs in general. For me, my something else came in my last semester of college, during my meetings with professors in their office hours on MIT’s visit weekend for prospective graduate students. MIT wasn’t the first graduate school I visited, and before those office hours, I just had the data points of names and rough areas people worked in: the amplitude and frequency of waves on the Charles. But during those meetings, what began to emerge was a new picture. I realized that graduate school is a long journey, but when it comes to what I work on, if I was doing math I was interested in and I had good communication with my advisor, the rest would sort itself out. I met my current advisor, Lawrence Guth, in one of these meetings, and in our conversation about Fourier restriction theory, I was able to connect my background and interests with a topic he is passionate about, foreshadowing the good interactions we would have for the years to come.
My advice for the graduate student pondering these kinds of questions? Solutions may come from unexpected directions, so be water, my friend!
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