Ultimately, on all of these walks, I reflected a lot on my time at MIT and my relationship with academia and (mental/physical) health. There’s a lot I could say, but I think I will just put the piece here. I am really proud of the piece and hope you like it.

Doesn’t Feel Real

“How did having a disability affect your time at MIT?”

*“It didn’t. Well, I mean. I couldn’t let it, so. It didn’t.”*

…

The flowers on cherry blossom trees, once bloomed, typically last about one to two weeks in springtime. At MIT, with these flowers comes the optimism of students. They signify a halfway point: between the cold winter days to the dreadfully hot summer nights. Between the “I can’t make it through this semester” to the “I can count the number of assignments I have left on my fingers.” The cherry blossom trees bear not only flowers, but freedom.

—

On the second floor of McCormick Hall in Room 239, where I stayed my freshman year, is a perfect unobstructed view of these trees. Having hardly left my room in the middle of a pandemic, I had a front row seat to see a late April thunderstorm wipe out the just-budding flowers before they even got to bloom. I watched the petals helplessly fall away. I wondered if they looked back at me with the same pity. And it was then, for the first time in the roughly two months I had been living on campus, that I decided to go for a walk with my friend Ana.

I was expecting a 20 minute stroll around MIT: 10 minutes going through the Infinite, 10 minutes walking back along the Charles river, and then parting ways so I could go back to my homework. Instead, when I met with Ana, she excitedly suggested we go for a walk along the Esplanade. I didn’t want to tell her no, and so off we went.

The first ten minutes of our walk I was nervous. I kept thinking about my laptop that I had left back in my room. My assignments, carefully organized using digital Post-it notes, screamed for my return. But I had also never done this walk before, and I hate not knowing where I am going.

My mom used to say that I’m directionless; not in life, but by foot. I could follow a path I’d taken a thousand times before, but not one slightly adjacent. But to be fair to my younger self, I come from a town of one-story buildings, deadends, and no discernable differences between this street (named Maple) and the next street over (also named a type of tree). In Fresno, California, most of the time you can’t even see the place you’re trying to walk to. So as a kid without a GPS in her pocket, I knew that deviating from a path I knew well could result in becoming woefully lost. And not only that, but the one time I attempted ‘a shortcut’ and got lost resulted in what will forever be known in my family as the “AutoZone incident”. So I stuck to the paths I knew, and I became labeled as directionless.

By the time Ana and I made our way off of the Harvard bridge and onto the footpath, my nerves eased up. The thoughts shouting “my laptop is in my room” became relaxing.

*My laptop is in my room.*

*It’s there, and I’m here.*

*Be here.*

It also becomes abundantly clear that the path we are taking is simple—you can practically see all the landmarks you’ll be hitting: Hatch Memorial Shell, Longfellow Bridge, and, of course, MIT. I can see where I am going, and I’m with nice company too. Outside of Room 239, the world feels so much bigger. The sky is so clear that you can see the tops of the skyscrapers.

…

It’s my senior spring, and I find myself walking around the Esplanade again. It’s been three years since the first time, and two years since Ana graduated. But today, most of the buildings are hidden through an uncomfortable fog. I wonder what it must feel like to work in those towers on days like today; is there a woman 100 floors up who can only see clouds when she looks out below? Does she feel like she’s on top of the world? Or secluded from it? I know things are happening up there, but I can’t see them, and because of that they don’t feel real.

It doesn’t feel like it’s been three years. As I walk along the Esplanade today, it’s as if a ghost of a younger version of myself walks besides me. A ghost materialized in the form of pure buzzing electrostatic *stress* from my freshman year brain. Stress over whether or not I was doing enough to get into graduate school, and whether or not I was making lifelong friends at MIT.

I look at this ghost beside me and feel a deep sense of pity. I wish I could go back and tell my freshman year self that things are going to work out. Beg her to slow down. But, in truth, I can’t help but feel like I’d do it all again. Who knows where I would be now had anything been different. Where I would be now had I worked less hard. It feels as though if I had done *anything* different, my career could’ve been nipped in the bud.

Today, I look ahead to graduate school and wonder if I will ever be able to slow down. The buzzing stress returns to my brain. My head feels… full. It’s like my brain is running at 1000 miles an hour while simultaneously being unable to process any of the internal or external information. I know things are happening up there, but I can’t *see* it, so it doesn’t feel real. Today, I empathize a bit too hard with the fogged-in woman on the 100th floor.

Something needed to be different. Something *needs* to be different. So why am I finding it so hard to change?

…

Once I find a familiar path, I like to keep on it, by foot and in life. In life, the path that I found was working hard on academics. I could put in the work, get the grades, and keep getting further and further along the path. And I’ve always loved school. But if you were to go dig up copies of my middle school report cards, you’d think I was lying, because in the span of the fall semester of eighth grade, I had missed more than a third of my classes.

The days after I had missed school, my teachers would always ask where I had been but I’d never really answer them. I’d say, “I’m sorry I missed class. Here’s my homework. It won’t happen again.” I didn’t want to tell them that the reason I was missing their classes was because I was in so much pain.

I had *tried *to tell them. Every day after lunch, my Algebra 1 teacher would ask how I was doing when I entered the classroom and I’d answer: “tired, but okay.” She didn’t think much of it. It simply became our routine. What I wanted to tell her is that I could start to feel a migraine forming in my left temple. That the slight aches in my neck, there from the beginning of the day, are no longer feeling so slight. That as much as I love school, I’m starting to feel like I can’t do this anymore.

I don’t tell her all of these things. I tell myself that there are a number of reasons not to tell her: that it doesn’t matter, that class starts in five minutes, that she doesn’t really care how I’m doing. But really, the reason I don’t tell her is because *I don’t know why* I’m in so much pain. And as such, in some weird way, the pain doesn’t feel real.

By the time I was diagnosed with a physical disability later that year, I was in and out of school nearly every day. I finally had a name to put to my pain: Ehlers-Danlos Syndrome Type 3. And because it had a name, I finally felt like I could ask for help.

I requested a 504 meeting with the guidance counselor, the vice principal, and most of my teachers to get some accommodations. In preparation, I wrote a letter. A letter describing all the pain that I had been experiencing and holding in for the past year, and advocating for any number of solutions we could implement so that I could stay in school. I read the letter aloud to my teachers, and tried to avoid the shocked looks on their faces and the pity in their eyes. When I was done reading the letter, my Algebra 1 teacher spoke up. She said that whatever they could do to help me, they would.

A day or two later, my accommodation request was denied. The vice principal had decided that because I was getting good grades, I couldn’t possibly need accommodations. As if to say: “The world isn’t built for you, so we won’t be either.” Painfully, the next day I went on independent study, and never saw my teachers again. I needed to focus on my health, so I did.

Something needed to change, and I changed it.

As I’ve gotten older, this has proven harder to do.

…

Things got better in high school, for a number of reasons. Physically, being on independent study for the spring semester was like a factory reset for my body, which meant that when I started high school my pain got better. Or at least, I had stopped focusing on my pain as much. And in high school I obtained academic accommodations for my disability. In this sense, things were improving.

However, I never used my accommodations. I didn’t like feeling *different*, and using accommodations to take breaks in the middle of the school day is not precisely blending in. From my perspective at the time, it seemed as though school (if nothing else, physically) was easy for others, and I wanted to be like them. It should be easy, shouldn’t it? But as the years went on (a brutal four years at that), my feet hit the pavement harder and harder. My bag began to feel heavier, and walking to and from campus proved difficult. All I could think to myself, as I refused to ask for help, was: “but shouldn’t this be the easy part?”

Over time I grew past this mentality. I started to accept that some things, like walking, are harder for some than for others. Which is why, I was shocked when I did the reading by Rebecca Solnit for the first day of the Wilds of Literature class. Solnit, off-handedly, states that walking is the “something closest to doing nothing” (5); “an amateur act” (4); an activity that “allows us to be in our bodies and in the world without being made busy by them” (5). I read this and didn’t just feel sad or angry. I felt *defeated*.

I *can’t *remember the last time I wasn’t made “busy” by my body as I walked around campus. I *don’t* feel like walking is an “amateur act.” And while sure, I concede that, for an able-bodied person, walking may be like the something closest to doing nothing, but to me it’s simply *not*. *“Why can’t things be easier?”* I ask myself.

It was then, in struggling to come to terms with the ableist language that Solnit was using, that I realized two things. Firstly, I decided to write about my physical disability for the Wilds of Literature class to, in some sense, put words to my struggling with Solnit’s piece. Secondly, I began to put words to the question “why am I finding it so hard to change?”

It is hard to change because: almost by definition, it can feel easier to stick to the status quo. It is hard to change because: who wants to be seen as different? It is hard to change because: I feel like I need to push myself harder and harder to get into a good school. And maybe these are bad reasons to not change. They probably are. But at least they are reasons; words I can put on the page and say “Here I am. My experiences are real. See me.”

Unfortunately, I am realizing this in the spring semester of my senior year of undergrad. Unfortunately, my freshman year self still has a long way to go.

…

On my freshman year walk, I asked Ana if we could slow down for a moment. I didn’t tell her at the time, but I had started to feel the hefty impact of my feet against the pavement with every step I took. I needed to slow down, and so we did. We found the perfect people-watching tree to sit on and for a while we just sat there.

After a couple of minutes, it dawned on me just how many people were walking along the Esplanade. For a moment, life felt normal. This moment, seeing dozens of people walking around on a clear sunny day with their friends, their kids, and their dogs. I thought: This is what I want out of my career. I want to end up in a place where I can go for a long walk simply because it’s nice out, or I need a break, or the buzzing returns to my brain. Even if it is in the middle of the day in the middle of the week.

But then it hit me that *it was the middle of the day*, and the moment felt more solemn. I realized that, when there isn’t a pandemic, most of these people would usually be at work. The kids at school. The dogs at home, waiting for their owner to return.

I watch people walk by and cynically think to myself that the world isn’t *like *this. The world doesn’t make space to be human. The world isn’t made for breaks, it’s made for breaking.

*Accommodation requests get denied.*

*Those at the top of the world find themselves fogged-in by the sky they hoped to touch.*

*Thunderstorms wipe out cherry blossoms before they ever get to bloom. *

All of these thoughts race through my head when I realize that, while this may be true, it doesn’t have to be. Sitting here with Ana is living proof of that. *I told her* that I needed to slow down and we did. I made space for myself, and Ana helped me do it. We sat there together and watched the world go by.

For the first time, I admitted to her and myself that something needed to change with how I was approaching MIT. The world might not make space for me, but I could, and needed to, make space for myself.

—

I wish I could say that I changed my ways during the rest of my time here at MIT, but I didn’t. I kept pushing myself harder and harder, and I kept learning that I couldn’t keep up with this. But I never changed. I kept telling myself that next semester I’d take it easier, or that this summer I would get some rest. Yet time and time again, I caved. I caved into a feeling of cynicism; a feeling that I wouldn’t get where I wanted to be in life by “slowing down.”

And maybe I was right to push myself so hard. By now, I’ve gotten into graduate school; in 6 years time, I will have a PhD in mathematics from MIT. It’s what I’ve always dreamed of, and I can’t help but feel like, had I done anything differently, things wouldn’t have worked out the way that they did. Surely there are alternate realities where I practiced better self care academically, but I can’t see it, and because of that it doesn’t feel real.

And yet, something needs to change.

…

I recently met with the professor who will be my advisor in graduate school. I’ve met with him countless times over the years, but this time is different. For the first time, I find myself telling him something I haven’t had the strength to before.

*“I have a physical disability.”*

“How did having a disability affect your time at MIT?”

*“It didn’t. Well, I mean. I couldn’t let it, so. It didn’t.”*

Tell him the truth. Tell him why you’re telling him *now*.

*“But, I need to make more space for this in my life as a graduate student.*

*That’s why I wanted to let you know.”*

It’s out there now. I’ve uttered the words I’ve refused to truly acknowledge for years. I’ve said it outloud. Now, it feels real.

…

Cherry blossom trees bloom once a year, and the flowers last for one to two weeks before the petals fall away making the way for new growth. The flowers signify a halfway point: between a nearly completed undergraduate experience, and the graduate experience that is yet to come. Between what does and doesn’t feel real.

Sitting beneath the trees, soaking in the springtime smells I know I will one day be nostalgic for, I am ready for things to start feeling real. My friends arrive and we make our way across the Harvard bridge. I’m not expecting this walk to be 20 minutes; if anything, I’m hoping the walk never ends.

Once across the bridge I went to that people-watching tree that Ana and I went to on our first walk. I made my way to the dirt and sat with my back against the tree. I brushed my fingers against the grass, rustling softly in the wind. I closed my eyes and heard the river water pass by. The sun begins to set, and I breathe in this moment.

*My backpack is in my room.*

*The buzzing in my head has subsided.*

*And the sky is so clear that, for a moment, I swear, I can see the 100th floor.*

Regardless, it’s 9:05am, and I am awake 9 minutes too early. I decide I will grab my iPad and go to the decision website in anticipation. It’s 9:06 when I think to myself: “I’ll just try putting in my username and password, just to confirm that I remember them.” So, I put in my information, and the webpage says something to the effect of: “WARNING: THE NEXT PAGE CONTAINS YOUR DECISION.” But surely the webpage is lying. The decisions won’t come out for another 9 minutes– no 8 minutes now. But, out of morbid curiosity, I click the button anyways. And I got in.

The beginning notes to *Another Believe*r by Rufus Wainwright begins to play in the background.
(0:00-0:37):01
to be clear these are time stamps in the song, I will use these this post. Also, this post is roughly modeled after my first Admissions post <em>I Hate the First</em>.
“Hello, I got something to tell you \\ But it’s crazy, I’ve got something to show you.”

I go to my kitchen where my mom was making breakfast. I don’t say good morning. All I do, is read her the first line of my offer letter. “Dear Paige, On behalf of the Admissions Committee, it is my pleasure to offer you admission to the MIT Class of 2024.” My mom is shocked and confused (hell, she doesn’t even know that MIT decisions are coming out today), and I start crying, and my guidance counselor on the phone tells me what I already know: my world will never be the same.

When I first started undergrad, I thought about what I wanted out of MIT; the things I wanted to take away, and the things I wanted to leave behind. There’s a world in which I just came here, took my classes, graduated, and left. No more, no less. But I didn’t want that. Or rather, I felt like I should want *more* than that. MITAdmissions saw *me*: a first-gen math nerd from
middle of no-where02
relatively
California. MITAdmissions took a chance on *me*.

(0:37-0:50): “So give me just one more chance, one more glance \\ and I will make of you another believer.”

There were certain obvious things I wanted to take away from MIT: making friends, learning math, etc. But when I *initially* thought about what I wanted to leave behind I came to the following conclusion:

I wanted to be a ghost.

I wanted to haunt the halls of MIT.

I wanted to be someone who was *here*. Someone you would remember when they’re gone. Not even necessarily
a legacy,03
(0:50-1:13): Guess what? You got more than you bargained. Ain't it crazy? You got more than you paid for. So give me just one more chance, one more glance, one more hand to hold.
but rather a memory. So that someday, when the light shined through the windows of the fourth floor of Building 2 just right, you’d remember that I was there.

And while in various senses I think I achieved that to some degree, these past few weeks my mind has been elsewhere. These past few weeks, my focus has shifted from what I am leaving behind, to what I am taking with me.

I used to hate the last week of classes. I hated how *empty *it felt. Either you knew folks who were graduating (or moving onto middle school or high school) and felt sad that you’d probably never see them again, or you didn’t and somehow that felt more hollow. Like, there were people who were living full and intricate and complicated lives, and suddenly they were just going to be gone.

It used to feel easier to ignore these feelings. To say “that’s just how life works” and move on. But that no longer feels true.

(1:13-1:45): “You’ve been on my mind, \\ Though it may seem I’m fooling. \\ Wasting so much time, \\ Though it may seem I’m fooling. \\ What’re we gonna do? \\ What are we gonna do about it?”

Coming into MIT I wanted to be a ghost, but that hasn’t been on my mind this past week or so since I turned in my last assignment. Rather, I’ve been hanging out with friends watching various Shrek movies and going on long walks and realizing that– my *friends* have become ghosts in my life. Or at least, they *will*.

I’m going to continue to stay in contact with my college friends, don’t get me wrong. But I keep having these dumb “realizations.” That, this will probably be the last time I am living in the same building with these people who have so deeply impacted me. That, I won’t be able to just walk down the hall and knock on my friend’s door and ask if they want to grab dinner (and for that matter, it seems like ‘real’ adulthood entails eating a nontrivial amount of dinners without your friends). That, though I will be here in two years to start working on my Ph.D, most of my friends will no longer be here.

It’s like I can already see their ghosts. I see them sitting at the table we used to play dungeons and dragons at. I see them in the math lounge eating sweet treats (because they worked hard today and they deserve it). I see them and I think: I don’t know how *this* (gestures vaguely) happened.

How did I get such good friends? At what point did they go from a person I said hi to in the halls to someone I will never forget? When did I stop thinking as much about how I was going to affect MIT, and started thinking about how MIT has affected/will affect me?

I don’t know. Perhaps I never will. Isn’t that how these sorts of questions always go?

(2:12-2:42): “So then, that is all for the moment. \\ Until next time, until then do not worry, \\ And give me just one more chance, one more glance, \\ and I will make of you, yeah I’m gonna make of you another believer.”

The music fades into the background.

So yeah, as past Paige predicted, I hate the last: the last weeks of undergrad, the last times seeing friends, the “last” blogpost. It’s silly: when I wrote my first MITAdmissions post, I knew that this one (roughly) would be my last. I knew that I’d want, more than anything, for this to not be how it ends.

*Take on Me (Acoustic)* by a-ha (0:00) starts to play in the background. The camera cuts to me writing this last post in the math department.

Technically, this isn’t going to be my “last” blogpost. There’s one or two more things I want to write about before I am no longer under the “Student bloggers” tab on the MITAdmissions website. I wanted this to be my official last post as an undergrad but that just. That just feels too sad. So I’m going to post one or two more things, and maybe write a blog or two for admissions when I am back as a graduate student. But I guess, this is it, more or less.

I can’t put into words how much this (gestures vaguely) is all I ever wanted. Hell, it’s more than I ever even dreamed of. I guess my guidance counselor was right: my world will never be the same. “How strange, and how lovely, it is to be anything at all.”- John Green

(2:40-3:04): “I’ll be gone \\ In a day or two… \\ In a day or two.”

The camera zooms out. The image of me sitting in the math department writing this post gets smaller and smaller as Killian court, and then the dome, and then nothing but the sky come into view.

]]>You can read the transcript here. This story is also accompanied by three strategies I thought were helpful.

There are a number of things I could discuss about this experience, both from the storytelling workshop (which was amazing and I highly recommend applying for next year if/when they run the program again) and about the story I told. But ultimately, I want to let this story speak for itself, so I think I will leave this here.

The one thing I will say is this: I got a lot out of writing about my experience at MIT– the ups and the downs. I am glad I got to be a blogger over the last two years, and am happy to have had the chance to share my garbage with the world. If you think you might like this experience too, and you will be an undergrad this upcoming school year at MIT, maybe consider applying to be a blogger when the application opens this summer :eyes: (an email will go out about it and it’s always advertised via the blogs).

]]>Take your favorite set of finitely many points out in the plane, i.e. in $\mathbb{R}^2$. Let’s call this set $A$. Now, suppose we look at the “shadow” of $A$ on the $x$-axis and the $y$-axis, and let’s call these sets $\pi_x(A)$ and $\pi_y(A)$ respectively. If it helps you to see it written down mathematically, we have the functions $\pi_x(x,y) = x$ and $\pi_y(x,y) = y$. See the below diagram for further clarity.

One question we can ask is the following: If we only know about the *size* (i.e. how many) points there are in the shadows of $A$, can we get a good bound on $A$ itself? The answer, in 2-dimensions, is quite immediate: Yes. For every point in $\pi_x(A)$, we can draw the corresponding *line* that maps to this point under the projection, called the fibers of this projection. Similarly, we can draw every fiber corresponding to $\pi_y(A)$, and we will notice that the set $A$ *must* lie at the intersection of the fibers from the $x$-axis and the fibers from the $y$-axis (depicted below).

To write this down mathematically, what we have proven is that \[A \subset [\pi_x^{-1}(\pi_x(A)) \cap \pi_y^{-1}(\pi_y(A))],\] and as such, we have that \[|A| \leq |\pi_x(A)| \cdot |\pi_y(A)|.\] Or, if you prefer more directly, we have \[A \subset \pi_x(A) \times \pi_y(A),\] (though this won’t generalize in higher dimensions– at least not what I am going to present in higher dimensions). I.e., the number of points in $A$ is bounded by the number of points in the projection onto the $x$-axis times the number of points in the projection onto the $y$-axis.

Okay, but how good *is* this bound? To understand how good this bound is, we can first consider *dimensional analysis*— a technique that is taught early on in 8.01 (Classical Mechanics) at MIT. The idea is the following: Suppose that we measured “volume” in $\mathbb{R}^2$ in square inches, i.e. $(\text{inches})^2$. If this were the case, then the shadows of $A$ onto the $x$-axis and the $y$-axis must be measured in inches. Therefore, looking at the above inequality (and replacing $A$ with its units and $\pi_x(A)$/$\pi_y(A)$ it their units), we get that \[(\text{inches}^2) \leq (\text{inches})\cdot (\text{inches}).\]

Given that we have inches squared on both sides, we might suspect that our inequality here is *very* good (for arbitrary finite sets in $\mathbb{R}^2$), and in fact it is! To see just how good out bound is, all we need is *one* example of a set $A$ where the inequality is an equality, and here is that example:

Take your set $A$ to be the set of integer lattice points, where each coordinate is an integer between $1$ and $N$ ($N$ here being some large number). Then, in total, we have that $|A| = N^2$ while $|\pi_x(A)| = |\pi_y(A)| = N$.

Hence, we have found a set where the inquality is an equality, and thus out bound is as good as one could’ve hoped for.

* Exercise: *Find an example of a set with roughly (up to some constant) $N$ points, whose projection onto the $x$- and $y$-axes has precisely $N$ points each. I.e., find an example of a set where the bound we got is really really bad.

Okay cool– that was a lot of math (and it is about to be quite a bit more in just a second), but let’s take a breath. What did we *do*? Well, essentially, we bounded the area of a set by the lengths of its shadows. And though this proof was fairly short, we might expect that the same thing holds in higher dimensions, which is in fact the case. In this post, the most we will do is study this in 3-dimensions, which we do now. **The proof will be a little bit lengthy, but that is kinda the point of this blogpost (see Part 2).**

Let $A$ be a finite set in $\mathbb{R}^3$, i.e. a bunch of points out in 3-dimensional space. Our goal is to (again) bound the number of points in $A$, i.e. $|A|$, by the number of points in its projections (i.e. projections onto the $xy$-, $yz$-, and $xz$-planes). Note: we denote these projections: $\pi_{xy}, \pi_{yz}$, and $\pi_{xz}$ respectively. More specifically, if it is helpful to see written down, we will have that \begin{align*} \pi_{xy}(x,y,z) &= (x,y) \\ \pi_{yz} (x,y,z) &= (y,z) \\. \pi_{xz}(x,y,z) &= (x,z).\end{align*}

Before we go into the proof of such a bound, let’s use dimensional analysis to figure out what this bound should look like. Namely speaking, we may expect to have a bound of the following form: \[|A| \leq (|\pi_{xy}(A)| \cdot |\pi_{yz}(A)|\cdot |\pi_{xz}(A)|)^{\alpha}\] for some constant $\alpha$.

Why might we expect this? Well, given our set $A$ is arbitrary, we should expect that *no particular* plane should be favored– i.e. there should be a symmetry to our bounds (as we have in the above formula). Secondly, and perhaps less rigorously– this bound *looks* the same as in the 2-dimensional case. So, for now, supposing that such a bound holds, let’s figure out what $\alpha$ should be by dimensional analysis. We measure $A$ in cubic inches and the projections (onto the 2-dimensional planes) in square inches. As such, we get that \[(\text{inches}^3) \leq [(\text{inches}^2)\cdot (\text{inches}^2)\cdot (\text{inches}^2)]^\alpha.\]

Hence, to get (at least through dimensional analysis) best bound we can hope for, we may expect $\alpha = 1/2$. In fact, this is the case:

**Theorem**: Let $A\subset \mathbb{R}^3$ be a finite set of points. Then, \[|A| \leq \sqrt{|\pi_{xy}(A)|\cdot |\pi_{yz}(A)| \cdot |\pi_{xz}(A)|}.\]

We will spend the rest of Part 1 proving this fact, but to do so we will need some new and better ideas. Notice that, it is in fact true that \[A\subset [\pi_{xy}^{-1}(\pi_{xy}(A)) \cap \pi_{yz}^{-1}(\pi_{yz}(A)) \cap \pi_{xz}^{-1}(\pi_{xz}(A))],\] and thusly we automatically have that \[|A| \leq |\pi_{xy}(A)| \cdot |\pi_{yz}(A)| \cdot |\pi_{xz}(A)|,\] but this bound is *worse* than the one we are trying to obtain (and thusly we will need better ideas).

The “better” idea we will need will be to write $|A|$ as a sum. In particular, consider the function: $\mathbf{1}_A$, which is 1 on the set $A$ and is 0 otherwise. I.e., \[\mathbf{1}_A(x,y,z) = \begin{cases}1 & (x,y,z) \in A \\ 0 & (x,y,z) \notin A\end{cases}.\] This is known as an indicator function, and it is really helpful for our problem because we have that \[|A| = \sum_{x,y,z} \mathbf{1}_A(x,y,z).\] I.e., we have written $|A|$ as a sum that we can now manipulate.

Not only that, but we can similarly define indicator functions for all of the projections of $A$, namely $\mathbf{1}_{\pi_{xy}(A)}$, $\mathbf{1}_{\pi_{yz}(A)}$, and $\mathbf{1}_{\pi_{xz}(A)}$. What can we do with this? Well, we can notice that for all $(x,y,z) \in \mathbb{R}^3$, we have \[\mathbf{1}_A(x,y,z) \leq \mathbf{1}_{\pi_{xy}(A)}(x,y)\cdot \mathbf{1}_{\pi_{yz}(A)}(y,z) \cdot \mathbf{1}_{\pi_{xz}(A)}(x,z).\]

Why is this true? Well, if $(x,y,z)$ is a point in $A$, then by definition $(x,y) \in \pi_{xy}(A)$, $(y,z) \in \pi_{yz}(A)$, and $(x,z) \in \pi_{xz}(A)$, so the left hand side (LHS) of the above inequality is 1 and so is the right hand side (RHS). *Note: there may be points where the RHS is 1 but the LHS is 0– Why??* Similarly, if the RHS is zero, then one of the three terms in the product must be zero (say, $\mathbf{1}_{\pi_{xy}(A)}(x,y)$). If this is the case, then it must be the case that $(x,y,z)$ is *not* a point of $A$ for any $z$! Therefore, when the RHS is zero, the LHS *must* be zero. This concludes the proof of the above inequality. Using it, we notice that we can use this in what we have proven thus far, namely: \[|A| = \sum_{x,y,z} \mathbf{1}_A(x,y,z) \leq \sum_{x,y,z} \mathbf{1}_{\pi_{xy}(A)}(x,y)\cdot \mathbf{1}_{\pi_{yz}(A)}(y,z) \cdot \mathbf{1}_{\pi_{xz}(A)}(x,z).\]

At this point, I want to use something known as the Cauchy-Schwarz inequality. There are a bajillion (or rather, at least 12) proofs of this inequality (none of which are particularly long, but that I don’t want to write down into this blogpost). Basically, the Cauchy-Schwarz inequality states that you can go from sums of products to products of sums (with certain exponents). In particular, we have that \begin{align*} &\sum_{x,y}\sum_z \mathbf{1}_{\pi_{xy}(A)}(x,y)\cdot \mathbf{1}_{\pi_{yz}(A)}(y,z) \cdot \mathbf{1}_{\pi_{xz}(A)}(x,z) \\ &\leq \left(\sum_{x,y} \mathbf{1}^2_{\pi_{xy}(A)}(x,y)\right)^{1/2}\left(\sum_{x,y} \left(\sum_z \mathbf{1}_{\pi_{yz}(A)}(y,z) \cdot \mathbf{1}_{\pi_{xz}(A)}(x,z)\right)^2\right)^{1/2} := I \times II.\end{align*}

Why was this helpful? Because one can see immediately that $I = \sqrt{|\pi_{xy}(A)|}$! *Note: Here, we used that $0^2 = 0$ and $1^2 =1$.* So now we just need to bound the term $II$, or rather $II^2$ so we don’t have to carry the squareroot around everywhere. To do so, we can expand the sum in $z$ into a double sum in $z$ and another (dummy) variable $z’$.

In particular, we have that \begin{align*} II^2 &=\sum_{x,y} \left(\sum_z \mathbf{1}_{\pi_{yz}(A)}(y,z) \cdot \mathbf{1}_{\pi_{xz}(A)}(x,z)\right)^2 \\ &= \sum_{x,y}\sum_z \sum_{z’} \mathbf{1}_{\pi_{yz}(A)}(y,z) \cdot \mathbf{1}_{\pi_{xz}(A)}(x,z) \cdot \mathbf{1}_{\pi_{yz}(A)}(y,z’) \cdot \mathbf{1}_{\pi_{xz}(A)}(x,z’). \end{align*} Try to see why the above bound is true! Using this, we have that \begin{align*} &\leq \sum_{x,z}\sum_{y,z’} \mathbf{1}_{\pi_{xz}(A)}(x,z)\cdot \mathbf{1}_{\pi_{yz}(A)}(y,z’). \end{align*} Again, try to see why! Rearranging these sums (now that they are independent of one another), we see that \begin{align*} &= \sum_{x,z} \mathbf{1}_{\pi_{xz}(A)}(x,z)\cdot \sum_{y,z’}\mathbf{1}_{\pi_{yz}(A)}(y,z’) \\ &= |\pi_{xz}(A)| \cdot |\pi_{yz}(A)|.\end{align*} Throwing back in the squareroot, we obtain the desired theorem.

* Exercise: *Prove that this bound is good by considering a similar example to the 2-dimensional case. Prove that this bound *can* be bad, i.e. that there is a set in which the inequality (while true) is WAAYYY overkill (for a more technical statement, see the analogous 2-dimensional exercise).

* Exercise: *Depending on your background in calculus, try proving a continuous version of this for “nice enough” sets in $\mathbb{R}^3$. I.e., can you use integration (as opposed to discrete sums) to get rid of the assumption the $A$ is finite?

* Exercise: *Try generalizing this in higher dimensions! This is known as the Loomis-Whitney inequality. The next difficult case to try perhaps is bounding the volume of a 4-dimensional finite set by the area of it’s projections onto (the 6!) 2-dimensional planes (i.e. the xy, xz, xw, yz, yw, zw-planes). If you find this interesting, consider looking into Alex Iosevich’s “A View From the Top” which is how I learned about this problem.

**Part 2a: okay fine I’m not done talking about math yet– but give me a sec.**

For those who skipped **Part 1**, the tldr of the math problem I was talking about was the following: Given some blob out in space, if you know how large it’s shadow is, can you know (roughly) how large the blob is? That’s is; that’s the problem (and it leads to what is known as the Loomis-Whitney inequality).

I first learned this proof (of the 3-dimensional) Loomis-Whitney inequality in the summer after my freshman year when I did a reading program with my (soon to be) research advisor Larry Guth and his graduate student Yuqiu Fu. Immediately, I fell in love with the problem (even if it was already solved by Loomis and Whitney all those years ago).

The reason I love this problem is two-fold. One: I love that this problem uses tools that I think are fundamentally interesting in how useful/simple they are (e.g., an inequality known as the Cauchy-Schwarz inequality, or using indicator functions). Two, and perhaps more fundamentally: I love that I can describe this problem in a single sentence: I want to understand some blob by knowing the shadows of the blob. Bam– that’s it.

And yet, when I look at the proof in all of it’s glory, I think to myself: *this proof is ugly as hell*. I mean sure! It works! It’s a complete proof! I can write it down completely from beginning to end. I can explain all the tiny steps that go into the overall proof. And I can admire it’s perfect and elegant use of simple inequalities. And yet. *And yet.*

I look at the proof and think: that was actually really hard to prove algebraically, especially given it was something I *expected* to be immediately clear.

—

People sometimes ask me how I got into the type of math that I do. And my response is always this story: In some classes, sometimes there is just something that irks me about a proof. I can write it down from start to finish. Examine all it’s details from beginning to end. And still, it feels too complicated. It seems to hard to be “the right way” to prove something. But alas it is. This is a feeling I have felt in many classes, especially at MIT.

Where sometimes, I would solve a homework problem in 18.701: Algebra I, look at it, and think “this must not be the simple way to prove this problem”. And most of the time I would be right. I’d turn to a friend who understood the subject better, and they’d give me a two-line, elegant solution. And then I’d stare at the solution and think to myself: well *this* seems to be hiding something behind the scenes. Like, there’s some magic that is happening that I don’t see in their two-line proof, and this (in turn) irks me more. Because in both cases, I don’t feel satisfied.

So imagine my surprise when I learn this proof in my reading course with Larry and Yuqiu, and I think to myself the same thing. But this time…. This time when I tell Larry that I think the proof *feels* too complicated for what we are proving– he says he agrees. That *he too* has thought about whether or not there is a simpler proof for this statement (and as far as I know hasn’t found one).

This was one of those few moments that really got me hooked into my area of mathematics: the fact that I could ask a question that I only barely knew how to put into words, and have it be something that others have been curious about before too.

**Part 2b: okay fine some actual life stuff**

I got into math for education. I mean, sure, I liked the subject myself, but part of what I liked most about mathematics was figuring out how to best explain/motivate a concept for other people (and thusly for myself too). This is why I ultimately decided that I wanted to become a professor.

And as a result, my freshman year I really felt this pressure in every class I took:

*I’m struggling now so that one day I can explain these concepts to my students.
*

At first, education felt like my anchor point. Something to aim for and aspire to. But somewhere along the way, the anchor point became a bit difficult for me to hang onto in quite the same way. What was once “Work through this to become a good professor” became

*Work through this to get into graduate school.
*

Now that I’ve gotten into graduate school– Now that I know where I will be for the next ~6 years of my life– Now that I am nearly ready to graduate from MIT–

—

As a mathematician, I look at the proof of the Loomis-Whitney inequality and think to myself: this is beautiful. As an educator, I’m conflicted. I mean sure– I can motivate why I like the proof *as a mathematician*, but I couldn’t *explain the proof* to someone on a walk. And because (for me) mathematics has always been deeply intertwined with education, that bothers me.

But what bothers me more than that, is that for the past couple of years at MIT I’ve felt like I’ve ignored *why* that bothers me. I’ve ignored the part of me that *wants* to think about how I present mathematics in a more accessible way (e.g., this blog). I don’t know. Perhaps that isn’t fully coherent.

All of which is to say, I am excited to stop ignoring that irksome feeling in graduate school. Because I made it. In Fall 2025, I will be returning to MIT to pursue my PhD in mathematics.

]]>For some reason, in a lot of buildings around campus, MIT would rather have pipes and wires and such along our ceilings than actual ceiling tiles and what not:

Insert obligatory meme response:

**Part 2: (metaphorical) There are no ceilings.**

When I was in high school, I knew (in some vague sense) that my undergraduate education was going to open many doors for me, regardless of where I ended up. In hindsight this was true, and I’ll circle back to this idea later. But when I got here, it didn’t *feel* like doors were opening.

Instead, it just felt like I was taking one step at a time. When I started my freshman year, I simply looked up classes I was interested in/ones that fulfilled requirements, I signed up for them, and off I went. Because I had attended a community college in high school, it didn’t really feel all that different; there is a goal in mind as a student (graduation, for one), and you just *start* working on that goal.

You just, take the next step.

And sure, you might have dreams and goals of what’s going to happen 100 or 1000 steps from now. From graduate school to being a professor, or from medical school to being a doctor, or becoming any 1 of millions of things you could possible be. But you can’t get to the 1000th step without taking the first. Or the second. Or the 999th.

When I started college, I used to think these steps would be *huge*:

- Go to college.
- Go to graduate school.
- ???
- Be a professor.

But in reality, each step can be incredibly, and somewhat unintuitively, small. Before I really got into blogging, I used to post my “steps” to Facebook, and they were simple, but meaningful. They still are.

I am now in my final semester at MIT. I am preparing for graduate school, which used to feel 100 steps away (when in fact it was almost 1,000,000), but now it’s here. Everything I’ve done up to this point has felt, at the time, like the right next step. And because it *felt* like the right thing to do at the time, in some sense it *was* the right thing to do at the time. But I look back on my time here, and I can’t help but *feel *like it was mostly luck.

I was lucky that I got to take the classes that I did in the semesters that I did them. I was lucky to have gotten research opportunities that made me a better mathematician (also don’t worry, the math research blogpost is coming out soonish I just wanted to write this post first). I was lucky to get the opportunity to be *here*.

And while luck may have played a role in some of these things, it isn’t the whole story.

I worked hard to take the classes that I did in the semesters that I did them. I cold-emailed multiple professors for my first research opportunity, and worked hard when I had one. I put in the work in high school, and took a leap of faith *and applied to MIT*.

*I *took the first steps, and *I* kept working hard.

Getting into MIT was one small step on my journey that got me to where I am today. But had that not happened, I would’ve kept walking. I would be somewhere different, and I almost certainly would’ve been a very different person (at least in the ways that were shaped by MIT). But I would’ve taken the next step.

I am now in my final semester at MIT. I am preparing for graduate school, and am realizing now that all of my goals feel 1,000,000 steps away. But as I reflect on my time here, looking down on all the stairs that I climbed to get to where I am today, I am left understanding for the first time that there are no ceilings. And I could go on a long rant about how that feels specifically true at MIT– the place with literally no ceilings, and with pipes and wires and such instead. But that doesn’t feel true.

There are no ceilings because I kept walking and will continue to do so. There are no ceilings because dammit I’m gonna keep pushing hard to get to follow my dreams. There are no ceilings because you can always take the next step. This isn’t Zeno’s paradox. You’ll keep moving forward.

In two days from now, MIT is going to release undergraduate admissions results. And though I don’t have any clue what’s going to happen next, I am certain you all are going to keep going along your individual staircases. And I, for one, can’t wait for you to take those next steps.

I, again, leave you with one of my favorite quotes.

“Believe in yourselves. Dream. Try. Do good.”

“Don’t you mean do well?”

“No… I mean, dogood.”

You’re going to do great things.

(special thanks to allison for getting me thinking about this topic)

]]>And it’s not like I haven’t had this question before. For example, I asked these questions when I was learning about how to multiply multiple-digit-numbers in elementary school, thinking to myself “surely mathematicians don’t sit in their offices all day multiplying bigger and bigger numbers”. But when I was younger, I knew that this couldn’t be the case. I just had no sense of what college-level maths looked like. So I ate my mathematical-vegetables and went on with my life.

So ending up in my first ever college class and asking the same questions was frankly a bit of a shock. And don’t get me wrong, calculus *is* beautiful, and (some) mathematicians *do* use calculus in their day-to-day life (including myself!). But calculus felt too… closed? I guess? I’m sorry, I am missing the exact right words for this, but to put it frankly: Calculus felt like, finished in some sense. It felt too polished. Like, what mathematician with a PhD is sitting in their office “researching” how to solve the integral

So anyways, I’m taking this calculus class and trying to figure out a) what maths after calculus looks like, and b) if I want to actually do that math. Then, I found a maths comedy special by Matt Parker.

Matt Parker has done *a lot* but you may have most commonly seen him on NumberPhile, or you may have read one of his books *Humble Pi* or *Things to Make and Do in the Fourth Dimension *(to name a few).
You may also know him from the Parker Square.04
listen I said I'd try not to mention it, not that I'd try that hard.

In this comedy special, Matt Parker talks about the concept of 4-dimensional space. I’ve written about 4-dimensional worms before (here and here), but let me discuss the concept here again too for those who may be unfamiliar (as I once was before I watched Parker’s comedy special).

You may have heard that we live in a three-dimensional world, but what does this mean? Well, one way we can describe this is that we can move in three directions: up/down, left/right, and forward/back. Alternatively, you can look at the corner of the room you are in, and see how many walls it takes to form the corner (i.e., three). The same concept is true in lower dimensions. For example, to picture two dimensions, simply imagine a piece of paper with a square on it. Notice how it takes two lines to “form a corner” of the square. Similarly, characters in 2-dimensional video games really only need two directions to describe where they are in space: up/down and left/right. You can picture an old-school 8-bit Mario moving left/right, or up/down, but there are no other directions for the 8-bit mario to move in (at least, not until Super Mario 64). Lastly, we can think of lines being one-dimensional (like the number line).

But what about four dimensions? We can’t quite necessarily picture 4-dimensional space well, but sometimes people refer to *time* as the fourth dimension. If we consider time to be the fourth dimension, then our four directions would be:

Up/Down, Left/Right, Forward/Back, and Future/Past.

Side note: viewing time as the fourth dimension is actually what my worm blogposts are about. In any case, when mathematicians talk about four dimensions, they typically mean something *slightly* more abstract and *slightly* more physical (in a way I will explain right now).

One way we can make dimensions more “physical” is to think about how we measure “volume” in those spaces. For example,

- 1D: Length: We can measure the length of a line segment.
- 2D: Area: We can measure the area of a rectangle via the formula base times height, where both the base and height are in some sense 1-dimensional (they are just the lengths of the sides of the rectangle).
- 3D: Volume: We can measure the volume of a rectangular-box via the formula base times height times width (notice that this requires knowing three 1-dimensional parameters…)
- 4D: ?????? It’s kinda unclear! But, if we wanted to find the volume of a four-dimensional box, it would make sense if it required four 1-dimensional parameters, like base, height, width, and then some other length in a different (fourth) direction.

Hopefully I’ve done a good enough job of trying to give a sense of what four (or more) dimensions kinda means.

In any case: Matt Parker. In Parker’s comedy special, he spends like an hour trying to explain how we can *try* to visualize the fourth dimension. And, sure, we can define the fourth dimension abstractly and mathematically, but actually trying to *visualize* higher dimensions is a bit of a difficult task! I think Matt Parker does a really good job of this, and highly recommend watching the video.

Upon watching this special, I started to realize that I really wanted to understand four dimensions better. I wanted to understand five dimensions; six; seven; hell, why not *n*-dimensions??? (by which I mean arbtirarily-many dimensions).

But I was only in single-variable calculus, and I didn’t *really *understand do mathematics in 4 dimensions, let alone *n*. So I forgot about it and moved on with my life.

Cut to my senior fall of high school when I’m taking multi-variable calculus and linear algebra, and *OHMYGOD I AM FINALLY DOING MATHS IN HIGHER DIMENSIONS*. Granted, most of the maths is in three dimensions, but doing calculus in three dimensions makes it clear how one would do it in higher dimensions.

But it was still unclear to me how I could *picture* five+ dimensions. I mean, because of Matt Parker’s comedy special I could somewhat visualize four dimensions in my head, but five+ dimensions seemed just out of reach. And surely, there wasn’t some way to use mathematics to picture higher dimensions.

All the while, I find myself applying to colleges for undergrad. For one of the application
essays,05
note that I applied through QuestBridge and this was a QuestBridge essay question
I found myself writing about a “topic you have explored simply because it sparked your intellectual curiosity”. In particular, I wrote about being inspired by Matt Parker and wanting to learn how to picture *n*-dimensions in college. But surely that wouldn’t happen.

Cut to being at MIT and doing maths research and *OHMYGOD I CAN USE MATHEMATICS TO PICTURE n-DIMENSIONS*. Most of my research at MIT for the past three years has been about trying to visualize *n* dimensions (in some sense). I love this research. I love that I can roughly explain my research to other people, and I love that I do this work in arbitrary dimension. In fact, most of why I like this type of research (in the field of maths known as analysis) is because I like being able to explain what it is I study to other people. And hell, this is part of the reason I fell in love with the concept of mathematics in the first place: Matt Parker’s ability to explain the concept of four dimensions to people in a *comedy* special is exceptional, and I wanted to one day be able to do this one day. But surely, I wouldn’t get the opportunity to explain what I research to other mathematicians until I was a professor.

Cut to being a senior at MIT, and *BEING INVITED TO PRESENT ON MY MATHEMATICS RESEARCH AT THE JOINT MATHEMATICS MEETING (JMM)*. For those who don’t know (and to be completely fair, I didn’t know what the JMM was until last year when there was one in Boston), the JMM is the largest mathematics conference in the world, and I was invited to present at two of the smaller sessions in my area of analysis! It was my very first JMM presenting material, and I presented on all the *n*-dimensional research I and others in my field have been working on. It was so much fun, and I got to meet a lot of really cool people.

One of those cool people I got to meet was MATT PARKER!!

During the JMM, there was a Mathematics Art Revue show with jugglers, dancers, poets, 06 Harry Baker is my favorite poet, give this poem a listen and then go listen to <em>Paper People</em> and more, and the show was being emceed by Matt Parker. After the show, he was in the lobby meeting people and signing people’s names in binary (he signed my JMM nametag!).

I told him about how much his comedy special influenced me. I told him how I wrote about it in my MIT application, how I was inspired to continue investigating mathematics, and how it led me to pursue mathematics research that I was presenting on at that year’s JMM. All because of a maths comedy show that I watched *~five years earlier*. How the time flies.

P.S.: If you’re interested in learning more about my research and what mathematics research looks like, my plan is to write about this in my next blogpost! Stay tuned :)

P.P.S: I used “maths” instead of “math” throughout this blogpost, and fun fact: when I applied to undergrad, I did used “maths” or “mathematics” instead of “math” in all of my essays.

]]>Earlier this year, Kano suggested we have a blogger retreat this semester. We really be in our romanticism era. One month later, with an Airbnb booked, tickets for the commuter rail bought, and our laptops ready for blogging, **we** **begin**.

(Also deep thanks to Kano for the planning and brilliant execution of this retreat, we are all very excited!!!)

*Also *about a month ago, in the middle of one of our blogger check-ins, Kano asked the question: What is liveblogging? The answer: (according to Wikipedia) a
“blog07
no shit
intended to provide coverage of an ongoing event in rolling text”. Apparently this used to be something that the bloggers used to do more often on MITAdmissions. Essentially, like a giant
X08
formerly known as Twitter
thread, we keep updating the blogpost throughout the day/event.

This idea sounded really cool to me, and when better a time to try liveblogging than on the blogging retreat? So, keep checking into this blogpost every now and then over the rest of the weekend to catch the latest updates!

Well as it turns out, many of us going on the retreat cannot drive, so we figured we’d take on the dark academia aesthetic of a train on a grey-skies-day. Five of us (myself, Kayode, Allison, Jessica, and Kano) grabbed lunch at West Garage and hopped on an Uber over to North Station where we had planned to meet Waly and Gloria.

Fun fact: starting today, the train going to our destination isn’t running due to construction on the line. Instead, there is a shuttle service into the city. Perhaps not as romantic as a train, but the retreat must go on. The five of us hopped on the shuttle to wait for the others to join and …. the shuttle took off lmao. Rip, we will catch up with Waly and Gloria soon enough.

**Waly**: I walked from MIT to North Station, where the MBTA commuter rail leaves from. Which is about a 59-minute walk according to Google Maps, but 40 minutes at a New York walking pace. I got there before everyone else on accident, so since I had about 30 minutes to burn, I went to a local cafe. I got a matcha latte and a sandwich, but there was no sugar in the matcha latte so that was a bit awkward o__o. I wanted a little unhealthy drink but it felt too healthy now, so I added some sugar packets. I was walking back to the station to meet up with everyone else when I got a message saying the train-turned-bus had unexpectedly ~~ditched me~~ left. So I went to North Station intending to take the next bus with Gloria who also didn’t make it with the first group. However, I proceeded to get lost in North Station because the MBTA had negative signage explaining where the shuttle bus was. Eventually, I found an MBTA worker who told me to go down a sus little corridor to find the shuttle, and I met up with Gloria to take the bus to our destination.

**Gloria**: I was actually awake since much earlier in the morning, but due to poor time management and lack of executive function, I didn’t start packing until 2:30PM (when everyone else left campus). It doesn’t help that I’m a chronic overpacker, spending a good 30 seconds pondering if I needed to bring extra AA batteries (??), a portable phone charger (the bus ride was like 40 minutes), an HDMI converter, etc. At 3, I Ubered from Maseeh to North Station, got directions to the shuttle loading area, and met up with Waly! Our bus pulled out of the station just as the sun began to dip below the horizon, casting horizontal golden rays across the bus interior. Our surroundings shifted from city streets to worn New England homes to coastline, and then we were in northern Massachusetts.

We had like a 30-45 minute meeting talking about our goals for the retreat and the blogs we want to work on. We began with the icebreaker: “Would you rather have feet for hands or hands for feet”, to which all but Waly agreed they would rather have hands for feet. We discussed blogger struggles:tm:, the (abridged) history of the blogs, and things we want to write this weekend.

Shortly after, we ordered pizza for dinner (pepperoni and hawaiian),09 much to Allison's dismay and began working on our various self-assigned projects.

It’s currently 3am, and half10 lmao just kidding, Gloria has now gone to sleep so 1/3 of the retreat remains awake of the retreat members are up and (mildly) active. We decided to turn on The Golden Bachelor about (checks watch) 6? hours ago? And it is still on. We are down to the final last three women, if that is of interest to you dear reader. Updates from tomorrow to come.

The Airbnb had checkout at 11am, which for those of us who were up at 4am was a bit a stretch but we made it lol. Kano, Gloria, and Jessica were chilling/writing/watching TV downstairs while the rest of us were asleep (or half asleep) in various locations around the house. At around 10:30 we nabbed a polaroid and a selfie, and departed to the train station to catch the shuttle back to Boston.

]]>every saturday or sunday my freshman fall, i’d go over to have dinner with my friends and their family.

every saturday or sunday my freshman fall—i can no longer remember which—my problem set for 18.701 (algebra I) was due.

—

i signed up for 18.701 because it was the right thing for me to do. at least that’s what everyone told me, and so that’s what i believed. at that point, i completed the introductory math courses i needed to take, and i knew what courses i wanted to eventually take, and so 18.701 became a way for me to start learning pure mathematics to the degree that i needed to learn it. and i had satisfied the prerequisites, and i had spent that summer trying to prepare my proof writing skills. i did everything right that i could do to prepare for this course. i told myself that i’d do it again.

in my preparation for this class, i read one of the problems in the first chapter, and i thought to myself “i can’t solve that.” and then i started taking the class. and i had to solve that problem. and i did. and this happened over and over again until one day i had completed the class. i did it.

there are moments where you can feel yourself growing as a person at mit: that problem you had no clue how to do suddenly is in your problem set, that topic you tried to independent study but were utterly confused, now makes sense. and at the end of every semester you’re just glad you made it to the end—but nonetheless you’ve made it—and now you have to make it again. 7 more times. 6. 5. 4. 3. 2.

—

there are moments where you have to make tradeoffs at mit.

that show you always watch every tuesday gets put on hold because your problem set is due, or you’re just too tired. that club you swore you’d join—that you desperately wanted to join—now conflicts with a class.

or, it’s saturday (or sunday), and you’re at your friend’s family’s place for dinner; except you’re not, because while everyone is hanging out downstairs, you have to hop on a zoom call to finish up the last few parts of this problem. it’ll be an hour at most. maybe 2. 3.

but it’s worth it, right? because now you’ve made it to the end. you’re a better person/mathematician/what have you now. and you’d do it again.

—

a week or so ago, my friends and i made apple cider and watched *over the garden wall* for those fall vibes. at the same time, my collaborators—friends—and i were wrapping up the first draft of a project we’ve been working on since this summer. and at one point, one of my collaborators asked if we could hop on a zoom call real quick to go through some final edits. while my friends are watching the adventures of wirt and greg, i was in my friend’s room on a zoom call working on these final edits.

and it’s nothing i hold against my collaborator—not at all—but when i hopped on the zoom call i felt such massive déjà vu back to my freshman year. off in some room in a house, while my friends are hanging out, and i’m taking a zoom call.

—

when i say i’d do it again, i mean it. if i hadn’t taken 18.701 that fall i wouldn’t have taken 18.702 (Algebra II) in the spring, or 18.102 in the spring, or 18.155 when i did, or—

where i would be right now in my life/my career/my education would be different. and that isn’t a bad thing. i like where i am at. i wouldn’t want to change that. but there are moments when you realize what parts of yourself you’ve sacrificed, or even worse you feel that you’re different and you don’t know how—and don’t know if—you want to change.

there are ways you can prevent these sacrifices. maybe you don’t need every point on this problem set; maybe it’s time for you to just go to bed and work on things tomorrow; maybe you should drop that class. these are easier said than done. these are lessons i’m just barely learning my senior year.

but even so. i’d do it again. Differently.

maybe i have grown.

Before MIT, I had never been on a plane before. Hell, I had barely left California, with the exception of one or two family trips to Las Vegas. So, when I went to the east coast for my freshman fall, I was terrified. I had no clue how things *worked*— even the supposedly “simple” things. I had no clue how to even board a plane, let alone how to get around
the city I was moving to.11
to be clear, my freshman fall was during COVID so I wasn't moving to campus, but I still decided to move to the east coast for reasons I'll get into later.

I was scared.

—

At one point in *The Curious Incidence of a Dog in the Nighttime *(one of my favorite books and plays), Christopher Boone decides to run away from home, but he never had to navigate the world on his own before. It scared him too:

And then more people came into the little station and it became fuller and then the roaring began again and I closed my eyes and I sweated and felt sick and I felt the feeling like a balloon inside my chest and it was so big I found it hard to breathe.

In the play, Christopher pictures his dad helping him. His dad says:

Watch what the people do. Watch how they get on and off the train. Figure it out. Count the trains. Get the rhythm right. Train coming. Train stopped. Train going. Silence. Train coming. Train stopped. Train going. Silence. Train coming….

—

When I got to the airport for my first ever flight, I watched what the people did, and I followed suit. And just like that, I wasn’t so scared anymore.

Because, sure, figuring out the world all by yourself can be scary– especially as someone who doesn’t really “go with the flow”. But following a sequence of steps? That– that I can do. And these ‘steps’ don’t just come out of thin air most of the time. More often than not, you can observe and extrapolate. Watch what people do, and take the next step.

The same thing is true in college… for the most part.

—

My freshman spring was the first time I ever came to campus, and the adults who drove my friends and I to MIT were *so* nervous. They wanted a detailed outline of what to expect when we arrived: how to check in, where to go for our COVID test, etc. etc.. But I wasn’t nervous at all. I knew how arriving to MIT would look like as a first year:

Go to check in, and go where they tell you to go next.

Lo and behold, that’s exactly what happened.

I feel like this is a good example of what most of your freshman year looks like. At least during the first semester. The Office of the First Year does a really good job of trying to make sure that first years aren’t lost/confused about what you are supposed to be doing and when you are supposed to be doing it.

But what do you do when you don’t know what to do (as a first year or otherwise)? Or when, if nothing else, you’re ready for something to be different?

The reason I moved to the east coast my freshman fall (even though I couldn’t move to campus) was that I wanted something different. In this past blog, I wrote that the pros and cons of staying at home were as follows:

**Pros: Safety and familiarity.
Cons: Safety and familiarity.**

I had spent all of high school getting ready to one day go to college, and suddenly COVID was supposed to stop me? I didn’t think so. I didn’t know much, but I knew that I needed something different, and I didn’t know what to do. In the end, I figured it out on my own; I took a leap of faith and moved to New York.

When I feel that way at MIT– when I don’t know what to do– I no longer feel like I need to figure things out on my own.

Most of the time in undergrad, there is *someone* who has been where you are at some point in their lives. This means that when you’re struggling with something, you can almost always turn to someone else and talk to them about it. Now, this doesn’t mean that it isn’t necessarily hard to do so; it can be hard to be that open and vulnerable with another person.

But I think it’s important to try and talk to other people about these things. If anything, it gives you a chance to *stop and reflect* upon the situation yourself. Even if sometimes you’re left still needing to take a leap of faith.

This summer, I’ve been reflecting upon the fact that soon I will be applying to graduate school. I’ve been looking back on the past three years and realizing that in less than a year undergrad will be over. And you know what? I’m ready for something to be different. I think, for the first time, I’m ready to take this next step. But it’s still terrifying.

So, I’ve been talking to people. I’ve been reaching out to alum friends and talking to them about the application process and how I’m *already* stressed about it. The end result of these conversations is always the same: I’m still ready to apply. But it makes it less scary. Less mysterious. More… human.

It’s wild to me that three years ago I was just starting out here at MIT. I was barely learning the numbers of buildings and classes. And now here I am, preparing for one day being ready to leave. It’s the natural motion of the world. It’s the rhythm of the trains.

Train coming. Train stopped. Train going. Silence.

At the end of the play, Christopher reflects on all the things he’s done, and dreams about all the things he will one day do. And he essentially says: “You know how I know I can do these things? Look at all I’ve already done.” As if to say: I’ve made it this far.

He then turns to the audience and says: “Does that mean I can do anything, you think? Does that mean, I can do anything?” And before someone can give an answer, the stage fades to black.

—

I’ve learned how to board a plane. I’ve struggled through three years at MIT. I’ve become comfortable in uncomfortability.

Does that mean I can do anything, you think? Does that mean, I can do anything?

Silence.

]]>Almost two years ago, I was feeling angsty and wrote this blogpost about being a sophomore but not really feeling “older”. I felt like I was still a first year, but without the FPOPs and orientation. I felt angsty because I felt like we (the class of ’24s) were supposed to just *get it*. Like we were supposed to just know where the lounge for our newly declared majors was, and like we were supposed to know how to advise actual first years on where they should live [as though we should have a semblance of what dorm culture was after spending our first years on Zoom].

Through all of this angst, was ultimately the realization that time was passing. I would never get to redo my first year at MIT. Whether I liked it or not, whether I felt like it or not, I *was* a sophomore. But of course, with the speed at which MIT operates, I got over this fairly quickly. There really isn’t anything to do other than accept the reality when you have to start turning in problemsets and the semester is in full swing.

Now, here we are two years later, about to become a senior this fall. Of course, one can argue over the semantics of whether or not the class of ’24 are seniors *right now,* [I mean, according to the student directory I’m already a fourth year….] but that’s besides the point. I’m entering my senior year– and it’s already beginning to have an (albeit mild) affect on my life.

For instance, the other day my mom and I were talking about when my graduation is. [*nervous laughter* it’s less than a year away] Another day recently, I had the realization that if I want hot cocoa/tea/coffee nights to be a thing on my wing of Next House, I would need to just make that *happen*. And jesus christ, MIT alum friends of mine are talking to me about applying for graduate school. Which, to be clear, I deeply appreciate. It just makes me feel Old. I can only imagine how it makes y’all feel.

I’ve just been realizing that we *really* aren’t kids anymore. [omg it’s the title of the post]

Recently (aka in the last blog posted), Fatima ’25 lamented that she is “halfway done with MIT and I still do not know how to MIT very well.” And even though I’m an [academic] year older, I can unfortunately say that this still rings very true to me. I think I’m beginning to realize that I don’t think I will *ever* know how to MIT very well– even after I eventually graduate.

Which of course, there’s an sense of irony to that sentiment; the idea of feeling like I won’t ever know how to MIT well, while simultaneously knowing I will one day have ‘done’ MIT. And yet, I think it’s the most honest sentiment I can offer at this time. I haven’t had a semester here where I felt like I’ve just ‘gotten’ it, and I doubt I’ll ever have one.

But being/becoming a senior, I’m going to have to act like I have my shit together. I’m going to have to run REX events. I’m going to need to offer advice to incoming first years who ask for it (whether it be in the role as an associate advisor, or in the role of ‘generally older person’). I’m going to have to “get it”.

Maybe that’s all there is to not being a kid anymore– playing pretend. I sure hope so, because beyond pretending, I don’t know who I’m supposed to be or what I’m supposed to be doing. I just know that I’m older and should act like it. And feeling angsty about that is no longer cute.

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