Vestigial

This was written for 21W.775 Writing about the Environment and Nature. It is highly likely that I edit this again and repost it. I also was experimenting with footnotes in this piece, which cannot be realized in this medium, so I've just left them as asterisks.

I am dropping 18.726 Algebraic Geometry II. I am dropping it after gritting my teeth, forcing my way through Algebraic Geometry I, getting a B, and telling myself that I'll try even harder in the second course to make up for the grade. I am dropping it after spending three hours every other day to understand the hour and a half long lecture, and an additional who knows how many hours spent on the problem sets. I am dropping it after believing for five years that this was what I was going to do for the rest of my life.

It's because I don't have enough background, enough context for the field. Of course, you're struggling with schemes and quasicoherent sheaves and Cartier divisors. They're algebraic analogues of manifolds, did you take a manifolds class? No, I shake my head. I was wasting my time taking computer science classes, back when I thought cryptography interested me. I was wasting my time taking music classes, back when I thought performance interested me.

Coincidentally, a friend from back home, Gavin, is visiting this weekend. He quit his job as a sled dog handler up in western Massachusetts** and is now in the New England area for a bit before making the drive back to Utah. Prior to this, I had been too busy with 726 to be a good host. Feeling rather guilty about it and with all this newfound time on my hands, I ask Gavin what he is planning on doing today and end up accompanying him to the Harvard Natural Science Museum.

** Slave labor, he calls it, but the reason he quit is not because he had to work 7 days a week from sunrise to midnight. It was because his boss – watch out for a Marla BB – is a genuinely evil person. She once made two middle school girls walk a mile through poison ivy because she suspected they fed the dogs food.

I am a bit surprised that he has this interest in museums. In high school, and I'll add the disclaimer that this was during the pandemic, I knew him for always trying to get me to play some video game (or sometimes smoke some weed) with him. That is not to say that he's not capable; no, he was in the same math classes as me until senior year, when he began working on his welding certification. He just didn't care much for school and always wanted a cool blue collar job. So here he is having the time of his life as a wildfire-fighter** in the summer and a dog musher in the winter. He looks quite the part too: he hasn’t cut his hair since high school graduation and put on quite a few pounds of muscle.

** Um actually, it's technically called a wildland firefighter, Gavin says, holding out his forefinger.

But all this about his appearance and his occupation does not change the fact that he is bouncing on the balls of his heels as we look at the first exhibit, a collection of glass flowers. He points at a yellow flower and tells me about how he sees them all the time in his deployments. He takes a picture of it and sends it to his firefighting buddies, explaining that they are trying to learn to identify the fauna they encounter in the wild. That sounds hard, I respond. I wanted to do that when I was younger.

I notice that most of the other museum-goers are families with excited children pressing their faces against the glass and tired parents staring blankly at the displays that their children are at. Looking at the expression on one father’s face, I will say that I have never so deeply resonated with a stranger before.
I used to have more interest in these sorts of things. I remember eating up every single word on every single panel and every single display at the Science Center Singapore. But now, as we walk through the rows of flowers, all I can think about is what I am going to do in place of 726 and how I am going to write off this failure in my graduate school applications.

Can we look at something else, I ask. The flowers look nice and all but I don't know, they're not all that interesting. I am beginning to lose my mind in the too crowded and too humid room. Gavin agrees. He says that he cannot wait to see the animals.

As we file out of the door, a need to explain my disinterest comes upon me and I confess to him that I am jealous of the childlike wonder he has. I seem to have lost it somewhere between the ages of 10 and 20. He turns around to face me and says with a smile: Luckily for you Angeline, it's a skill you learn.
We walk through a winding hallway, reconstructions of animals in their natural habitats snarling at us from both sides. In his little snow cabin in the woods, Gavin explains, there wasn't much to do except read. And so he read more books this winter than he did in his life. He recommends me the book Humankind: A Hopeful History by Rutger Bregman.

We stop in front of a fluffy red fox, its hackles raised and teeth bared. In 1959, a team of Soviet scientists tried to breed domesticated foxes by selecting the ones that had the least aggressive behavior to humans out of each batch. In as little as five generations, the foxes were noticeably smaller than their wild counterparts and some exhibited downward-pointing tails and floppy ears. In the book, Bregman claims that us Homo sapiens are also a domesticated species. We are smaller than our hominin cousins, with more juvenile features and a tendency to play, qualities of domesticated animals, because we simply bred out (as in, didn't mate with) the ones who had more aggressive traits.

The world is a kind place, and we are born to be good people, Gavin concludes. I agree with his sentiment, as I have always and will always be an optimist.** You should read the book, Gavin looks me firmly in the eye, you will like it.

** The thief thinks everyone steals, Gavin remarks after noting that Marla believes that everyone is evil.

The hallway opens up to a more expository part of the museum, the part that teaches you about evolution and whatnot. Next to the poster that every museum has of Darwin's Galapagos finches is a display about convergent evolution. The skeleton of a hare, frog, grasshopper, and wallaby stand side by side. These animals with vastly different ancestry all evolved to have a similar hind leg structure: three main bones and long toes folded like a spring, able to store plenty of energy for a powerful jump.

Does this mean that this structure is the optimal structure for jumping? Maybe alien species that jump will also look like this. I catch myself. No, that's a rookie mistake; just because this structure converges on a local maxima of some kind of loss function that models jumpability does not mean that it is the global maxima. That only happens when the loss function is convex.**

** Nota bene: I assume natural selection works via a hill-climbing algorithm: at each iteration x, we look at values that are close to x (these nearby values represent mutations) and pick the value y in this set that maximizes f(y) (jumpability). I think it's reasonable to assume that jumpability is continuous, so in this case hill-climbing will always converge (get arbitrarily close to) a local maxima.

After satisfying myself with this conclusion, I look up from the skeletons and notice Gavin giving me a questioning look, probably wondering what I was muttering under my breath about. I was just wondering if their jumping mechanisms are optimal, I explain. He shrugs.

I look back at the legs. No, the more I think about it, the less likely that the loss function would be a nice convex. Nature selects against those that are not good enough to survive. It does not select for optimality. These animals just jump well enough to survive, that's all. That’s why there are variations in their analogous structures. I’ve been thinking about it wrong all this time. The world is a kind place, and it does not demand all of us to be optimal.

Gavin has wandered to the other side of this display and I move to join him. It is a panel on vestigial structures. Vestigial structures are so cool, Gavin says with a gleeful smile on his face. Dolphins have hip bones because they used to walk on land, he explains**. I begin to read the words on the wall. Dolphins began as semiaquatic dog-like creatures that slowly learned to swim better than they can walk, and now we have the clade of Cetacea, which consists of modern day whales and dolphins.

** Earlier on, when we stopped in front of a taxidermized wild boar, Gavin told me about how many feral pigs were in Texas and that you could pay to get on a helicopter and shoot at them with a machine gun (what a true Texan experience). I asked Gavin where he got all these animal fun facts from. He showed me an Instagram Reel. I suspect that his feed is a lot more educational than mine.

I look at the image of the dolphin’s hip bone on the wall. The hip bones are vestiges of the dolphin's past life on land. These are useless bones left in the animal, its purpose now solely to inform scientists of its ancestry. Remarkably, they have not disappeared despite having no use. Yes, they have not disappeared because Nature does not demand perfect optimality from its creatures. The world is kind, after all.

It is not entirely true that vestigial structures are useless. More precisely, vestigial structures are defined to be body parts that have lost most or all of their original function. Keyword most, not all. The hip bones in whales, and I’ll assume by extension dolphins, affect a male’s sexual success.

I do not need to know only what is strictly necessary for my discipline. I mean, the goal of an undergraduate education is to have a survey of the different areas of mathematics. These form the foundation for which further topics are built on, and one field often draws from many, many different fields. I thought algebraic geometry only requires knowledge of algebra, but it helps quite a lot to have a good intuitive understanding of the geometry of manifolds (a scheme is built much like a manifold), which comes along with plenty of topology and differential geometry (especially when we talk about tangent spaces and vector bundles). Outside of that, there are plenty of analogues to number theory, too (the Picard group and the ideal class group coincide in the case of Dedekind rings, and both are constructed quite similarly), and the language of categories (a sheaf is a functor from a topological space to a set) is a must.

Any moment spent learning mathematics is a moment well-spent. Even if they seem to me merely vestiges of past effort, there will be a day that that experience comes in handy. I have learned to think of solving problems in a different way after taking algorithms. Music has taught me how to try and try and try again until I can play that note, after which it comes easier and easier until it becomes natural. I apply the same principles when I’m learning difficult math topics. I have to believe that daily continued effort will lead me to be a better mathematician.

We end in the rock room. Despite the moment of enlightenment I experienced in front of the dolphin’s hip bone, I am still very tired and very ready to go home. We have been in the museum for at least three hours, and I really do not want to walk through this too large and too brightly-lit room of rocks. Nevertheless, Gavin drags me through every single row of crystals. I stare at a pristinely cut gemstone, wondering how people could tell them apart from tinted plastic. What is the value of gemstones to me if I could not tell them apart from cheap plastic?

Beside me, Gavin has his phone in his hand and brings it up to one of the crystals to snap a picture. I ask what he is doing. For my next mining trip, he explains. My buddies and I went mining last fall for an engagement ring. We mined red beryl, a hundred and fifty thousand times rarer than diamonds. He brought it to the jeweler and had it put in its natural, uncut form to propose to his girlfriend. Gavin shows me a picture. The little red rock at the center did not look like it was worth more than diamond, or plastic while we’re at it.

But this rhetoric will lead me nowhere. If I do not give such gestures values, then I will certainly not be happy in life. These thoughts stem from the tiredness that I feel, not from what I actually think to be true. Who wouldn’t be happy if their partner went to a cave and mined a ring for them? What a beautiful gesture, I must believe even when I am tired.

Anyways, Gavin continues, let me know if you see any cool stones. He points at the placard next to the crystal he took the picture of: Farmington, Utah. I shoot him a barrage of questions. How did you get into the mines? They were on public land. So anyone can walk in and mine at the rocks? Yup. You don’t need some kind of certification? Some license? It’s public land. We walked in with pickaxes and a case of beer and started swinging. I guess you can’t bring automated tools, but yeah, anyone can go in and carve out a couple pretty gems.

Maybe I can carve my own path, too. Surely I will not gain much out of the second class in a sequence after fudging my way through the first class. I have always felt that I needed more time to work through the concepts, and here I am with more time. I will take the semester to review and reinforce my foundations. The vestiges of my past knowledge will surface in due time and teach me, and I will learn until it becomes second nature to me. It might feel like slow progress, but Nature is slow, and forgiving, and kind. Yes, I will carve out my own path.

On the bus ride back, Gavin tells me about the Granite Mountain Hotshots. Hotshot crews are the cream of the crop of wildland firefighters, deployed on the largest and most dangerous fires around the nation. The crew members must complete a long list of certifications on wildfire behavior, safety protocols, and specialized firefighting techniques; have many hours of field experience; and demonstrate their physical ability and teamwork through field exercises before earning their hotshot status.

Pencil-whipping is what firefighters call signing off on certifications without strictly ensuring that all the requirements were met. Granite Mountain gained their hotshot status quite quickly due to some amount of pencil-whipping. In 2013, they were deployed to a fire in Yarnell, Arizona. A young crew ready to prove that they earned the hotshot status, they threw themselves at the fire. A combination of not following protocol, fatigue, and unexpected conditions killed 19 out of 20 of the Granite Mountain Hotshots.