UChicago Essays

Since I got in, I figured I'd upload the essays I used.

Prompt:
How does the University of Chicago, as you know it now, satisfy your desire for a particular kind of learning, community, and future? Please address with some specificity your own wishes and how they relate to UChicago.

I’m a person who really dislikes closed-mindedness. My parents, as immigrants, were inexperienced with the nuances of English, creating a struggle for me to communicate particularly complex and thorny ideas to them. Despite my knowledge of this and continual failures, I continue to try and show them my perspective. But they were content with what they knew and didn’t want to complicate things. My failed attempts contributed to my high valuation of being able to reflect on your position and embrace intellectual challenge/change despite the discomfort. Thus the University of Chicago’s facilitation of “strong disagreement” and the “questioning of stubborn assumptions” (to quote the Report of the Committee on Freedom of Expression) is alluring to me. In constructing my current belief system, I spent many sleepless nights pondering and struggling to articulate my thoughts for my blog. I would love the opportunity to put my philosophy through the wringer to (hopefully, but unlikely) see it pay off and still be standing through it all. And even if it turned out to have holes, I’d be grateful to have sussed them out sooner rather than later.
I also find UChicago to promote what I want of an institution of learning: learning. I seek an education that involves both the familiarization with agreedupon facts and the stitching together of these facts into a beautiful and brilliant quilt. A lot of my classes in high school were mostly just rote memorization, but through my rigorous experience at PROMYS, I discovered a delight in the creativity of math. Thus the CORE curriculum will feed my bonfire for learning. But of course, if these new thoughts aren’t expressed, they can never turn over old ideas from the well-trod, beaten path. For that reason, I greatly admire UChicago’s staunch defense of free speech.
With respect to my career interests, UChicago offers a unique buffet of math opportunities from the REU to the DRP to the Paris study abroad program. UChicago provides a thriving community of future mathematicians as well—something I never experienced in my high school, as engineering and cs (especially due to the Austin area) attracted many people. Admittedly, I lack exposure to the many subfields of math. Still, I’ve enjoyed pure math thus far for its beauty and interconnectedness. And if I continue on this path, I’d be eager to interact with the impressive quantity and quality of algebraic geometers at UChicago, like Bảo Châu Ngô. In addition, the undergrad course offerings in pure math is unusually vast, with classes like algebraic number theory and representation theory on groups.

Prompt:
What can actually be divided by zero?

Zero. OK, admittedly, this isn’t the “0” you’re probably familiar with; it’s the “0” of the 0 ring. A ring is essentially a list of things (which we in the industry call “numbers”) where adding and multiplying, to simplify, “make sense” and “0” is common notation for the number such that 0 + a number = that number. For example, the integers have sensible addition and multiplication, so they are a ring, but the economy isn’t since multiplying a dollar by a dollar doesn’t make much sense. Sounds pretty reasonable, right?
What’s the deal with a thing over zero? (cue Seinfeld music) Well, let’s try and pretend it’s legal to divide by 0, so something / 0 = something else. Then something else × 0 = something. But, if multiplication makes sense, addition does too, so we have a ring, and in all rings, anything times 0 is 0 (to save on gory details, just trust me on this). Thus something else × 0 = 0 = something. So something has to be 0! But then, something else else times zero is zero, so something else else is equal to the something else. But then then, if something else is something else else for all things, then every “thing” in the “list” is just the same! This property is what the 0 ring is: a “list” with just “0”. So something / 0 = 0 / 0 = 0.
So we just divided by zero, much to all my schoolteachers’ (imagined) chagrin. But that’s no fault of their own: math just isn’t cut and dry. That’s part of what makes math so interesting; you can delve into crazy universes where the rules are different, like the world where we divided by 0. The only limits are your imagination and logic (although even logic isn’t binary. There are debates [called LEM] over whether every statement is exactly either true or false [as an exercise, consider “this statement is false”]). Many see math as an icy, sterile land of uptight authorities dictating what you can and can’t do, deciding what is true and what isn’t. But math is full of creativity and freedom in the novel, stunning approaches to problems, the world you explore from what you choose to be true (like LEM), and how you use language.
An amusing, often-overlooked, side of math is inventing notation. In math, you can essentially invent your own words—quite the contrast with the sciences, where there is considerable consternation over conventions like Pluto’s planetness. For instance, once I was working on this exercise (a math one, not a physical), and I wanted some notation for a certain concept. So, I asked my friend—who had his laptop out, as usual of his love of learning—to search up the notation for it. To my surprise, it didn’t exist! So I just made something up. Albeit it was imperfect; it had non-related uses in other contexts. That is, at least as far as I know—who knows, maybe all of math is related. This freedom to do whatever you want (within reason) in math is a breath of fresh air in the stifling atmosphere of authority.
Notation in mathematics can also be profound. There is a specific notation for constructing special kinds of rings using other rings that I learned about last year during PROMYS. But it wasn’t until about a year later that I realized what was actually going on behind this notation—it was delightfully concise and elegant. I guess what allows notation like this to become canon in mathematical lore is the conversation that goes on in math. It’s like the tension between my parents’ conservatism and my younger self’s strong idealism and progressivism. 1 Through conversation, some of my more extreme beliefs began to be tempered by my parents’ beliefs, settling me to my current more moderate and less-biased position. The conversation begins with a discovery connecting two branches. Then, through papers, mathematicians of each branch propose ways to highlight the beauty and truths in their respective branches. Finally, the conversation starts to end as the better idea flourishes. This convergence towards the truth arises from rational and free choices, unlike my decision to have short hair, which was influenced by my Asian heritage and gender (don’t get me wrong, I like it; it just wasn’t made completely freely). Yet, in the face of peer pressure to use the new notation, the mathematician can still freely use the “worse” idea (and they do at times, such as in teaching), so long as say that they are using the “worse” idea.
But like the discovery that, in a sense, 1 + 2 + 3 + 4 + . . . = −1/12, maybe in the future, dividing by zero will make sense in a different way. If that day comes, I would be one of the first people embracing and exploring that new universe.