CATEGORY THEORY (2/2)
by greenturtle3141, Aug 13, 2023, 7:43 PM
IV: Among the Stars
V: Category Theory
I think most people would agree that category theory is not a very useful subject. However, much of it revolves a fascinating principle that everyone ought to take to heart: To study an object, or to study the object's relation to everything else --- these are one and the same.
This is a tenet that can be found everywhere, hidden in plain sight. A real-life object is fully determined by all its properties: How it looks from each angle, how it feels, how it reacts to chemicals, etc. Numbers such as
and 
are unexciting until we relate them to other numbers, using a connection called functions.
Even if the two halves of the object/relations duality are not quite equivalent, it cannot be denied that examining something's relationship with everything else is crucial to understand it. Vector spaces are uninteresting, but we can learn so much from them by studying linear transformations. Instead of a studying a studying a function
, we can probe it by considering 
for every 
. That is, we see how 
gets along with all other functions 
, and this will give a great deal of information about .
In a modern, Western-dominated age where individualism reigns, it is more important than ever to recognize and appreciate our connections with others. The equivalence of our identity and self may be intuitive, but there is so much more to one's identity than oneself. The relationships you have with others make up a great part of who you are. In some cultures, one's identity is even considered to be primarily such relationships!
Treasure each and every one of your connections, whether they are a friend or more. They are a part of you, whether you know it or not.
I met E in kindergarten. We grew up together, survived the worst of times, and experienced the best of times over the next
years. Our friendship wasn't always idyllic, and at times it took some turbulent turns, usually due to my own mistakes. But we got through it, and she is indisputably my best friend today.
Ever since the end of middle school, I kept my friend group small --- just E, S, and me. I met S through E, and although we were quite an unlikely duo, we eventually warmed up to each other and became great friends. The three of us had a fantastic time in high school together, texting each other every day and hanging out whenever we could. Though, thanks to my young insecurities, I was often plagued with intrusive thoughts:
How do I know that they actually care about me?
What if they're just bothering to spending time with me so I don't feel bad?
If there's one essential piece of wisdom I've learned over the years, it is that an affinity and talent for clever things is of little use if it is not applied to the intricacies we experience in real life. Navigating a world of people may feel like a skill that is woefully divorced from analytical minds. But I've found that, more often than we may expect, mathematically-inspired approaches can solve much of the issues we encounter in our social lives and more. I first realized this when I discovered the beautiful solution to my insecurity.
Suppose that my friends did not care about me. Then, since they talk to me and spend time with me, they are only doing so so that I don't feel bad. This implies that they care about my feelings, and so therefore they care about me, contradiction.
Corollary. E and S are true friends.
For a good number of years, I bought into something that I've dubbed the friendship nuking paradox --- a line of reasoning which essentially proves that there is no reasonable approach for getting a relationship besides praying. Here is the paradox:
She is the most brilliant person I know, second maybe only to W. But that's like comparing apples and oranges. Whereas W's brilliance stems from mathematics and the analytical world, her brilliance involves a skillfulness in many things, combined with an insatiable curiosity and a life philosophy that would shatter my own. I find that much of the thought processes I have at the present are well-inspired by her thinking. In a way, she has become a major part of who I am today.
The thought of asking crossed my mind. Could I justify the risk? She was a good friend and I knew her well enough to know that she handles social issues quite well, especially given her views on relationships. She'd often say, for instance, that much of the world's problems would be solved if people would simply be direct about what they want to do with one another. What this entailed was that
for a small 
. I was about 
certain that she would say no, which put the expected value of asking at
![$$\mathbb{E}[\text{I ask}] = A\mathbb{P}(\text{She says yes} | \text{I ask}) - B\mathbb{P}(\text{Friendship ruined} | \text{I ask}) \approx \frac{A-9B\varepsilon}{10},$$](//latex.artofproblemsolving.com/1/4/9/149aa4c64ec43518feaeced03eb4f0721607004f.png)
where 
is large and 
is a somewhat large but smaller value. I was confident that 
--- after all, I knew her well. Thanks to the smallness of 
, I deduced that the expectation ![$\mathbb{E}[\text{I ask}]$](//latex.artofproblemsolving.com/4/c/0/4c0035baadc62fdbad0077a378f989162271fa45.png)
was positive --- the first time it had been positive for anyone.
I asked on Valentine's day. She said no, as I expected. Naturally, I was worried about nuking the friendship and tried to backpedal. But she's a difficult person to fool, and reassured me that I did the right thing: "You should always ask."
I learned a lot from those words, and with time, I eventually came up with a resolution to the friendship-nuking paradox, inspired by her.
VI: Mathematician's Path
VII: UNITY
Shoot for the moon. Even if you miss, you will land among the stars.
Senior year was a new era for me. I had already achieved my dream, and with W's graduation, I was indisputably the dominant mathematical force at Syosset. What more was left to accomplish?
I really had no idea about the miracles that would happen that year.
Besides applying to college, I was involved in quite a number of activities. I played chess, volunteered at a local chess program, played the violin somewhat competitively, was the president of the chess club, was the president of mathletes, was president of the non-profit math teaching organization started by W, and was also working on math research. On top of all that, I was still active in math competitions --- I led several teams of fellow Syosset mathletes to PUMaC, CMIMC, and HMMT, and I was selected to be Nassau A's team leader for that year's NYSML and ARML.
I was a pretty terrible chess club president since I wasn't too involved in pushing for much-needed improvements. Much of my energy went towards running mathletes and the teaching organization. They were both great ways for me to give back to the mathematical community, and I loved making handouts and teaching. I would spend hours crafting the perfect handouts, and sometimes I'd go off the rails and make things like this. Even though I don't really consider myself much of a leader, I found it natural in the end and I learned a lot from being in the spotlight.
As for research, it wasn't exactly a huge priority. The research coordinator at Syosset told me that for the Regeneron STS competition (which I was required to apply for), I was going to either stick with my junior year project (The Light's Out puzzle and its variations) or come up with a new project. I went with the latter --- There was a silly game called "Twiddle" on one of my puzzle phone apps that I wanted to try and mathematically work out. The project would be incredibly niche and essentially useless, so I was sure this would have very little chance in research competitions. I was just doing this for fun.
I spent the summer at CMU for a "pre-college" program, where I took two classes, met new friends, and worked on my research project, which I dubbed as the "Number Rotation Puzzle" (NRP). Due to how chaotic the moves were, my efforts went mostly to making educated guesses on sequences of moves that could be helpful in, say, making a 3-cycle.
Eventually, I found a 3-cycle algorithm that worked on all square boards! It wasn't too bad, and proving that the specific 3-cycle algorithm could induce any 3-cycle I wanted (subject to "Parity Restrictions" (PRs)) was not hard by using a sort of "whirlpool" algorithm that brings numbers to the center and expands them outwards, with correctness justified by a taxicab monovariant. I wrote up my paper and that was that.
Mere days before school started, disaster struck. I was googling around to make sure that my work was original, i.e. I was the first to solve the NRP. I had done this before numerous times, and didn't find anything substantial... until that day, when I discovered an obscure forum post on a speedcubing website, in which a certain "Ravi Fernando" had beat me to it! My project was in jeopardy. There was only one way to save it: Generalize my solving algorithm to rectangular boards.
Now, this might not sound like a big deal to you, so let me emphasize: This was horrible. Sure, the
NRP with 
rotating blocks was pretty bad due to lack of mobility. But the rectangular 
NRP with rotating blocks is a dozen times more insurmountable. There are only two legal moves (two less than the minimum you could get for square boards), and every move screws up like of the board. I had good reason for avoiding this line of inquiry for my research --- it was frankly impossible. And now I had no choice but to do the impossible... in two months.
Playing around with the puzzle was not proving to be helpful, so I resorted to a desperate play: Programming. I wrote up a Python script that can simulate sequences of NRP moves for the board with rotating blocks, and implemented a breadth-first search that would seek moves which induce permutations that become 3-cycles when repeated a sufficient number of times by examining their cycle decompositions. Soon, I found three separate algorithms that did the job, each for different values of 
, but whose union would cover all 
! I didn't bother proving their correctness --- I was certain that some lousy induction would do it. The first hurdle was over.
Next, I needed to prove that the ability to execute this specific 3-cycle could induce any other 3-cycle that respects PRs. If I could do that, then the NRP is solved in the general case. I quickly came up with an algorithm that can send any number to any location, called "Spiral". With some fidgeting, I could modify Spiral so that it can send any two numbers to any two locations. But this isn't enough --- I needed to send any three numbers to any three locations, and there just didn't seem to be enough mobility to grant me the ability to this.
That's when a disgusting idea hit me: That summer, while playing around with various algorithms, I found an 8-move sequence, called "Cycle", that only moves numbers along an outer "band". Could this give me the mobility I needed? With some thought, I realized that I could use Cycle and alternate it with the modified Spiral algorithm whenever I needed to free up one of the rotating blocks for use. The resulting convoluted six-stage concoction was dubbed the "Spiral-Cycle Algorithm".
It remained for me to prove the algorithm's correctness --- particularly, Spiral. Spiral operated by "spiraling" numbers closer and closer to the center, so I knew that I needed some monovariant. I tried a taxicab monovariant, but I was puzzled to find that it didn't work. What could I do? Out of desperation, I tried the straightline distance, which felt incredibly unnatural since the game takes place on a grid. I was shocked to find that in considering the straightline distance, I stumbled upon some basic Olympiad geometry that could help study the change in this distance. By constructing a magic point, I could induce similar triangles and reduce the analysis of the distance change to a computation that uses the Law of Cosines (!!!). By some miracle, it all managed to work out, and with that, the general case for the NRP was felled.
I was feeling pretty ecstatic! I still had a reasonable amount of time before the Regeneron STS application, and I had only one hurdle left: Special cases on smaller boards that couldn't be covered by the general algorithm. Many of them were incredibly simple, but one in particular --- the
board with 
rotating blocks --- was devilish. There seemed to be some magic lurking in the background of this case: When I solved the left column, I always seemed to end up solving the rest immediately without having to move the left column again. And, with some combinatorics, I discovered and verified that if I solve the numbers in one parity, the other parity would always automatically be solved. What was going on?
I was completely stumped. In disbelief, I wrote up a Mathematica script that could verify my observations, and indeed it confirmed that the orbit of the group of achievable permutations on this board was
. I had no idea why.
But then I came up with an insane idea. Even though some proofs I had looked at before took a part in suggesting this approach to me, it was a long shot and nothing short of divine inspiration. See here for the details. With that, my NRP project was done.
Was it useful? No. But was it fun? Was I satisfied and proud of myself? Absolutely. I submitted the paper to the STS and didn't think much of it.
I was sitting in my AP Physics C class when the research coordinator came in to announce who in our research class was among the 300 STS semifinalists. A large fraction of the advanced research program was in this physics class, so it made sense for her to announce it there. I listened with practically zero investment. After all, why would a niche and useless project make it anywhere in a national research competition?
She called out four names. I almost fell off my chair when I heard mine.
Two weeks later was the day before they would announce the finalists. I kept getting calls from a mysterious number called "Washington D.C." throughout the day. I dismissed them as spam, though a small thought crossed my mind: They call the finalists the day before the official announcements, right? But my project obviously wasn't finalist worthy, so it was definitely spam.
I'm sure you know where this is going. That night, our household was getting calls from "Washington D.C." too. That's when I finally considered the possibility. "Isn't today the day when they call the finalists?", I suggested to my mom. Right then, the phone rang. My mom picked up the phone, handed it to me, and my life was thrown into a beautiful chaos.
The Regeneron STS Finalists Week was incredible. I made new friends, got to meet Chuck Schumer and Francis Collins, I took a picture with Adam Conover... I was roommates with Vincent flippin' Huang! We got to explore D.C., do escape rooms, eat fantastic food, and in the end there was a huge gala where they announced the winners. It's a week that I will treasure forever.
Alright, before you start thinking that I'm some kind of legend that excels at everything, I'll finish with a funny story. As part of the Finalists Week, they had scientists and/or mathematicians interview us to assess our knowledge and competence across science and math. The assessment would consist of questions such as "Look at this graph of a faucet's water's temperature over time, can you explain the fluctuations you see?" and various others that I won't spoil to try and maintain the interview's integrity. For the most part I did alright, with the glaring exception of one question: "About how many atoms are in a grain of sand?" I gave them an estimate of
, which was received with... judgmental stares. I genuinely thought it sounded right! You, the reader, likely know better.
Senior year was a new era for me. I had already achieved my dream, and with W's graduation, I was indisputably the dominant mathematical force at Syosset. What more was left to accomplish?
I really had no idea about the miracles that would happen that year.
Besides applying to college, I was involved in quite a number of activities. I played chess, volunteered at a local chess program, played the violin somewhat competitively, was the president of the chess club, was the president of mathletes, was president of the non-profit math teaching organization started by W, and was also working on math research. On top of all that, I was still active in math competitions --- I led several teams of fellow Syosset mathletes to PUMaC, CMIMC, and HMMT, and I was selected to be Nassau A's team leader for that year's NYSML and ARML.
I was a pretty terrible chess club president since I wasn't too involved in pushing for much-needed improvements. Much of my energy went towards running mathletes and the teaching organization. They were both great ways for me to give back to the mathematical community, and I loved making handouts and teaching. I would spend hours crafting the perfect handouts, and sometimes I'd go off the rails and make things like this. Even though I don't really consider myself much of a leader, I found it natural in the end and I learned a lot from being in the spotlight.
As for research, it wasn't exactly a huge priority. The research coordinator at Syosset told me that for the Regeneron STS competition (which I was required to apply for), I was going to either stick with my junior year project (The Light's Out puzzle and its variations) or come up with a new project. I went with the latter --- There was a silly game called "Twiddle" on one of my puzzle phone apps that I wanted to try and mathematically work out. The project would be incredibly niche and essentially useless, so I was sure this would have very little chance in research competitions. I was just doing this for fun.
I spent the summer at CMU for a "pre-college" program, where I took two classes, met new friends, and worked on my research project, which I dubbed as the "Number Rotation Puzzle" (NRP). Due to how chaotic the moves were, my efforts went mostly to making educated guesses on sequences of moves that could be helpful in, say, making a 3-cycle.
Eventually, I found a 3-cycle algorithm that worked on all square boards! It wasn't too bad, and proving that the specific 3-cycle algorithm could induce any 3-cycle I wanted (subject to "Parity Restrictions" (PRs)) was not hard by using a sort of "whirlpool" algorithm that brings numbers to the center and expands them outwards, with correctness justified by a taxicab monovariant. I wrote up my paper and that was that.
Mere days before school started, disaster struck. I was googling around to make sure that my work was original, i.e. I was the first to solve the NRP. I had done this before numerous times, and didn't find anything substantial... until that day, when I discovered an obscure forum post on a speedcubing website, in which a certain "Ravi Fernando" had beat me to it! My project was in jeopardy. There was only one way to save it: Generalize my solving algorithm to rectangular boards.
Now, this might not sound like a big deal to you, so let me emphasize: This was horrible. Sure, the
NRP with 
rotating blocks was pretty bad due to lack of mobility. But the rectangular 
NRP with rotating blocks is a dozen times more insurmountable. There are only two legal moves (two less than the minimum you could get for square boards), and every move screws up like 
of the board. I had good reason for avoiding this line of inquiry for my research --- it was frankly impossible. And now I had no choice but to do the impossible... in two months.Playing around with the puzzle was not proving to be helpful, so I resorted to a desperate play: Programming. I wrote up a Python script that can simulate sequences of NRP moves for the
board with rotating blocks, and implemented a breadth-first search that would seek moves which induce permutations that become 3-cycles when repeated a sufficient number of times by examining their cycle decompositions. Soon, I found three separate algorithms that did the job, each for different values of 
, but whose union would cover all 
! I didn't bother proving their correctness --- I was certain that some lousy induction would do it. The first hurdle was over.Next, I needed to prove that the ability to execute this specific 3-cycle could induce any other 3-cycle that respects PRs. If I could do that, then the NRP is solved in the general case. I quickly came up with an algorithm that can send any number to any location, called "Spiral". With some fidgeting, I could modify Spiral so that it can send any two numbers to any two locations. But this isn't enough --- I needed to send any three numbers to any three locations, and there just didn't seem to be enough mobility to grant me the ability to this.
That's when a disgusting idea hit me: That summer, while playing around with various algorithms, I found an 8-move sequence, called "Cycle", that only moves numbers along an outer "band". Could this give me the mobility I needed? With some thought, I realized that I could use Cycle and alternate it with the modified Spiral algorithm whenever I needed to free up one of the rotating blocks for use. The resulting convoluted six-stage concoction was dubbed the "Spiral-Cycle Algorithm".
It remained for me to prove the algorithm's correctness --- particularly, Spiral. Spiral operated by "spiraling" numbers closer and closer to the center, so I knew that I needed some monovariant. I tried a taxicab monovariant, but I was puzzled to find that it didn't work. What could I do? Out of desperation, I tried the straightline distance, which felt incredibly unnatural since the game takes place on a grid. I was shocked to find that in considering the straightline distance, I stumbled upon some basic Olympiad geometry that could help study the change in this distance. By constructing a magic point, I could induce similar triangles and reduce the analysis of the distance change to a computation that uses the Law of Cosines (!!!). By some miracle, it all managed to work out, and with that, the general case for the NRP was felled.
I was feeling pretty ecstatic! I still had a reasonable amount of time before the Regeneron STS application, and I had only one hurdle left: Special cases on smaller boards that couldn't be covered by the general algorithm. Many of them were incredibly simple, but one in particular --- the
board with 
rotating blocks --- was devilish. There seemed to be some magic lurking in the background of this case: When I solved the left column, I always seemed to end up solving the rest immediately without having to move the left column again. And, with some combinatorics, I discovered and verified that if I solve the numbers in one parity, the other parity would always automatically be solved. What was going on?I was completely stumped. In disbelief, I wrote up a Mathematica script that could verify my observations, and indeed it confirmed that the orbit of the group of achievable permutations on this board was
. I had no idea why.But then I came up with an insane idea. Even though some proofs I had looked at before took a part in suggesting this approach to me, it was a long shot and nothing short of divine inspiration. See here for the details. With that, my NRP project was done.
Was it useful? No. But was it fun? Was I satisfied and proud of myself? Absolutely. I submitted the paper to the STS and didn't think much of it.
I was sitting in my AP Physics C class when the research coordinator came in to announce who in our research class was among the 300 STS semifinalists. A large fraction of the advanced research program was in this physics class, so it made sense for her to announce it there. I listened with practically zero investment. After all, why would a niche and useless project make it anywhere in a national research competition?
She called out four names. I almost fell off my chair when I heard mine.
Two weeks later was the day before they would announce the finalists. I kept getting calls from a mysterious number called "Washington D.C." throughout the day. I dismissed them as spam, though a small thought crossed my mind: They call the finalists the day before the official announcements, right? But my project obviously wasn't finalist worthy, so it was definitely spam.
I'm sure you know where this is going. That night, our household was getting calls from "Washington D.C." too. That's when I finally considered the possibility. "Isn't today the day when they call the finalists?", I suggested to my mom. Right then, the phone rang. My mom picked up the phone, handed it to me, and my life was thrown into a beautiful chaos.
The Regeneron STS Finalists Week was incredible. I made new friends, got to meet Chuck Schumer and Francis Collins, I took a picture with Adam Conover... I was roommates with Vincent flippin' Huang! We got to explore D.C., do escape rooms, eat fantastic food, and in the end there was a huge gala where they announced the winners. It's a week that I will treasure forever.
Alright, before you start thinking that I'm some kind of legend that excels at everything, I'll finish with a funny story. As part of the Finalists Week, they had scientists and/or mathematicians interview us to assess our knowledge and competence across science and math. The assessment would consist of questions such as "Look at this graph of a faucet's water's temperature over time, can you explain the fluctuations you see?" and various others that I won't spoil to try and maintain the interview's integrity. For the most part I did alright, with the glaring exception of one question: "About how many atoms are in a grain of sand?" I gave them an estimate of
, which was received with... judgmental stares. I genuinely thought it sounded right! You, the reader, likely know better.V: Category Theory
Yoneda Lemma wrote:
Let 
be a locally small category and let 
be a functor. Let 
be an object and consider the Yoneda embedding 
. Then
be a locally small category and let 
be a functor. Let 
be an object and consider the Yoneda embedding 
. Then
I think most people would agree that category theory is not a very useful subject. However, much of it revolves a fascinating principle that everyone ought to take to heart: To study an object, or to study the object's relation to everything else --- these are one and the same.
This is a tenet that can be found everywhere, hidden in plain sight. A real-life object is fully determined by all its properties: How it looks from each angle, how it feels, how it reacts to chemicals, etc. Numbers such as
and 
are unexciting until we relate them to other numbers, using a connection called functions.Even if the two halves of the object/relations duality are not quite equivalent, it cannot be denied that examining something's relationship with everything else is crucial to understand it. Vector spaces are uninteresting, but we can learn so much from them by studying linear transformations. Instead of a studying a studying a function
, we can probe it by considering 
for every 
. That is, we see how 
gets along with all other functions 
, and this will give a great deal of information about .In a modern, Western-dominated age where individualism reigns, it is more important than ever to recognize and appreciate our connections with others. The equivalence of our identity and self may be intuitive, but there is so much more to one's identity than oneself. The relationships you have with others make up a great part of who you are. In some cultures, one's identity is even considered to be primarily such relationships!
Treasure each and every one of your connections, whether they are a friend or more. They are a part of you, whether you know it or not.
I met E in kindergarten. We grew up together, survived the worst of times, and experienced the best of times over the next
years. Our friendship wasn't always idyllic, and at times it took some turbulent turns, usually due to my own mistakes. But we got through it, and she is indisputably my best friend today.Ever since the end of middle school, I kept my friend group small --- just E, S, and me. I met S through E, and although we were quite an unlikely duo, we eventually warmed up to each other and became great friends. The three of us had a fantastic time in high school together, texting each other every day and hanging out whenever we could. Though, thanks to my young insecurities, I was often plagued with intrusive thoughts:
How do I know that they actually care about me?
What if they're just bothering to spending time with me so I don't feel bad?
If there's one essential piece of wisdom I've learned over the years, it is that an affinity and talent for clever things is of little use if it is not applied to the intricacies we experience in real life. Navigating a world of people may feel like a skill that is woefully divorced from analytical minds. But I've found that, more often than we may expect, mathematically-inspired approaches can solve much of the issues we encounter in our social lives and more. I first realized this when I discovered the beautiful solution to my insecurity.
Suppose that my friends did not care about me. Then, since they talk to me and spend time with me, they are only doing so so that I don't feel bad. This implies that they care about my feelings, and so therefore they care about me, contradiction.
Corollary. E and S are true friends.
For a good number of years, I bought into something that I've dubbed the friendship nuking paradox --- a line of reasoning which essentially proves that there is no reasonable approach for getting a relationship besides praying. Here is the paradox:
- Suppose we ask out
. There are two cases.
- is not a friend.
- Then there is no good reason to ask out , contradicting the premise.
- Then there is no good reason to ask out
- is a friend.
- Then would likely be off-put, and the risk of losing as a friend or marring the friendship is too great. Hence asking out is a poor idea in this case as well.
- Then
- By exhaustion of cases, and by arbitrariness of , we conclude that relationships cannot be formed.
.She is the most brilliant person I know, second maybe only to W. But that's like comparing apples and oranges. Whereas W's brilliance stems from mathematics and the analytical world, her brilliance involves a skillfulness in many things, combined with an insatiable curiosity and a life philosophy that would shatter my own. I find that much of the thought processes I have at the present are well-inspired by her thinking. In a way, she has become a major part of who I am today.
The thought of asking crossed my mind. Could I justify the risk? She was a good friend and I knew her well enough to know that she handles social issues quite well, especially given her views on relationships. She'd often say, for instance, that much of the world's problems would be solved if people would simply be direct about what they want to do with one another. What this entailed was that
for a small 
. I was about certain that she would say no, which put the expected value of asking at![$$\mathbb{E}[\text{I ask}] = A\mathbb{P}(\text{She says yes} | \text{I ask}) - B\mathbb{P}(\text{Friendship ruined} | \text{I ask}) \approx \frac{A-9B\varepsilon}{10},$$](http://latex.artofproblemsolving.com/1/4/9/149aa4c64ec43518feaeced03eb4f0721607004f.png)
where 
is large and 
is a somewhat large but smaller value. I was confident that 
--- after all, I knew her well. Thanks to the smallness of 
, I deduced that the expectation ![$\mathbb{E}[\text{I ask}]$](http://latex.artofproblemsolving.com/4/c/0/4c0035baadc62fdbad0077a378f989162271fa45.png)
was positive --- the first time it had been positive for anyone.I asked on Valentine's day. She said no, as I expected. Naturally, I was worried about nuking the friendship and tried to backpedal. But she's a difficult person to fool, and reassured me that I did the right thing: "You should always ask."
I learned a lot from those words, and with time, I eventually came up with a resolution to the friendship-nuking paradox, inspired by her.
- Suppose we ask out . There are two cases.
- is not a friend.
- Then there is no good reason to ask out , contradicting the premise.
- Then there is no good reason to ask out
- is a friend.
- It would be natural for to be off-put for a bit. But if the friendship actually ends up significantly marred, then perhaps they were not worth asking in the first place, nor were they necessarily an ideal friend either.
- It would be natural for
VI: Mathematician's Path
“Aren’t these all just puzzles?”, I muttered to myself, bewildered, amidst one quiet but fateful night. I continued scrolling down the webpage of Olympiad math problems, my eyes darting across the screen, anticipating the worst. Were these really the “proof problems” that I had been dreading? Were they just like the puzzles I loved to think about when I was younger? Was I on the right website?
Yes, yes, and yes. I leaned back and breathed a sigh of relief. The USA Mathematical Olympiad was not the unfamiliar territory that I had imagined it to be. In fact, a childhood of solving Rubik’s cubes, sudokus, and enigmatic riddles had been preparing me for the USAMO all this time. This realization manifested into my strongest belief: Mathematical problems are puzzles, just as puzzles can be mathematical problems. Ever since that moment, I knew that I was destined to become a mathematician.
Fast forwarding to college, I was struck with a major dilemma: What kind of mathematician do I want to be?
I had to decide between getting a PhD or getting a "real" job in industry / Wall street. There were countless factors that plagued me: Do I want another 4+ years of school? Do I like research? Would I be wasting my potential if I don't go to grad school? Shouldn't I get as much money as I can to support myself and family? Friends and professors alike were urging me to go to grad school as well, but it was an uncertain path.
In my junior year, I discovered my answer in an unexpected place: A Technological ethics class.
That ethics class was the only class I've ever gotten a B in, ruining my perfect record. Nevertheless, it was one of the most important classes I've ever taken, despite the high workload and harsh grading. The class concerned itself with the various ethical issues that can arise with technology, from technological risks to privacy, and the policies that could be enacted to curtail these issues. Throughout the course, my eyes were opened to two truths:
To maximize my potential to use mathematics to help people, I should study mathematics to the fullest.
Yes, yes, and yes. I leaned back and breathed a sigh of relief. The USA Mathematical Olympiad was not the unfamiliar territory that I had imagined it to be. In fact, a childhood of solving Rubik’s cubes, sudokus, and enigmatic riddles had been preparing me for the USAMO all this time. This realization manifested into my strongest belief: Mathematical problems are puzzles, just as puzzles can be mathematical problems. Ever since that moment, I knew that I was destined to become a mathematician.
Fast forwarding to college, I was struck with a major dilemma: What kind of mathematician do I want to be?
I had to decide between getting a PhD or getting a "real" job in industry / Wall street. There were countless factors that plagued me: Do I want another 4+ years of school? Do I like research? Would I be wasting my potential if I don't go to grad school? Shouldn't I get as much money as I can to support myself and family? Friends and professors alike were urging me to go to grad school as well, but it was an uncertain path.
In my junior year, I discovered my answer in an unexpected place: A Technological ethics class.
That ethics class was the only class I've ever gotten a B in, ruining my perfect record. Nevertheless, it was one of the most important classes I've ever taken, despite the high workload and harsh grading. The class concerned itself with the various ethical issues that can arise with technology, from technological risks to privacy, and the policies that could be enacted to curtail these issues. Throughout the course, my eyes were opened to two truths:
- I don't need to pursue an incredibly high income and be rich. I just need enough to be happy.
- Various mathematically-oriented jobs may be morally questionable (see Weapons of Math Destruction by Cathy O'Neil).
To maximize my potential to use mathematics to help people, I should study mathematics to the fullest.
VII: UNITY
21-269 Vector Analysis is the most brutal freshman-year course you can take at CMU. Back then, considering its difficulty, my impression was that only the most competent students in analysis should be a TA for the course. So, I was quite surprised when I received the offer to TA the course after I applied. I loved teaching and I knew I had much to offer if I were to become a TA, though I was worried about my severe anxiety getting in the way of my abilities. It wouldn't be a walk in the park. But I decided to shoot for it.
What can I say? It was one of the best decisions I've ever made. I once again got to apply my silly antics in my handouts and recitations, while also giving intuition here and there that I thought would be helpful. The students liked me a lot, and I got to be their TA for another two semesters, for the next two analysis courses in the sequence.
When I first attended a CMU Math Club meeting back in my freshman year, I told myself that one day, it would be me up there giving a math club talk, mostly out of arrogance. Three years later, that would become a reality! But I would never have anticipated the circumstances leading up to my math club talk.
In my junior year, the current Math Club president was S. S and I knew each other for a bit at that point, since we've collaborated on various homeworks in the past. When the time came to pass the torch to a new Math Club executive board, she highly encouraged me to apply for Vice President External (VPE).
The VPE's jobs entailed sending out the weekly email, inviting speakers to give a math talk every Wednesday, ordering pizza for said talks, and introducing the speakers. None of that seemed up my alley whatsoever: Sending an email to 1000+ people was nerve-wracking, emailing professors to give talks seemed daunting, and I sensed that I would be awkward when introducing speakers. Since I wanted to bring the fractured CMU math community together, I kinda wanted to run for the other VP position instead (the Vice President Internal), but N wanted that and it would be unwise to run against her.
These various factors made the decision difficult. However, S told me that the roles didn't need to be so clear cut, meaning that as VPE I would have the power to work towards my goal of unity. Moreover, the then-VPE told me that he would give ownership of the Math Club Discord server to whoever the next VPE was going to me. The Discord server? The one that's dead and practically a ghost town? To many, it didn't seem appealing. But I saw opportunity.
I applied for VPE, running against two other candidates. I ran my campaign around reinvigorating the dead Discord server to help facilitate a stronger sense of community within CMU's math majors, among other things such as my sense of humor and my "extremely low IQ". Thanks to my silly antics as a TA, I got my recitation to vote for me, and I also actively campaigned in my various math courses.
When the voting period concluded, my slice of the voting pie chart looked like Pacman.
On 2022, April 4th, I became the owner of the CMU Math Club Discord server. Y'know, the dead one. I knew it would be an uphill battle, but I had no intention of letting it stay dead.
First, I knew that to CPR the server back to life, I needed to give chest compressions by encouraging daily activity, even if the reason for the activity is contrived. My idea: A Problem of the Day. It was the perfect solution: Every (other) day, I could get users to look at the POTD channel and post solutions and/or thoughts. And, as a former Olympian, I had an unlimited supply of problems.
On Monday, April 25th, I started the CMUMC POTD with a beautiful, simple problem.
The POTD was my great outlet for sharing the best problems I had ever seen. I didn't want to just spam contest problems. Rather, I tried to share only the problems I thought were beautiful, brilliant, or exuded elegance.
I took it also as an opportunity to share problems that I thought were too obscure, and deserved more attention.
Today, I am no longer running the POTD, having continued it for over a year, ending with
problems. It is one of my proudest initiatives to date, having played a pivotal role in revitalizing the server. And indeed, as of this writing, the Math Club Discord server is far from dead, even in the midst of summer.
As VPE, one of my jobs was to fill every Wednesday with a speaker. In the weeks before my senior fall semester, I was scrambling to find speakers for the first meeting.
I could fill the several Wednesdays after the first one, but none of the professors I contacted seemed to be able to speak on the first Wednesday. It was just too short of a notice. I realized that I had no choice: The first speaker had to be me.
In a blatant abuse of power, I asked myself if I could be the speaker for the first week, and he reluctantly said yes. And so I dusted off an old project --- the NRP. With the mathematical insights that I gained over the past three years, I was able to improve the project: By playing around a bit, I discovered that my incredibly long 3-cycle algorithm could be shortened greatly, and consequently I stumbled upon a constant time (!) 3-cycle algorithm that works on all board sizes. I quickly modified my presentation to incorporate the new findings, and I added tidbits at the end that explore a connection between the NRP with rotating blocks and group theory that I learned from emailing Ravi Fernando years ago. I also had programmed a neat Python script that could play the NRP and execute the solving algorithm I outlined in my paper, complete with pretty rotation animations. I rehearsed a bit, streamed a trial run to the puzzle illuminati, and... I was ready!
On 2022, September 7th, I gave my math club talk to an audience of around 60-70 people. I ended my talk with a live demonstration of my Python solver. It went perfectly.
My only small regret is that I did not quite order enough pizza.
In January, I received a PhD offer letter from UCLA. Needless to say, I was surprised. UCLA was a Top 2 school for me, I didn't think I actually had a chance. Maybe it was a fluke.
Two weeks later, I received an offer letter from my top school: NYU. I couldn't believe my eyes.
Maybe I should start believing in myself more.
April 19th, the day of my Master's thesis defence.
I had finished my 70-slide presentation in the nick of time. I had 25 minutes to get through all of them --- an unreasonable task for most, but with my presenting style it was plausible. After a quick rehearsal, I was semi-confident that I could do it.
I headed to the room I had reserved early. I was not the first one there --- one of the students in my recitation was there early as well. The thesis defences are public, so anyone could show up and watch me get grilled by the thesis committee. I set up my laptops, opened up my presentation, and waited.
Another recitation student arrived. Then another. And another. Practically all of them showed up.
My college friend group arrived too, with A, C, J, and K all coming to watch me give a talk that they wouldn't understand. She came too.
Leoni, my advisor, arrived as part of the thesis committee. He seemed surprised at just how many people were there already. Soon, Tice arrived, followed eventually by the esteemed Fonseca.
The math club arrived, with C', S'', C'', E', S''', and at least three others. I really don't know how many came, I lost count. The room was packed with upwards of 25-30 people who came to support me in what seemed to be one of the most crowded master's thesis defences in the math department's history. I was astonished.
If I were ever asked "What is your favorite quality of yours?", my answer would have nothing to do with my mathematical knowledge or my talent and affinity for clever things. No. The aspect of myself that I most value is the connections that I can foster with others --- the shining keystone of Category Theory. What good are the fruits of my labor if they affect no one else, if there isn't someone to bear witness, if I have nobody to celebrate them with? My gift for mathematics is a blessing that I treasure and do not intend to undermine, but I would not be where I am today without the support of all the countless people in my life, from the love from my parents, to the love and wisdom from my best friends that has become a part of me, to the appreciation shown by my recitation students, to the guidance and thoughtfulness of my excellent professors and advisor, to the camaraderie of my classmates and the math club board members I closely collaborated with. The size of the audience that day is the greatest testament to how important people are in my life.
And as I marveled at the sea of assorted familiar faces from all parts of my college life, I thanked everyone for coming before I began one last presentation during my time at Carnegie Mellon and secured my master's degree in mathematics.
What can I say? It was one of the best decisions I've ever made. I once again got to apply my silly antics in my handouts and recitations, while also giving intuition here and there that I thought would be helpful. The students liked me a lot, and I got to be their TA for another two semesters, for the next two analysis courses in the sequence.
When I first attended a CMU Math Club meeting back in my freshman year, I told myself that one day, it would be me up there giving a math club talk, mostly out of arrogance. Three years later, that would become a reality! But I would never have anticipated the circumstances leading up to my math club talk.
In my junior year, the current Math Club president was S. S and I knew each other for a bit at that point, since we've collaborated on various homeworks in the past. When the time came to pass the torch to a new Math Club executive board, she highly encouraged me to apply for Vice President External (VPE).
The VPE's jobs entailed sending out the weekly email, inviting speakers to give a math talk every Wednesday, ordering pizza for said talks, and introducing the speakers. None of that seemed up my alley whatsoever: Sending an email to 1000+ people was nerve-wracking, emailing professors to give talks seemed daunting, and I sensed that I would be awkward when introducing speakers. Since I wanted to bring the fractured CMU math community together, I kinda wanted to run for the other VP position instead (the Vice President Internal), but N wanted that and it would be unwise to run against her.
These various factors made the decision difficult. However, S told me that the roles didn't need to be so clear cut, meaning that as VPE I would have the power to work towards my goal of unity. Moreover, the then-VPE told me that he would give ownership of the Math Club Discord server to whoever the next VPE was going to me. The Discord server? The one that's dead and practically a ghost town? To many, it didn't seem appealing. But I saw opportunity.
I applied for VPE, running against two other candidates. I ran my campaign around reinvigorating the dead Discord server to help facilitate a stronger sense of community within CMU's math majors, among other things such as my sense of humor and my "extremely low IQ". Thanks to my silly antics as a TA, I got my recitation to vote for me, and I also actively campaigned in my various math courses.
When the voting period concluded, my slice of the voting pie chart looked like Pacman.
On 2022, April 4th, I became the owner of the CMU Math Club Discord server. Y'know, the dead one. I knew it would be an uphill battle, but I had no intention of letting it stay dead.
First, I knew that to CPR the server back to life, I needed to give chest compressions by encouraging daily activity, even if the reason for the activity is contrived. My idea: A Problem of the Day. It was the perfect solution: Every (other) day, I could get users to look at the POTD channel and post solutions and/or thoughts. And, as a former Olympian, I had an unlimited supply of problems.
On Monday, April 25th, I started the CMUMC POTD with a beautiful, simple problem.
Problem 1 wrote:
Good Morning. I have a cup of tea and a cup of milk, in equal quantities.
I take a spoonful of the tea and stir it into the milk. I then take a spoonful of the milk/tea mixture and stir it into the tea.
Which cup is more contaminated?
I take a spoonful of the tea and stir it into the milk. I then take a spoonful of the milk/tea mixture and stir it into the tea.
Which cup is more contaminated?
The POTD was my great outlet for sharing the best problems I had ever seen. I didn't want to just spam contest problems. Rather, I tried to share only the problems I thought were beautiful, brilliant, or exuded elegance.
Problem 72 wrote:
Do there exist uncountably many pairwise-disjoint subsets of a plane, each homeomorphic to the letter Y?
I took it also as an opportunity to share problems that I thought were too obscure, and deserved more attention.
Problem 55 wrote:
I have a rectangular piece of paper. It fits on my circular plate without hanging off the edge. Prove that if I fold the paper along some straight line, then it will still fit on the plate.
Today, I am no longer running the POTD, having continued it for over a year, ending with
problems. It is one of my proudest initiatives to date, having played a pivotal role in revitalizing the server. And indeed, as of this writing, the Math Club Discord server is far from dead, even in the midst of summer.As VPE, one of my jobs was to fill every Wednesday with a speaker. In the weeks before my senior fall semester, I was scrambling to find speakers for the first meeting.
I could fill the several Wednesdays after the first one, but none of the professors I contacted seemed to be able to speak on the first Wednesday. It was just too short of a notice. I realized that I had no choice: The first speaker had to be me.
In a blatant abuse of power, I asked myself if I could be the speaker for the first week, and he reluctantly said yes. And so I dusted off an old project --- the NRP. With the mathematical insights that I gained over the past three years, I was able to improve the project: By playing around a bit, I discovered that my incredibly long 3-cycle algorithm could be shortened greatly, and consequently I stumbled upon a constant time (!) 3-cycle algorithm that works on all board sizes. I quickly modified my presentation to incorporate the new findings, and I added tidbits at the end that explore a connection between the
NRP with rotating blocks and group theory that I learned from emailing Ravi Fernando years ago. I also had programmed a neat Python script that could play the NRP and execute the solving algorithm I outlined in my paper, complete with pretty rotation animations. I rehearsed a bit, streamed a trial run to the puzzle illuminati, and... I was ready!On 2022, September 7th, I gave my math club talk to an audience of around 60-70 people. I ended my talk with a live demonstration of my Python solver. It went perfectly.
My only small regret is that I did not quite order enough pizza.
In January, I received a PhD offer letter from UCLA. Needless to say, I was surprised. UCLA was a Top 2 school for me, I didn't think I actually had a chance. Maybe it was a fluke.
Two weeks later, I received an offer letter from my top school: NYU. I couldn't believe my eyes.
Maybe I should start believing in myself more.
April 19th, the day of my Master's thesis defence.
I had finished my 70-slide presentation in the nick of time. I had 25 minutes to get through all of them --- an unreasonable task for most, but with my presenting style it was plausible. After a quick rehearsal, I was semi-confident that I could do it.
I headed to the room I had reserved early. I was not the first one there --- one of the students in my recitation was there early as well. The thesis defences are public, so anyone could show up and watch me get grilled by the thesis committee. I set up my laptops, opened up my presentation, and waited.
Another recitation student arrived. Then another. And another. Practically all of them showed up.
My college friend group arrived too, with A, C, J, and K all coming to watch me give a talk that they wouldn't understand. She came too.
Leoni, my advisor, arrived as part of the thesis committee. He seemed surprised at just how many people were there already. Soon, Tice arrived, followed eventually by the esteemed Fonseca.
The math club arrived, with C', S'', C'', E', S''', and at least three others. I really don't know how many came, I lost count. The room was packed with upwards of 25-30 people who came to support me in what seemed to be one of the most crowded master's thesis defences in the math department's history. I was astonished.
If I were ever asked "What is your favorite quality of yours?", my answer would have nothing to do with my mathematical knowledge or my talent and affinity for clever things. No. The aspect of myself that I most value is the connections that I can foster with others --- the shining keystone of Category Theory. What good are the fruits of my labor if they affect no one else, if there isn't someone to bear witness, if I have nobody to celebrate them with? My gift for mathematics is a blessing that I treasure and do not intend to undermine, but I would not be where I am today without the support of all the countless people in my life, from the love from my parents, to the love and wisdom from my best friends that has become a part of me, to the appreciation shown by my recitation students, to the guidance and thoughtfulness of my excellent professors and advisor, to the camaraderie of my classmates and the math club board members I closely collaborated with. The size of the audience that day is the greatest testament to how important people are in my life.
And as I marveled at the sea of assorted familiar faces from all parts of my college life, I thanked everyone for coming before I began one last presentation during my time at Carnegie Mellon and secured my master's degree in mathematics.
March 2025
November 2024