I didn't participate so much in competition math. I made it to the AIME. I did well on the AIME, but not enough to make the Olympiad. Other than that, I did the Ross program after my sophomore year, and then I took many college classes at a nearby college, and I did some research with a professor there.

Oh that's such a hard question. I think most of the people who do well in college math and research are not super competitive or deep into the math competition world. It depends on what you mean by competition math. If it's people who are doing well, like locally, that's one thing. But there's a difference between people who make MOP and the people who win MOP. So the people who win everything are just generally good at things, and they can do whatever they want, whatever they set their mind to. So I think in that sense, there's a correlation at the very top. But it only goes one way. And that's because it's only a correlation. There are so many other factors you can't really divorce from each other. Like, these people are already very good.

I enjoy seeing people understand things. When someone gets something that I explained it feels good. Also, I enjoy explaining things that I'm interested in at the moment, because then it gives me a reason to study it carefully. It helps me with my own work.

So far, I think I've been very fortunate. I have a frustrating moment that's not so much frustrating. But when you leave college, I mean, I'm starting grad school in the fall, and you see all your friends making a lot more, like, a lot mroe money, and living a very nice lifestyle. You start to wonder whether you should do that too. And a lot of people think about this a lot earlier, but I just happen to be a little late to the game. So I mean, I'm considering that there are a lot of things I didn't try when I was in college, like I didn't intern anywhere. I just did research on the summers. The question is framed negatively, but I think doubt is a good thing.

I did some in high school. I reached out to a professor at my local university. I just showed him the stuff that I did at the Ross program, so then eh knew that I was, like, legit. And then this was very, very lucky. He was just very gracious and happened to have something to think about, something very hands on. So we worked on that for a bit. I mean, it wasn't anything huge, but in college, I think this whole process is a lot more structured and in some sense, less competitive. So in high school, there are maybe a few programs around the country that formalize doig research for high school students. Like at PROMYS, there are some research labs, or there's RSI, there's MIT PRIMES, things like that. But at the undergraduate level, the government proivdes a lot of funding for REU programs. These are research experiences for undergraduates. These are scattered around the country, lots of different subjects that you can choose, and I think many people also just opt to reach out to professors at their schools and do that. There's also the world of industry research which I wish I'd explored more, but there's probably a lot more funding beind that too.

So this is my first year at PROMYS. I did the Ross program in hgih school which is very similar. As a student, I learned how to work, like a lot, and I learned why rigor is interesting or important or even possible in the first place. A lot of people don't really think about rigor before they come to places like this, very carefully. And it gives you, I guess, a slight edge in terms of just understanding formal arguments. As a counselor, you learn to read things very well. Like you learn to read your students' work carefully and efficiently, and you get a lot better at spotting when there's an error, or on the other hand, when there's something interesting going on. And you get faster. And I notice that when I try to read other things now I apply a similar critical viewpoint, and then that's really nice as a counselor.

So I'm starting graduate school in the fall, planning to study number theory. I have a lot of time to figure out what exactly I want to do, maybe stay in academia or research or something, if I'm good at it, and if I can make enough money and live in a place I like. But other than that, I would consider, probably industry positions that are higher paying and well located.

Um, so I went to a small charter school, and nobody at my school really did math, very seriously. I did some competition math. I was never really that good, nor was I really motivated to get good. Because I had to go to another place to take the competition and I didn't know anybody who did it either. I'm from Singapore, and in Singapore, I also did math. There were more people doing math there. And when I moved to America, I kept doing math, but not as much anymore. So I applied to PROMYS , and I did the PROMYS application p-set and I was like, wow, this is so much fun. This is way better than competition math So I got into PROMYS as a rising junior and so I did PROMYS every year. I love PROMYS . So that's like, the main thing I did in high school. I didn't really find the motivation to do a lot of competition math or, like, math in general during the school year, so I really just, like, worked on my classes and stuff.

Yeah, I think the correlation exists. I think if you're good at competition math, you do have the skills that help you be good at higher math, but it's very different. You can do very, very well at competitions, like IMO, gold medalists but that doesn't mean you're guaranteed to do well on like, I know an gold medalist who got a B or a C on an algebra test, right? So it's like, it definitely helps. Like these kids, they pick up things pretty fast. I definitely feel like I spend more time doing readings and stuff compared to them, so they're really quick. But I feel like math, like higher level math, is a lot of like, you just keep working at it. And you just keep thinking about it and just working with examples. And like, the more intuition you build, the more experience you build with it. It just translates to being better at math, right? That being said, it definitely helps to be good in a competition math, but it's different. You have to also put in the time.

Um, I don't know. I think mainly because when I was early on in high school, I didn't like math a lot because I was not good at competition math. So I hated that. So then I feel like worrying about competition math. The thing with comp math is you need to get the quick insight a lot of the time. And I just didnt get it, but there's a lot of cool things about math. And when I hear people talk about not liking math, it's like you just haven't seen a lot of math. It's the wrong impression. And I do want to expose people to that kind of math first before they make judgments about whether or not they want to do math. But I feel like so many people miss out, like I'm talking like generic maybe not like math math people, but like generic people. They hate math because they think it's just computation. There's so much more to it. But yeah I guess it's because I like math, and I think people have the wrong impression about it. You know, you just want to correct that.

Oh, I mean, so many, so many times. Even when you're solving problems and you don't really get it. That's always really bad. What helps me? Perhaps it's like, not really the best. It's not healthy. But I just go. I'm not a quitter, and I can't have people know that I'm a quitter. You know, it's like how you go on a run with someone else and you both run further because you guys, like mutually, don't want to be the first one to stop. That's the thing I love. There was one problem set, I remember in my college that was really bad. A lot of people dropped the class because of that p-set. But then I was like, I'm not a quitter. I cannot live this down if I don't do it. And then, you know, I think, especially when you're young, it's easy to think, I can't do this immediately, I can never do it. But you'll be so surprised by like, you just keep working at it. It falls into place. And also, it's okay to ask people for help. And you understand it later.

I guess my main experience with research was at PROMYS when I was a student, I actually didn't like doing research labs, because I wasn't very into the topic, like, I think there's very limited math that you can do when you are when you don't know that much math. A lot of math research requires having the foundation. So I didn't like it. I didn't like the math I was working with, because a lot of it is, like combo problems here, like bounding, oh my god. Bounding is so un-fun. So yeah, I don't like it that much. But also, I'm excited to do research, like this coming year, when I have more background and I can do more work into the stuff that I'm interested in. It's really a lot of you get a problem, and you just sit and play with it for a while, and you ask questions about it Like, how you approach a PROMYS problem, but a very long term skill. And, yeah, it's also something I'm more interested in. My research lab was actually quite interesting, and I might think about it more sometime in the future. But, I don't know, after a while, if the problem isn't something that interests you on a deep level, you really do get bored of it because you think about it for such a long time, and this is also my issue, because I get bored of problems pretty quickly. Yeah, I like certain kinds of math, and it wasn't really that.

Oh, um, just knowing what pure math really looks like. Because everybody really knows what competition math looks like, and pure math is nowhere like that. And I think math is really cool. I mean, PROMYS has changed me a lot in the sense that I used to not like math, and then now I really do like math, but, yeah, you find a new way of doing math, and also how to approach problem solving. I think it helps with math in general, and this is like jump started at PROMYS. But as I worked through the math, I learned that a lot of problems, like if you just put time and effort into it, and your progress is definitely not linear, like it's not clear that you're getting closer to your to solving the problem, because you're still stuck on the problem, and then suddenly you're unstuck, because a lot of the growth is hidden. And so, after doing PROMYS, like having this experience, I learned so much math every year during PROMYS, and like the maturity that you get from it. I definitely believe way more in hidden progress. And like, any work you do is good work, even if it's not getting anywhere.

Yeah, I'm not really sure what I'm gonna do. Career wise, money is great and in academia it's hard to get good sources of money, good or even consistent sources of money. Um I mean, I think I'm treating math more as like right now I'm majoring in it, and it's like, if I'm gonna work in something, then math is gonna be my hobby more than something that's I'm directly gonna be used in my job. But like, as I said earlier, I learned a lot of life lessons from doing math, and so I think that's very important. But I'm also maybe going to academia, because I really enjoy teaching. So if I go into academia, I can very directly apply the math that I know into that, but I'm definitely not going to need that if I'm going to go into coding or CS or something like that.

Yeah, I think math in high school, okay, in high school, it's very formulaic, and you can lay out all the steps you do, or there's a way to make it very formulaic. And I think that was what I found grade school math to be. I think it's fair that people don't like math because math is difficult but we shouldn't not like something because it's difficult. That's not the mentality to lead life with. I think in school, it focuses a lot on the computational aspects, but not very much on the prettiness. It really doesn't give you time to explore. And I think math is a lot about getting results. And this is what my algebra professor said about research. It's like, you don't really pick a problem and then solve the problem. More often than not. When you're doing research, you pick an object of study, like I'm gonna study this specific group, I'm gonna study this specific thing in math, and then you study, you understand it a lot, and when you do that, results pop out. With math in school, the emphasis is on getting the problem solved. Math is, you get an object and you study it, and then results pop out from studying. It's about the journey not the destination. That's what math is in the end. And yeah, and it's dumb that they don't really talk about the journey. They just care about the destination. And to be fair, it's all the kids fault anyways, because they're like, oh, teacher, when am I ever gonna need this in my life? And then the teacher gives them a problem where they need to apply in their real life, and then they cry about it.

In middle school I started doing a lot of competition math. In high school I continued doing so and I was very engaged, and that was probably the thing that took up most of my time outside of school. I stopped being so engaged after, say, junior year of high school, but I guess after my 10th grade, I went to PROMYS, which is where I am now, and I went back to PROMYS for the summer after end of the summer after my senior year. So I went back three years, and I think that was a very good exposure to what math is like in college and beyond.I think the main place where I had social capital in, you know, my old middle and high school was from me being smart. I was the person that people could ask for homework help. I could be the person that, like, they asked me to know, like test answers or whatever, that I could be, the person that they could rely on to, you know, do well in math competitions or whatnot. And I think having the opportunity to engage with people who had similar interests to me, and who I could talk about math with, and who I could just interact with without being so self conscious about, like, what I provide to others and how others perceive me was very, was just, was very refreshing. And although a lot of these people weren't from my area, I talked a lot with them, like, online, and I'd meet up with them, like, whenever we go to, like state competitions, and when we travel to like national competitions. A lot of people from those I'm very close with now, for example, one of my closest friends from North Carolina starting in middle school, we actually had dinner yesterday because he's around because he goes to MIT, and so having that community was very solid for my upbringing.

I think there is very little correlation. In fact, there might be negative correlation. The reason being is that I think if you do a lot of competition math, you gain the impression that you're very good at math. And so I guess I'm speaking for myself. But once I got into college, I realized that there are a lot of people who had much great, much greater exposure to college math than I had, and I thought I had a good amount, because I went to PROMYS for three years, and I felt like a lot of people who went to PROMYS before me, you know, excel very well in terms of math. But I went there and I was, I suddenly felt very behind. And a lot of these people had many, many years of just like, I don't know, like studying group theory or studying, linear algebra before me. Like even some high schools offer these classes at their high school, it's just like, if you've already taken these classes before, how am I supposed to compete with that? And I think competition math teaches you a lot of math, but it is not the same kind of skill set as pure math in a lot of ways. I think there's some intersection perhaps but I think generally, the most important thing of pure math is is language which is something that, like Glenn says you want, you want to express a certain idea, and so you try to, like formulate the proper definitions and the proper language in order to express that idea. And then you develop the idea very abstractly from the ground up. Whereas in competition math, you have these tools in place already. So you're not generating anything new, but you're applying like that, that the novelty in competition math is you're taking these ideas and trying to apply it. You're trying to combine them in a more creative way. And so there's a small subtlety there, but it's very profoundly different, like they're very fundamentally different.

And I think, for example, like the way of thinking for competition, math, maybe more in line with, say, I don't know, like computer science or something, but you have these algorithms in place, and you're aware of these algorithms. And the question is not how can I construct a new meaning, like, research, of course, exists in these areas, but especially within the college classroom setting, it's not like construct these new things, but rather, okay, I've learned this algorithm now, how can I apply it to these problems in very unique ways? So yeah, I think actually, a lot of my competition friends end up not staying in pure math. And the people who do stay in pure math are the ones who, the vast majority of them are the ones who had a lot of more pure math experience when they were in high school, which makes sense, like if you studied a lot of a certain subject in high school, then you're bound to study it more in college. It's just that I think competition math should not be equated to pure math. And so, if you see them as two different subjects it makes complete sense. And a lot of people like, they think that it's like, the same thing. Or me, I thought it was the same thing, and then I, you know, I went in with this idea that I could, just, you know, do well, and then it didn't turn out that way.

Math is designed so precisely that in a lot of instances I think it's just difficult to not teach it in a compelling way. Like so many things in math, are interesting. So as long as like, you are able to convey this excitement to your students, then it can be extremely, extremely eye opening. I'm sure this is true for other subjects, but I think the ideas that you find in math and the way the subject has been developed for such a long time, there's so much richness to it. And, you know, teaching someone a really slick proof, or a really full result which comes out of nowhere. It's just so, so exciting.

I guess the most memorable time was after freshman year. So at the end of my freshman year, I got kicked out of college because of COVID, or like we all got kicked out of not just I got singularly kicked out of college. We all were forced to move off campus because of COVID. And during COVID, I decided to take a break from particularly math, or it wasn't even particularly math. I just wanted to explore everything I was interested in to see whether or not I really wanted to do math in the first place, because I felt like in my freshman year I'd taken a lot of these, required classes that are needed to take other more advanced classes. Is in, in the STEM area. So I took, , you know, the intro statistics series, the intro computer science sequence. Took , you know, the intro, , the math class that freshman people take. And I don't know, I didn't feel very particularly attached to any of them, whereas, a lot of my friends were like, Oh, this class is, extremely interesting when I want to, continue pursuing this or Oh, I've already found a lab to do research, alongside the topic of this class and I didn't feel the same conviction towards anything. So I was like, Okay, we're in this pandemic right now. I don't really want to take online classes. A lot of classes I want to take, I don't think would translate well into an online setting. for example, I was interested in exploring interested in, exploring music, music. Like, online classes for music is just I think it's terrible, that's stupid. And so I decided to take a gap year and a legal absence and just explore everything that I wanted to do. And this was very refreshing. at the end of the day, I I kept close track of my time with the philosophy that Okay, I would not give myself any super hard of responsibilities, but I would just explore. I would just do what I want to do, and I keep track of my time. So I had this whole spreadsheet, and I would keep track of my time what I did, every 30 minutes of the day, and then this is just, very, just very clear and very raw data of what I want to do. And my belief was that once I looked at the spreadsheet and I saw what I invested my time in the most, that's the thing that I should be doing, because that's the thing I'm inclined to do, and very shockingly, or maybe not shockingly at that time, math was definitely not up in that list, and it was a lot of non stem stuff, and that got me very confused for a very long time. So then when I came back sophomore year, I decided to take a lot of these classes, and these humanities fields, and some of the classes kind of let me down, but a lot of them were very, very good and, you know, very thought provoking. And I was engaging in these classes a lot more than I was doing the math classes I was taking, just with all my requirements, and so I was very seriously thinking about away putting math altogether, once I got into my sophomore year.

But I think the big turning point for me was actually coming back to PROMYS as a counselor, because after talking with some people here, I realized that maybe my frustration with math was not because I was disengaged with it, but because I didn't see the bigger, the greater vision of why to study math. I guess when you're taking math there so many prerequisites that you need to fulfill in order to get to classes that are interesting. All the undergrad classes you take are these classes so you can understand these graduate level classes, and do these graduate level classes so you're able to understand I don't know, these advanced grad topics courses. And then from there you can choose your research topic and go forward. This is not the same for biology. For biology, you can take an intro class and find a question that interests you, and you can join a research lab, which many people do, and they just work on lab all of their undergrad and you know what the problems are, and you know what the motivation is. For math, this is very obscure. You spend a lot of time in order to even understand what the big research questions are in your field. So I just didn't see that big picture. But I think the timing was just really right, because I learned enough math to kind of understand the bigger picture at a very high level. So it was very intriguing to me. And through talking with other counselors, going to counsel seminars and stuff, I realized I was able to see, oh, this is why these things are so important. This is why this subject is one you teach such early on in your curriculum. Galois theory was so important. It seemed so stupid to me. I thought it had no application, it's just so random. But in fact, it's one of the most fundamental things that comes up in many areas of math. And just seeing this kind of motivation for all these things was greater motivation. And now I have ,now I know, where I'm going, moving forward. Now I have something I want I want to reach, and some topic I want to study.I think that really shifted my perspective on math. I still have qualms now of whether or not I should stay in STEM or pure math. You know, the job prospects are pretty low. I've done this actual math research once, and it was pretty difficult for me. And so I'm still not convinced that math academia is the right space for me. But at the time being, it's the thing that I enjoy very much, and so I'm sticking with it.

Don't be discouraged. Don't be discouraged if you don't understand something, because it's learning, and it's very, very rare to understand things for the first time. And so I think people come in with the expectation that you know you should understand that the first time around. And it's very discouraging. And so if you just take the if you really are interested in math, then you should figure it out. If someone is interested in studying math and the motivation is already there, which is a great thing. I would also encourage people to you know, do competition math, as long as you have a healthy relationship towards it which I guess I talked about in our panel, but yeah, I think you should explore things. what helped me in college was I started to just study things when I thought it was important to me and there was a very strong reason for me to study it. And this just comes from studying things just in depth. Once you learn enough things, natural questions will arise, and this thing keeps coming up over and over again, this issue over and over again, and then you try to address it. So you study things when it comes out of necessity. Um, I think that's the most impactful way and the most efficient way to learn. Previously, I would try to learn things where I don't know, this is a good example. Oh, I'm taking this class next semester. I don't really know much about it, but a lot of my friends are taking it, and I want to prepare beforehand. So I'm going to read the first few pages of the textbook. I can get a head start. That's pretty good motivation. That's a pretty good reason to study, but I don't think it's a strong reason, and that can lead to discouragement in a lot of cases. So I think you should think about why you want to study math, and have a strong conviction for it before you study math. I think this is true for everything. Just be very mindful of why you want to study and why you want to study it. If your motivation is strong enough, then whatever you do to study, whether you just say, search up on the internet to find some resources, ask some people around you, I think anything will be pretty fulfilling.

So I guess I did math research in PROMYS for two years, and that was back then, I didn't know anything, so it was a bit difficult to make any progress. But I think thinking about questions that you know, no one has really, you know, no one really knows the answers to, was an extremely cool and extremely novel thing. At the same time, I knew that the advisors, right, the people who proposed the projects, already had an outline in mind, and so we were just trying to chase that idea. And so it felt both very novel, but at the same time very just we're basically just doing someone else's work that they already kind of know what to do, but they're just letting students do it, which I think is a great, great thing, but it wasn't all you know, all shining stars or whatever. I did math research at a REU, a research experience for undergrads program last year. I worked with one other person who actually went to PROMYS in 2019 and 2020. Yeah, I was working with him. And then our mentors, they proposed this project, and they build it as you know, being accessible, and undergrads would be able to understand it. And I had taken several grad courses at a time to say, okay, this should be manageable, but then, yeah, it was just hard swinging out the gate, you know, very, very deep into the details, I guess, yeah, and I personally found it pretty hard to follow along, let alone contribute any ideas on my own. But it gave me a taste of what actual math research would look like. I think maybe the project wasn't the best fit for me . I wasn't prepared for it enough. But, yeah, I'm still trying to think about my experience more. And I guess I'm still working on the project, so I'm still very frustrated by it, but yeah, I think that the TLDR is I was not prepared for the project, and I'm still trying to prepare for it right now, trying to gain, not only the necessary background, but the necessary intuition to move forward. But I think if you're working on a project, and you have this kind of intuition, and this kind of you see a problem, and you don't really know what to do, but you have this gut feeling of what things should look and how things should arise, then it becomes very interesting. And this was, there's one point in the project where I was doing this smaller problem, and I did get this feeling, and it was very, very satisfying. Either you keep doing it and your gut feeling is right and things turned out the way I expected, in this really nice way, or things don't turn out the way that that you expect, which is very frustrating. At the same time, then there's something interesting going on here.

But at least my impression in math, at least, the things I study in classes, slash this project, is that a lot of math research can be very distilled. You made do this paper that contains say five pages of just say one proof for one theorem. Or you have all these definitions, and then you set up this lemma, and then you prove this theorem. But, you're able, at the end of the day, you're able to distill all these ideas. It's in this five page proof you can identify, usually, at max, three, usually two things which would make this proof work. And the rest is just filling in details and understanding. What is it necessary for this proof to work? Then things are very illuminated, I guess. In contrast, for instance, I did after my freshman year, I did this, quantitative social science research. It's basically statistics for differential privacy. It's you have this database, and then you want to be able to access all the all the relevant statistical values, the mean, the standard deviation, whatever. But you don't want each row to be traced back. You don't want anyone's identity to be revealed. And so the idea is that you add a bunch of random noise, and it's well, how much bias is there on? The problem can be easily stated. You know you can understand the project, but I felt that during the whole time I was just doing a bunch of these computations that didn't really have any motivation, and you're just making these estimates. And that was not super satisfying to me, because there was, no bigger picture. Just Oh, you're just fidgeting with numbers to make things work. To me, that isn't really math. That's just computation. I think those are two different things.

I guess the mantra a promise is to think deeply as simple things. And I forget to do it a lot, but I think it's a very important thing to do. Thank you. I You, especially in math, but also, in other fields. Yeah, I think it's a very, I've been very influential in the ways that I go about I don't know reflection of myself, thinking about, what I want what I want to do, and what's best for me. That's, if you, if you think hard enough about, certain ideas, then, you'll be able to distill it into, I don't know, to, a spareessentials, let's say say you come up with a problem, Oh, I'm feeling down right now, right, but I don't really know why I'm feeling kind of down. Then you think about okay, what are the possible things that was making me feel sad and then this leads on. And then the more you think about this, I think the true underpinnings, , will come up. And so, yeah, there's a whole framework about just really the largest amount of information and a thing that really affects everything comes from the simplest ideas is a super important, super important framework. I will say that I spent too long trying to I don't know, in a way, axiomatizing my life. I was such a believer of the PROMYS experience I thought that I could create an inventory of how I should act kind of a inventory for my morality, and then that could dictate how I do everything. I was very insistent on making this possible. But I think such a thing is very, very unreasonable, but if you're not so hard, driven. On making everything, super precise. Then if you think about things deeply enough, then you can get your understanding about yourself and about your surroundings.

I'll just keep studying it until I don't enjoy it anymore, in which case, I'll stop doing math. I think that's my general philosophy with what I do is just, I'll keep doing it until I don't enjoy it, and then I do something else. Yeah, maybe this is , very not, um, not responsible me, but I don't know. I feel it's difficult to plan things so far ahead if I don't really know , because stuff changes so quickly. Next month will be very different from me now, right? So it's, I think, planning to precisely what I'm going to do is, kind of, kind of difficult, but for the time being what I'll do next year is a master's program, and then I'm going to do a PhD program in math. I'm going to Sorbonne in Paris, and then I'm applying to PhD programs next year. I'm applying to only PhD programs in the US. So if I do go to PhD, I'll go back to the US. But, yeah, we'll see where life takes me.

Okay, I didn't do any competition math. I tried it in middle school. I didn't really like it. It was in sixth grade. In high school, my situation was pretty unique. In my freshman year, I did AP Calc because I had already been taking high school courses during middle school, and then in the middle of my sophomore year, I was taking BC Calc. But then a lot of stuff happened, and I ended up changing schools to this thing called the Met school in Providence. It's, I don't know, very non traditional, project based learning all of this stuff. And there were a lot of reasons for that, but one of them was just that I was out of math programs at my old public school. Another was that I was kind of not doing well in non-math courses. I don't even know how to describe it. It's a long time ago. Whatever. I don't need to get into it. It's not about math. But at any rate, I finished taking BC calc did well on both of the exams, and then I my dad was teaching at Brown at the time during my junior year, so then I was auditing some classes at Brown University, and I also did that my senior year, and that was my first real exposure to pure proof based math and I really liked it. And also because, the Met school was this project based thing, I was also supposed to do an internship, and I ended up doing a computer science internship. I'm no longer in computer science, but I did that during my senior year, and I was also taking more math classes. And after I finished, because my high school experience was kind of rocky, I decided that before I went to college, I wanted to take a gap year, and I just continued my internship in Boston until then. Then the pandemic happened. Then I went to southern Connecticut because it was close to our house. And then I went to Boston University

Um, I think something that I really like about math and specifically about talking about math is there are a lot of different ways to express the same concept a lot of the time, and teaching it in and of itself, and thinking of how it should be taught, can help reframe your own conception of what kind of thing you're trying to teach is, and teaching someone and answering their questions and thinking about how best to represent something is also just a fun exercise for me.

I did research at an REU at Yale, and our group was looking at hyperbolic topology. Specifically we were supposed to find the minimal spanning surface of the ideal octahedron, which sounds intimidating, but it's essentially you have the shape in hyperbolic space that looks kind of an octahedron, except it's kind of curved and its point go out to infinity, because that's something that happens. And then you want to cut it in two, and you want to find the smallest area of the hexagon that cuts it too. Anyway, the problem with it was, I think it was just kind of a poorly picked problem, because if we didn't have a background in hyperbolic topology or hyperbolic geometry, sorry, we kind of learned it over the course of the thing. We learned some stuff, but not enough to tackle this unsolved problem. And what we ended up doing was, instead of tackling it from a more conceptual, proof based standpoint, we kind of just brute forced what this hexagon should look using simulations. We just did a lot of formulas, did a bunch of calculus, and made a regression that should simulate what it is. It was still cool to work on, but I feel it wasn't a full research experience. I also went to Isaac's talk about sphere packing, and I feel that's something that's probably a lot easier to approach than something like hyperbolic geometry. But I guess I just hope I get another chance in the future, but it was still fun.

Well, I was there in 2023 and I felt last year it was still really rewarding, because I just enjoyed teaching. if the REU was my first exposure to research, this was my first real exposure to teaching. And I definitely enjoyed that. It was a lot of work. I also did not know almost any number theory. I had taken a group theory course that covered some of it, but there was still a lot of new ground. So I feel last time I was kind of more reclusive. I didn't really go to talks. I just did grading because I was struggling to keep up with the number theory myself. And I still think that was a really valuable experience for me, because I just hadn't done something like that before. And this year, it's still hard for me, I think to attend all these talks, because number of the counselor seminars is legitimately absurd. There were five today. Then each of them is an hour, a lot of the time they go over. Yeah, I can't do it. I haven't done it anyway. Yeah, I think this year I've definitely been pushing myself to try to experience more new math. And I've also realized that specifically, this format of counselor seminars just hasn't really worked for me. And there have been a couple other things that the counselors did that has worked. And we were actually talking about this today, and we're going to try some other stuff. But I think more effective ways of learning this stuff for me would be more informal discussions and things that, and watching a video together and trying to understand it, or reading something together and trying to understand it. But anyway, I'm kind of rambling. All in all, it's been a very valuable experience.

So I may have mentioned this before, maybe I didn't, but I was actually a double major in math and Japanese and I've always been interested in math from when I was a really little kid, and when I started BU I was probably going to do math and CS, and then I did Japanese language credits. I really liked it. I started doing it more. One thing led to another, and here I am. So now I'm in this situation where a lot of my hobbies and things I do in my free time are related to Japanese things like the visual novels I read, the games I play, the fighting game players I talk to even. But the math is something that I don't really do in my free time as much and something that I probably should be doing more. But my reasoning for going to grad school for math is because I could envision myself doing either math or Japanese translation as a career is. I think if I did Japanese as a career, then I probably just wouldn't do math as a hobby. And it feels lke I'm keeping more of a nice balance for myself, if I have math as a career and then Japanese as a hobby, but it's still going to be hard to balance. And anyway, in terms of the math question itself, I'm going to this master's program, and then I'm going to go to a PhD. most likely academia, because so far, my favorite experience with math has been teaching, but it's all still very up in the air, even though I'm a graduate student now, so don't be scared if you can't make a decision.

Yeah, I honestly, because of , how revolutionary learning group theory was in my what was it would have been junior year. That was just mind blowing to me. I personally think the lack of proof based math in grade school is just kind of tragic. You have geometry even and then everyone hates geometry. And you don't get to see a lot of the really cool proofs. It's just putting things in an order and knowing the definitions. Even around my dad, I felt I wasn't really exposed to something that required rigor until I started taking university classes. But just the existence of rigor and proofs and honestly, just an appreciation of math for math sake. Some proofs are just really cool, and I think people should see them, and that'd be awesome.

I did not participate in competition math, but I had, let's say, a very privileged math upbringing. My parents realized that I was really a fan of the stuff since I was a very little kid, and my dad bought me tons and tons of math books ever since I was like seven, and I'm immensely grateful for that. But what that meant was that by the time I entered high school, I actually knew a lot more math than most of my peers, so the school agreed to do a special thing where I would have one on one classes with the physics teacher, Mr. Klotz, who would just the two of us do a math class, and he would teach me things like multivariable calculus and linear algebra and differential equations and sort of, I think the last year or the second to last year of high school, he said to me, Hey, I remember you saying that you didn't feel as strong in geometry. So I got this book. I don't really know what it's about, but it's this book called Riemannian geometry by Brazilian mathematician Manfredo Perdigão do Carmo, and I'm probably mispronouncing that, which is not really the sort of geometry that perhaps either of us were expecting. Riemannian geometry is all about the study of curved surfaces in many dimensional space. And I think I spent a lot more time with the book than he did, and it eventually was just me explaining what was happening to him. But yeah, I mean, that's not a very typical thing. I did not really experience the high school math curriculum that almost all high schoolers in the country experience. I have no idea what the difference between algebra one and algebra two is, just because I had this weird arrangement where I skipped all of that, which is not to say that that's a prerequisite for you know, becoming a mathematician is starting really early. I had a friend named Davidowski, who was not into math in high school, a classmate of mine my same year, who was not the same level of math kid that I was at the time, but he fell in love with math in college and is now doing math or, I forget if it's Masters or PhD in UW Madison, and knows a lot more of algebraic geometry and related fields than I do. He was here not too long ago for the Mordell conference, which was at MIT. And, you know, I only was able to make it for one of the talks, just because of the way PROMYS and timing worked out. But you know, I have very little confidence that I would be able to understand most of the talks, and it was his field, so he felt, you know, very at home in it, even though, you know, he did not start before high school the same way I did.

I mean, they are very different skills, right? I want to, I don't want to say that one is useless, you know, I think a lot of people say they're afraid that training for competition math only lets you be good at competition math, which, there is some truth to that. It is very different from research. I think it's a very fun thing, and, , you know, a very mentally demanding thing, the same way that learning to solve a Rubik's Cube very fast, or memorize a deck of cards in a shuffle deck of cards in under a minute is very impressive. But I don't think that it has the same relation to college math and research math that it might seem. I feel there's competition questions, textbook questions and research questions, and they all have very different flavor, right? A textbook question, you are expected to be able to solve it with the techniques that happened in the chapter. All right? A competition question, you don't know what the tools you're supposed to bring to the table are, but you know that there is a solution that someone else has figured out, that this is a good level of difficulty for people who know about you know use these tools, and there exists a solution, and we have a few in this document over here. And then research math is you have no idea if the question you're asking is solvable. You have no idea if you're even asking the right questions. And it's a lot more exploratory, right? Because research happens on the order of months, if you're lucky, very often, right? And you know, competition happens in the order of half an hour. I don't know if there's a particular time scale for textbook questions, I will say competition questions very often there it's good to grab one from an old contest and stick with one problem for half an hour, or do a session with a community on one problem for half an hour, and just absorb everything you can get from it, because a lot of them are deeper than you are able to appreciate in the time constraints that these problems have. For what it’s worth I was, you know, and as I said, I never did competitions, but I did have a hand in setting up a competition once a high school competition. So Yale has hosts something called MMATHS, a high school math competition. Also Girls in Math is another thing that happens at Yale. And when I say Yale hosted, I mean it's entirely student-run. The students come up with the questions, run out of time to play test the questions, because they're students and have other work to do. And then the questions end up being accidentally too hard, which is definitely what happened when I was involved I came on. This is 2022. I think I came on to that very late, and they said, hey we do not have enough questions for this competition that's happening in just a few weeks. We need to just brainstorm and come up with a bunch of random stuff as quickly as possible. Five of the things I came up with ended up on there. I have no idea if they were the right level of difficulty, because we couldn't get people to test them before it happened. Actually, that's a lie. I gave one to someone I knew on Discord who happened to be an IMO medalist, He solved it in the span of 15 minutes, which is pretty impressive. But that test, the time controls were such that you only really had five minutes for question, which probably told us that wasn't the right move. It ended up being the max score was 8 out of 12 on a bunch of rounds. Po-Shen Loh's daughter won it.

I mean, I get excited by this stuff. I like to get other people excited about the same same stuff stuff I'm excited by. There's also a selfish level to it, which is that I understand something a lot better if I explain it to someone else. Right? The minicourses that I gave, right? So the counselors can give minicourses to the students, generally around an hour. The only reason I was able to give those mini courses was because I had tried to explain those same topics one on one to people in the past, and I saw where they got stuck or where I realized that I didn't fully understand it as much as I thought I did, and that informed how I was able to talk about it, and that sort of informed how my lesson plan was for proving that Pi is irrational, or explaining what homology is. I have this online community. I'm a part of a Discord server focused on math, and any time I explain to someone on there what an injunction is in category theory or what have you, I find I understand the topic by the end of it way better than I did at the start.

I'm kind of in the middle of one now, and have been for a while. Well, I mean, here's the thing, when I was younger, because, as I mentioned, I started studying math pretty young, because my parents sort of put a lot of actual money into buying a lot of books for me. And, you know, my prize, the one possession I have, material possession, I have on this earth that I value above everything else is my bookshelf, partly for sentimental value. Anyway, I remember at one point , when I was really young, deciding that I had hit a brick wall with differential equations, that I had learned calculus before most poeple learned calculus, and then I got to differential qeuations. I could get it, and that was fine. That was where my wall was. And then eventually I learned differential equations, and I think that was just this mental thing for me, of okay, maybe this concept of a wall doesn't actually exist. So one thing I've been trying to learn for a while is algebraic geometry and commutative algebra, and for whatever reason, every time I've picked up a book to try to learn about it, I've gotten stuck. I've got a tiny bit into the book and said, okay, I can't understand what's happening here anymore. And I was reading all these algebraic geometry books, and was like ,okay, I think the issue is that I need to brush up on a prereq. Maybe I need to learn some commutative algebra first. So I got a commutative algebra book, a big fat one written by someone named Eisenbud. I was like okay, this is too much. I'll give this back to the person I borrowed it from. I'm going to get a smalled one called Atiyah and MacDonald's. That's sort of where I am. And right now it's more that particular book is more exercises than exposition, basically a small amount of text and then a lot of exercises. And I was thinking I would be able to get to them, some of them during PROMYS. And it turns out I just didn't have any time. But I've just been frustrated with this one subject of commutative algebra for a long time. And I'm just persevering because I know that I've hit a wall in the past and been able to get through it.

I was involved in an REU in college. So REU stands for research experience for undergrads. This is during COVID. It was entirely virtual. The other two people on my team understood the topic way better than I did. I did not really have a great time, and I don't think I got much out of it, which is a shame. I mean, I've talked to other poeple since then, and they said that REUs are overrated, which may be true. I can't really say from where I'm sitting. But that's the only formal reserach experience I've had. But I don't know. I'm constantly thinking about random stuff. , right now I'm thinking about these random probability puzzles. I'm going to explain this one niche thing. Choose four random points between zero and one, sort them from least to greatest, label them A, B, C, D, from least to greatest. And then you can ask things, like what's the probability that A squared plus D squared is bigger than B squared plus C squared? So sum of the outer squares is bigger than the sum of the inner squares. Or you can ask, what's the probability that one over A plus one over D is bigger than one over B plus one over C, same thing but with reciprocals. Well it works out that with the first one with squares, the probability is exactly the natural logarithm of 2. And with the second one, the probability is exactly pi squared minus 9. And there's a whole family of generalizations of these problems which are leading me to some wacky integrals that I don't really know how to solve. But hopefully the internet does so I've been asking some questions on math StackExchange. I was just synthesizing other stuff I'd been thinking about in the past. There's a certain geometric shape you can get by intersecting three hyperboloids that ends up, surprisingly, to have a volume of exactly log two, for some reason. And then I was thinking about some other probability stuff, and then I realized, hey, wait a minute, that first thing, the volume thing, can be recast into the language of probability, and then it becomes this weird puzzle kind of, yeah, I don't know. There's another topology thing that I was thinking about a while ago which, I was trying to ask people online, hey, does anyone know the solution to this one very specific topology question? And then nobody ever saw those posts I made, and then I solved it myself, because no one else was and that came from thinking about a random thing in a book I read once, or I read another thing in a book more recently, which was, this number theory thing. And I realized, hey, if you change that two to a three, what happens, and then that led me to discovering this number sequence that seemingly did not exist online, but had a bunch of really weird properties. So I ended up submitting that to the OAS, which is this online database, or it's called the online encyclopedia of integer sequences. Anyway, just stuff I'm reading about and realizing hey, that makes me think about something else I saw once. I don't know, but that's not formal research, in the sense that, , I never turned any of them into a paper. I never really wrote any of this stuff down, except bits and pieces of it on Stack Exchange and on Discord servers or a Facebook group called actually good mathematics problems.

I really teaching, and so this is my second year as a counselor. I've never been here in any other capacity than as a counselor. So I've never been here as a student, but I was here last year. Last year, I taught three mini courses. Well, two minicourses and one counselor seminar. This year, I taught too many courses, and I just had a lot of fun doing that. It was just explaining this one topic of my choice to a classroom full of people is something I don't very often get the chance to do so I'm grateful for PROMYS, for the opportunity to do that, also just the opportunity to hang out with these incredibly, mathematically talented kids who are as excited about this subject as I Am and willing to learn and all that. How exactly did you phrase the question, what unique insights or understanding have you gained? I mean, I know PROMYS is a lot of work on the student side, you know, as someone who's never been a student, it seems a lot of work to do all of these problems every night, even though you obviously don't have to complete the whole thing every night, and sort of doing the exploration on your own first before the lecture gets to it. I think that's a very great way to do pedagogy, to do teaching that doesn't really have the opportunity to exist in a lot of other places because, it's such a full focused thing that you can't really have that sort of experience taking a lecture course in university. For example, just because, if you're taking a lecture course at university, you're also taking four or five others, right, or three or four others, and you're splitting your time. And this is condensing a whole, semester long or a year long course into the span of a month and a half, and asking students to focus on this one thing extremely deeply, just the fact that that can exist and it can happen well, is a very nice thing about PROMYS.

I plan to remain in academia. I teaching. I want to stay, you know, in a situation where I can do teaching. So, my career goal is to become a professor at a university or potentially teaching at high school, I personally would prefer the university route, but, you know, depends on how things shake out. You know, I don't know that I have full control over that. And yeah. So I guess if the question is, how does, how do you plan to use math as a math teacher that doesn't, yeah, yeah. But, you know, hopefully I could create something that's actually useful to someone who, you know, who wants math more than just for the sheer pleasure of learning it, you know, hopefully I can help with something that has, you know, applications to some algorithm or something I don't know, I don't know that I really have full control over that either, but I don't know. It's just math has been my passion and my hobby for my entire life, and I want to turn that into my career.