A Transformative Summer at PROMYS India
In early January, my middle school mathematics teacher sent me a link to a math camp with an annoyingly long name: Program in Mathematics for Young Scientists, commonly called PROMYS. I might not have even seen it — math hasn’t been my thing for years — if she had not sent the application problem set along with the link. I opened the PDF, and the problems were really interesting.
Over a month, I solved them for fun, returning to them every now and then. As I was interning and 11th grade was ending, I couldn’t give this too much time. I submitted the application 5 minutes before the deadline — with six out of the eight problems completed. I wasn’t at all confident I would get in — I was pretty honest in my application that math was far from the passion of my life, and my performance in the problem set was less than stellar. I had a brilliant friend who had mathematics for every meal (I once bumped into him one Saturday morning at the local library — I was going to read a novel, he had come to read math books for fun). If that was the competition I was facing, I was sure I was cooked. Still, the problem set had been interesting to solve, and I always enjoy writing essays.
Near the end of March, I got in.
I was thrilled! For six weeks, I’d be exploring a field I liked a lot, and in general, I was sure it would be a deeply educative and enjoyable experience (I was not wrong in thinking this). It was an easy decision to make, and I confirmed my seat minutes after I got the email.
I didn’t have much time to think about PROMYS before the event — my internship was ending, 12th grade started, and I wrote the SAT. One Sunday evening, though, I was there — rather nervous and having no idea what to anticipate. I knew that we were going to do math — specifically, something fancy called number theory, but I didn’t know what that was or what exactly we would be doing.
I expected a lot of free time and packed accordingly. Lots of books, along with a truckload of school work that Indian schools cleverly assign to be done in the “holidays”.
Before dinner on orientation night, I bumped into a couple of people and we started talking. The first thing I noticed was that these people were insanely smart and truly loved math — they knew all sorts of stuff, both about math itself and about mathematical community things (famous people, olympiads, colleges). The second thing was they were really interesting people. My appreciation for the depth of the people at PROMYS would only increase.
The First Half
PROMYS India was directed by Professor Ila Varma, from the University of Toronto — and the assistant director was Professor Aditya Karnataki, from the Chennai Mathematical Institute. Here’s how the program works: in the morning, there’s an hour and a half of lecture by one of the two Professors. In the end, they distribute a problem set that we work on for the rest of the day, and then submit it to our “counsellor” — undergraduate math students from prestigious universities, who support and grade the work of their three students. My counsellor happened to be the only one from IISc itself, which was quite useful. He’s a deeply knowledgeable and patient man, and I learnt not only a lot of math but a lot of what it means to be a good teacher.
The first week was slightly rocky — I kept alternating between feeling very out of place and enjoying it tremendously. I made a couple of close friendships, which I felt really good about — historically, I’ve been a very introverted person, and I’ve been trying to change that for a few years.
That weekend, I genuinely saw the point of the program. I spent hours trying to prove something arcane: the fact that every two integers can be expressed in the form a = qx + r, known as the division algorithm. This led me down a rabbit hole — every proof I came up with depended on something even more fundamental that my counsellor said I could prove.
It all finally boiled down to proving that 1 is the least natural number — that there is no integer between 0 and 1. After hours with my counsellor, I came up with a beautiful proof. It wasn’t much, the proof — quite basic, in fact, especially in light of what I’d go on to discover and study in the coming weeks — but that feeling was so overwhelmingly good that all the frustration for days was worth it.
In the second week, we started something called exploration labs. Groups of three or four students do something akin to research in a specific subject area. My team and I investigated why there is a “period” in shuffles — when you shuffle cards repeatedly, why does the same ordering return? How can we predict this ordering? What does it have to do with the type of shuffling? These are fascinating questions, and the expo labs, as we colloquially called them, remain one of the highlights of the PROMYS experience.
We spent the whole week on one trajectory only to discover that we’d misunderstood the question — a pretty embarrassing and frustrating situation. I was reminded of teachers back at school emphasizing “read the question paper!” If only we had.
The problem sets seemed harder, but the concepts we were exploring, while intimately related to integers as they were in the first week, were more foreign and interesting. The idea of this half of the program was to reteach — more precisely, rediscover — integers (and natural numbers) with their properties. We finally boiled down everything we know about integers to eight core statements, all seemingly unimpressive or unrelated. From them, however, we derived absolutely everything else. You wouldn’t believe the things we had to prove -2 is a prime, -1 is not a natural number, ab = a implies that b is 1.
Professor Karnataki took lectures this week. His style of teaching was very different from Professor Varma’s, and yet very enjoyable — he incorporated a lot more humor. One of the great mysteries at PROMYS was how the Professors finished whatever they wanted to. The lectures seemed extremely relaxed and impromptu — some student would ask a random question and we’d go down a new route for half an hour. And yet the classes always ended on time.
I was slowly adjusting to the fact that I was academically at the bottom of this group. It was still frustrating at times, however, to be systematically and repeatedly thrashed by people who would solve a question that I spent hours on in two minutes. I couldn’t ask for a less competition-oriented program than this one — the faculty and counselors actively worked against comparison, and encouraged and celebrated individual growth. Yet, here I was, pretty sorely disappointed by how weakly I was performing, when absolutely nobody but me cared. I wondered how it would be if I were in a program where competition between participants was acknowledged and mattered a lot, like a traditional college program.
During this period, Malcolm Gladwell’s talk on why you shouldn’t attend Harvard kept coming to my mind. One of Gladwell’s central arguments is that our self-esteem isn’t based on rationality — but based on quick, almost unintentional, comparisons between ourselves and the people we interact with in our daily lives. Harvard’s weak students don’t think that they’re at the bottom of one of the most prestigious student bodies on the planet — they just think they’re at the bottom. When I listened to his talk, I was like: nah bro, I’m smarter than that. Surprise, surprise — it turns out I’m not.
The US Dream Lives
In general, PROMYS offered a few strong points on why I shouldn’t aim to study at top universities in the US — but it offered even more reasons why I should. Perhaps most importantly, the people — I’ll say it multiple times, but the people at PROMYS are what made it special. The conversations we held, the experiences we shared, the paths we travelled — it was all brilliant. These people reflected what I’d long stated was a core reason behind my aiming for universities like Yale: they were all genuine learners. People who learn for the sake of learning. Not for money, not for prestige — but genuine, thirsty, learners.
Something more surprising is that success, or depth, in one area, seems to translate to depth in multiple areas. Perhaps this is because success is an attitude to life, or perhaps it’s something else. Whatever the reason is, it’s an extremely interesting phenomenon. Chris Guillebeau did mention it in his manifesto A Brief Guide to World Domination, but I didn’t believe it — until I saw it. Another funny thing was mathematical thinking seems to translate to having the same interests. So many people there liked politics and CS — both topics that close to nobody likes in a random distribution of high schoolers. For the first time in my life, I was with a group of people for a non-coding reason, all of whom code.
The level of accomplishment in that community was awe-inducing. The Professor had studied at Caltech, my RA at MIT. Early on in the program, I was speaking to one of the counsellors and she casually told me that when she attended PROMYS, she solved every single problem in every single problem set, along with taking the advanced seminar. I took her to be a role model — if she could do it, why not I? I learnt later that she represented India at EGMO (the second most prestigious international math olympiad, arguably equivalent to the first — International Math Olympiad — but for girls alone) and won a silver medal. My fellow students were no less impressive for their age — they spoke calmly about going to their state’s training camps, or even going until the last round before the national training camp (known officially as “getting a merit certificate” for IMOTC). Board exams’ (10th and 12th exams in India are conducted by a state or national organization) results came during the camp — with average scores at 95+, people barely cared. I’ve met people who complain about high marks — oh, it’s not good enough — I’ve met people who don’t do well — but I rarely meet a person who does exceedingly well and then barely cares. Like I said — true learners, not fake nerds running after money.
At this place, insanely good performances are celebrated, but not worshipped — they’re acknowledged as a result of tremendous hard work… and then people move on. This attitude was refreshing — hard work and outstanding results are the norm here, not the exception.
Professors genuinely care — both about their subject area and about your learning. When they speak about the subject, you can visibly hear the love and excitement in their voice. They’re willing to discuss their field outside of their class — all day, you wanted to.
I’m really happy about experiencing all this, though — regardless of what happens in college decisions later this year, I think I genuinely care much less. I see that there is a rather dark side to going to top universities, and I also see that my reasons for wanting to attend them are watertight. Whatever happens, it will not be as good or bad as it originally seemed. (While it’s easy for well-meaning elders to pass off general advice, the specific and real-life learning of this is pretty valuable.)
Moving On
The third week contained a midterm exam. I liked how caring they were — they gave us sample tests, and review lessons. I was quite scared, but the paper turned out to be easier than expected. The format of the test was interesting: you write as much as you can in two hours, and then spend the whole weekend rewriting the same paper completely. It tests two different skills in a very nice way. We spent most of the weekend working on the exam. Unlike normal problem sets, we weren’t allowed to collaborate on this one — which made it harder but also had a different feel of reward, as everything I did was mine alone. Still, I couldn’t solve many questions — highlighting the fact that collaboration has its virtues.
Expo labs were progressing quickly, and we’d come across many interesting discoveries. One of the biggest mathematical takeaways from the program is how deeply networked mathematics is. I’ll never be able to see mathematical branches as separate again — everything relates to everything else in the most unexpected ways possible. In this case, what we were researching — how to predict the period of a shuffle — turned out to be directly predictable by a concept we’d studied in the previous weeks about the set of all numbers coprime to a given integer.
Week 4 was hard. By now, it felt as if PROMYS was dragging me along. Week 5 opened with the election, and everybody was hooked. It was rather enjoyable. If in the 4th week, PROMYS was dragging me along and succeeding, even if I was quite bruised — in the 5th, I felt PROMYS was failing to even do that. We were now exploring number systems apart from integers — some normal ones like Gaussian integers (which are complex numbers with integral coefficients), and some very weird ones like Z[√-5].
Regardless, the second half was beautiful in a way the previous one wasn’t. What made the first half attractive was how we solidified our foundations and understood the basics really well. In the second half, we solved — for me, at least — terribly challenging problems — that alone was worth it in the few times I succeeded. Also, we were working in unusual number systems — and seeing the similarities — and more astonishingly, the differences — was pretty darn cool.
The last week came much faster than anybody anticipated — the second half just ran through like a blur. We had our final exam — again, I was terrified; again, it went better than I anticipated. The last two days after the final passed in a blur. We did a lot of non-math activities; I tried to spend as much time as possible with the wonderful crowd that I’d soon be leaving behind.
PROMYS was probably the high point of my high school journey (pun not intended). It taught me a different level and a different kind of mathematical thinking. I don’t think I’ll ever think of math the same way again, or ever think that there’s a part of math that’s impossible for me to learn. The community was just brilliant — I can’t emphasize this enough — and I’ll foster most of the relationships I made there for as long as I can. The ability to do hard and focused work was a long overdue lesson.
I also learnt a lot of other lessons — honestly, the math wasn’t the biggest thing I gained from this, not even close. With a very limited set of clothes, in less-than-ideal housing conditions, and rather mediocre food, I was the happiest I’d been in a very long time. Hard, meaningful, work; enlightening and invigorating friendships, and an unhurried life might just be the only things one needs.
Will I pursue math further? Possibly — probably — not. Was attending PROMYS India 2024 worth it? Definitely.
