Program in Mathematics for Young Scientists (PROMYS) 2023

My journey to PROMYS India began with a slight detour—arriving a few days late due to my AP exams. By the time I reached the Indian Institute of Science (IISc) campus late at night, the exhaustion from travelling had settled in, but so had a growing excitement. I dragged my luggage through the dimly lit, sprawling pathways of the campus, following directions to the Mathematics Department building.

The first glimpse of the building was awe-inspiring. Its imposing architecture, combined with the stillness of the night, left an impression of intellectual grandeur. The ground floor was illuminated, and through the glass-walled classrooms, I spotted a small group of students. It was hard to know then, but these would become the very classrooms where I’d spend the majority of my time, and the people I saw that night would grow to be some of my friends.

Though it was late, I was welcomed warmly by the counsellors and students who were still awake. They introduced themselves, shared snippets of their day, and helped me feel at ease despite my tardy arrival. After some introductions, a counsellor kindly walked with me to my dormitory, the Jawahar Guest House (JVH). Along the way, they gave me a brief orientation about the programme schedule and what to expect over the coming weeks.

Room 207 at JVH became my assigned space for the next six weeks. Initially, I shared the room with a roommate, but he left after a week due to illness, leaving me with the room entirely to myself. This unexpected arrangement turned out to be a blessing. Room 207 became an unofficial headquarters for my friends, serving as a place for late-night discussions, problem-solving sessions, and light-hearted moments.

PROMYS India accommodations were split between JVH for the boys and the more luxurious Hoysala House for the girls. Hoysala was conveniently closer to the Mathematics Department, while JVH was a solid 1.5 kilometres away. The daily walk from JVH to the department became a routine, often repeated multiple times a day. Though tiring, these walks became opportunities for reflection and camaraderie, especially when joined by friends or engrossed in lively discussions about P-sets.

A Typical Day at PROMYS India

Mornings at PROMYS were reserved for intellectual nourishment. Each weekday began with a lecture on number theory by Professor Ila Varma. Her teaching style was rigorous yet engaging, and the topics she covered laid the groundwork for the mathematical challenges we would tackle each day. The lectures were precise, rich with insights, and often interwoven with questions that left us pondering long after the sessions ended.

Following the lecture, we were handed the daily Problem Set (P-set), which became the focal point of the day. The problem sets were unusually challenging and more demanding than anything I had encountered before. Each question encouraged us to think creatively, challenge assumptions, and embrace the unknown. Unlike traditional classroom problems, these P-sets often asked us to "Prove, Disprove, or Salvage" mathematical statements, a format that was entirely new to me. Still, they soon became my favourite aspect of the programme.

Since I had joined the programme late, I had to solve a few P-sets before arriving on campus. These were handed to my counsellor upon arrival. The early P-sets were challenging but manageable, offering a gentle introduction to the programme’s intellectual demands. However, as the days progressed, the P-sets grew significantly harder. By the later weeks, solving even one or two questions a day felt like an accomplishment.

The programme often demanded moments of epiphany, when problems that initially felt insurmountable suddenly became clear. One such epiphany for me came while working through the proof of quadratic reciprocity. After several days of effort, the argument finally made sense, and the experience was deeply satisfying.

The Exploration Lab

In addition to the daily problem sets, the First Year Exploration Lab provided me with an opportunity to engage more deeply with mathematical research. The lab focused on sustained work on a single topic, culminating in an expository write-up and a presentation to the PROMYS India community. It was a different pace from the problem sets and required more effort spread out over 6 weeks.

At the start of the lab, each participant was given a list of ten possible topics, spanning geometry, combinatorics, number theory, and algebra. These were not meant to be surveyed briefly but explored in depth. After reviewing the options, I chose to work on Pick’s Formula and Euler’s Theorem, drawn by the way the topics linked geometric intuition with algebraic structure.

One of the defining constraints of the lab was that external resources, such as books and online material, were not allowed. Every definition, argument, and connection had to be developed independently or through discussion within our group. I worked in a group of five, and the differences in how each of us approached problems became one of the most valuable aspects of the experience. Discussions often led us to reconsider assumptions and refine ideas we had initially taken for granted.

At first, Pick’s Formula appeared relatively straightforward. As we worked further, however, we began to uncover deeper connections, particularly its relationship to Euler’s Theorem for polyhedra and its broader implications. Without external references, we had to carefully test each idea and justify every step. Much of this work happened during long evenings in the Mathematics Department or back in my room at the Jawahar Guest House.

My counsellor played an important role throughout the process, offering guidance through probing questions rather than direct answers. This approach was quite challenging but effective, closely mirroring how mathematical research often develops through gradual refinement.

Looking back, the Exploration Lab served as a condensed introduction to mathematical research. It emphasised collaboration and persistence, and it provided a valuable contrast to more structured coursework.

At the end of the programme, I received copies of all ten Exploration Lab topics, each containing a set of questions and prompts. I later typeset all ten in LaTeX, which allowed me to revisit the problems and appreciate both their depth and the care with which they were designed. All ten areas of inquiry are included below.

Memories

PROMYS India was not only defined by mathematics but also by the people I experienced it with. By the end of the first week, I had found a close group of friends, made up of both first-year participants and returning students. That group quickly became my support system throughout the programme.

Our group was known for its playful shenanigans. One of our most memorable pranks was convincing others in the programme that a friend in our group had been accepted to Harvard. He had not. On another occasion, we collectively skipped the “mandatory fun” activities organised by the counsellors and instead played cricket in the field next to the Jawahar Guest House. The decision was not particularly well received by Professor Varma, but it remains one of our favourite memories.

Weekends offered a brief pause from the programme’s intensity. Saturdays were often spent at a nearby mall, eating, talking, and doing very little of consequence. Sometimes we stopped at the Sarvam Complex for dosas. These small excursions provided a welcome break from the academic routine and helped turn a demanding programme into a shared experience.

The IISc campus itself was an inspiring environment. Its lush greenery, serene pathways, and architectural beauty provided a perfect backdrop for learning. One of my favourite places was the JRD Tata Memorial Library, which I managed to access despite not officially being allowed. Armed with a fabricated “student ID” (the first few digits of my phone number), I convinced the watchman to let me in. Once inside, I spent hours immersed in thought, poring over books, and finding solace in the quiet corners of the library.

By the final weeks, the programme reached its most demanding point. The problem sets became increasingly difficult, often requiring long hours of discussion and careful thought. At the same time, this challenge was what made PROMYS engaging. It reinforced the idea that mathematics is as much about the process of discovery as it is about arriving at a solution.

Looking back, PROMYS India stands out as a formative academic experience. It sharpened my problem-solving skills and introduced me to a community of peers who shared a serious interest in mathematics.

Beyond the coursework, the programme emphasised persistence, collaboration, and intellectual curiosity. The friendships I formed and the habits of thought I developed have stayed with me well beyond the summer.