Program in Mathematics for Young Scientists (PROMYS) 2023

Arrival and First Impressions

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 closest 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 group of 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 P-sets were not just exercises; they were a deep dive into mathematical discovery. 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 but 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’s structure demanded epiphanies—moments of sudden clarity where solutions to seemingly insurmountable problems revealed themselves. One such moment for me was the proof of quadratic reciprocity. After days of grappling with the concept, the pieces finally fell into place, and the sense of accomplishment was unparalleled.

The Exploration Lab: A Deep Dive into Pick’s Formula and Euler’s Theorem

In addition to the daily problem sets, the First Year Exploration Lab offered a unique opportunity to immerse ourselves in mathematical research. This component of the programme encouraged us to explore a single topic in greater depth, ultimately culminating in an expository write-up and a presentation to the entire PROMYS India community. It was an experience that not only expanded my mathematical understanding but also pushed me to approach problems with a spirit of curiosity and self-reliance.

At the beginning of the lab, each participant was presented with a list of ten mathematical topics, ranging from geometry and combinatorics to number theory and algebra. These were not merely areas to skim through; they were potential doorways to deeper exploration. After carefully reviewing the options, I chose to work on “Pick’s Formula and Euler’s Theorem,” drawn by the intriguing interplay between geometry and algebra that these topics promised.

The lab posed a unique challenge: using online resources or books for assistance was strictly prohibited. This meant that every concept, proof, and connection had to be discovered independently or collaboratively within our assigned groups. My group consisted of five people, each bringing a different perspective and approach to the task. The diversity in our backgrounds and problem-solving styles enriched the process, turning our collaboration into an intellectually rewarding experience.

Pick’s Formula, which relates the area of a lattice polygon to the number of its interior and boundary lattice points, initially seemed straightforward. However, as we ventured deeper, we uncovered intricate links between Pick’s Formula and Euler’s Theorem on polyhedra, as well as surprising applications in other mathematical areas. Without external resources to rely on, we were forced to question, analyse, and rigorously test every conjecture, often working late into the night in the Mathematics Department or even in my room at JVH.

Our exploration was guided by counsellors who provided insightful feedback without revealing answers, steering us toward self-discovery. This method, while challenging, was transformative—it mirrored the process of genuine mathematical research, teaching us to be patient and persistent in the face of uncertainty.

Writing the expository paper was both a rigorous and rewarding exercise. It required me to not only synthesise the ideas we had developed but also articulate them clearly and logically. I had to connect the dots between geometry and algebra, presenting our findings in a structured and accessible manner. The process of drafting, revising, and finalising the paper sharpened my skills in mathematical communication—an essential aspect of research that I had not fully appreciated before.

The culmination of the lab was a group presentation to the entire PROMYS India cohort, including the faculty and counsellors. Each member of our group took on a specific aspect of the topic, ensuring that our collective understanding was reflected in the presentation. The experience of explaining our work to a broader audience was both nerve-wracking and exhilarating. It deepened my appreciation for the power of collaborative learning and the importance of conveying mathematical ideas effectively.

Reflecting on the lab, I am grateful for the skills it helped me develop—independent thinking, collaboration, and the ability to persevere through challenges. It was a microcosm of mathematical research, condensed into a few intense weeks.

At the end of the programme, I received physical copies of all ten topics that were initially offered for the Exploration Lab. These documents, containing the questions and prompts for each topic, served as a testament to the depth and variety of the mathematical challenges we encountered. Later, I typeset all ten problems in LaTeX, a process that allowed me to revisit and reflect on the elegance of the questions and the programme’s emphasis on presenting mathematics with precision and clarity. I have given all ten of these areas of inquiry bellow, feel free to give them a look.

Friendships and Adventures

PROMYS India was not just about mathematics; it was also about the people who shared this journey with me. By the end of the first week, I had found my circle of friends—a mix of first-year participants and returning students. This group became my support system, and our shared experiences fostered a bond that transcended the programme’s academic challenges.

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). Another time, we collectively decided to skip the “mandatory fun” activities—group events organised by the counsellors—and instead played cricket in the field next to JVH. This act of rebellion did not sit well with Professor Varma, but it became one of our fondest memories.

Weekends offered a brief respite from the intensity of the programme. On Saturdays, we often visited the local mall, indulging in food and laughter. Occasionally, we stopped at the Sarvam Complex for dosas—a simple but delightful treat. These outings provided a welcome break and allowed us to bond outside the confines of the campus.

Reflections and Challenges

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’s intensity peaked. The P-sets became formidable, often requiring hours of collaboration and contemplation. Yet, these challenges were precisely what made PROMYS so rewarding. It taught me that mathematics is not just about solving problems but about embracing the process of discovery.

A Lasting Impact

Looking back on my time at PROMYS India, I am struck by how transformative the experience was. It deepened my love for mathematics, strengthened my problem-solving skills, and introduced me to a community of like-minded individuals who shared my passion.

Beyond the mathematics, PROMYS taught me resilience, collaboration, and the joy of intellectual exploration.

It was a programme that demanded hard work but offered unparalleled rewards. The friendships I formed, the lessons I learned, and the memories I made will remain with me as I continue my journey in mathematics.

PROMYS was more than a summer programme; it was a gateway to a world of possibilities, where curiosity met rigour and discovery became a way of life.