Mathematics at PROMYS India
"I really loved that no matter how much effort I put in, I would never be done with my Psets! I could keep on finding more conjectures and try to find a more rigorous version of my solutions, generalize statements." Enakshee Mondal, PROMYS India 2023
In mathematics, maybe more than in any other science, research is an activity of the mind. The primary goal of the mathematician is to understand - to discover essential ingredients of complex systems in order to render them simple, to find order within apparent chaos, to draw analogies between different structures, and to find connections between seemingly disparate branches of mathematics and science. To make interesting new contributions in the field of mathematics requires a healthy mix of creativity, experience and hard work.
Number Theory
Through their intensive efforts to solve an assortment of unusually challenging problems in Number Theory, participants will practice the art of mathematical discovery. The problem sets encourage students to design their own numerical experiments and to employ their own powers of analysis to discover mathematical patterns, formulate and test conjectures, and justify their ideas by devising their own mathematical proofs. Some students choose to work alone; others prefer to work in collaboration.
The daily Number Theory lecture is designed to help the students synthesise and formalise what they have largely discovered for themselves through their work on the daily problem sets. For this reason, the mathematical content of the lectures lags at least three days behind the mathematical content of the daily problem sets.
First-Year Labs
First-year students may also choose to meet in small groups to work on projects called First-Year Exploration Labs, guided by a counsellor and faculty. At the end of the summer, the students write up their work and also make a presentation of their research to the assembled PROMYS community.
The first-year students at PROMYS India 2024 undertook six exploration labs.
Supplementary Lectures and Minicourses
The regular programme activities are supplemented by a wide range of lectures by faculty and guests of the programme. These lectures introduce participants to related scientific fields and include discussions of a broad range of mathematical and maths-related topics. Independently, the counsellors also organise minicourses on topics of their own choosing.
The following Guest Lecture were given at PROMYS India 2024:
- Professor Jaclyn Lang (Temple University): Are we there yet?
- Professor Purvi Gupta (Indian Institute of Science): Who cut my cheese?
- Professor Caleb M Shor (Western New England University): Rounding sequences and counting by congruence classes
- Professor Tulsi Srinivasan (Azim Premji University): Counting Potato Print Patterns
- Professor Raman Parimala (Emory University): Sums of Squares
- Professor Rehana Patel (Wesleyan University): What are Ramsey Numbers?
The following Guest Lectures were given at PROMYS India 2023:
- Professor Parthanil Roy (Indian Statistical Institute Bangalore): Could Euclid Trisect Every Angle?
- Professor Kiran Kedlaya (UC San Diego): Cyclotomic Polynomials and Roots of Unity
- Professor Alina Bucur (UC San Diego): Zeta Functions and Euler Products
- Professor Apoorva Khare (Indian Institute of Science): Divisibility Tests, Recurring Decimals and Artin’s Conjecture
- Professor R Sujatha (University of British Columbia): Prime Numbers
- Professor Padmavathi Srinivasan (Brown University/ICERM): Using Unique Factorization to Solve Diophantine Equations.
Mathematics for Returning Students
Students who find their PROMYS India experience especially worthwhile may be invited to return for a second summer to participate in more advanced mathematical activities.
Advanced Seminars
To ensure that returning students and counsellors find their experience intellectually stimulating, PROMYS India offers advanced seminars as well as the mandatory Number Theory lectures.
2025 Advanced Seminar to be conducted by Professor Radhika Ganapathy, Indian Institute of Science (IISc) Bangalore
2024 Advanced Seminar on Probability Theory
Conducted by: Professor Parthanil Roy, Indian Statistical Institute, Bangalore Centre
Description of the Course: This course will discuss basics of probability starting from its axiomatic definition, and venturing into various properties and notions (e.g., law of total probability, Bayes' theorem, random variables, expectation, variance, covariance, etc.). Many models of statistical physics have strong connections to probability theory. These connections will be explored through examples and exercises.
2023 Advanced Seminars
Quadratic Forms (Professor Aditya Karnataki, Chennai Mathematical Institute)
Projective Geometry (Professor Philip Engel, University of Georgia)
Returning Student Labs
In addition to the advanced seminars, returning students undertake research projects, supported by counsellors and mentored by professional mathematicians.
The returning students at PROMYS India 2024 presented their results to the PROMYS India community at the end of the summer session and wrote up their findings as a research paper.