# 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 2023 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 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.*

*
*(Click

**here**and

**here**to see recent guest lectures and counsellor talks given at PROMYS Boston in the U.S.)

## 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. (Click **here** to learn the advanced seminars offered at PROMYS Boston between 2010 and 2023.)

#### 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 2023 presented their results to the PROMYS India community at the end of the summer session and wrote up their findings as a research paper.

Click **here** to learn the mentored research projects undertaken at PROMYS Boston in 2015-2023.)