# Mathematics at PROMYS India

"PROMYS has been the most mathematically intense experience of my life,"

Shubham Aggarwal, Mehta Fellow to PROMYS 2015 and 2016

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.

## 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. (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.

### Advanced Seminars in 2020

##### Projective Geometry

**Professor Philip Engel, University of Georgia**

Haven’t you always been annoyed that parallel lines don’t intersect? Well they do, in the projective plane! We will discuss this and many other reasons why the projective plane is so great. For instance, the number of solutions to two polynomial equations *f(x,y)=g(x,y)=0 is* exactly degree(*f*) ∗ degree(*g*) - if you count them correctly! We’ll also talk about symmetries of the projective plane, and how these symmetries can be used to prove theorems from antiquity. For example, maybe you’ve heard that *x²+y²=1* is a circle. Well, in the complex projective plane it's actually a sphere! What about *x³+y³=1 *? Come to find out.

##### Quadratic Forms and Beyond

**Dr. Aditya Karnataki, Beijing International Center for Mathematical Research (BICMR)**

A quadratic form is simply a polynomial where every monomial term has degree 2. In one variable, the only quadratic forms are *f(x) = cx²* where *c* is a constant. In two variables, we have *f(x,y) = ax² + bxy + cy²*. Nevertheless, the theory of quadratic forms is lucrative enough to have been extensively studied by all past, and indeed present, masters of number theory. Why does such a simple premise attract so much attention? Surprisingly, it has opened up many doors in and guided the evolution of number theory. We will try opening some of those for ourselves and have a look inside.

## Returning Student Labs

In addition to the advanced seminars, returning students undertake research projects, supported by counsellors and mentored by professional mathematicians. The students present their results to the PROMYS community at the end of the summer session and write up their findings as research papers. (Click **here** to learn the mentored research projects undertaken at PROMYS Boston in 2002-2019.)