Vigyan Yuva awardee Mahesh Ramesh Kakde: Studying equations, not solving them
The number theorist has proved various conjectures relating functions and other objects attached to polynomial equations
Mahesh Ramesh Kakde, winner of this year’s Vigyan Yuva Shanti Swarup Bhatnagar award for Mathematics, is a number theorist at the Indian Institute of Science, Bengaluru, who studies polynomial equations. He explains how, rather than look for solutions to the equations, he studies the relations between different objects attached to the equations.
What I do
My area of research is algebraic number theory. The main goal of number theory is solving polynomial equation for integer solutions. The most familiar equation of this kind is the Pythagoras theorem, which can be expressed as x² + y² = z², and we know there are infinitely many integer solutions for (x, y, z), which are known as Pythagorean triplets.
Other equations of this kind, however, can be very difficult. For example, take the Fermat equation ‘xⁿ + yⁿ = zⁿ’. Today, it has been proved that no integer solutions exist for the Fermat equation if n is greater than 2. But to find out whether a solution exists or not, mathematicians could not have proceeded by just checking in the hope of hitting upon a solution. Since there are infinitely many integers, we cannot simply substitute values of x, y and z and keep checking if there is a solution. Even if we checked for 100 billion values and found no solutions, it would not mean there are no solutions.
It is known that there is no algorithm into which you could feed a polynomial equation so that it would tell you, after a finite number of steps, whether there is an integer solution or not. And that is where our motivation comes from.
How I do it
What we need is a conceptual way of studying these polynomial equations. It is not easy to describe these concepts in very simple terms, but we study these equations by attaching different mathematical “objects” to them. Objects are different kinds of mathematical structures that can be algebraic, analytical, or complex functions called L-functions.
One can attach some geometric objects and some algebraic objects to the same equation. Now, rather than look for integer solutions, we observe these objects. If there is a solution, these attached objects will satisfy some properties, so you would also expect these objects to be related to each other. And that’s the kind of thing that I work on—proving the relationships between different kinds of objects.
Some of the objects we attach to equations are L-functions, class groups, and Selmer groups. There are very general conjectures relating L-functions and class groups, or Selmer groups. I have proved some of these conjectures over the years.
A subtopic that I work in is called Iwasawa theory. It is a systematic way of proving conjectures relating to L functions and arithmetic. My first major work proved the non-commutative main conjecture of the Iwasawa theory. More recently, I proved (jointly with Samit Dasgupta from Duke University) the Brumer-Stark conjecture.