Winner of the the Infosys Prize 2022 in Mathematics for his contributions to algebraic number theory, Mahesh Kakde, professor of mathematics at the Indian Institute of Science, Bengaluru, has spent years working on modern number theory, especially in the field of conjecture, a mathematical statement, usually involving a pattern in numbers, that has not yet been proven through rigorous testing. Prof Kakde has worked on the noncommutative Iwasawa main conjecture, the Gross-Stark conjecture (with Samit Dasgupta and Kevin Ventullo), and the Brumer-Stark conjecture (with Samit Dasgupta). His research has helped resolve these outstanding conjectures, which are at the heart of modern number theory. “In mathematics, the role of these highest peaks is played by the great conjectures — sharply formulated statements that are most likely true but for which no conclusive proof has yet been found. These conjectures have deep roots and wide ramifications. The search for their solution guides a large part of mathematics. Eternal fame awaits those who conquer them first,” writes theoretical physicist Robbert Dijkgraaf in an article in Quanta magazine.
In this interview with Lounge, part of a series of interviews with leading Indian scientists working at the cutting edge of Indian research, Prof Kakde talks about mathematics and abstraction, why conjectures are like ‘guesses without proof’, and how modern technology interacts with the work of mathematicians.
I work in an area of mathematics called ’number theory’. Number theory is study of integers. We all learn about counting and integers pretty much as soon as we start talking and certainly before going to school and then we learn how to add them and how to multiply them. Perhaps that is why we have a (false) sense that integers are easy. But they are not easy when we ask questions combining addition and multiplication. For example — a counting number p bigger than 1 is called a prime number if it is divisible by only 1 and itself. So the first few primes are 2,3,5,7,11,13,17,19, 23…. But 15, for example, is not a prime because 3 divides it. Thus, prime numbers are defined using multiplication on integers. If you ask a question about adding two primes, then one of the simplest statement one can think of is open - is every even number larger than 2 a sum of two prime number? For example - 4 = 2+2, 6 = 3+3, 8 = 3+5, 10 = 3+7, 12 = 5+7 etc. This is known as Goldbach conjecture and has been open for over 250 years. Another example is the twin prime conjecture that says - are there infinitely many prime numbers p such that p+2 is also a prime? For example, the first few twin prime pairs are - (3,5), (5,7), (11,13), (17,19), (29,31), (41,43) …. We don’t know if this list stops somewhere or goes on forever.
Similarly, if we have equation and we want integer solutions for them, then it is in general a very hard question. For example, can you add two perfect nth powers to get another perfect nth power - x^n+y^n = z^n? The answer is yes if n=1 (easy!) and n=2 (not so easy but known for about 2000 years). However, when n>3, the only way this equation can be satisfied is if one of x, y or z is 0. This is known as Fermat’s last theorem and proved by Andrew Wiles in 1995. It used incredible amount of hard and very technical 20th century mathematics.
Modern number theory does not necessarily look at one equation at a time. We can attach various other mathematical objects to an equation; these objects are geometric or algebraic or analytic. We then study these objects and try to get some information about the original question or relate the original question to properties of these objects.
My own interest in number theory goes back to high school. I happen to come across some “elementary number theory” books. The problems were easy to understand. Solutions were sometimes easy and sometimes very tricky. But they always seem to involve finding a nice pattern or property of solutions. Thus, even before starting my undergraduate studies I had decided I would study number theory. Of course, back then I had no idea how sophisticated the subject has become. But as I realised this it in fact became more and more appealing because it leads to understanding deeper patterns, some of which can probably not even be stated without the vast theory and the vast language that number theorists have built over the last couple of centuries.
Conjecture is basically a guess that is made without proof. For example, above we saw a ‘guess’ that every even number bigger than 2 can be written as a sum of two prime numbers. We do not know if this is true, but we guess that this may hold.
However, let me hasten to add that conjectures in mathematics are not made arbitrarily. Conjectures that stand test of time are often made after brilliant people spend a lot of time understanding structure or pattern of the mathematical objects they are studying. Conjectures in mathematics are like lighthouses… they give us a direction that one should take and every now and then we can also reach one.
It is almost in the name as it is often called “abstract algebra”. I wouldn’t call it elusive. One aim of algebra is abstractification (if will allow me to use that word) of certain properties that we have found useful elsewhere. For example, if we simply take a set with two operations called “addition” and “multiplication” with properties similar to what we are used to seeing in integers, then we get a mathematical object called “ring”.
As a number theorist, I view algebra as a tool and as such if you do not know what you are building/buying a tool for, then it will seem useless or elusive. Having said that it is also a language and initially one does need some patience and perseverance to learn it before being able to use it well.
Modern technology (even going back 60 years) has certainly helped in some ways. One example is the use of computers by Birch and Swinnerton-Dyer to guess what is now known as the Birch and Swinnerton-Dyer conjecture, one of the Clay math millennium problems. There are other such examples. Modern technology through fast travel, emails, internet has helped mathematicians collaborate easily and with many more people. I don’t think mathematics has become easier, or harder than it was 200 or 400 years ago. It may be different. Most people are more specialised since techniques used are more specialised. It may have gotten easier in one sense — it is now possible for a large number of people to be pure mathematicians and make a living out of it. Pure mathematics has been valued more in the last 80 years than it perhaps ever was.
I study mathematics just for the sake of it. My reasons are very simple — I would rather do mathematics for the sake of it than otherwise, and fortunately I live in the times where I can make a living out of it. The timelessness of an abstract mathematical theorem appeals me. Whereas applied mathematics, through simplifying hypothesis, always seems like a compromise. Applied mathematics is too hard for me.
I am not sure that it is harder than other scientific or creative pursuits at high level. The only difference seems to be that in other fields outcome is on a statement on paper which looks like it is written in an alien language. It can be touched, seen, felt or listened. The proportion of people who excel in any field is always very small. Having said that it is a good idea to raise overall mathematical literacy just as one would say that it is good if most children learn to play one or two sports. On the question of how to do it effectively — I don’t know. I haven’t really thought about it.
