Chasing (a) Conjecture: Chandrashekhar Khare on proof, persistence, and the poetry of mathematics

Chandrashekhar Khare, the acclaimed mathematician who proved one of the most important conjectures in number theory, does not fit the stereotype of a man lost in abstraction.

Speaking from his home at UCLA, where he chairs the mathematics department, his conversation is a tapestry woven with threads of Hindustani classical music, Marathi literature, and the philosophy of the creative process. His memoir, Chasing (a) Conjecture, is an intriguing map of the mind, a journey into the “divine madness” that compels a human to spend a decade making a mathematical conjecture real.

“A conjecture is just an outgrowth of wanting to know an answer without knowing the answer, guessing the answer,” he explains, demystifying the term that governs his world. “But on the other hand, it never is on firm ground until you manage to prove it. In mathematics, the gold standard is proof.”

For Khare, the pursuit is a delicate dance between two forces – the flash of creative intuition and the scaffold of logical rigour. He describes it with the reverence of a musician discussing riyaz.

“When you come to research, you’re asked to solve problems where you may not find a solution for months or even years… You need a new idea. And that comes from saturating your mind, immersing your mind in the situation and the problem, and then hope that that process might generate an insight which we call intuition. Sensing something. It’s like catching a scent of something. And then you follow your intuition and follow that with analytical and rigorous logical thinking.”

This process, he admits, is addictive and all-consuming. “Mathematics is addictive because it is in the mind… It draws certain people in. It is a very addictive form of thinking. And in some ways you’re really hassled because you’re not being able to solve the problem. So it’s like an irritant. It’s a little bit like an oyster. The way it makes a pearl. It has to be. It gets irritated and then it produces that pearl.”

The path is strewn with failure, a theme Khare addresses quite candidly. He recalls his graduate student days at Caltech, a period of such struggle he nearly quit. “To be able to concentrate on something, you need to be a little bit good at it. If you’re getting nothing out of something, it’s impossible to keep pursuing it. You have to cross that critical line at which the subject starts rewarding you.”

His advice to young students, especially in India’s high-pressure academic environment, is: “Don’t be discouraged by how brilliant other people are… In a creative endeavour, there is scope for a lot of different kinds of being brilliant. The person who’s quickest will not necessarily do the best.”

“I’ve heard this quote of some famous Japanese mathematician: if you cannot hit that bullseye, just expand the bullseye. So then you’ll have hit it. Have a broader sense of what achievement is, what success is,” he adds

For him, the qualities needed are : “Certainly persistence. Persistence because most things of value cannot be done overnight. They require a journey. Persistence of interest, strong interest in the subject. And a certain rigour about oneself.”

After proving Serre’s conjecture, Chandrashekhar Khare embarked on another long quest, this time for the Leopoldt Conjecture.

At the heart of Khare’s philosophy is a quote from the mathematician Charles Hermite that he holds dear: “It is the person, and not the method, who solves a mathematical problem.” “What it says is that in mathematics… it’s not the technical skills you have which will in the end result in your success in solving a problem. It is your personality, all aspects of yourself. Your persistence, your doggedness, your sense of beauty, your viewpoint… Everything what you are informs yourself as a creative person.”

This belief illuminates why his book and conversation are so rich with references beyond mathematics—to the bhakti saints, to Charles Dickens and Virginia Woolf, to the music of Kishori Amonkar and Kumar Gandharva. “I like to have things where I’m not being told specific things. I like an atmosphere being created and music does that… In mathematics, the results I like the most are where you somehow start with something you don’t know quite what you’re doing… and create a structure starting with nothing.”

Khare dismantles the lonely genius myth. While the deep work happens in solitude, mathematics is now a profoundly collaborative enterprise. “The image of a person sitting in a room and just thinking is not the way mathematics is done currently. You are always using ideas and insights of people doing mathematics all over the world.”

After proving Serre’s conjecture, he embarked on another long quest, this time for the Leopoldt Conjecture, a journey into another “wilderness” that has yet to yield its secrets. “I’m not confident I’ll ever solve this conjecture. But then to go in a wilderness, go on a journey, because then your failure kind of sensitises you to certain aspects of mathematics.”

Today, he is driven less by the need for another monumental proof and more by the pure joy of the search. “The joy is in the discovery, the collaboration… I do mathematics for the pleasure of it. It is a romantic quest.”

As our conversation ends, he returns to the core of his memoir’s mission: to pull back the veil on this quest. “My target audience is someone who’s not a mathematician… I want to convey that mathematical research is a creative endeavour. It has a lot of similarities to someone who wants to be a musician or artist. The struggle is quite similar.”

In Chasing (a) Conjecture, Khare has done more than narrate a personal triumph. He has rendered the abstract palpably human, proving that the most profound proofs are not just about numbers, but about the timeless, maddening, and beautiful persistence of the human spirit.