Monk returns to mathematics and solves problem that stood for two decades

For six years, Mahan Mj stepped away from mathematics.

Fresh from earning his PhD from the University of California, Berkeley, in 1998, the mathematician left research behind to become a monk of the Ramakrishna Order. He did not publish research or pursue mathematical problems during those years.

Then, in 2004, a bout of "moderately idle curiosity" brought him back to the subject.

What followed eventually led him to solve a mathematical problem that had remained open since the early 1980s.

Mj, now a professor at the Tata Institute of Fundamental Research (TIFR), had been working on questions involving chaos, symmetry and geometry. After returning to mathematics, he came across two papers by Brian Bowditch and Curtis McMullen and began trying to reprove and generalise their results using ideas from his doctoral work.

That effort gradually opened a new path.

" In hindsight, it was almost as if the mathematics and the structure behind it was leading one by the nose," Mj says.

For him, the experience also challenged the conventional idea that mathematics is something one simply "does".

"This convinces me that the best and most satisfying mathematics 'happens' and is not really 'done' in the usual sense that we understand these words," he says.

From electrical engineering to mathematics

Born in West Bengal in 1968, Mj was drawn to mathematics from an early age. Yet social expectations led him to study electrical engineering at IIT Kanpur.

By his third semester, he knew he wanted something else.

"The mathematical content in engineering was both sparse and not of a fundamental kind," he recalls.

He applied to transfer to mathematics, despite being a semester late for the usual process. Support from the institute's academic leadership and faculty in mathematics and physics helped make the move possible.

The decision changed the course of his life.

At IIT Kanpur, topology courses taught by Prof. Shobha Madan particularly captured his imagination.

"She was an incredibly inspiring and dedicated teacher and made the subject come alive," he says.

He also credits the intellectual environment around him, remembering classmates including Rajesh Gopakumar, T Senthil and Anurag Singh, whose commitment to their respective fields was, he says, "absolutely transparent".

The mathematics behind the breakthrough

The problem Mj eventually helped solve involves an intricate relationship between geometry, symmetry and chaotic dynamics.

One way to visualise it is to imagine infinitely many symmetries acting on the surface of a sphere. With so many transformations involved, the resulting dynamics can become highly chaotic.

The Cannon–Thurston map provides a mathematical framework for making sense of that complexity.

William Thurston and Jim Cannon established the result in a test case in 1985, while Yair Minsky later generalised it in the early 1990s. During his PhD, Mj developed a new proof and extended the result to groups with negative curvature, a framework associated with mathematician Mikhail Gromov.

When he returned to the subject in 2004, Mj spent about two years working through increasingly broader special cases.

Eventually, he recognised that the most general of those cases appeared capable of covering the problem Thurston had posed.

A model for the general setting had already been developed by Minsky and his collaborators, Jeff Brock and Richard Canary. Mj's task was to establish that their model satisfied the conditions required by his special case.

It did. The long-standing question was answered in the affirmative. Mj's mathematical interests remain rooted in what he describes as "the visual and intuitive kind dealing with shapes, symmetries, and movement on shapes."

He compares different mathematical traditions to the gharanas of Hindustani classical music and cites Thurston, Gromov and Dennis Sullivan among the mathematicians who have continually influenced him.

Where mathematics and monastic life meet

For Mj, becoming a monk and returning to mathematics were never contradictory choices.

"One does not live in disjoint compartments," he says.

He believes monastic life has also created a certain freedom from concerns over money, recognition and status.

"This leaves more mental space to get back to the basic point of doing mathematics: the curiosity that drives it, the sheer joy one gets out of it, and the joy one gets out of sharing that knowledge with others."

That perspective also informs his response to the KRS Sastry Prize, which has been awarded for the first time this year.

Mj says he is grateful for the recognition, but sees the award in practical terms too. Three of his collaborators are at the Indian Institute of Science (IISc), and the prize gives him an opportunity to spend more time there.

"I hope to visit IISc more and this is a really good excuse," he says.

A warning for students in the age of AI

Mj's message to young mathematicians is equally focused on curiosity and understanding.

He sees the current excitement around artificial intelligence and mathematics as something that needs to be viewed carefully.

A "dust storm" of claims surrounds AI's ability to solve mathematical problems, he says, but believes the picture is more complicated.

"This is only partly true," he says, arguing that the fundamental reasons humans have pursued mathematics for roughly 2,500 years remain largely untouched by the AI revolution.

At the heart of mathematics, he says, are clarity and understanding.

A precisely framed question can open the door to an entire world of ideas. Without that conceptual purpose, he argues, mathematics risks becoming a series of binary questions without meaning.

"Removing the conceptual motivation and retaining only the binary questions is like having a bunch of doors in an empty field leading nowhere."

His advice to students is therefore simple: preserve the human desire to understand.

Hold on to the "very human aspiration for clarity and understanding", he says, while recognising that AI can also create "deceptive short-cuts" that may bypass the deeper process of learning and discovery.