From the memoir: How a mathematician was inspired to think creatively at Cambridge University

Our first-year lectures were held in a massive lecture hall with more than two hundred students squeezing into one room. The lectures were university-wide, bringing together students from different colleges, and all the maths students entering Cambridge had to attend mandatory courses. The air would become fetid, and there were bouts of sneezing and coughing. Once a girl even fainted in the lecture hall and the students had to be evacuated. We had to take both pure and applied mathematics courses, and I did not enjoy being made to study fluid mechanics and mathematical physics. In fact, the applied mathematics courses made me almost regret not going to Yale or another US university in which one did not specialise right away, and where I could have done pure maths courses and supplemented them with other courses in literature and philosophy and languages, subjects much more to my taste.

Some of our lecturers were charismatic, like Dr JME Hyland, who taught linear algebra, and was a logician. He imbued some of the material with a philosophical and mysterious air, and even now these concepts carry for me the charge he endowed them with. He riffed on the notion of “dual of a vector space V” and the fact that “the dual of the dual of a vector space V is canonically isomorphic to V”. Isomorphism between two objects means that they can be identified with each other. Confusion and controversy raged amongst students about what “canonical” meant. It had a religious charge to it and stood for the mathematical ideal of being independent of all choices.

Many of the students were befuddled by Hyland’s lectures on duality. He was probably guilty of playing up and exaggerating greatly the depth of things on which he discoursed. But the lectures took on the mad comic air of an episode of the 1980s political sitcom Yes Minister with Hyland tying the students up in knots of confusing, tongue-twisting statements reminiscent of Cabinet Secretary Sir Humphrey Appleby confusing his boss, Minister Jim Hacker, with his excessively complicated formulations. Like Appleby, Hyland invariably came out on the right side of debates.

He managed to point out something an objecting student had not considered in a putative counterexample to his assertion. I enjoyed the drama of Hyland’s lectures, the back and forth between him and his students and the confusion he sowed in our minds that led to animated discussions. His polar opposite was Professor Alan Baker, who did not engage with students at all, and took to his heels after every lecture, almost fleeing in fear from any student who might want to ask him a question. He was a very distinguished mathematician, someone who had won a Fields Medal (as exalted as the Nobel prize for mathematicians) for his work in an area that, believe it or not, is called transcendental number theory.

The university-level lectures were complemented by college “supervisions” in which pairs of students went to meet with a supervisor to discuss homework assignments and have their work checked. There were varying levels of sincerity among the supervisors. Some barely engaged with their students while others went above and beyond, and enthused students with their infectious love of the subject. The supervisions took us into the untidy homes of young couples with children bawling in the bedroom, at other times to basements with the footfalls of passing pedestrians falling like summary judgements.

In my second year, I got to do more pure maths and my most memorable experience was learning group theory. The course was taught by Professor John Thompson, an American and an éminence grise in group theory. One of the most prominent mathematicians in Cambridge, he had won a Fields Medal for his work in group theory in the 1970s. It was an introductory course in group theory, and so the material was totally trivial for him. Yet he mumbled at the board and copied onto the blackboard from a textbook that he held open before his eyes. Students grumbled and complained about him.

In spite of his lecturing style, I found Professor Thompson very inspiring. He had a heavy beard and a fierce intensity in his eyes and persona. I would see him sometimes cycling down King’s Parade with his head full of groups, I imagined. Though most of his audience probably found him excruciatingly boring, I found myself listening to him carefully. In the mumble of the lecture, he would sometimes drop a phrase that blazed in its evocativeness. “You have to be really careful of this. It seems innocuous, but at times it can bite!” This was apropos of a definition that was easy to misunderstand, and could lead one into making a grievous error. Something that seemed rather trivial in the text would become portentous: “This simple thing can really build on itself, and the whole shebang could impose itself on you.” I imagined him in the American prairies, looking at distant horizons across limitless acres of cornfields, his thoughts resonating in those vast empty plains.

A good definition brings together many examples under one roof, identifying key structural elements which are common across them. The mathematical definition of “group” is economical, and yet it allows for a rich array of examples. Long before groups had been defined formally in the first half of the nineteenth century, they had already been an implicit, unnamed presence in many mathematical developments. Fermat, Euler, Legendre and Gauss had used what one now recognises to be group-theoretic ideas, related to multiplication of numbers in mod N arithmetic. Groups had arisen informally in Galois’ study of symmetries of roots of polynomials and became one of the master topics of modern mathematics. Group theory studies symmetries for their own sake and is a vital area of pure mathematics.

After groups were formally defined, there arose the taxonomical task of classifying them. This turned out to be too hard a problem to solve completely, but by the 1980s, after decades of strenuous efforts by mathematicians working all over the world, the classification of all finite groups that further have a special property called simple (which does not mean “easy”!) was completed. Without defining ‘simple group’ formally, we can just say that they are to groups what prime numbers are to numbers. Simple groups are the atomic elements in group theory, and all groups are made by ‘combining’ simple groups in a certain way. Around the time I studied there as an undergraduate, the massive programme of ‘classifying finite simple groups’ had been completed, with significant contributions by Cambridge mathematicians. One of Professor Thompson’s landmark theorems is the Feit-Thompson theorem, a cornerstone in the classification of finite simple groups. It says: “A finite group G whose size is an odd composite number is never simple.”

The proof of this one-sentence theorem takes up more than two hundred pages. The Feit-Thompson theorem reflects a surprising interaction between the size of a finite group G and a property like it being simple which is internal to the group G. The axioms defining a group are spare but, as illustrated by this theorem, lead to an unexpectedly rich structure.

I thought of a basic question in group theory and wrote a note to Professor Thompson about it. I went and dropped it off in his pigeonhole near the main office of the Department of Pure Mathematics and Mathematical Statistics (DPMMS) on Mill Lane. I must have given him my address as, to my delight, there was a note for me a few days later in my pigeonhole at Trinity College. Professor Thompson suggested that I go meet him in the common room of the DPMMS.

When I went to the common room, I felt like an impostor. There were models of the five regular polyhedra discovered by the Greeks in antiquity – cubes, tetrahedrons, octahedrons, dodecahedrons and icosahedrons – dangling from the ceiling.

On worn sofas and low tables, people sat drinking tea or coffee and playing games like chess and Go. The best minds in the country were locked in bloodless combat. Knots of people formed around an ongoing game, watching and commenting. This scene before me seemed like the Kingdom of Symmetry with citizens sipping cups of tea, consuming orange-flavoured Pims biscuits and absorbed in board games.

Professor Thompson retained his mystique at close quarters. So many decades later I remember neither the question nor his answer, although at that time it must have been burned into my mind. The atmosphere in the common room of the mathematics department on Mill Lane reminded me of the chess club in the compound of Zandu Pharmaceuticals that I had frequented in Bombay. In both places one felt a narrowing down of the world to something hard, intricate and specific, which needed and consumed all of one’s attention.

Excerpted with permission from Chasing a Conjecture: Inside the Mind of a Mathematician, Chandrashekhar B Khare, Juggernaut.