Music of the spheres
C.S. Seshadri (1932-2020), the algebraic geometer of international stature, was instrumental is setting the stage for the re-emergence of Chennai as a vibrant centre of mathematical research.
IN the early decades of the last century, Srinivasa Ramanujan famously pursued his interest in mathematics while working as a clerk in the Madras Port Trust. Less wellknown is the fact that at around the same time a group of professional mathematicians thrived in Madras (now Chennai).
This “Madras School” of mathematics, which made the city the centre of Indian mathematics, was led by Ananda Rau and R. Vaidyanathaswamy—both products of Cambridge—and Fr Racine, a French Jesuit priest who taught at Loyola College in Chennai. Among the products of this school were S.S. Pillai, S. Minakshisundaram, K. Chandrasekharan, Ganapathy Iyer, Kesava Menon and K.G. Ramanathan.1
Many of these people figured in the life of C.S. Seshadri, whose extraordinary life and achievements I now turn to.2
Seshadri’s career is a case study in how family background, peer group, mentors, institutions and luck combine with innate ability and personality to determine life outcomes. Innate ability is the least quantifiable factor, and with Seshadri this was not apparent immediately. This is because of his personality, which was deliberate and confident but understated. (He shunned ostentation in all matters, whether it be the furnishing of his office, his clothes or even the notebooks in which he prepared his careful lectures.) Those familiar with the depth and breadth of his work, however, will vouch for his ferocious intelligence. As for personality, he demonstrated by example that being calmly focussed on the matter at hand—whether it be mathematics, music or administration—is what it means to be true to oneself and to society.
Seshadri was born in Kancheepuram on February 29, 1932, in a family of 11 children, in relatively affluent circumstances. He was a good student. An uncle (who had been trained as a chemist at the Institute of Science, Bangalore) kindled his interest in mathematics. In 1948, Seshadri joined Loyola College, where he spent the next five years, first in the intermediate class and then in the B.A. (Hons) course. Fr Racine had arrived in Loyola College in 1939.3
By all accounts he was not a great teacher, but he had studied in France with the legendary mathematicians Elie Cartan and Hadamard and, to quote M.S. Raghunathan,“.... Racine naturally had an excellent perspective on mathematics, which he
brought to India with him. He began weaning some Indian mathematicians away from traditional Cambridge-inspired areas and Minakshi[sundaram] was his first big success; and there was a galaxy of brilliant students to follow…. To mention a few names: K.G. Ramanathan, C.S. Seshadri, M.S. Narasimhan, Raghavan Narasimhan, C.P. Ramanujam.”
Of these, M.S. Narasimhan was in the same “batch” as Seshadri. After graduating from Loyola, the two of them went to Bombay (now Mumbai) to join the Tata Institute of Fundamental Research (TIFR) on the advice of Fr Racine.
THE EARLY YEARS AT TIFR
TIFR was the brainchild of Homi Bhabha, the outstanding theoretical physicist and institution builder and father of India’s nuclear programme. Homi Bhabha was convinced of the centrality of mathematics. It is no accident that the TIFR letterhead said: National Centre of the Government of India for Nuclear Science and Mathematics. In 1949, K. Chandrasekharan, then 29 years old, was invited by Homi Bhabha to join TIFR. Chandrasekharan (known as K.C.) had obtained his doctoral degree in Madras under the guidance of Ananda Rau and was at the Institute for Advanced Study (IAS) at Princeton, United States, when the invitation came from Homi Bhabha.
Over a period of 16 years, K.C. built an outstanding school of pure mathematics at TIFR. (He moved to ETH Zurich in 1965.) It was his extraordinary luck that Narasimhan and Seshadri arrived at TIFR in 1953 as graduate students; in turn it was their good fortune that K.C. had prepared the ground for them, as well as for those who came later.
K.G. Ramanathan (known as K.G.R.) joined K.C. in 1951 after earning his PHD under Emil Artin in Princeton. Over the years, a stream of outstanding students went from Madras to TIFR, many of them from Loyola College (and a similar group from Vivekananda College); together they formed an outstanding peer group. (In later years, this pattern changed, and TIFR began to attract students from elsewhere in India.) K.G.R. and K.C. together anchored the mathematics academic programme at TIFR. They devised a programme of visitors from abroad—mostly from France but also from elsewhere in Europe and the U.S. (and later also Japan) who gave courses of lectures that rapidly took the cohort of bright students from the basics to the cutting edge of mathematics. The names of the lecturers is a roll call of outstanding researchers of the era, and the notes of the lectures—typed, “cyclostyled” and bound securely in an elegant large format—acquired a legendary status around the world.
Modern mathematics is characterised by having a lot of information distilled into definitions. A good point of view gives rapid access to deeper properties of the mathematical objects under study and also gives good reasons to prioritise some over the other. If that is too nuanced, let us just say that good taste is important in mathematics, and more so now because it is a thriving subject and there is so much of it. This is why “learn from the masters’’ is particularly good advice. This is the luxury that students at TIFR had—those who grasped this opportunity made outstanding careers for themselves. Seshadri was one of them, as also Narasimhan.
Among the fields that flourished in TIFR—AND the tradition continues to this day—is algebraic geometry, which has its somewhat fusty origins in coordinate geometry. But during the first half of the 20th century, it underwent a series of revolutions, above all at the hands of Alexandre Grothendieck, and came to occupy a central role in mathematics in the second half of the century.
THE BREAKTHROUGH YEARS
During the 30 years that Seshadri spent at TIFR (from 1953 to 1984), he grew from graduate student to an algebraic geometer of international stature. Before we turn to the work itself, two remarks must be made. First, because of the relatively improvised nature of the training at TIFR, there was a certain amount of casting-around-for-direction that went on, even with the most gifted students. When they did find a direction, it was often by themselves and by accident. In the case of Seshadri, he eventually found a direction—algebraic geometry in the French mode, as formulated by his “guru” Claude Chevalley and later Grothendieck, and a set of problems relating to “moduli theory”, which is the study of families of algebro-geometric objects initiated by Chevalley and Andre Weil.
(A parabola is an example of an algebro-geometric object. It is defined by an algebraic equation; the “constants” in the equation are in mathematical argot “moduli”, and as the moduli change we get different parabolas. The set of moduli is a “moduli space”.)
Seshadri’s first substantial piece of work, however, did not involve moduli or, in fact, much geometry. This was his solution of a conjecture of Jean-pierre Serre in the simplest non-trivial (two-dimensional) case. Pavaman Murthy, Seshadri’s first student, would later solve this in three dimensions. Later, Quillen and Suslin independently proved the conjecture in general. This whole line of work, on projective modules, is a major strand of the field known as commutative algebra, and one where Indian researchers, mostly from TIFR, continued to make significant contributions.
In the early 1960s, inspired by ideas of Andre Weil, Narasimhan and Seshadri began studying families of “irreducible unitary representations of the fundamental group of a Riemann surface”. By a wonderful concatenation of circumstances, the American mathematician David Mumford was simultaneously engaged in resurrecting ideas of David Hilbert to construct a “Geometric Invariant Theory” (GIT) with a view to using this to construct moduli spaces in algebraic geometry. As an example, he considered “moduli spaces of stable vector bundles on an algebraic curve”, and Narasimhan and Seshadri had the epiphany that their space of irreducible representations
could be identified, after some hard work, with Mumford’s space of stable vector bundles. The Narasimhan-seshadri theorem, as it came to be known, is a cornerstone of modern geometry and has been generalised in many directions, most notably with the work of Atiyahbott, Donaldson, Hitchin and Uhlenbeck-yau.
Seshadri continued to work on foundational aspects of GIT. As for moduli theory, he introduced the notion of parabolic bundles and (jointly with Vikram Mehta) proved a version of the Narasimhan-seshadri theorem that holds for these objects. Parabolic bundles have many facets, not only do they give the most natural examples of a subtle phenomenon in GIT, the “variation of stability with respect to a parameter”, they also crop up in the mathematics of string theory.
In the late 1970s, in departure from his earlier work, Seshadri launched a major programme to develop a modern theory of “standard monomials”. He was joined in this by his students (and later collaborators) C. Musili and V. Lakshmibai. This was a tour de force of algebra, geometry and combinatorics that yielded substantial new results and (in the hands of Peter Littelman) led to unexpected connections with the works of Kashiwara on mathematical objects called crystals.
Along the way, he discovered basic results in algebraic geometry that are codified in his ampleness criterion and the definition of the “Seshadri constant”.
To give the layperson some idea of the level of all the work described in broad brushstrokes above, Grothendieck, Serre, Quillen, Mumford, Atiyah and Yau were all winners of the Fields Medal, and Uhlenbeck was an Abel laureate.
Seshadri’s achievements were recognised by the Indian and international community. He was elected to fellowships of the major Indian academies, the Royal Society, the U.S. National Academy of Sciences and The World Science Academy based in Trieste, Italy. Seshadri was awarded the Bhatnagar Prize, the
Trieste Science Prize and the Padma Bhushan.
I remember hosting a party at IAS when the news of his election to the Royal Society was announced. (I was a “postdoc” there. Seshadri was visiting, as were some other younger colleagues from Bombay.) Seshadri was very pleased, and told us of a message of congratulations he had received from an Australian mathematician remarking that the time taken for this recognition was proportional to the distance from England.
CHENNAI MATHEMATICAL INSTITUTE
In 1984, Seshadri moved back from Bombay to Madras and joined the Institute for Mathematical Sciences (IMSC). He persuaded his younger colleague and well-known number theorist R. Balasubramanian to join him. In 1985, he welcomed P.S. Thiagarajan, who had spent many years in Europe, to start a group in theoretical computer science at IMSC. This set the stage for the reemergence of Madras as a vibrant centre of mathematical research. In 1989, Seshadri moved with Thiagarajan to the SPIC Science Foundation and started a School of Mathematics, working out of a modest set of offices and lecture rooms in a building in T. Nagar. The first set of graduate students joined, and a nucleus of a high-level research in mathematics and theoretical computer science was formed.
By 1998, the new institution had
matured into the Chennai Mathematical Institute (CMI) and started its now flagship undergraduate programme. Soon afterwards, R. Sridharan, who was retiring from TIFR after a long and distinguished career, joined CMI as a senior faculty member.
By 2006, the institute had become a deemed university and was housed in elegant low-slung buildings with an extraordinary open architecture, at the heart of the software park SIPCOT, some 20 kilometres south of Chennai along the Old Mahabalipuram Road. This growth was facilitated and sustained by the support from the Department of Atomic Energy, the University Grants Commission (UGC) and, more recently, the Department of Science and Technology (DST). There was significant private funding from both individuals and private institutions. The Infosys Foundation made a major donation, and the Shriram group has been a consistent supporter.
CMI has had to be continuously mindful of the need to raise funds; this has resulted in a frugal culture. Seshadri bore the associated stresses with patience, for the most part. He was self-deprecating during meetings with potential donors, often quoting the Kunjan Nambiar poem, which he had learned from his colleague S. Ramanan: “Deepasthambham mahashcaryam, namukkum kittanam panam.” (The lamp post is wonderful, we also need money.)
Seshadri’s model for his institute was the great modern universities of the West, particularly the U.S., with their campuses alive with debate, music, theatre, literature, art, science and mathematics, where active practitioners pass on their passion and skills to the younger generations. Constraints of funding and availability of faculty have meant that for the moment activity in CMI is restricted to (pure) mathematics, theoretical computer science and theoretical physics. (Recently, a master’s programme in data science has been added.) Nonetheless, the students are exposed to a culture where research coexists with learning and the arts are accorded their due. They go on to achieve successful careers in academia and industry.
Music, like mathematics, ran like a golden thread through Seshadri’s life. His family had deep connections with Carnatic music, in particular, the school of the legendary Naina Pillai. From an early age, he was immersed in it, listening and singing by ear until he could reproduce complicated compositions. Formal training started rather late in life when he was 24, and perhaps because of this, he was never a concert performer. “Music is also not an easy game, as it calls for early commitment and complete surrender,” he said in a recent interview.
Back in Madras, he formed a number of close friendships with musicians and serious aficionados of music. Among them was Shri (“Spencer”) Venugopal. They had regular sessions where they shared their thoughts on music and sang together. Venugopal talks of Seshadri’s approach to music, which was “not only aesthetic, but also educated”, and praises the laya-sruthivak suddham of his singing. They shared an enthusiasm for the Dhanammal bani (school of music); Venugopal remembers occasions when T. Brinda (Dhanammal’s granddaughter) sang for an audience of two. Seshadri was open to musical experiences of various genres and once even expressed a wish to be reborn as a Dhrupad singer.
A GIFT FOR FRIENDSHIP
If you encountered Seshadri, you would have described him as “warm and friendly”, irrespective of your age or position in life. This was not an artifice. To those who could recognise it, there was a special enthusiasm that turned on when he encountered someone with great talent, passion and anything interesting to say. It is not surprising, therefore, that he formed a number of warm friendships with gifted individuals of all stripes. David Mumford was a close friend, as also M.S. Narasimhan and a number of other intellectuals in India and around the world, including younger colleagues, among them Pavaman Murthy and Lakshmibai. Particularly deep were the friendships built on love for music. S. Parthasarathy of the SPIC Centre for Energy Research and R. Thyagarajan, industrialist and founder of the Shriram Group, were others with whom Seshadri shared his passion for music. Thyagarajan became one of CMI’S staunchest supporters.
Seshadri’s closest friend, without doubt, was his wife and companion, Sundari. She was a talented singer with a passion for classical music as well as Hindi film songs of the golden age, which included Asha Bhosle’s item numbers, which she could belt out with gusto. Possessed with joie de vivre, she was the centre of a joyous circle of Seshadri’s family and friends.
Seshadri had been plagued by a variety of ailments in the last decade. He endured them with habitual grit and good humour. Sundari’s passing in October 2019 was a heavy blow to him, and his health began to deteriorate. The end, when it came, was due to a heart attack, late in the evening on July 17, 2020.
Seshadri is survived by his sons Giridhar and Narasimhan. m T.R. Ramadas recently retired as Distinguished Professor at Chennai Mathematical Institute and is now Adjunct Faculty there. Earlier, he was Head of Mathematics at ICTP, Trieste.
Notes
For wonderful accounts of these lives, see Raghunathan, M.S. (2003): “Artless innocents and ivory-tower sophisticates: Some personalities on the Indian mathematical scene”, Current Science, 85, pp. 526–536.
https://bhavana.org.in/ proofs-transcendence-cs-seshadri/. An interview with Seshadri wherein he speaks at length about his life and work.
http://gaddeswarup. blogspot.com/2008/09/ remembering-fr-racine.html for an appreciation of this extraordinary man.