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"description": "On May 15, 2021, the eminent Indian mathematician Mudumbai Seshachalu Narasimhan passed away at his home in Bangalore.\nHis work in the field of geometry is internationally recognised, having deep connections with different branches of mathematics and theoretical physics.\nNarasimhan spent much of his career at the Tata Institute of Fundamental Research (TIFR) in Mumbai, where he was a key figure in the creation and development of the internationally acclaimed modern Indian school of algebraic geometry.\nAfter retiring from TIFR, from 1993 to 1999, Narasimhan was Head of the Mathematics Section of the International Centre for Theoretical Physics (ICTP) in Trieste, an institution created in 1964 by the Pakistani 1979 Nobel Laureate in Physics Abdus Salam.",
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"references": [
{
"type": "Article",
"id": "bib-bib1",
"authors": [],
"title": "\nJ.-M. Drezet and M. S. Narasimhan,\nGroupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques.\nInvent. Math.97, 53–94 (1989)\n"
},
{
"type": "Article",
"id": "bib-bib2",
"authors": [],
"title": "\nG. Gallego, O. García-Prada and M. S. Narasimhan, Higgs bundles twisted by a vector bundle.\narXiv:2105.05543 (2021)\n",
"url": "https://arxiv.org/abs/2105.05543"
},
{
"type": "Article",
"id": "bib-bib3",
"authors": [],
"title": "\nG. Harder and M. S. Narasimhan, On the cohomology groups of moduli spaces of\nvector bundles on curves.\nMath. Ann.212, 215–248 (1975)\n"
},
{
"type": "Article",
"id": "bib-bib4",
"authors": [],
"title": "\nM. S. Narasimhan and K. Okamoto, An analogue of the Borel–Weil–Bott theorem for hermitian symmetric pairs of non-compact type.\nAnn. of Math. (2)91, 486–511 (1970)\n"
},
{
"type": "Article",
"id": "bib-bib5",
"authors": [],
"title": "\nM. S. Narasimhan and T. R. Ramadas, Factorisation of generalised theta functions. I.\nInvent. Math.114, 565–623 (1993)\n"
},
{
"type": "Article",
"id": "bib-bib6",
"authors": [],
"title": "\nM. S. Narasimhan and S. Ramanan, Moduli of vector bundles on a compact Riemann surface.\nAnn. of Math. (2)89, 14–51 (1969)\n"
},
{
"type": "Article",
"id": "bib-bib7",
"authors": [],
"title": "\nM. S. Narasimhan and S. Ramanan, Deformations of the moduli space of vector bundles over an algebraic curve.\nAnn. of Math. (2)101, 391–417 (1975)\n"
},
{
"type": "Article",
"id": "bib-bib8",
"authors": [],
"title": "\nM. S. Narasimhan and C. S. Seshadri, Stable and unitary vector bundles on a\ncompact Riemann surface.\nAnn. of Math. (2)82, 540–567 (1965)\n"
},
{
"type": "Article",
"id": "bib-bib9",
"authors": [],
"title": "\nM. S. Narasimhan and R. R. Simha, Manifolds with ample canonical class.\nInvent. Math.5, 120–128 (1968)\n"
},
{
"type": "Article",
"id": "bib-bib10",
"authors": [],
"title": "\nM. S. Narasimhan, The Collected Papers of M. S. Narasimhan, two volumes,\nedited by N. Nitsure, Hindustan Book Agency (2007)\n"
}
],
"title": "Mudumbai Seshachalu Narasimhan (1932–2021)",
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"id": "S1",
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"content": [
"1 Life and career"
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"content": [
"Narasimhan was born on 7 June 1932 in Thandarai, a small town in Tamil Nadu (India), to a prosperous farming family.\nAlthough their circumstances were somewhat reduced after his father passed away when he was only thirteen, his family encouraged him to do what he wanted.\nFrom a young age he showed a great interest in mathematics and already in school he decided to become a researcher, even before really knowing what that meant.\nHe completed his first university studies at Loyola College in Madras, in the heart of British India.\nThere, he had as a teacher the French Jesuit Father Charles Racine, who was in contact with legendary figures of mathematics such as Elie Cartan, Jacques Hadamard, André Weil and Henri Cartan.\nRacine introduced him to modern mathematics, unknown in India, and, in particular, to the great French school.\nAt Loyola College Narasimhan met C. S. Seshadri – also deceased in 2020 – who would later become one of his main collaborators."
]
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"Following his studies at Loyola College and on the advice of Father Racine, Narasimhan moved in 1953 to the newly created TIFR in Bombay to do his doctorate under the direction of K. Chandrasekharan, one of the founders of the centre’s School of Mathematics.\nThere he was able to interact with first-rate mathematicians who came as visitors to teach courses of two or three months.\nAmong them was Laurent Schwartz – Fields medallist in 1950 – who would have a great influence on Narasimhan and would be his mentor during his three-year stay in Paris in the late 1950s, where he would also coincide with Seshadri.\nIn the initial period of his stay in Paris he could not completely concentrate on mathematics as he was hospitalised due to a sickness.\nHowever, he used that time to read the paper of Kodaira and Spencer on deformations of complex structures which eventually played a great role in his future work.\nDuring his time in France he also collaborated with Japanese mathematician Takeshi Kotake, who was also in Paris to work with Schwartz."
]
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"When he returned to TIFR in 1960, Narasimhan and Seshadri started an intense collaboration that resulted in the famous Narasimhan–Seshadri theorem, published in 1965.\nA bit later, he began his long and fruitful collaboration with S. Ramanan.\nAlong with Ramanan, who was his first student, Narasimhan’s student roster includes other such illustrious names as N. Nitsure, R. Parthasarathy, V. K. Patodi, M. S. Raghunathan, T. R. Ramadas and R. R. Simha, who have made essential contributions to various areas of mathematics.\nNarasimhan’s presence at TIFR was indeed a source of inspiration to several generations of young mathematicians."
]
},
{
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"id": "S1.p4",
"content": [
"During his time at TIFR, Narasimhan had also important administrative activity.\nIn particular, he was the first Chairman of the\nNational Board for Higher Mathematics, which was set up in 1983 by the Government of India, under the Department of Atomic Energy, to foster the development of higher mathematics in the country.\nTogether with S. Ramanan, who acted as Secretary, Narasimhan undertook the task of setting it up in the initial years.\nHe was also a member of the Executive Committee of the International Mathematical Union (IMU) during the period 1983–1986, as well as President of IMU’s Commission on Development of Exchange."
]
},
{
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"id": "S1.p5",
"content": [
"After retiring from TIFR, Narasimhan was the Head of the ICTP Mathematics Section from 1993 to 1999.\nIn this position, he carried out in particular important work in supporting young mathematicians from developing countries.\nWhen he retired from ICTP, he continued to be an adviser of ICTP and served as a member of its Scientific Council.\nIn 2020, he was awarded the Spirit of Abdus Salam Award by the family of the ICTP founder at a ceremony where numerous mathematicians from around the world showed him their great admiration, respect and affection."
]
},
{
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"id": "S1.p6",
"content": [
"After his stay at the ICTP, Narasimhan spent three years at SISSA (Trieste), before returning to India, where he continued his mathematical activity at the Indian Institute of Science in Bangalore."
]
},
{
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"id": "S1.p7",
"content": [
"Narasimhan’s work earned him many prestigious awards, including the Shanti Swarup Bhatnagar Prize (1975), Third World Academy of Sciences Prize for Mathematics (1987), the Srinivasa Ramanujan Medal (1989), the French Ordre National du Mérite (1990), the Padma Bhushan Award by the President of India (1990), the C. V. Raman Birth Centenary Award of the Indian Science Congress (1994), and the 2006 King Faisal International Prize in Science that he shared with Sir Simon Donaldson.\nHe was also a Fellow of the Indian National Sciences Academy, Indian Academy of Sciences, the Royal Society of London and the Third World Academy of Sciences."
]
},
{
"type": "Paragraph",
"id": "S1.p8",
"content": [
"Narasimhan was a great fan of detective novels, and literature in general, in Tamil, English and French.\nHe also liked Indian classical music, as well as Western classical music."
]
},
{
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"id": "S1.p9",
"content": [
"Narasimhan was married to Sakuntala Narasimhan, a renowned Indian classical music singer and journalist.\nThe couple had a daughter, Shobhana Narasimhan, a physics researcher and professor at the Jawaharlal Nehru Center for Advanced Scientific Research, and a son, Mohan Narasimhan, who, after obtaining an MBA and having worked in the US for several years, returned to India, where he teaches martial arts."
]
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"Narasimhan made important contributions in several areas of mathematics, including algebraic geometry, differential geometry, representation theory of Lie groups and analysis.\nHere, we will focus mostly on his work in algebraic geometry, and specially in the theory of moduli spaces of vector bundles on Riemann surfaces, with particular reference to works that are more familiar to the author.\nFor details, one can consult the Collected Papers of M. S. Narasimhan [",
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"The theorem of Narasimhan and Seshadri"
]
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"Upon his return to TIFR in 1960, Narasimhan embarked on an intense collaboration with Seshadri that resulted in the famous Narasimhan–Seshadri theorem, published in 1965.\nThis theorem captures the interconnection between various branches of geometry, topology and theoretical physics, and was the basis for later fundamental works by some of the greatest mathematicians of our time such as Michael Atiyah, Raoul Bott, Simon Donaldson, Karen Uhlenbeck, Shing-Tung Yau, Nigel Hitchin and Carlos Simpson, among others."
]
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"content": [
"The problem of classifying holomorphic vector bundles over a compact Riemann surface ",
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" of genus ",
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" is a central one in algebraic geometry.\nThe set of equivalence classes of holomorphic line bundles on ",
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" higher rank holomorphic vector bundles were classified by Grothendieck (1957), and in a different fashion by earlier work of Birkhoff (1909).\nThe case of elliptic curves (",
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" the problem is much harder.\nInspired by some remarks in the 1938 paper of A. Weil on “Généralisation des fonctions abéliennes”, Narasimhan and Seshadri started looking in 1961–62 at unitary vector bundles.\nA unitary representation ",
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"A breakthrough came with the work of Mumford on Geometric Invariant Theory.\nIn the 1962 International Congress in Stockholm, he introduced the notion of stability of a vector bundle on a compact Riemann surface, and proved that the set of equivalence classes of stable bundles of fixed rank and degree has a natural structure of a non-singular quasi-projective algebraic variety, projective if the rank and degree are coprime.\nLet ",
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"After they became aware of Mumford’s work, the relation with unitary bundles was clear to them.\nNarasimhan and Seshadri proved that an irreducible unitary bundle is stable.\nFor arbitrary degree they showed that the stable vector bundles on ",
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".\nFrom this, one deduces that a reducible projective unitary representation of the fundamental groups corresponds to a direct sum of stable holomorphic vector bundles of the same slope (what is nowadays referred as a ",
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"The Narasimhan–Seshadri theorem has been a paradigm and an inspiration for almost 60 years now for many important developments.\nThe theorem was generalised by Ramanathan (1975) to representations into any compact Lie group.\nThe gauge-theoretic point of view of Atiyah and Bott (1982), using the differential geometry of connections on holomorphic bundles, and the new proof of the Narasimhan–Seshadri theorem given by Donaldson (1983) following this approach, brought new insight and new analytic tools into the problem.\nIn this approach a projective unitary representation of the fundamental group is the holonomy representation of a unitary projectively flat connection."
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"The case of representations into a non-compact reductive Lie group ",
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"-Higgs bundles.\nThis correspondence was generalised by Simpson (1988) to any complex reductive Lie group (and in fact, to higher dimensional Kähler manifolds).\nThe correspondence in the case of non-compact ",
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".\nIt is perhaps worth pointing out that this theorem is a twisted version of an existence theorem of harmonic maps of Riemannian manifolds proved by Eells–Sampson (1964) pretty much around the same time as the theorem of Narasimhan and Seshadri.\nCorlette’s theorem, which holds for any reductive real Lie group, can be combined with an existence theorem for solutions to the Hitchin’s equations for a ",
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"-Higgs bundle, given by the author in collaboration with Bradlow, Gothen and Mundet i Riera (2003, 2009) to prove the correspondence for any real reductive Lie group ",
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"There is another direction in which the Narasimhan–Seshadri theorem has been generalised.\nThis is by allowing punctures in the Riemann surface.\nHere one is interested in studying representations of the fundamental group of the punctured surface with fixed holonomy around the punctures.\nThese representations now relate to the parabolic vector bundles introduced by Seshadri (1977).\nThe correspondence in this case for ",
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" was carried out by Mehta and Seshadri (1980).\nA differential geometric proof modelled on that of Donaldson for the parabolic case was given by Biquard (1991).\nThe case of a general compact Lie group has been studied by Bhosle–Ramanathan (1989), Teleman–Woodward (2003), Balaji–Seshadri (2015), Balaji–Biswas–Pandey (2017) and others, under suitable conditions on the holonomy around the punctures."
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"It was in the mid 1950s that the first papers of Narasimhan appeared.\nThey were devoted to the study of the Laplace operator on Riemannian manifolds (1956) and certain extensions of elliptic operators (1957).\nAfter these, he wrote a paper giving a new approach to the construction of Green’s function of an open Riemann surface (1960), and another paper studying the local properties of variations of complex structures on a relatively compact subdomain of an open Riemann surface (1961)."
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"Together with T. Kotake, Narasimhan proved a theorem characterising real analytic functions by Cauchy-type inequalities satisfied with respect to powers of a linear elliptic operator with analytic coefficients (1962).\nThis result was used in the original proof of the Atiyah–Bott fixed point theorem, and has been generalised in several directions by many authors, including Lions–Magenes, Bouendi–Goulaouic, Bouendi–Metvier and Bolly–Camus–Mattera."
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"], Narasimhan made an important contribution to the theory of representation of Lie groups.\nIt had been suggested by Langlands (1966) that, in analogy to the Borel–Weil–Bott theorem for compact groups, the Harish-Chandra discrete series of a real semisimple non-compact Lie group defining a symmetric space of Hermitian type could be realised as square-integrable harmonic forms in certain holomorphic vector bundles.\nThe work of Narasimhan and Okamoto was the first breakthrough in the proof of this conjecture.\nAlthough Narasimhan did not pursue this any further, his student Parthasarathy has contributed in an important way to this field."
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"] proved that the moduli space of vector bundles on a curve is locally factorial and determined the Picard group, showing that this is isomorphic to the integers.\nTheir results enable one to define a generalisation of the Riemann theta divisor of the Jacobian.\nThe famous Verlinde formula gives the dimension of the space of sections of powers of the theta line bundle (generalised theta functions) on the moduli space.\nTsuchiya–Ueno–Yamada (1989) had proved factorisation theorem and the Verlinde formula in the context of Conformal Field Theory.\nNarasimhan and Ramadas [",
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"In joint work with S. Kumar (1997), Narasimhan extended his result with Drezet to the moduli space of principal bundles over a compact Riemann surface with a simple, simply-connected connected complex affine algebraic structure group.\nAnd with S. Kumar and A. Ramanathan (1997), using the relation between principal bundles and infinite Grassmanians, they elucidated the relation between conformal blocks and generalised theta functions.\nThis enables one to compute the dimension of the space of generalised theta functions using the Verlinde formula.\nThis was also proved by Beauville–Lazlo (1994) in the vector bundle case."
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"Narasimhan and M. Nori (1981) proved that there are only finitely many smooth curves having a given abelian variety as the Jacobian.\nI. Biswas and Narasimhan (1997) studied Hodge classes of moduli spaces of parabolic bundles on general curves.\nWith Y. I. Holla (2001), Narasimhan proved a generalisation of a theorem of Nagata on a ruled surface to the case of a bundle of flag varieties associated to a principal bundle."
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"We were very lucky to have Narasimhan in Madrid on several memorable occasions.\nIn 2006 he participated in a panel, jointly with Sir Michael Atiyah, Jean-Pierre Bourguignon, Philip Candelas, José Manuel Fernández de Labastida, and Shing-Tung Yau on “New Interactions between Geometry and Physics”, organised in the context of a conference in honour of Nigel Hitchin for his 60th birthday, that took place in Madrid soon after the International Congress.\nIn 2012, the Instituto de Ciencias Matemáticas (ICMAT) in Madrid organised a conference in his honour for his 80th birthday, and later in 2017 he was invited as a special guest for a conference that ICMAT organised celebrating Ramanan’s 80th birthday.\nOn that occasion he participated in a special panel jointly with Antonio Córdoba, Nigel Hitchin and S. Ramanan on “Mathematics in India and Europe”.\nA photographic exhibition on “Kolam, an Ephemeral Women’s Art of South India” by photographer and anthropologist Claudia Silva was opened after the panel."
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"Over the years, we had many discussions on the possibility of establishing a scheme for mathematical collaboration between India and Europe in our research field.\nThere had been some bilateral programmes between France and India, and we were contemplating the idea of bringing that to a larger context.\nIt took a long time, but eventually we established a collaboration programme involving four nodes in Europe (Aarhus, Madrid, Oxford and Paris) and four in India (Bangalore, Mumbai and two in Chennai).\nThis was the Indo-European Project on Moduli Spaces that was operating during the period 2013–2017, involving more than eighty mathematicians, funded under the Marie Curie Programme by the European Commission, and coordinated by ICMAT in Madrid.\nNarasimhan played an important role in the gestation of this project."
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"In addition to discussing mathematics and scientific collaboration, Narasimhan and I very much liked to enjoy a glass (or two!) of good red wine, very often in company of our common friend and collaborator Ramanan, and other good friends.\nMy wife and I were very fortunate to enjoy his great hospitality and that of his wife Sakuntala and daughter Shobhana, at his home during our many visits to Bangalore over the last few years."
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"From left to right: Guillermo Barajas, Guillermo Gallego, Gadadhar Misra and M. S. Narasimhan, Bangalore, 2020"
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"I last saw Narasimhan in person in Bangalore in February 2020, during an activity on Moduli Spaces organised at the International Centre for Theoretical Sciences (ICTS).\nOn that occasion we also had the opportunity to have a very nice dinner, accompanied as usual by good red wine, with our friend Gadadhar Misra and other friends.\nAfter the ICTS meeting, I went to Chennai for a few days, for a visit to the Chennai Mathematical Institute (CMI), where as a matter of fact I also saw C. S. Seshadri for the last time.\nI had actually met Seshadri in the late 1980s when I was still a graduate student at Oxford, where he gave a talk on parabolic bundles, a subject of great interest at the time in relation to Jones–Witten theory and the Atiyah–Segal approach to topological quantum field theory."
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"My students Guillermo Barajas and Guillermo Gallego also came to the workshop at ICTS in February 2020, and after that, while I was visiting the CMI, they went to the Indian Institute of Science to discuss with Narasimhan for a week.\nAs always Narasimhan was extremely generous, spending a lot of time talking with them and, together with Gadadhar Misra, entertaining them (Figure ",
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").\nThe last paper of Narasimhan, written jointly with Gallego and the author [",
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"] appeared just a few days before his passing.\nThe discussions with Barajas were very useful in connection with a joint paper of Barajas and the author (2021), which generalises to principal bundles and Higgs bundles the Prym-type construction given by Narasimhan and Ramanan (1975)."
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"In addition to being a great mathematician, Narasimhan was a wonderful human being.\nHe was kind, generous and sympathetic, and is very much missed by many people who loved him."
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"\nOscar García-Prada is a CSIC research professor at the Instituto de Ciencias Matemáticas – ICMAT in Madrid.\nHe obtained a DPhil in mathematics at the University of Oxford in 1991 and had postdoctoral appointments at the Institut des Hautes Études Scientific (Paris), the University of California at Berkeley and the University of Paris-Sud, before holding positions at the University Autónoma of Madrid and the École Polytéchnique (Paris).\nIn 2002, he joined the Spanish National Research Council (CSIC).\nHis research interests lie in the interplay of differential and algebraic geometry with differential equations of theoretical physics, specifically in the study of moduli spaces and geometric structures.\n",
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