From the Editor's Keyboard

FEW EVENTS associated with modern award of this prize has become a char- science have such a long and continu- acteristic feature at the ICMs, keenly ous history as the International Con- looked forward to by the delegates. gresses of Mathematicians. The idea More prizes and various new fea- of such meetings germinated with the tures have been added at the recent "world congress" of mathematicians congresses. and astronomers organized as part of the World's Columbian Exposition The ICMs are also unique among aca- held in Chicago (The Chicago World's demic events, as their organization is Fair) in 1893, and the first official Inter- looked after by a body representing national Congress of Mathematicians the world mathematical community, was held in 1897, during August 9–13, that regenerates itself regularly via in Zurich. The constitution set down at new inputs through elections – the Zurich envisaged having a congress at International Mathematical Union intervals of three to five years, and it (IMU). In the current form the IMU was was also decided to hold the second established in 1951 (renewing the IMU congress in Paris in 1900 on the oc- that was in existence during 1919 to casion of that city's Exposition Univer- 1936) and it has been regularly orga- selle marking the end of the century. nizing the ICMs since then, apart from The famous problems formulated by undertaking various activities pro- Hilbert that have since played an enor- moting international co-operation in mous role in the mathematical devel- mathematics. opments to-date, were a highlight of the Paris Congress. ICMs have since The ICMs have served as an occasion then taken place with remarkable to review the major developments in regularity at intervals of 4 years, except the subject, and to look into possible during the World War years; the reader future directions in various branches, may refer [1] for many interesting de- and above all have played a major role tails about the ICMs until 1986. In 1936 in unifying the practitioners of mathe- the Fields medals were instituted and matics into a global mathematical over the years announcement and community. Many talks at the ICMs

1 have been trend-setters in various The Hyderabad Intelligencer is orga- topics. The participation in ICMs has nized into 4 sections. We begin with grown enormously over the years, with welcoming messages, with some very the growth of the subject itself and enthusiastic and enlightening words increasing globalization of the com- of hospitality, from Professors László munity. Until 1986, the venues of the Lovász, President, IMU, M. S. Raghu- congresses had been in the western nathan, Chair of the Local Organiz- world. In 1990 the congress was held in ing Committee of ICM 2010, and S. E. Kyoto and in 2002 in Beijing, covering Hasnain, Vice Chancellor of the Univer- new ground. ICM 2010 in India makes sity of Hyderabad which is co-ordinat- a new landmark in the global spread ing the ICM. We invited also Professor of the event. Of course we have still a S. R. S. Varadhan to join this "recep- long way to go and undoubtedly del- tion committee", so to speak, and he egates at the Congress, and mathema- graciously agreed. Lovász places the ticians around the world would like to ICMs in perspective from the point see it happen in their neighbourhood, of view of the mathematical com- if not in their own country. munity. Raghunathan brings out the Indian context for ICM 2010. Hasnain It was indeed thoughtful of Springer would convince you of the allure that Hyderabad holds. Varadhan reminisces publishers to have come up with the over his India years and the changes in idea of putting together an "Intelli- a mathematician's life over time! In the gencer", on the occasion of the ICMs. context of their messages I may recall While venturing into a new territory here for the reader's benefit Raghu- and cultural setting, which would be nathan's article "India and The World the case with a large number of del- of Mathematics" [8] and an interview egates at any ICM, an introduction to of Varadhan by Rajendra Bhatia in the place, in terms of the surround- The Mathematical Intelligencer [2] for ings as well as its mathematical iden- related reading. tity, would go a long way in putting the guest at ease. It is commendable We then have a section on ancient that Springer has been providing Indian mathematics. It is common such a service to the delegates on a knowledge that India has had a long complimentary basis, at congress af- and fairly uninterrupted tradition in ter congress, since the 1978 Helsinki mathematics. However, on the whole Congress. there is not much familiarity with the details of the mathematical develop- It was with some trepidation that I ac- ments in the ancient times, even in In- cepted the invitation to be Editor for dia. We bring to our readers a detailed the Hyderabad Intelligencer, awed perspective on ancient Indian math- by the responsibility it would hold to ematics by Professor Kim Plofker, who convey to the delegates in a limited is one of the plenary speakers at ICM issue the essential features of the 2010 (a confession: she was invited to mathematical identity of the country, write the article before it was known that I considered a part of the task. that she was to give a plenary talk!). In I gradually felt relieved, as I received connection with the article it would very good co-operation from in- be befitting to draw the reader's at- vited authors on the one hand, and tention to the author's book [6], and the technical editor Dr. Priti Nanda, a its review by David Mumford [4]. One medical doctor now devoted to the of the current hot topics in the stud- science publication activity (!) and the ies in ancient Indian mathematics is staff of Springer, and it is my earnest discoveries in Calculus made in Kerala hope that what we have put together starting from the 14th century. Profes- would be gratifying to the delegates sor Ramasubramanian brings to us at the Congress and other readers of some of the mathematical details of this issue. the discoveries.

2 Mathematics in the modern era is the We conclude the section with a spe- theme of our next section. I have tried cial tribute to the legendary Indian to trace the rise of modern mathemat- mathematician Srinivasa Ramanujan, ics in India. It is a pleasure to especially bringing to our readers various quota- acknowledge the indebtedness to tions recreating his genius. [5], [7] and [3] (apart of course from the individual sources noted in the The following section is intended references with the article) and also to present to the reader glimpses of to bring them to the reader's atten- the mathematical life in India. Profes- tion for further reading. The period sor David Mumford has had a long covered extends until the middle of association with the mathematical the 20th century. Coming to the more community in India, having spent recent times we present an article by extended periods in the country and Professor Ramadas on the rise of Geo- having made many friends among In- metry in India. There is of course a lot dian mathematicians. We have here an more to Indian mathematics of our article by him recounting his experi- times. Mathematical activity in India ences, that covers a broad spectrum of grew multi-fold in the second half of life in India. Professor Bhatia, an author the 20th century. The Tata Institute of of several books, throws light on the Fundamental Research, with centres at Mumbai and Bangalore, has been mathematical life in India through a at the forefront in research in a vari- discussion on books that nourished it ety of areas of mathematics. Various over the years. Professor Ramaswamy other Institutes, the Indian Statisti- dwells on the gender issue, describing cal Institute (with centres at Kolkata, the Indian experience in women's par- Delhi, and Bangalore), the Institute of ticipation in mathematics, based on Mathematical Sciences, Chennai, the the narratives by women mathemati- Harish-Chandra Research Institute, cians in the book Lilavati's Daughters, Allahabad, the Indian Institute of Sci- of which he is one of the Editors. ence, Bangalore, have also contributed significantly. Many "Indian Institutes of Apart from introducing the mathe- Technology" have been established matical community with these articles and while their main thrust has been we also bring to the reader a variety of on Technology and Engineering they information introducing Hyderabad. have also contributed to mathemati- A historical portrait of the city by Mr. cal research in the country substan- Narendra Luther, a former civil servant tially. In the competitive scenario for who is a noted authority on the history research, the universities have been and culture of Hyderabad, takes the at a disadvantage, in terms of research reader on a tour of the city through environment and infrastructure, but time and space. We also include use- nevertheless there are many math- ful information on various aspects of ematicians at various universities who interest to the visitors. have made worthwhile contributions. Indian mathematicians have made I would like to thank all the authors, their presence felt in almost all major the editorial staff involved in the pro- branches of mathematics. It would be duction of the articles, and also nu- unrealistic to try to convey even in a merous people, mathematicians as nutshell the achievements in differ- well as others, whose comments and ent branches, in an issue like the pres- suggestions led to improvements in ent one. It is hoped that the article of the articles (including mine!). Above Ramadas would serve as a sample in all, special thanks to Springer India communicating the strength and glo- ry of contemporary Indian mathemat- for providing me this wonderful ics and that the reader will get to learn opportunity. about the achievements in other areas S. G. Dani through other means. Editor

3 References [5] R. Narasimhan, Coming of age of Math- ematics in India, Miscellanea Mathe- [1] Donald J. Albers, G. L. Alexanderson, matica, Ed. P. Hilton, F. Hirzebruch, and and Constance Reid, International R. Remmert, Springer-Verlag, 1991. Mathematical Congresses, An Illustrated [6] Kim Plofker, Mathematics in India, History, 1893–1986, Springer-Verlag, Princeton University Press, Princeton, 1987. NJ, 2009 (xiv+357 pp). [2] Rajendra Bhatia, A conversation with S. R. S. Varadhan Math. Intelligencer [7] M. S. Raghunathan, Artless innocents 30 (2008) (No. 2), 24–42. and ivory-tower sophisticates: some personalities on the Indian mathemati- [3] Joseph W. Dauben and Rohit Parikh, cal scene, Current Science 85 (No.4), 25 Mathematics in India (Unpublished). August 2003.

[4] David Mumford, Mathematics in India [8] M. S. Raghunathan, India and the World (review of the book Mathematics in of Mathematics, available at: http:// India by Kim Plofker), Notices of the www.icm2010.org.in/mathematical- American Math. Soc. 57 (2010), no. 3, miscellany/indian-world-mathemat- 385–390. ics\end{thebibliography

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4 László Lovász Eötvös Loránd University, Budapest President, IMU

ICMs and the Mathematical Community László Lovász

WELCOME TO the International Congress embody the main purpose for our of Mathematicians, 2010, in Hyderabad! being here, enlighten us on the latest developments. The Prize Committees The ICM is the single most important and various experts contribute their event in mathematics, held once opinion towards the decisions about in four years. Its organization is the the coveted prizes, awarding of which most important task for the Executive has been a major highlight of the Committee of the International Congresses. Editors and publishers Mathematical Union, and indeed for of the Proceedings help us retain the whole community. the record of the event for posterity. Numerous mathematicians and Let me highlight the tasks performed journalists endeavour to ensure that by various entities in the organization the general public learns about the of this mammoth event. The local significance of the Congress. Apart organizers set up the venue for the from these, the Congress is also programme of the lectures and special enriched by many colleagues who meetings, arrange social events, wrote letters with suggestions for coordinate housing and travel, and special events and ideas about their attend to many-many more details scope, who sent nominations for towards successfully conducting the prizes and lectures, who have come event. The Programme Committee to attend the Congress, as also those holds the responsibility of addressing who sent their students to the the enormous task of selecting plenary Congress. and sectional speakers, with the objective to represent, as broadly as Arriving at the Congress venue and possible, the recent developments looking for the lecture halls may be in mathematics. The speakers, who a frightening experience, especially if

5 you are attending a congress for the and well recognized already, but to first time. You might walk down the young people and newly obtained crowded halls for a long time without results, thereby calling the whole seeing a familiar face, and perhaps community's attention to these you will be able to follow only a small budding mathematicians and their fraction of the talks. A lot of effort, has achievements. actually been made to ensure that the invited talks, especially the plenary It is a special pleasure to announce talks, be accessible to a general mathe- that a new prize, the Chern Medal matical audience, but it still would be Award, will be given for the first time difficult to follow so many ideas from at the 2010 Congress. This award, different branches of mathematics established in memory of the out- within such a short time. No wonder standing mathematician S. S. Chern, that one often comes across skepti- will be given for lifelong contribution cism about holding the Congresses. to the field of mathematics. I trust But if you talk with someone from you will find it a special privilege to physics or computer science, or from be present at the first awarding of other branches of science, you will the Chern Medal. find that he or she is envious of the fact that we mathematicians have ICM 2010 has a lot more to offer, the benefit of such a special event including various special events, and where one can listen to carefully of course, most of us would also like to selected speakers describing the latest take advantage of this opportunity to developments, witness awarding of imbibe the beautiful scenery, historic the most important prizes, have panel monuments, culture, and art that discussions on important issues, etc. India offers.

The Fields Medal and the Nevanlinna I wish you a pleasant and rewarding Prize are unique in their scope: they stay in Hyderabad. award the highest recognition not to senior people whose work is known Email: [email protected]

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6 M. S. Raghunathan TIFR, Mumbai Chair, LOC, ICM 2010

An Indian Endeavour M. S. Raghunathan

IT GIVES me immense pleasure to wel- India has had a long engagement come, on behalf of the Indian math- with the pursuit of mathematics. The ematical community, all delegates to Pythagoras theorem figures in Baudh- ICM 2010, in this city of Hyderabad. ayana Sulvasutra, a work going back As most of you would already know, to the 8th century before Christ – two this is the first time that India is host- centuries before Pythagoras. The con- ing the Congress; and it is only the ception of zero as a number and its third time that the Congress is being use in the place-value system seems held in Asia. The first Congress held in to have originated from here in the Asia was organised by the Japanese early centuries after Christ. Aryabhata in Kyoto in 1990. I had attended the (476 – 550 CE) and Brahmagupta (598 2002-Congress in Beijing, and like all – 668 CE) were mathematicians of the other delegates was greatly impressed first rank who among other things, by the superb way in which the Con- laid the foundations of Algebra. There gress was organised. It is our hope to were two other mathematicians of be able to emulate our Chinese col- note—both going by the same name leagues and that our guests will carry of Bhaskara belonging to the 7th and back very pleasant memories of this 12th centuries. The 14th and 15th cen- event. turies saw the emergence of a great school of mathematics in south west The subcontinent has fostered struc- India near the present-day town of tured intellectual activity from be- Calicut. Madhava, the leading figure fore the advent of the Christian era. of that school had, in essence, discov- Taxila (in Pakistan) and Nalanda (in the ered calculus two centuries before Indian state of Bihar) were great centres Newton and Leibnitz. Except for that of learning in ancient times, organised last flicker, the 15th century which saw along lines not dissimilar to modern the renaissance propel Europe into universities. In the deep South there a leadership role in practically every was the Tamil Sangam, a forum for po- human endeavour, seems to have ets and scholars resembling modern been the start of a decline of intellec- literary academies. tual pursuits in south Asia.

7 This country resumed its tryst with the Union in 1952. Marshall Harvey Stone, queen of sciences after a break of a few the man whose vision and extra- centuries with the advent of Srinivasa ordinary efforts resulted in the "new" Ramanujan in the second decade of IMU thanks among others, the Indian the 20th century. There were other bril- National Committee for its help in liant figures that emerged who – even drawing up the statutes of the IMU. if they were not in the same league K. Chandrasekharan, an eminent as Ramanujan – kept the mathemati- Indian mathematician served on the cal scene lively. Two institutions, the Executive Committee of the union with Indian Statistical Institute of Kolkata distinction for 24 consecutive years with branches in Delhi and Bengaluru (starting 1954), five of them as Secre- and the Tata Institute of Fundamental tary and four of them as President. Research of Mumbai with a center in Bengaluru have excellent schools of We look forward to a mathematically mathematics and have helped im- stimulating Congress. We have orga- mensely in establishing India firmly on nised programmes that will present the mathematical map of the world. to you, our guests, some glimpses of our cultural heritage by way of music We are delighted that we have been and dance. We do hope that you will given this opportunity to host the ICM find these a rewarding experience. The by the International Mathematical city of Hyderabad is half a millenium Union. The ICMs have been exemplary old and is steeped in history. It is one fora for international collaboration in of our fast growing metros and is a mathematics; and India, ever since it strong competitor to Bengaluru as an emerged as an independent nation, IT hub of the country. India has a lot to has been an active and ardent sup- offer to the tourist and I urge each of porter of international co-operation you to take this opportunity to spend in diverse areas. India had formed a more time in this country and savour "National Committee" which actively some of its touristic delights. participated in the meetings that were the run-up to the formation of the (present) International Mathematical Email: [email protected]

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8 Seyed E. Hasnain Vice-Chancellor University of Hyderabad, India

Allure of Hyderabad Seyed E. Hasnain

India has had a glorious tradition of the country. Hyderabad represents a mathematical discoveries. Aryabhata blend of tradition and modernity and (476–550 CE) made outstanding con- is often seen as a macrocosm repre- tributions in various areas including senting the entire nation. With India’s astronomy, plane and spherical trigo- largest Convention Centre located in nometry, algebra, and linear diophan- the city, it is no wonder that Hydera- tine equations . . . From the invention of bad continues to attract mega confe- the place value system to the brilliant rences not only from within India but work of Srinivasa Ramanujan there has also from abroad. been a continuous stream of contri- butions from Indian mathematicians The University of Hyderabad is a Cen- some of which had reached Europe tral University (an institution of the fed- via the Arab world. It is therefore only eral Government of India), established natural that one of the ICMs should be in 1974 by an Act of the Indian Parlia- held in India too. ment. In the relatively short span of three decades, it has acquired a lead- The city of Hyderabad has emerged ing role in the academic life of India as the most happening place in the through excellence in research, com- country. The city can boast of more mitment to education of the highest than 40 institutions of higher learning quality, and efforts to harness knowl- R & D including three Central Univer- edge for development. The Ministry of sities, the Indian School of Business Science and Technology, Government (ranked as the No. 12 business school of India has ranked the University of in the world) and a large number of R Hyderabad amongst the top two uni- & D and production centres of phar- versities in the country and the univer- maceutical industries. The city also has sity has been awarded the ‘Promotion the world’s second largest production of Universities Research & Scientific Ex- facility of recombinant DNA products cellence ‘PURSE’ Award’. The University and can rightly be proud of being the has been reaccredited by the NAAC largest producer of generic drugs in and has been awarded a cumulative

9 grade point average (CGPA) of 3.89 on from the University, both academic as a scale of 4.00 at ‘A’ grade, which is the well as those involved with providing highest (97%) ever given to any uni- infrastructure have contributed enor- versity. According to the SCOPUS da- mously to the efforts, and I would like tabase analyses of multi-disciplinary to take this opportunity to record my areas and over-all excellence, the Uni- appreciation for all of this. versity has been ranked number 1 in the country. Having successfully com- I am now greatly pleased to welcome, pleted the ‘University with Potential for on behalf of the University and on my Excellence Programme’ (2002–2007) own behalf, the many delegates from sanctioned by the UGC, the University all over the world who have come to is looking forward to being declared participate in the ICM. I am sure that as a “University of Excellence” by the this ICM will become a landmark event UGC. in the mathematical calendar of the new decade, thanks to interaction with It was but natural, therefore, for our the best mathematical minds over the Department of Mathematics, and the nine days. It is also a great pleasure University as a whole, to endeavour to welcome delegates to the Interna- towards hosting the International tional Conference of Women Math- Congress of Mathematicians and it ematicians (ICWM), the first ever such was a matter of great pleasure and event linked with an ICM. The ICWM celebration for us when the General which celebrates the achievements of Assembly of IMU accepted the pro- women in mathematics, is also being posal at its meeting in 2006 in Spain. supported by the University. During the four years since then, the University has provided strong sup- I trust our efforts will ensure that the port in various respects to the efforts stay will be comfortable and fulfilling of the Local Organizing Committee to our guests. Visitors are encouraged towards organization of the event and to visit the University of Hyderabad, the Department of Mathematics of the known for its sprawling and idyllic University has been the hub for all the campus, which is not far from the ven- preparations. I would especially like to ue of the Congress. I urge our interna- compliment Professor Rajat Tandon, tional guests to take full advantage of who served as Secretary of the Execu- the Indian hospitality, to experience tive Organizing Committee (EOC), and the traditions and culture of our coun- Professors T. Amarnath and S. Kumare- try and to feel our vibrant and dynamic san who were members of the EOC society. and also various subcommittees, for the roles played by them in prepara- tions for the event. Many more people Email: [email protected]

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10 S. R. S. Varadhan New York University, USA

Then and Now S. R. S. Varadhan

I AM greatly pleased to welcome you history, that is in many ways symbolic to India, where I was initiated as a of the country as a whole. mathematician. It is a wonderful feel- ing for me to be here at this first ICM in For the mathematical community, India, a homecoming for such a unique the ICMs serve to look back and take and prestigious event. I have been in stock of the achievements in various contact with the mathematical life in branches of mathematics and to peep India through my visits to various insti- into the future to the extent possible. tutions, participation in conferences, While we do that collectively, I find it and witnessing the developments in also fascinating to contemplate on the the country over the years, and this changing everyday life of a mathema- ICM seems like a crowning event to tician. I have spent the last 50 years as celebrate together with the Indian a research mathematician and the way mathematical community. These con- we carry on with our daily academic gresses provide us all a great opportu- routine has changed a great deal. I re- nity to learn about the latest advances member the first time we needed to in various areas, many other than get hold of a publication not available our own and, meet and interact with in our library at the Indian Statistical mathematicians from different parts of Institute, Kolkata. The Indian Math- the world – a whirlwind of experiences ematical Society had a good library, packed in a period of a few days. located at the Ramanujan Institute of Mathematics in Chennai, and we While my own roots in India are largely could get the required journal shipped connected with Madras (now Chennai) to us. It took about 10 days for the mail where I grew up, and Calcutta (now to arrive and the library was asked to Kolkata) where I got my Ph.D. – the copy the article of interest. This would first stamp as a mathematician – I am involve the Institute's photographer nevertheless happy to welcome you taking pictures of the article page by here at Hyderabad, a thriving, mod- page, developing it, printing it, drying ern, multicultural city with a colorful the print, and then binding it into a

11 booklet that would be catalogued and from blackboard and (white) chalk, to placed as an item in the library. It took whiteboard and colored pens, to pre- anywhere between a month and 2 sentations with computers that pro- months for all of this to happen. Then vide a world of possibilities to make came the days of photocopiers, initially the presentations more elegant, easier from Xerox, a term which became syn- to follow, and enjoyable. onymous with photocopying. Life be- came easier, but mind you in the early But the way we present our research, days there used to be constraints and i.e., the way we write our work for pub- rigorous accounting of how much one lication has not changed in any funda- could photocopy at the department. mental way over the last 50-years. We But now the internet and google have still write linearly for the print medium. made almost everything available in- Some are good at it and some are not. stantly – just a mouse click away. And A long paper of 50 pages or more per- yet, the connection can be slow and haps will have a certain small number the downloading may take a few ex- of key ideas that are important and tra seconds, so we still have reasons to other logical steps needed to estab- get annoyed and complain! lish these key facts. This is more like a tree with logical steps reaching down The world of writing and publishing to the root, which is the main result. which is an integral part of a research While we cannot exhibit this in a con- career has changed too. In the good ventional publication, it is possible in old days one wrote an article by hand a purely electronic medium. A good and gave it to the mathematical typ- expositor can convey this in a lecture, ist; if you were a graduate student, but it is very hard to achieve this in a or junior faculty without much clout, linearly ordered text. the work could linger at the end of a queue for ever so long. Now many This has not happened yet, but per- of us, a growing tribe, key in the text haps will, in the near future; an ani- directly on our laptops and, thanks to mated imagery with diagrams, words Knuth, format it with Tex or such other and equations may be used to explain software. We can make changes so a concept much better than a printed easily to correct mistakes, as long as book. I am sure we will get there. a correction is possible! The way we present our work has changed too, Email: [email protected]

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12 Kim Plofker Union College, Schenectady, NY USA

A Perspective on Ancient Indian Mathematics Kim Plofker

The Indian mathematical tradition was an important trading partner is closely intertwined with many of contemporary urban societies in historical, linguistic, and sociocultural Mesopotamia. We can infer from its factors that are unique to classical trade connections and from standar- Indian civilization. At the same time, dized weights and measures found in it constitutes an important node its archaeological sites that it possessed in the global web of pre-modern some form of commercial and practical mathematical knowledge, through arithmetic. Unfortunately, however, which Indian science maintained no legible documentary record like connections with its counterparts all that found in the cuneiform tablets over the ancient and medieval world. of Sumer and the Akkadian empire Here we briefly trace the development survives from these early urban Indian of mathematical ideas in India, placing cultures. them in the wider cultural context. The oldest mathematical knowledge 1. Searching for the beginnings to have been passed down to the era of recorded history in South Asia Modern humans have lived in South is attested (very incompletely) in the Asia for at least 50,000 years, and earliest hymns in the corpus of sacred their first known urban civilization in compositions known as the Vedas. the region (comprising the various These hymns were composed and Harappan and Indus Valley cultures, orally transmitted in Vedic or Old Indo- in the northwest of the subcontinent) Aryan, an archaic form of Sanskrit, emerged around 5,000 years ago. the chief classical Indian language. From at least the mid-third millennium Old Indo-Aryan is an Indo-European BCE onwards, this complex of cultures language related closely to Old Iranian

13 (Avestan, the sacred language of 2. The allure of large numbers Zoroastrianism), and less closely to Latin, Greek, Lithuanian, and the other Later Indian sacred texts appearing members of the Indo-European family. in the millennium (or thereabouts) This linguistic relationship is one of following the early Vedic hymns also the few available keys for dating the involve various sequences of number Vedic texts, which were not recorded words, some of which are protracted in writing until many centuries after to surprisingly large magnitudes, as their composition, and which contain in Yajur Veda 7.2.20, which invokes no known references to historical successive powers of ten up to a trillion. events or other unambiguous internal Such vast numbers could have had no indicators of their date. However, since practical application in the context it is still quite uncertain when, where, of Vedic culture, but were apparently and how the different Indo-European deemed significant for more abstract languages diversified, the age of the reasons. This fascination with the Vedas also remains uncertain, as do precise specification of unrealistically the relationships between the Indo- immense quantities persisted into Aryan cultures that created them and the post-Vedic period of the great the (apparently) earlier urban Indus Indian epics. For example, the famous cultures. Most scholars assign the Mahābhārata, compiled probably oldest parts of the Vedic canon to the sometime in the early first millennium mid- to late second millennium BCE, CE but containing material many and the younger parts to various dates centuries older, describes a cosmic within the subsequent thousand years time-span over four billion years long, or so, based on intertextual references made up of 360-year periods called and modifications in the constantly “divine years”: evolving form of the language. The sages say twelve thousand The content of the Vedic texts is of these [divine years] is a yuga devoted to hymns, prayers, spells, ritual [cycle]; a thousand elapsed [cycles] practice and exegesis, cosmological are called a day of Brahman speculation, etc. Their reflections of (Mahābhārata 12.224.28). the mathematical thinking of the culture that created them are mostly The literature of Buddhism and Jainism, fleeting and incidental. Among the which emerged about the middle of scattered mathematical concepts the first millennium BCE as religious appearing even in the early hymns is and philosophical systems distinct a fully developed system of decimal from the Vedic tradition, reflects similar number words, attested in verses like interest in the potential vastness of the the following: world of numbers. The major account of the life of the Buddha, the Lalitavistara devahitim.  MXJXSXU GYÂGDĈDV\D. Utum. ... codified perhaps in the early first JDYÂP. man. d. ĖNÂ GDGDWDK. ĈDWÂQL millennium CE, describes the young VDKDVUDVÂYHSUDWLUDQWDÂ\XK. Buddha’s victory in a mathematics competition where he could not They have kept the order of the only solve any problem posed to twelve-month as ordained by the him but also recite from memory the gods . . . By giving hundreds of names of 23 successive powers of cows, the frogs have prolonged 100, beginning with a hundred times life in a thousand soma pressings ten million. The Buddha went on to 1 (R. g Veda 7.103.9–10) . describe even greater sequences of

1 The Sanskrit text of the hymn is transliterated from the edition in (Kashyap and Sadagopan [5]); the translation of the first excerpt from the famous “Frog Hymn” is from Doniger [2], p. 234. A transliteration is included only in this instance to give the reader a flavour of the original; in the sequel, in the illustrative examples only (close) translations are given.

14 numbers that could only be conceived designing and building baked-brick by mind-boggling comparisons with, altars for ritual sacrifices, the oldest for example, the number of grains of of which date back to perhaps 800 sand in the beds of a hundred million BCE or thereabouts. For reasons not Ganges rivers. completely clear after the lapse of so many centuries, it had become an Such cosmic speculations were important goal among Vedic people carried even further in some Jaina to perform certain sacrifices on altars texts, which explored abstract quanti- that conformed accurately to various tative concepts such as enumerability geometric figures such as circles, and infinity. Quantities were classified rectangles and squares, triangles, as enumerable, innumerable, or and isosceles trapezia. (The forms infinite, with various sub-classifi- of altars could also be zoomorphic, cations in each category. Attempts resembling a falcon, tortoise, etc., or were made to describe inexpress- derived from constructed objects ibly large numbers by approximate such as chariot wheels; generally the paraphrases in physical terms and, at required proportions in the figures least by the mid-first millennium CE, and number of the bricks to be used by deft mathematical manipulation were also stipulated). The manuals of concepts akin to exponents and contained instructions in brief Sanskrit logarithms. For instance, the smallest prose aphorisms for constructing innumerable number of the lowest these figures or transforming one order is defined as the number of such figure into another while mustard-seeds contained in a pile with preserving its area. The tools used for For reasons not completely clear a circular base whose radius is defined these constructions were stakes and after the lapse of so many centuries, by laying out at given intervals all the cords, whence the name Śulba-sūtra it had become an important goal mustard-seeds contained in a (smaller) (“cord-rules”) for this genre of texts. among Vedic people to perform pile whose radius is defined by laying out at given intervals all the mustard- Even the oldest of the “cord-rule” certain sacrifices on altars that seeds contained in a (still smaller) pile works reflect a sophisticated geo- conformed accurately to various whose radius is defined . . . and so on, metric understanding, including stan- geometric figures such as circles, dard technical terms for particular where the number of steps in the rectangles and squares, triangles, recursion is the cube of the number of quantities and figures as well as mustard-seeds required to build such awareness of abstract relationships and isosceles trapezia. a pile covering the entire inhabited like the “Pythagorean” theorem. The region of the earth in the Jaina cosmos. following excerpts give some idea Jaina scholars used a technique called of the mastery of geometric and “multiple-multiplication” (essentially, arithmetic manipulation that these raising a number to its own power) to manuals reveal (Sen and Bag [10]; the find an approximate lower bound for quoted excerpts are from pp. 18–19). this quantity, which is about ten to the The diagonal cord of a rectangle power of ten to the power of ten to [literally “long quadrilateral”] the 45th power. produces [a square area equal 3. Ritual geometry and other to] what both the corresponding mathematical pursuits long side and cross-side produce separately (Baudhāyana-Śulba- In the late Vedic and post-Vedic periods sūtra 1.12). after the turn of the first millennium Increase the [given] amount by BCE, priests, officials and scholars one-third and [increase] that [third] composed the earliest surviving by one-fourth less its own 34th Sanskrit sources that contain not just [part]. [That is] the diagonal [of a references to number words but also square with side equal to the given detailed expositions and attestations amount]. (Baudhāyana-Śulba-sūtra of mathematical activities. Probably the first such texts were manuals for 2.12).

15 In other words, the diagonal of a Greek astrological and mathematical square with side s is considered to be concepts to form the basis of later Indian mathematical astronomy. Authors on music and prosody used combinatorial techniques to enume- or approximately 1.414216 s. rate and order the different poetic During the so-called “second urbani- meters that could be created by zation” phase in South Asian history, combining “heavy” and “light” syllables beginning around the middle of the in different patterns. Although there first millennium BCE with the growth are unfortunately no extant instruc- of cities from which rulers controlled tional treatises pre-dating the middle extensive kingdoms, other significant of the first millennium CE that applications of mathematics left their expound these subjects in detail in traces in contemporary literature. the way that the specialist “cord-rule” For instance, we know from the manuals elucidated ritual geometry, Arthaśāstra, a Sanskrit treatise on the surviving hints are sufficient to infer political and military administration the thriving state of the development composed around the 3rd century of mathematics in India during this BCE, that the administrative staff at period. a royal court included scribes and “calculators”, who computed and 4. Written numerals and reported on financial transactions. mathematical notation Priests specializing in the study of astronomy calculated calendric cycles Ancient Sanskrit learning, particularly to synchronize the ritual observances its corpus of sacred texts, was traditio- required by sacred texts when the sun nally passed on by oral recitation and and the moon were in certain posi- memorization. There is no surviving tions; after the invasion by Alexander’s indisputable evidence for writing in army and the establishment of Indo- any Indic language before the inscrip- Greek petty kingdoms in the north- tions of the emperor Aśoka in the west, these calendric computations 3rd century BCE (although there is a (jyotis.a in Sanskrit) merged with some plausible argument that writing may

This "17" is found on the walls of a man-made cave on the ancient trade route from the coast around present Mumbai through the Nana Ghat, in Western India, to the interior. It dates from roughly 100 BCE. It records a donation of 17 waterpots to a nearby temple. There should be little doubt that our current decimal digits derive from this culture. (Courtesy: Bill Casselman)

16 This is copied from G. R. Kaye's 1930 edition of the Bakhshali manuscript. It is Folio 46 recto. The manuscript was found in 1881 in what is now Pakistan, near the town of Mardan. Dating it is problematic, but Takao Hayashi dates the manuscript itself to around 800 CE and the mathematics in it somewhat earlier. There were no decimal fractions available to the author, but as in other Indian mathematics originating in this period, the author is used to – perhaps even enjoys – handling rational numbers with almost incredibly large numerator and nominator. The computation is part of the verification of a solution of a quadratic equation, and this analysis allows one to reconstruct the entire fraction (which continues off to the right on the missing part of the birch bark fragment) as 50, 753, 383, 762, 746, 743, 271, 936 / 7,250, 483, 394, 675,000,000 . (Courtesy: Bill Casselman) have been introduced somewhat or perhaps it resulted from an earlier, in the “second urbanization” independent modification of the period mentioned above). These Indian non-place-value decimal nume- and other inscriptions prior to the rals that were already widespread in 5th century CE attest to systems of Aśoka's time. non-place-value decimal numerals composed of separate symbols for Ironically, it was the persistent tradition the numbers 1–9, 10–90, and at least of orality in Sanskrit scholarship that some multiples of 100 and 1000. gave rise to some of the best available evidence for the early development of Inscriptional evidence for the fully place-value decimal numerals in India. developed place-value decimal nume- Since verse compositions that could ral system (including the zero symbol) be memorized and recited remained that eventually spread from India to the ideal in Sanskrit didactic literature, the whole world dates from a later scientific texts were composed in period. The earliest known inscriptions verse, and their authors needed containing such numbers are no older flexible ways of expressing numbers than the late 7th century CE. However, that could fit into the metric structures we can be sure that Indian place-value of verse. One of the solutions they numerals are significantly older than devised was a sort of verbal encoding these inscriptions, dating probably to where ordinary words stood for the centuries around the turn of the numbers associated with them, as Common Era whose mathematical “eye” or “hand” for two and “void” for activity, as we saw in section 3, is so zero. Sequences of such words strung sparsely documented in surviving together represented larger numbers, texts. This lack of documentary details and could be reliably interpreted denies us the ability to reconstruct only as a sequence of digits (which exactly how or when the ten-digit by convention began with the least system emerged: Perhaps it was significant): thus the sequence “void- originally inspired by the almost eye” would inevitably mean “zero units complete place-value system used and two tens” or 20, a meaning that for counting-rod numerals in China, depended crucially upon the place-

17 value principle. Since verbal number abacus-like device where a row of sequences of this type are found in small pits held tokens or counters verse compositions beginning as early to represent the digits of a number; as the 3rd century CE, it can be deduced written calculations, on the other that mathematical practitioners were hand, seem to have been performed fully familiar with the idea of place by scratching numbers on the ground value by that time. Other hybrid or on a dust-board. We know from verbal-numerical systems, invented texts as early as the 5th and 6th probably no later than the 4th and 5th centuries CE that unknown quantities centuries CE, also attest to knowledge in calculations, frequently called of the place-value principle: They yāvattāvat or “as much as so much”, assign to the sounds of the Sanskrit could also be designated by names of alphabet the decimal digits, enabling colors or by a word meaning “bead”, encoding of numbers by stringing which may indicate an earlier practice the combinations of sounds together of using small tokens (of different to form sequences of syllables, which colors?) along with written numbers as may or may not take the form of actual a sort of proto-notation. At least by the Sanskrit words; (Plofker [7], pp. 43–57 start of the 7th century, a purely written -A plausible argument that the Indian notation for arithmetic and algebra place-value decimal system was manipulations had been developed, known even by the end of the first employing equations whose two millennium BCE is given in Sarma [9]). sides contained various combinations of terms, with abbreviations standing At least by the start of the When written out in numeral form, for the names of unknowns and seventh century, a purely written place-value integers in Indic left-to- operations. Perhaps around this time, right scripts appeared with the most the abbreviation for “subtracted” notation for arithmetic and algebra significant digit first, that is, on the morphed into a symbol somewhat manipulations had been developed, left side. It is clear from textual and equivalent to our minus sign, in the employing equations whose two inscriptional evidence that by the late form of a small perpendicular cross sides contained various combina- first millennium CE, fractions were immediately following the designated written vertically with the numerator negative number. (Later texts indicated tions of terms, with abbreviations over the denominator, and zero was negative numbers with an overdot.) standing for the names of represented by a circle or round dot. There were conventional formats unknowns and operations. Sexagesimal or base-60 fractions for laying out and manipulating the (originally adopted from the Greco- quantities in certain types of problems, Babylonian sources mentioned in such as reducing algebraic equations, section 3) were written as sequences multiplying multi-digit numbers, of decimal numbers like our own extracting square and cube roots, and hour: minute: second notation, but solving problems in the “Rule of Three the separator between successive Quantities” or simple proportion. Many sexagesimal places was a vertical of these features appear in the earli- stroke, the standard punctuation est surviving Sanskrit mathematical mark of written Sanskrit. All these manuscript, a collection of birch-bark notational conventions are probably pages which may date back as far as much older than the earliest firmly the 7th or 8th century; see Hayashi attested documentation of them, but [4] for a comprehensive study and we cannot state precisely when they facsimile reproductions. originated. 5. Textbooks on mathematics and Except for the forms of the numerals mathematical astronomy themselves, we have very little direct evidence about how mathematical The earliest surviving complete computations were physically or Sanskrit works that were intended graphically represented before the as comprehensive textbooks of middle of the first millennium CE. mathematical and astronomical Occasional mentions of merchants’ practice date from the start of the “counting pits” suggest a simple 6th century CE onward. Like almost

18 all Sanskrit didactic treatises and more than exhaustive expositions. The other literature of this period, they job of explaining or demonstrating were composed in metrical verses the formulas in detail was left to that were usually quite elliptical and the commentators or to individual sometimes positively cryptic. The first teachers. The following excerpt of these textbooks, of which the most from a commentary of Bhāskara (a famous are the Āryabhat.īya (ca. 499) contemporary of Brahmagupta) on a of Aryabhat.a and the Brāhmasphut.a- verse from the Aryabhat.īya describing siddhānta (628) of Brahmagupta, how to solve what we would call treated mathematics in general linear equations of the form ax + c = (gan. ita or “calculation”) in separate bx + d for the unknown x, illustrates chapters within expositions of a typical interplay between text and astronomy. Mathematics chapters commentary.2 dealt with such diverse subjects as basic arithmetic operations, geometry, In order to show an example of series, and commercial arithmetic, as equations, he [i.e., Āryabhat.a] well as ones that were seldom used states: outside astronomical contexts, such “Divide the difference of two men’s as trigonometry. This should not be coins by the difference of [their] taken to imply that only budding beads; the quotient is the price of astronomers and astrologers were a bead, if the acquired wealth [of taught mathematics; probably there both men] is equal.” [Āryabhatīya The verse treatises themselves, were many works designed for more . 2.30] general mathematics teaching, in in keeping with the ideals of the Sanskrit as well as vernacular langua- Sanskrit oral tradition, were With the word “bead” is designated ges, that were eventually supplanted an object of unknown price supposed (at least in theory) to by later texts and vanished from . . . “By the difference of beads”: be concise epitomes for students the manuscript tradition. (The meaning, by the difference of Āryabhatīya and the Brāhmasphuta- to memorize, so they resemble . . the objects of unknown price. siddhānta were considered the condensed study guides more than “Divide the difference of two founding texts of two important men’s coins”: “of two”, meaning, exhaustive expositions. The job astronomical schools, which may of only two; that indicates this of explaining or demonstrating partly explain their greater longevity). operation is not for three and so the formulas in detail was left to on. “Difference of coins”: and by this The corpus of textbooks on the commentators or to individual is understood the wealth whose mathematical science also included amount is known. “Coin” means teachers. works treating only arithmetic dinars and so on . . . Example: . . . (“calculation with knowns” or “[dust-] Seven unknowns and seven coins board calculation”), or only algebra are equal to two unknowns [and] (“calculation with unknowns” or “seed- twelve coins. How much is the calculation”), or only mathematical amount of one unknown? astronomy (jyotis.a-gan. ita). As was usual for Sanskrit verse compositions, 7 7 all these works acquired commentaries Setting down: 212 (mostly in prose, and sometimes in vernacular languages), written usually Procedure: As before, the difference by later scholars but occasionally by of the beads [or] unknowns, the work’s own author, to explain and subtracted [with the minuend] illustrate the verses’ meaning. The above, is 5. The difference of coins, verse treatises themselves, in keeping subtracted [with the minuend] with the ideals of the Sanskrit oral below, is 5. The quotient from tradition, were supposed (at least division of the difference of coins in theory) to be concise epitomes by the difference of unknowns is for students to memorize, so they the amount of the unknown, 1. resemble condensed study guides

2 The translation is based on that of Keller [6], vol. 2, pp. 121–123.

19 Brahmagupta’s mathematics chapters, and finding solutions to second-order one on arithmetic and one on indeterminate equations are also algebra, contain a topical structure treated here. that indicates a well-developed classification of mathematical fields. These two chapters do not come He begins the arithmetic chapter close to exhausting the mathematical with the so-called “eight operations” knowledge packed in the Brāhma- (addition, subtraction, multiplication, sphut.asiddhānta. In the astronomical division, squaring, square-root, cubing, portion proper, Brahmagupta uses and cube-root) for both integers and many of the above gan. ita topics and fractions, and continues with methods also draws upon selected aspects for reducing fractions and finding of trigonometry (including the sine, unknown quantities in proportions, cosine, and versed sine), spherical starting with the basic Rule of coordinates, interpolation by first and Three Quantities, which later Indian second-order algebraic functions and mathematicians held to be the basis by trigonometric functions, closed- of calculation in general. The following form and iterative approximation sections refer to specific “practices” formulas, and combinatorics. The impact of the remarkable or applications that seem to reflect technical treatises composed by - a much earlier traditional division of Aryabhata, Brahmagupta and mathematical knowledge. (Plofker [7], The impact of the remarkable technical . treatises composed by Āryabhata, their successors was not confined pp. 59–60, 141.) They are divided into . algorithms for dealing with mixed Brahmagupta and their successors to India. Arabic- and Persian- quantities (for example, shares of was not confined to India. Arabic- and speaking mathematicians during profits in a joint commercial venture), Persian-speaking mathematicians the expansion of the Islamic various kinds of series, geometry during the expansion of the Islamic world encountered various Sanskrit world encountered various (including Brahmagupta’s renowned results on the geometry of cyclic works, and adopted many of their Sanskrit works, and adopted quadrilaterals), and some specialized features such as decimal place-value many of their features such as applications of solid geometry, such numerals (now the so-called “Arabic” decimal place-value numerals as finding the dimensions of ditches, numerals), trigonometry of sines, and computational techniques. Some (now the so-called "Arabic'' piles of bricks, sawn timber, and heaps of grain. The last practice involves of this knowledge filtered through numerals), trigonometry of sines, computation of shadows, which to Byzantium and the Latin West in and computational techniques. would be useful for practical land medieval and early modern times. Some of this knowledge filtered mensuration and timekeeping with a Indian mathematical astronomy and the elaborate astrology that through to Byzantium and the gnomon; it employs only similar right triangles, not trigonometric functions. accompanied it, borne by pilgrims, Latin West in medieval and traders and colonists, also significantly early modern times. Algebra in Brahmagupta’s work influenced scientific traditions in China is largely concerned with the so- and Southeast Asia. (Plofker [7], pp. called “pulverizer”, or solutions of 181–182, 255–259.) linear indeterminate equations. In addition to an extensive discussion 6. The emergence and expansion of the pulverizer, he gives the first of the medieval Indian surviving set of rules in Sanskrit for mathematical canon computation with negative quantities and zero, including the interesting The vigorous advance of Indian claim that a quantity divided by zero mathematics is often considered to has the special status of “zero-divided”, have culminated with the magisterial where multiplying the zero-divided treatises of Bhāskara (Bhāskara II to quantity by zero restores the original distinguish him from the 7th century quantity. Methods for computing Bhāskara mentioned above) which with the square roots of non-square became enduring classics in almost numbers, solving various kinds of all branches of mathematics and equations in one or more unknowns, astronomy. In Bhāskara’s Līlāvatī on

20 arithmetic, Bījagan. ita on algebra, and 7. Innovations and foundations in Siddhānta-śiroman. i on mathematical the “Kerala school” astronomy, all composed with accompanying expositions by the Probably the most dazzling post- author, students and scholars found Bhāskara innovations in Sanskrit a model of mathematical learning on mathematics were the discoveries which they based many later works. made by a series of scholars in the small Bhāskara’s organization of the basic coastal region of Kerala (the “Malabar content of arithmetic and algebra coast”) from about the 14th to the was in the main very similar to 17th centuries. Their results dealing Brahmagupta’s, but his treatment with infinite series (especially for was generally more comprehensive trigonometric functions), calculation (although there were a few subjects, with infinitesimal quantities, numerical such as the results on cyclic quad- approximations, and proofs of earlier rilaterals, where Bhāskara was less results in geometry were in many cases well-informed than his predecessor). undreamed-of elsewhere in the world, until the development of calculus Bhāskara’s astronomy treatise and the subsequent explosion of also followed the basic outline of mathematics in early modern Europe Brahmagupta’s for its computational led to their rediscovery. algorithms, but its overall design emulated a somewhat later model: The mathematical researches of this It was split into two parts, the first group of scholars appear to have being a detailed set of procedures sprung from the work of a genius for predicting and computing named Mādhava who worked in Bha-skara’s mastery of almost astronomical phenomena (the a village near modern Kochi (or all subjects in the realm of so-called gan. ita section), and the Cochin) starting probably in the second containing a more theoretical late 14th century. Mādhava left gan.ita as a whole was such that description of the spherical geometry several surviving works on aspects of he became known in later years on which those procedures were astronomical computation, but his as ''Bha-skara the Teacher'', and based (the gola section, from the remarkable ideas on infinite series and Sanskrit word for sphere). Bhāskara’s related topics are preserved only in a his works were more copied and mastery of almost all subjects in the few verses quoted by later members emulated than any others. realm of gan. ita as a whole was such of his school. Mādhava’s seminal that he became known in later years as influence is apparent in the careful “Bhāskara the Teacher”, and his works study and exegesis of his results by were more copied and imitated than his mathematical progeny, and in the any others. respectful epithet “Gola-vid” (“one who knows the sphere”) that they However, the “standard” Indian used in referring to him. However, mathematics based on Bhāskara’s we know very little about his own writings was by no means a stagnant scholarly background or sources of or fossilized corpus. Later writers inspiration, beyond the fact that he may have imitated the construction and his successors were followers of and content of his works, but they the astronomical tradition founded also contributed many new results by Āryabhat.a (although they were in subjects ranging from algebra fully conversant with important works to combinatorics to approximation in the Brahmagupta tradition as well, rules to geometric demonstrations. such as the treatises of Bhāskara II). Through the Indo-Muslim empires arising in the early second millennium Mādhava’s few surviving verses on and afterwards, Indian scholars also mathematics constitute probably the learned of “foreign” ideas like Euclidean most densely packed compendium geometry, tables of ephemerides, etc., of sheer brilliance of any existing and numerous works in Sanskrit reflect mathematical corpus. In a scant few varying degrees of experimentation hundred syllables he recorded infinite with such ideas. series expressions for the circum-

21 ference of the circle (later rediscovered the Kerala school’s fascination with by Leibniz as a formula for π), for an explaining and justifying the mathe- arbitrary arc of a circle in terms of matical techniques they worked with. its sine and cosine (now generally associated with the name of James 8. Modern efforts at synthesis, Gregory and the arctangent function), and ultimate amalgamation and for the sine and cosine of a given arc (the so-called “Newton power The Sanskrit mathematical tradition series”), as well as an approximate value for many centuries had smoothly of π accurate to eleven decimal places assimilated into its own lively growth and a rule for computing a correction a diverse collection of foreign sources, term to finite series approximations on subjects ranging from Greco- of the circumference. The following Babylonian sexagesimal fractions to translation of Mādhava’s expression Hellenistic trigonometry to Greco- for the circumference series gives an Islamic geometric diagrams. All of these idea of the conciseness and elegance subjects and many others merged with of his formulas (Kriyākramakarī on uniquely Indian developments in a Līlāvatī 199; Sarma [8], p. 379): shared textual environment of Sanskrit verse and commentary, and found a Add or subtract alternately the niche in the standard classifications of diameter – multiplied by four Sanskrit learning. However, traditional and divided in order by the odd Their results dealing with infinite Indian mathematical science could numbers like three, five, etc. – to not survive the impact of modern series (especially for trigonometric or from the diameter multiplied by mathematics in European colonial functions), calculation with four and divided by one. administration and education in the infinitesimal quantities, numerical nineteenth and twentieth centuries. approximations, and proofs of These alternately subtracted and added terms define the circumference The spread of printing in India from earlier results in geometry were of a circle to be in many cases undreamed-of the 19th century onward brought many Indians into contact with elsewhere in the world, until the textbooks and journals based on development of calculus and the European mathematics, which was subsequent explosion of mathe- where D is the diameter of the circle, promoted by colonial administrative matics in early modern Europe an expression exactly equivalent to policy as the preferred medium for the familiar series for π . mathematical education. This contact led to their rediscovery. 4 inexorably drew India into modern global mathematics and away from Extensive commentaries on Mādhava’s its traditional scientific literature. mathematical verses by his successors Nonetheless, several ingenious atte- provide unusually detailed exposi- mpts were made to synthesize the tions and demonstrations of his indigenous Indian discipline of gan. ita discoveries. The infinite-series results with the new knowledge contained were explained by rationales involving in European journals and textbooks. infinitesimal quantities, where a tiny The following excerpt is taken from arc of a circle’s circumference was one such effort at hybridization, a approximated by its chord or sine. short treatise composed in 1873 Ingenious geometric and trigonome- in mixed Sanskrit verse and prose, tric arguments were used to combine which combines traditional geometric the expressions for these tiny arcs as technical terms with newly coined terms of recursively extended series. ones to expound the basics of conic These elaborate derivations reflect sections.3

3 An early 19th century colonial administrator who supported this scientific synthesis movement states his case for it in (Wilkinson [11]). The author of the quoted excerpt (Dvivedī [3], p. 3), was a student of Wilkinson’s own chief pupil and collaborator.

22 Now when a regular cone [lit., “even [4] Hayashi T. The Bakhshālī manuscript needle”] is cut by a plane at an (Egbert Forsten, 1995). unequal distance from the base- [5] Kashyap R. L. and Sadagopan S. Rig circle, then [the figure in] the place Veda Mantra Samhita (SAKSIVC of that cutting is an ellipse [lit., “long Bangalore, 2005). circle”]. [6] Keller A. Expounding the mathemati- cal seed: A translation of Bhāskara I Scattered works such as these mark on the mathematical chapter of the the final stage before the long Āryabhat.īya, 2 vols. (Birkhäuser, 2006). tradition of Indian mathematical [7] Plofker K. Mathematics in India research – ingenious, adaptable, and (Princeton, 2009). Sanskritized to the last – was merged and absorbed into the culture of [8] Sarma K. V. Līlāvatī (Punjab University, international science. 1975). [9] Sarma S. R. “Śūnya in Pi˙ngala’s Chandah. - References sūtra”, in (Bag & Sarma 2003), pp. 126–136. [1] Bag A. K. and Sarma S. R. The Concept [10] Sen S. N. and Bag A. K. 'The Śulbasūtras' of Śūnya (Indian National Science (Indian National Science Academy, Academy, 2003). 1983). [2] Doniger W. The Rig Veda (Penguin, [11] Wilkinson L. 'On the use of the 1981). Siddhántas in the work of native [3] Dvivedī S. Pratibhābodhaka (Sampur- education', Journal of the Asiatic Society nanand Sanskrit University, 1985). of Bengal 3, 504–519, 1834.

Email: [email protected]

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23 S. G. Dani TIFR, Mumbai

India’s Arrival on the Modern Mathematical Scene S.G. Dani

THANKS TO a unique combination of students; in 1855 the college was circumstances, India turned out to renamed as Presidency College and be one of the first countries outside brought under government control. of Europe to enter the realm of mod- The Presidency College in Madras was ern science in a big way. Though the started in 1840 by Eyre Burton Powell, modern educational system set up by a professor of mathematics. In 1857, the British colonial rulers was aimed the first modern universities were primarily at training Indians to serve in established, by the British government, the subordinate civil service positions in Calcutta (the then capital), Bombay in India, the intellectual urges in the and Madras. Though the universities Indian society, deeply rooted in tradi- were modelled on the University of tion and waiting to manifest them- London, with several decentralised selves at the first opportunity, broke colleges where the faculty was largely through in the new environment and engaged, onerously, in disseminat- soon transformed the scientific scene ing curricular material and had little in the country. This transformation opportunity to influence students process was also fostered by some en- into an enlightened approach to lightened British individuals, though knowledge, in course of time, various few in number. individuals and institutions steered the academic life of the country onto Modern educational institutions be- a positive track. As the capital, Calcutta gan to emerge in the first half of the offered Indians the best and sustained 19th century. The Hindoo College opportunities to be exposed to mod- was founded in Calcutta in 1817 with ern science, leading eventually to the expressed intention of provid- a stream of eminent scientists, J. C. ing modern education to the Indian Bose, P. C. Roy, C. V. Raman, S. N. Bose,

31 M. N. Saha, . . . . Around 1920, Calcutta Karnataka, Andhra and Tamil Nadu; was one of the most intense sites of in terms of vocation the members scientific activity outside of Europe included teachers and administrators ([3], p. 37). of mathematics and education, engineers, students, as well as civil Mathematics progressed in parallel, in servants and other bureaucrats (these various parts of the country. Everyone too in significant numbers). Posing knows of the legendary genius Sriniva- problems for solutions and presenting sa Ramanujan (1887–1920), and given solutions in the subsequent issues the mystique of the prodigy, to many was one of the notable features in the minds it may seem that modern Indian early volumes. There were also many mathematics began with him. While no short notes on a variety of topics in doubt the indirect impact of the great geometry, trigonometry, arithmetic, man on Indian mathematics is immea- etc. Many members contributed surable, the development of the sub- regularly to this activity, including ject in the country has a separate tale M. T. Narayaniyengar and S. Narayana of its own, manifesting a collective and Aiyar who edited the journal, and R. P. conscious effort by the then emergent Paranjapye (known also as Wrangler society, starting well ahead of the ad- Paranjapye with his Cambridge title vent of Ramanujan. The founding of acquired at the Tripos examination) two major mathematical societies, the who was an Honorary Member of Indian Mathematical Society and the the Society, and Honorary Librarian Calcutta Mathematical Society, before of the Society’s Library located at Ramanujan had left the shores of India Poona (now Pune). At the end of for Cambridge where he received his the 4th issue in Volume 1 there is an first professional recognition, stands as extract included from ‘The Times of ample testimony to this. Indeed, the India’ (a daily newspaper published two mathematical societies played an from Bombay to-date) of 10th July important role in the early develop- 1909 titled 'Mathematics in India' and ment of mathematics in India (see [7] it is hard to resist the temptation to for a detailed account in this respect). quote the following segment from the article: “In these days no science The Indian Mathematical Society was can be considered to deserve the established in 1907 (as ‘The Analytic name unless it takes the neat form of Club’, which in 1910 – a 100 years mathematics; and the very presence ago from now – was turned into the of some masters of mathematical ‘The Indian Mathematical Society’) by science at Bangalore would have V. Ramaswamy Aiyar, a civil servant who produced there such a mathematical was the Deputy Collector of Gooty (in atmosphere. Without mathematics the Anantapur district of the present science is very often apt to degenerate state of Andhra Pradesh). The Club into pure empiricism or a barren record subscribed to various international of facts.” How contemporary! journals to promote an interest in mathematics, and also ran a magazine While the situation had not matured of its own, which in 1909 was formally enough to help Ramanujan very much styled as a journal. The membership of with his mathematics, the interest in the Club listed in the first issue, as also mathematics and the formation of a in later issues, of the journal (which at fraternity of mathematically inclined that time had the name Journal of the people helped Ramanujan materially Indian Analytic Club) shows a broad until he found his way to G. H. Hardy, spread in many respects: geographical and onward to fame. locations of the members ranged all the way from Lahore and Amritsar in The Calcutta Mathematical Society was the north, Trichinapally in the south founded in 1908. Asutosh Mukherjee and Rangoon in the east, many cities who was the Vice Chancellor (equiva- in (present) Maharashtra, Gujarat, lent of President in the US system, the Madhya Pradesh, Uttar Pradesh, Chancellor in Indian universities is

32 typically a figurative head for the uni- awarded the first ‘Doctor of Philoso- versity from the Government) of the phy’ in Mathematics from the Calcutta Calcutta University from 1906–1924, University, for a thesis titled, ‘Paramet- was instrumental in the society being ric Coefficients in the Differential Ge- established, and he was the President ometry of Curves in an N-space’, which until 1924. Though a lawyer by profes- earned him the Griffith Memorial Prize sion, who later became a High Court for 1910. Hadamard is known to have judge, he had a profound interest in had a high regard for the work of mathematics, and published some Mukhopadhyaya ([16], p. 246), and papers on algebraic curves and differ- stated in the Memoirs of Collège de ential equations. The Calcutta Math- France that the new methods intro- ematical Society also started a journal, duced by Mukhopadhyaya would be Bulletin of the Calcutta Mathematical the subject of discussion in their Col- S. Mukhopadhyaya Society in 1909. The early issues of the loquia. Mukhopadhyaya also made (Courtesy: Calcutta Mathematical journal show a considerable contact some interesting contributions in non- Society) with French and Continental European euclidean geometry. A collection of mathematics of that time; this is signifi- his papers in geometry was published cant especially against the backdrop of and its review in Nature [13] states the British colonial context of the era. that the papers in it are “of special in- The first issue, dated April 1909, opens terest both on account of the original with a translation of an article by Emile methods employed and the results Borel 'The Method of Poincaré' from La obtained.” Mukhopadhyaya was one of Revue du Mois of March 1909. Each is- the founding members of the Calcutta sue carried a section ‘Notes and News’ Mathematical Society. He served on in which many interesting pieces of its Council and was elected Vice Presi- news were reported; Hilbert’s solution dent in 1917 and President in 1935; (his of Waring’s problem on represent- tenure as President was short, as he ability of positive integers by a fixed passed away in 1937). He was a regular number of nth powers, a report on contributor of research papers to the the Fourth International Congress of Bulletin of the Society. Practically all his Mathematicians held in Rome (in April work until 1927, including in particu- 1908), information on awards of priz- lar, his four vertex theorem mentioned es, prize problems announced by the above, was published in the Bulletin. Academies of Sciences from France, In later years, he published also in Belgium, Denmark, the Royal Society of the Mathematische Zeitschrift and the Naples, the Prince Jahlonowski Society Tohoku Mathematical Journal. After his of Leipsic, may be cited as some exam- retirement in 1932, Mukhopadhyaya ples. Each issue had a section ‘Societ- went to Europe to study methods of ies and Academies’ listing talks held at education, and on returning to India various learned bodies in Europe and wrote a series of memoirs about this America. (see [4] for more details).

Asutosh Mukherjee is also recognised Another notable name from this pe- for promoting mathematics in Cal- riod is Ganesh Prasad (1876–1935). cutta by bringing in several talented He was the first D.Sc. from the Allaha- mathematicians. One of his finds was bad University. After his initial work in Syamadas Mukhopadhyaya (1866– India, in 1899 he went to Cambridge, 1937), well-known for his theorem in England and then to Göttingen where global differential geometry, the “four he came in contact with Klein and Hil- vertex theorem”, that a simple closed bert. Soon after his return to India, be- strictly convex curve in the plane, oth- tween 1905 and 1912 he published 3 er than the circle, must have at least papers in the Mathematische Annalen four points (called vertices) at which on differential geometry, concerning the curvature has a local extremum surfaces of positive curvature. Later on (maximum or minimum) [12]. He ob- he made several important contribu- tained his M.A. degree from the Presi- tions to potential theory and summa- dency College, Calcutta, and was later bility of Fourier series, and also wrote

33 11 books, among them ‘A Treatise on few remarks, mainly in relation to the Spherical Harmonics and Functions Indian mathematical community; for of Bessel and Lame’ which became a a general reading on Ramanujan, ref- classic. He worked in Allahabad and erences [11], [1], and [2] are especially Benaras before moving to Calcutta in recommended. Discovering him was 1923, where he was to continue until undoubtedly, one of the proudest his death. He was a keen promoter of achievements of the Indian Mathe- research and guided many students. matical Society (see [14]). Ramanujan’s Apart from mathematical research he contributions to the Society’s journal also had a great interest in the history began to appear in 1911 and his first of mathematics. His book ‘Some Great article ‘Some properties of Bernoulli’s Mathematicians of the 19th century: numbers’ attracted considerable atten- Their Lives and Their Works’ (Benaras, tion. Ramanujan contributed several 1933–1934), in 2 volumes, adding to papers to the journal, including some S. Ramanujan over 700 pages, containing portraits after he went to England. Interestingly, (Courtesy: Bruce Berndt) of 16 celebrated mathematicians of an account of his work done in Eng- the 19th century is a valuable contri- land, written by G. H. Hardy was pub- bution. This book as well as his earlier lished in the journal of the IMS in 1917. book ‘Mathematical Research in the He returned to India in 1919, after a Last 20 Years’ (Berlin, 1923) have been long delay caused by the first world cited in [20](p. 186 and p. 216 respec- war, to receive a hero’s welcome, with tively). He endowed a fund to enable the newspapers publishing special ar- the Calcutta Mathematical Society to ticles announcing his arrival. A stipend award a regular prize for work related of 250 pounds had been arranged for to the history of Indian mathematics him at the University of Madras. There (however the organization in charge he continued to work on q-series and of it never actually made an award). produced his “lost notebook” ([1]). Under his influence, two of his doc- Much as he would have indeed liked toral students Avadhesh Narayan to, his condition did not permit him Singh and Bibhutibhushan Datta to play a direct role in the develop- who, even though their doctoral work ment of mathematics in the country. was in pure and applied mathemat- Moreover, his health deteriorated rap- ics, respectively, turned to the study idly and he passed away in April 1920, of ancient Indian mathematics, to within a little over a year after his re- which they made profound contribu- turn. However, he left a huge legacy tions, and in particular wrote a book inspiring generations of students in which is an important reference in India by infusing a romanticism in the the area, even today. While in Benaras, pursuit of mathematics. he founded the Benaras Mathemati- cal Society. In 1924, he succeeded While Madras was not destined to Asutosh Mukherjee as President of the have the benefit of Ramanujan in a Calcutta Mathematical Society. guiding role, within a few years of his passing away a duo of mathemati- These are but a few pointers to the cians was to jointly build a congenial upward trajectory in the develop- atmosphere for the advancement of ment of mathematics in India during mathematics there: K. Ananda Rau the early decades of the 20th century. (1893–1966) and R. Vaidyanathaswamy Then came Ramanujan. The rather (1894–1960). Being only 6 and 7 years unique life story of Ramanujan, with younger than Ramanujan, they were his remarkable mathematical abilities, both, interestingly, products of the his finding a benefactor in G. H. Hardy, times when career choices were yet to his unhappy family life, and his sad be influenced by the saga of Ramanu- demise at the early age of 32 due to jan. Like Ramanujan, Rau worked with a mysterious disease, have all been oft Hardy at Cambridge, and as a student quoted and well documented cover- he won the coveted Smith prize for his ing a variety of aspects and nuances; results extending H. Bohr’s theorem we shall not go into it here, save for a on orders of functions defined by the

34 Dirichlet series. He was an outstanding way in establishing the journal. He analyst and had important results to was President of the Society during his credit on the summability theory 1940–1942. He was also a Founding and functions of a complex variable. Fellow of the Indian Academy of Sci- He established very general “Taube- ences. Besides mathematics, he was rian theorems” which had resisted the keenly interested in Carnatic music efforts of some of the best analysts. In and yogic sadhana. He was a follower his later years, he worked on modular of Sri Aurobindo and held weekly ses- functions and representations of in- sions interpreting certain Vedic texts tegers in terms of quadratic forms. A (see [15]). theorem named after him appears in Hardy’s classic book ‘Divergent Series’. Annamalai University, situated on the He served as professor at Presidency outskirts of a small town called Chi- College, Madras from 1919 until 1948 dambaram, an ancient pilgrimage when he attained the then mandatory site of Hindus about 150 miles south K. Ananda Rau age for retirement. He was from all ac- of Madras, was home to A. Narasinga (Courtesy: Ramanujan Institute for counts a very inspiring teacher, and Rao (1893–1967) during the years Advanced Study in Mathematics, gave clear and impressive expositions 1929–1946; he then moved to Andhra University of Madras) of topics (see [18]). Many of his stu- University at Waltair, on the east coast, dents went on to do good work and and later to the Indian Institute of achieve renown: T. Vijayaraghavan, Technology (IIT), Madras in 1951. Rao S. S. Pillai, K. Chandrasekharan, M. V. worked on Euclidean geometry and Subbarao, and V. Ganapathy Iyer. was also interested in other fields such as aerodynamics. He was an inspira- Vaidyanathaswamy also went to U.K. tion to young mathematicians. He for higher studies, but it was after a was the founder editor of The Mathe- period of teaching and 4 years as re- matics Student, a journal founded by search scholar at Madras University. He the Indian Mathematical Society in obtained his Ph.D. from St. Andrew’s 1933, following a decision taken at the University, and worked with H. W. Silver Jubilee of the Society in 1932. Turnbull, E. T. Whittaker and H. F. Baker The journal, which continues to be during his stay of 3 years. He ventured published to-date, is aimed at math- into new areas such as symbolic logic, ematics teachers and students, who lattice theory, and topology. Returning are also encouraged to contribute to India in 1925 he joined the Universi- articles to it. Rao worked as the editor ty of Madras in 1927, after a stint at the almost single-handedly with great de- Benaras Hindu University. Based on votion for 18 years, until 1950, placing his courses of lectures delivered at the it on a firm footing. He also promoted University of Madras, he published in good research. He instituted a medal, 1947 the book ‘Treatise on Set Topology’, named after himself, which is awarded the first book on the topic in English; through the IMS in recognition of qual- the first edition was published by the ity research work by young authors. Indian Mathematical Society; a second edition with some improvements was Elsewhere in India also, mathematics published in 1960 by Chelsea Publish- was flourishing simultaneously, with ing, New York, and it was reprinted in contributions to various topics. B. N. 1999 by Dover Publications, NY. He Prasad (1899–1966) who was at the retired from the university in 1952 but University of Allahabad through most worked later for a few years at the In- of his career, published widely on dian Statistical Institute (ISI), Calcutta Fourier analysis. He went to England, and Sri Venkateswara University, Tiru- where he was a student of Titchmarsh, pati. He was actively involved with the and to Paris where he interacted Indian Mathematical Society. He was with Denjoy, and his work was well Editor of the Journal of the Indian Math- thought of. In 1958 he founded the ematical Society over the prolonged Allahabad Mathematical Society and period 1927–1950, and his concerted started a journal. At Delhi, Ram Behari R. Vaidyanathaswamy efforts and innovations went a long (1897–1981), a student of J. L. Synge at (Courtesy: Indian Mathematical Society)

35 Cambridge and Dublin, was engaged under the guidance of Syamadas in work on differential geometry (he Mukhopadhyaya and published some was at St. Stephen’s College to begin papers relating to convex curves. His with and moved to the University in career course changed after joining 1947). He served as Secretary to the ISI, not quite to statistics, but some- Indian Mathematical Society during thing related. In 1939 he wrote his 1936–1942. Rabindranath Sen (1896– celebrated paper on the construction 1974) at Calcutta also made numerous of balanced incomplete block designs contributions to differential geometry, answering some problems posed by especially on parallelism. Nikhilranjan the renowned statistician R. A. Fisher Sen (1894–1963), who was brought during his visit in the previous year as to Calcutta by Asutosh Mukherjee in President of the Indian Science Con- 1917 worked on applied mathematics gress, in Bombay (see [10] for a discus- S. S. Pillai (theory of relativity and also fluid me- sion on the genesis and impact of this (Courtesy: Indian Mathematical Society) chanics). During 1921–1924 he was in paper, including some quotes from Europe, on study leave from the uni- R. C. Bose). He went on to prove sev- versity, where he came in contact with eral important results on design the- stalwarts like Einstein, Planck, Som- ory and the theory of error-correcting merfield, and de Broglie. Apart from codes. This work involved algebra and applied mathematics he also studied number theory, concerning especially, probability theory and topology. Both the general linear groups over finite N. R. Sen and R. N. Sen were active in fields. Many of his early papers (as the Calcutta Mathematical Society also later ones) continue to be cited and both held the position of Presi- frequently in recent literature. In the dent during certain years. 1940s, he was also part of the Univer- sity of Calcutta, when a post-graduate In 1931 the ISI was founded by P. C. department of Statistics of the univer- Mahalanobis (1893–1972), himself an sity was started in ISI. In 1947, he left accomplished statistician known for for the United States. ISI was to later the “Mahalanobis distance” (a statisti- produce several more excellent math- cal measure gauging proximity or oth- ematicians in various areas of math- erwise between group populations), ematics. The renowned Abel Laureate some important techniques relating S. R. S. Varadhan is also a product of ISI. to sample surveys, and pioneering studies in anthropometry; Mahalano- The 1930s are memorable for Indian bis also started the journal Sankhya – mathematics, also for the achieve- The Indian Journal of Statistics, which ments of S. S. Pillai and T. Vijayaragha- also published mathematical papers, van. S. S. Pillai (1901–50) was a student especially in areas like probability, of Ananda Rau, and was later a lecturer design theory, combinatorics, etc. In at the Annamalai University during post-independence India, as member 1929–1941, where his creativity was of the Planning Commission, he con- seen in full bloom. He is well-known tributed significantly to the five-year for his work on Waring’s problem, con- plans. The renowned statistician C. R. cerning determination of the smallest Rao worked at the ISI for almost four integer g(k), given a natural number k decades (including as Director during greater than 1, such that every natu- 1972–1976), until leaving for the United ral number can be written as a sum States in 1979. There was also a sizable of at most g(k) integers that are k-th group at ISI actively pursuing diverse powers themselves; the existence areas of statistical practice and theory. of such a number g(k) was proved While primarily focussed on statistics, by David Hilbert in 1909. By the fa- ISI has contributed greatly since the mous classical theorem of Lagrange early days to the development of ad- g(2) = 4, Wieferich and Kempner vanced mathematics. R. C. Bose (1901– proved in 1912 that g(3) = 9. Pillai gave 1987) is one of the prominent names a remarkable, almost complete solu- associated with the Institute, from its tion towards determination of g(k) early times. Bose started his research for k > 7; g(k) was proved to take the

36 expected value 2k + [(u)k ] – 2 (square and again, like Ramanujan he sent brackets denote the integral part), un- some of his work to Hardy. Indeed, as der a condition on the fractional part in the previous case this led (though in of (u)k that is valid when k is even, and a less dramatic fashion) to his proceed- very likely holds for all k. He showed ing to join Hardy, then at New College, also that g(7) = 143, and in a later Oxford. His Oxford years, 1925–1928, paper that g(6) = 73.1 Pillai’s achieve- were very fruitful and greatly influ- ments were overshadowed to an ex- enced his later work. He proved some tent by Dickson’s independent work impressive theorems on summability, on the problem around the same time; especially Borel summability. On his Dickson also gave the value of g(k) in return to India he held appointments case the condition alluded to above at Annamalai and then at Aligarh Mus- fails; it is however not known even lim University where he had close in- today whether the condition actually teraction with André Weil; the latter fails for any number. Recognition did spent two years, 1930–1932, there as come eventually to Pillai for his out- Professor of Mathematics and Head of T. Vijayaraghavan standing contributions to the theme. the Department. In 1931 Vijayaragha- (Courtesy: Indian Mathematical Society) He worked also on other problems in van left Aligarh in protest when the number theory, especially diophan- Vice Chancellor offered him professor- tine approximation. In the latter area ship from which he planned to oust he proved the beautiful theorem that Weil ([17]) and moved to Dhaka (now given natural numbers a, b, m and n in Bangladesh). In a paper published in and δ > 0, for integers x, y if amx–bny 1932 he disproved, in a very ingenious is non-zero then its absolute value ex- and elegant manner, a conjecture of ceeds m(1–δ)x for all sufficiently largex . E. Borel about the growth of solu- Based on the theorem he proposed a tions of non-linear ordinary differential conjecture, now known after him, on equations ([8], [6], [16]). Impressed by the finiteness of the number of inte- the work, G. D. Birkhoff arranged to ger solutions of exponential diophan- have Vijayaraghavan invited as Visiting tine equations, which still remains Lecturer of the American Mathemati- unresolved. He had an invitation to cal Society in 1936. visit the Institute for Advanced Study, Princeton for the year 1950–1951, and Vijayaraghavan is also well-known was to participate in the International for his work on diophantine approxi- Congress of Mathematicians-1950 at mation, from around 1940, relating Harvard. However, on his way to the especially to the asymptotics of the US, tragedy struck, and Pillai died in fractional parts of αθn as n tends to an air-crash near Cairo on 31 August infinifty, α and θ being fixed real num- 1950; for various details the reader is bers, in which he generalized a result referred to [5]. of Hardy. This work led him to study a class of numbers which came to be T. Vijayaraghavan (1902–1955) was in- known as Pisot-Vijayaraghavan num- fluenced by Ananda Rau and began bers.2 In 1946 he moved to Andhra to work independently even as a stu- University, but left it in 1949 for dent at the Presidency College. Like Madras (now Chennai) to take over as Ramanujan he had problems sailing Director of the Ramanujan Institute of through a formal educational system, Mathematics, newly founded by the

1 It was shown by Jing-run Chen in 1965 that g(5) = 37 and the remaining open case k = 4 was settled in 1986 in the work of R. Balasubramanian, a plenary speaker at ICM-2010, jointly with Deshouillers and Dress, proving g(4) = 19.

2 These numbers were also studied by Pisot in his thesis and are also referred as PV numbers or Pisot numbers.

3 The Institute later became part of the University of Madras, in 1957, following a period of financial difficulties and the demise of Dr. Chettiar; initially it was independent of the Department of Mathematics of the university, but in 1966 the two were merged.

37 businessman educationist Dr. Alagap- paper with Pleijel continues to be pa Chettiar.3 Vijayaraghavan served cited regularly in contemporary lit- as Director of the Institute until he erature – truly, a feat for a paper over passed away in 1955, at a young age, sixty years old. During a visit of a few following a heart attack. Vijayaragha- months in 1950 to the Tata Institute, van was also active in the Indian Math- Minakshisundaram authored a book ematical Society; he was the Secretary jointly with K. Chandrasekharan, titled during 1947–1951 and President dur- ‘Typical Means’ which was very well ing 1951–1953; he served also as the received. He delivered an invited talk Librarian of the Society for four years. at the International Congress of Math- ematicians of 1958, held at Edinburgh, About a decade younger to on Hilbert algebras. He was however, S. Minakshisundaram (Courtesy: Indian Mathematical Society) Vijayaraghavan and Pillai was an- not happy in the university setting and other stalwart of Indian Mathematics though he liked teaching and tried S. Minakshisundaram. He was indeed, hard to generate interest for research a very gifted mathematician and his in his students, he felt the system to work has had a lasting impact. He was be rather stifling and longed to be at initially influenced by Ananda Rau and a research institution. He was eventu- worked on the topic of summability of ally appointed Professor at the newly- series, a passion for which was to con- created Institute for Advanced Study at tinue through his life. His first paper Simla, and was happy to move there. was a striking generalisation of a re- He had embarked on writing a book sult of Ananda Rau. However, he soon on spectral theory, but unfortunately, came under the influence of Fr. Racine, in 1968, passed away before complet- a Jesuit missionary about whom I will ing it. say more later, and another math- ematician M. R. Siddiqi from Osmania By the 1940s there were quite a few University, Hyderabad and pursued mathematicians around the country, the area of differential equations, to publishing works of high quality in vari- which he made many contributions ous branches of mathematics. In num- of the highest quality. He obtained ber theory it would be appropriate to his doctorate in 1940 for his work on mention Hansraj Gupta (1902–1988) non-linear parabolic differential equa- and S. Chowla (1907–1995), both from tions. Some portions of the ensuing Punjab. Gupta worked at the Govern- period was very arduous and he had ment College, Hoshiarpur, which sub- to survive by doing some private sequent to the independence of India coaching of students for university ex- was merged into the present Panjab aminations, but he stuck to pursuing University. His most important work is mathematics despite the adverse cir- on partitions. Chowla studied in Cam- cumstances (see [19]). He later got an bridge under Littlewood. He worked appointment as lecturer in the Andhra at the Government College at Lahore University, Waltair. In 1946 he received during 1936–1947, but following the a visiting membership at the Institute partition of India he left for the United for Advanced Study, Princeton where States. Apart from these two, Panjab he spent 2 years. The Princeton years University, now situated at Chandi- resulted in what is considered his best garh, was to see in the later years work, in collaboration with the Cana- another stalwart in number theory, dian mathematician, Pleijel, on the R. P. Bambah (born 1925). Thanks to eigenvalue problem of the Laplace op- the efforts of these experts, a school erator on a compact Riemannian man- in number theory flourished at Panjab ifold. He introduced the idea of using University and important results con- the heat equation in the study which tinue to come out of there. turned out to be a powerful tool. It featured as an important component C. T. Rajagopal (1903–1978) from in a new approach to the celebrated the Ramanujan Institute, Madras, index theorems for elliptic operators, who succeeded Vijayaraghavan as due to Atiyah, Singer and Patodi. The the Director of the Institute after the

38 latter’s death, contributed greatly to ing 1937–1949 made many contri- the area of summability theory, pub- butions to questions of elasticity and lishing in several leading journals in- fluid dynamics. In 1950 he moved to cluding the Annals of Mathematics. R. the Indian Institute of Technology (IIT), S. Varma (1905–1970) who was on the Kharagpur (near Calcutta) where he faculty of the University of Lucknow developed a school of applied math- from 1938 to 1946 worked on inte- ematics. V. V. Narlikar (1908–1991), gral transforms and special functions; who was at Benaras Hindu Univer- Varma later moved to Delhi to be part sity, worked on relativity; he was well- of the Defence Science Organisation known around India and popularised where he was engaged in problems the subject. One of his early students of ballistics and operations research. P. C. Vaidya is also well-known for his S. M. Shah (1905–1996), who was at achievements in the area. B. S. Mad- Aligarh (a junior faculty when A. Weil hava Rao (1900–1987) most of whose was there) until 1958 when he left for professional life was spent at the Cen- the United States, worked extensively tral College in Bangalore worked on on functions of a complex variable, various topics in mathematical phys- dealing especially with properties ics; after retirement from Central Col- of entire functions. Ganapathy Iyer lege he went to the Defence Science (1906–1987), who was brought to An- Organisation at Pune where he is said namalai by Narasinga Rao was another to have been very effective. P. L. Bhat- excellent mathematician working on nagar (1912–1976) was at Delhi during entire functions. Both Shah and Iyer 1940 to 1955, and was engaged dur- were knowledgeable on the Nevan- ing that period in pursuing astronomy linna theory. Many of Ganapathy Iyer’s and astrophysics; a Ph.D. from the papers involve application of methods University of Allahabad, he holds the of functional analysis to spaces of en- distinction of a two-part Ph.D. thesis, tire functions. D. D. Kosambi (1907– one on astrophysics under A. C. Ba- 1966) who was also a colleague of nerjee and the other on summability A. Weil at Aligarh and worked at Fer- under B. N. Prasad. He later moved to gusson College, Pune from 1932 to the Indian Institute of Science, Banga- 1946 made important contributions lore, as Head of the Department of ap- during that period to differential ge- plied mathematics, where he not only ometry, dealing especially with path made important contributions to fluid spaces; during 1946-1962 he worked dynamics but was also instrumental in at the Tata Institute of Fundamental developing the subject at the Institute Research, Mumbai – in later years he and around the country. got interested in statistics and also nu- mismatics and history which became The picture of the mathematical sce- his mainstay. nario in India during the 1940s would be incomplete without depiction of Notably, there was at least one woman the role of a few foreign mathemati- to have attained prominence on the cians. A. Weil’s stint at Aligarh during Indian mathematical research scene 1930–1932 was noted earlier. On ac- by the 1940s. S. Pankajam, a student count of the brevity of the stay and the of Vaidyanathaswamy who worked at overall context it did not really have the University of Madras, published much effect, except indirectly through many interesting results in the 1930s his influence on Vijayaraghavan; Weil’s and 1940s on logic and foundations of ideas were to have a much greater mathematics; another woman student influence on mathematics at the Tata of Vaidyanathaswamy, K. Padmavally, Institute of Fundamental Research worked on analysis and topology in during a later stage, which I will not be the 1950s. going into here (see [17]). There were however, two mathematicians from Considerable work was also done abroad who were in India over a long in areas of applied mathematics. period of time and influenced Indian B. R. Seth who worked at Delhi dur- mathematics substantially: F. W. Levi

39 (1887–1966) and Father Racine (1897– Mathematische Annalen, the Quar- 1976). Levi was a German mathema- terly Journal of Mathematics (Oxford), tician who worked at Leipzig during Tohoku Mathematical Journal. Contri- 1920–1935, but had to leave the coun- butions were also made to the Pro- try on account of his Jewish ancestry ceedings of the National Academy of when the Nazi’s took over. He came to Sciences, USA as also to Nature. Many Calcutta as a professor in 1935 and was high quality papers were also pub- there until 1948 when he moved to the lished in journals published from Tata Institute in Mumbai at the invita- India. Apart from the Journal of the tion of Homi Bhabha; he later returned Indian Mathematical Society, the Bul- to Germany in 1952. Levi’s work was letin of the Calcutta Mathematical So- in group theory and generalisations, ciety, and Sankhya mentioned earlier and he had a great deal of influence in various other journals had begun and introducing algebra (then called mod- established themselves with good ern algebra) in India, especially in the professional standards. The Indian university curricula; he is sometimes Academy of Sciences was founded in referred to as “father of algebra in India” 1934 and soon started its own journal, (see [9]). His seminar at the University the Proceedings of the Indian Academy of Calcutta on algebra and geometry of Sciences; the mathematics papers seems to have influenced the work of appeared in Section A of the Proceed- R. C. Bose (see [10]). He was an active ings, which was later changed to a se- promoter of mathematics and was the ries of the Proceedings with the name President of the Indian Mathematical suffixed with “Mathematical Sciences”. Society for 5 years, during 1942–1947. Many of the established mathemati- Father Racine did his doctorate in Paris cians contributed regularly to these in 1934. He had joined the Jesuit order journals. There were also other jour- in 1929 and after his doctorate, was nals brought out by various universi- sent to India to work in St. Joseph’s ties or local institutions. The Journal College, Tiruchirapally. From there he of the Annamalai University in which moved to Loyola College, Madras in some of Pillai’s results appeared, Pro- 1939 where he stayed until his death ceedings of the Lahore Philosophical So- in 1976, even though he had retired in ciety in which a large number of notes 1967. Father Racine had studied under by S. Chowla appeared, Journal of the Elie Cartan and Hadamard and had University of Bombay in which some many friends among leading French papers of Kosambi and Narlikar are mathematicians of the time. With that found, Journal of the Mysore University background, he greatly influenced stu- to which Madhava Rao contributed, dents into acquiring a modern view of etc. Many of the journals also published mathematics. Minakshisundaram was papers by established mathematicians his first great success in this respect, from abroad. While publishing in local and many more followed in later years journals precluded reaching out to a (see [17]). wider readership, and in some cases this may have cost the authors much, As this overview of the growth of math- it contributed greatly in generating a ematics in the country indicates, India lively mathematical environment. The had well and truly arrived on the mod- contributions of the good mathemati- ern, global mathematical scene by the cians, Indian as also foreign were help- 1940s. A large number of papers were ful in sustaining the journals. Journal published by the researchers from publication was carried out with great India in leading journals around the commitment; for instance, publication world, including the Annals of Mathe- continued even when there was pa- matics, American Journal of Mathemat- per shortage during the second world ics, Duke Journal of Mathematics, the war [16]. Bulletin, Proceedings and Transactions of the American Mathematical Soci- The rest I would say, modifying the ety, the Journal and the Proceedings common adage, is contempo- of the London Mathematical Society, rary history. In 1947, India became

40 independent. Under the leadership of [8] H. Davenport, 'T. Vijayaraghavan', Pandit Jawaharlal Nehru great store J. Lond. Math. Soc. 33, 252–255, 1958. was placed on the development of [9] L. Fuchs and R. Göbel, 'Friedrich science. The Tata Institute of Funda- Wilhelm Levi, 1888–1966', Abelian mental Research (TIFR) was founded groups (Curaao, 1991), 1–14, Lecture by Homi Bhabha in 1945, and thanks Notes in Pure and Appl. Math. 146, to his enlightened views about math- Dekker, New York, 1993. ematics and the efforts of K. Chan- [10] Harald Gropp, 'The birth of a drasekharan, a strong School of Math- mathematical theory in British India, ematics was built up in Mumbai. Many Sets, Graphs and Numbers' (Budapest, new institutions emerged in the post- 1991), 315-327, Colloq. Math. Soc. János Bolyai 60, North-Holland, Amsterdam, independence era, including the IITs 1992. (Indian Institute of Technology), and new universities. Others were restruc- [11] Robert Kanigel, The Man Who Knew Infinity,Scribner’s , New York, 13, 1991. tured and rejuvenated. TIFR and ISI es- tablished new centres. The impact of [12] S. Mukhopadhyaya, 'New methods all this on the mathematical develop- in the geometry of a plane arc', Bull. 1 ments in the country would be a much Calcutta Math. Soc. , 31–37, 1909. broader topic, and would deserve to [13] Syamadas Mukhopadhyaya, Collected be addressed some day. For now, it is geometrical papers, Parts 1 and 2, time to celebrate the first International Calcutta University Press, 1929 and Congress of Mathematicians happen- 1931 resp.; reviewed in Nature 4 April 1931 and 8 July 1933. ing in India! [14] Prof. Naraniengar’s Reply, J. Ind. Math. References Soc. 20 (Jubilee Commemoration Volume), p. 40. [1] Bruce C. Berndt and Robert A. Rankin, [15] A. Narasinga Rao and V. Ganapathy Ramanujan: Letters and Commentary, Iyer, 'R. Vaidyanathaswamy (1894- American Mathematical Society, 1995; 1960)', The Mathematics Student 29, Special Indian Edition: Affiliated East- 1–14, 1961. West Press Pvt. Ltd., 1997. [16] R. Narasimhan, 'Coming of age of [2] Bruce C. Berndt and Robert A. Rankin, Mathematics in India', Miscellanea Ramanujan: Essays and Surveys, mathematica, (Ed.) P. Hilton, F. American Mathematical Society, 2001; Hirzebruch, and R. Remmert, Springer- Indian edition published by Hindustan Verlag, 1991. Book Agency, 2003. [17] M. S. Raghunathan, 'Artless innocents [3] William A. Blanpied, 'Pioneer scientists and ivory-tower sophisticates: some in pre-independence India', Physics personalities on the Indian math- Today, May 1986. ematical scene', Current Science 85 (No.4), 25 August 2003. [4] G. B., 'Syamadas Mukhopadhyaya', Bulletin of the Calcutta Math. Soc. 29, [18] C. T. Rajagopal, 'K. Ananda-Rau', 115–120, 1937. J. London Math. Soc. 44, 1–6, 1969. [5] K. Chandrasekharan, 'S. S. Pillai', [19] K. G. Ramanathan, 'S. Minakshi- J. Indian Math. Soc., 15, 1–10, 1951. sundaram', J. Indian Math. Soc. 34, 135–149, 1970. [6] K. Chandrasekharan, 'Obituary: T. Vijayaraghavan', Math. Student 24 [20] Dirk J. Struik, A Concise History of (1956), 251–267, 1957. Mathematics, Fourth edition, Dover Publications, New York, 1987. [7] Joseph W. Dauben and Rohit Parikh, 'Mathematics in India', Current Science, (to appear). Email: [email protected]

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41 T. R. Ramadas Abdus Salam ICTP Trieste, Italy

Rise of Geometry in Modern India T.R. Ramadas

Introduction and Seshadri at the Tata Institute of Fundamental Research (TIFR). I will, Geometry has long preoccupied the for the most part, concentrate on this Indian mind. More than three millen- work and the themes that emerged nia ago elaborate geometric construc- from it, though other developments tions were used in the designing of and personalities will also figure in sacrificial altars. The Pythagoras theo- the story. rem was known to Indians in the 8th century BCE, prior to Pythagorean To be sure, the Indian contribution to times. Over the centuries, the study of modern geometry did not begin with geometry progressed in conjunction TIFR. In 1909, Mukhopadhyaya (work- with that of astronomy.1 When one ing in Calcutta) proved one of the first turns to Indian mathematics of the global results in differential geometry modern era, the phenomenal persona – that a simple closed strictly convex of Srinivasa Ramanujan and his discov- curve in the plane must possess at eries in number theory spring to mind. least four points (“vertices”) where In the second half of the last century, the curvature has a local maximum however, there has been an explosion or minimum (unless the closed curve in the interest in geometry. It is the happens to be a circle). With the ex- aim of this article to recount some of ception of the work of Minakshisun- this glorious history. daram – which we will come to at the end of this account – this was the The story has a clear focus (at least in high-water mark in Indian geometry hindsight) – the work of Narasimhan until the sixties.

1 For more on this -- for example, the discoveries made in Kerala in the sixteenth century – see the other articles in this issue – Editor.

42 In January 1992, an International Colloquium on "Geometry and Analysis" was organised at TIFR to mark the 60th birthdays of Narasimhan and Seshadri. This photo was taken during one of the sessions. In the first row are (from left to right) Siu, Narasimhan, Ramanan, Raghunathan, Seshadri, Procesi and De Concini. (Courtesy: Tata Institute of Fundamental Research, Mumbai)

The Tata Institute A man of broad interests and concerns and with extensive contacts, A few words about the genesis of the Chandrasekharan persuaded an School of Mathematics at TIFR will help extraordinary procession of people set the stage for what follows. The his- – Laurent Schwartz and Carl Ludwig tory of TIFR deserves to be recounted Siegel to name but two – to visit at length and in scholarly depth, but and teach at the young institute. The my aim is to sketch the background. lectures found a receptive audience; The Tata Institute was founded in 1945 among the students were Narasimhan and built up by Homi Bhabha. An ac- and Seshadri. complished theoretical physicist from the Parsi aristocracy of Bombay, he Narasimhan and Seshadri had access to the most eminent sci- entists of his time, and equally impor- Narasimhan and Seshadri were stu- tantly, the ear of Jawaharlal Nehru – dents together at Loyola College, whose enlightened views on Science Madras, where they both caught the as a vehicle of progress reflected the eye of Father Racine, himself a stu- optimism of the age. In 1949, Bhabha dent of Elie Cartan and Hadamard. (who was familiar with the Institute for It was natural therefore, that they Advanced Study at Princeton) took the went together to TIFR, where their advice of Hermann Weyl and von Neu- mathematical tastes evolved in par- mann and charged Chandrasekharan allel. Seshadri’s tended towards the with the task of building up a School of algebraic and Narasimhan’s towards Mathematics at TIFR. (Chandrasekha- the analytic. Inspired by their read- ran was at that time an assistant to ing of Kodaira-Spencer and Weil, Weyl.) Chandrasekharan, a classical they decided to study the space of analyst now in retirement in Zurich, unitary representations of the funda- is the father of the School. He was mental group of a Riemann surface. primarily responsible, together with “I read the I.C.M. (1962) talk of David K. G. Ramanathan, for the flowering of Mumford where he defines stable mathematics that followed. bundles on a curve and announces

43 The Narasimhan-Seshadri Theorem Given a Riemann surface Σ, and a considerably subtler to demonstrate. representation ρ of its fundamental In fact, they extended this result to group on an n-dimensional complex stable bundles of nonzero slope as vector space, one can naturally well. define a holomorphic vector bundle Eρ on Σ. Given a holomorphic vector In the eighties, the Narasimhan-- bundle E, its slope is the ratio of its Seshadri theorem was interpreted in degree (the first Chern number) to differential geometric terms by Atiyah its rank. (The bundle Eρ has degree and Bott: A holomorphic bundle zero, therefore its slope is zero). on a Riemann surface is poly-stable D. Mumford defined a vector bundle if and only if it admits a hermitian E to be stable if the slope of any metric such that the corresponding proper sub-bundle is strictly less Chern connection is projectively than that of E. Narasimhan and flat. Soon afterwards Kobayashi and Seshadri discovered that if ρ is Hitchin independently formulated irreducible and unitary, then Eρ the corresponding result in higher is stable, and they also proved dimensions, and this was proved by the converse statement, which is Donaldson, Uhlenbeck and Yau.

that stable bundles form a quasi- Seshadri asked this question in the projective variety”, Narasimhan once mid-seventies, and the subtle answer told me. “Since I was aware that Se- (due to Mehta and Seshadri) was that shadri knew Mumford, I asked him such a representation yields a bun- to write to Mumford asking for more dle on the compact (unpunctured) information/details.” It soon became surface, but with a “parabolic structure” clear to the young collaborators that at each of the punctures. When the (irreducible) unitary bundles and insights of conformal field theory stable bundles should be the same came to the attention of geometers in objects. the eighties, parabolic bundles lay at the heart of their geometric interpre- Although this work was immediately tation. Higher-dimensional analogues recognised, it took several decades for found applications to the topology of the full significance of this discovery to manifolds. become clear. On the one hand, it gives a differential geometric characterisa- Vector bundles on curves: tion of an algebro-geometric property The study of moduli spaces (stability) – and this has been gener- alized to higher dimensions through In the late sixties, Narasimhan and the work of Atiyah, Bott Kobayashi, Ramanan embarked on a systematic Hitchin, Donaldson, Uhlenbeck, Yau study of the geometry of moduli spac- and others. On the other hand, it is es of vector bundles on a curve. (I say a prototype of such transcendental “curve” rather than “Riemann surface” criteria in other contexts. because from the outset they used the full power of modern algebraic It was left to Nigel Hitchin to discover, geometry.) The results that came out two decades later, the beautiful of this investigation proved basic generalisation of the Narasimhan- when these moduli spaces became Seshadri theorem to the case of non- the focus of much attention in the unitary representations. nineties. Narasimhan and Ramanan also lovingly explored the geography What corresponds to unitary repre- of the moduli spaces in low genus, sentations of the fundamental group and discovered surprising connec- of a punctured Riemann surface? tions with classical geometry.

44 In 1973, Harder and Narasimhan spec- z The work of Nori on algebraic cy- tacularly applied Weil conjectures to cles and Zariski’s conjecture (on the compute the Betti numbers of these fundamental group of the comple- moduli spaces by using an analogue ment of a curve in a surface). of a classical formula of Siegel to count points over finite fields. Equally impor- z The work of Paranjape and tantly, they introduced the Harder – Srinivas on cycles and the study of Narasimhan filtration of a non-semi- the topology of varieties by Gurjar, stable bundle. This filtration and its Shastri and others. analogues have become a basic tool in algebraic and arithmetic geometry. z Contributions to commutative algebra, by Mohan Kumar, Geometric invariant theory Bhatwadekar and others.

The study of moduli problems was z The extensive theory of ‘standard one of Seshadri’s main preoccupations monomials’ developed by Sesha- from the outset (this motivated his dri and his students Lakshmibai early work on Mumford’s conjecture and Musili. on geometric reductivity), and it was thus natural that he became one of z The collaboration between Mehta the principal architects of Mumford’s and A. Ramanathan that pro- geometric invariant theory. Indeed, duced the restriction theorem – parabolic bundles provided the first a basic tool for studying sheaves (and still, the most natural) context in in higher dimensions – as well as which different “polarisations” become the wonderful tool of “Frobenius relevant. splitting”. Indian algebraic geometry – depth and breadth z Foundational work on the differential-geometric aspects of gauge theories and connections Before turning to the phenomenon between conformal field theory of Patodi, I stress that much has been and vector bundles on curves. (As left out of this account. In particular, Narasimhan’s student, I was fortu- I have restricted myself to work that nate enough to be part of this). was done by mathematicians working in India. It would take a much longer Geometry continues to be the most article to cover the depth and breadth active area of research at TIFR. Alge- of work that came out of the Indian braic cycles, principal bundles, the school of geometry. To list but some of topology of algebraic varieties, and the highlights: arithmetic geometry are the foci these days. Chennai – where Seshadri z R. Narasimhan’s work in several complex variables. moved to in the eighties and eventu- ally founded the Chennai Mathemati- z Seshadri’s work on Serre’s con- cal Institute – has emerged as an active jecture (about vector bundles on center of algebraic geometry. Groups affine space), aspects of which of geometers have formed elsewhere continued to engage the atten- as well – in Bangalore, for example, tion of many at Bombay, Pavaman and at the Indian Institute of Technol- Murthy, Parimala, Ramanathan ogy in Bombay. and Raghunathan among them. Differential geometry z The vanishing theorem of Ramanujam – the prototype for As we already saw, the Narasimhan- the theorems of Kawamata and Seshadri theorem bridges differential Viehweg. and algebraic geometry. During the

45 early days, through the lectures of and Ramanujan the third. During the Ambrose, Koszul and Schwartz, the past few decades, the trend towards language of bundles and connec- encapsulation of geometric and tions became thoroughly familiar topological data in series has revealed to many in Bombay, and one ma- a novel kind of ability – not given to all jor result that came out of this was – that has the algebraic and geometric the theorem on Universal Connec- intuitions intertwined. (In recent years, tions (1961) due to Narasimhan and for example, it has been discovered Ramanan. One could also argue that that certain generating series in enu- the work of R. Parthasarathy, a student merative geometry define modular of Narasimhan, on Dirac operators and forms and in some cases even “mock representation theory was aided by theta functions”, something that this differential-geometric viewpoint, would have, no doubt, delighted Ra- as was Raghunathan’s thesis. In fact, manujan). Patodi possessed this skill to the phrase “Differential Analysis” was an extraordinary level, and it is to the invented at TIFR in order to achieve credit of his advisors (Narasimhan and an appropriate title to cover the con- Ramanan) that they not only recogn- tents of what took the form of the In- ised this, but were able to help Patodi ternational Colloquium on Differential complete his heat equation proofs of Analysis (1964). basic Index theorems.

[This is a good place to interject a Before his life was tragically cut short few lines on the remarkable series of by illness, Patodi went on to collabo- International Colloquia that started in rate with Atiyah, Bott, and Singer in TIFR in the fifties. At the first of these a series of fundamental papers on Colloquia (1956) Selberg introduced aspects of the Index theorem. “. . a general relation which can be considered as a generalization of the To complete the circle, I must mention so-called Poisson summation formula. the work of S. Minakshisundaram with This relation we refer to here as the A. Pleijel (1949) on the asymptotics ‘trace formula’.” Four years later, at of the heat equation on Riemannian the Colloquium on Function Theory, manifolds, which prefigured the work he announced his results and con- of Patodi. I quote from Thangavelu, jectures on the rigidity of lattices in who in turn refers to K. G. Ramanathan’s Lie groups. The Differential Analysis obituary note on Minakshisundaram: meeting had contributions by (among “After taking the D.Sc. degree from Ma- others) Atiyah, Bott, Hormander, Mal- dras University in 1940, Minakshisun- grange, Milnor, Smale and Thom. In daram found himself without a job. 1964 came the Colloquium on Alge- Thanks to the timely help of Fr. Racine braic Geometry, with Weil and Groth- who was professor at Loyola College, endieck among the participants – this he could earn a living by coaching stu- was when Grothendieck presented his dents for the university examinations. “Standard Conjectures” to the world]. During these years he and Fr. Racine organised a weekly Mathematics Sem- By the mid-sixties, the focus of geom- inar which attracted many enthusias- eters at TIFR had shifted to algebraic tic participants like Chandrasekharan geometry and algebraic groups, with and Ramanathan”. Marshall Stone, who one spectacular exception – Vijay encountered them there, was respon- Kumar Patodi. sible for getting Chandrasekharan and Minakshi-sundaram to the Institute for Traditionally, a distinction has been Advanced Study, where Minakshisun- made between geometric, analytic, daram and Pleijel met! and computational intuition. One could think of Poincaré as represent- ing the first, Weierstrass the second, Email: [email protected]

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46 David Mumford Brown University, USA

Passages to India David Mumford

MY CONNECTION with India started throu- The world has become such a small gh an unexpected letter. I had written place in the intervening 40+ years a short paper for the Proceedings of the that it is hard for anyone today to International Congress at Stockholm in recall (if they were then alive) or be- 1962 on the idea of “stability” of orbits lieve (if they were not) how exotic of a representation of an algebraic this seemed then. India, even today, group and included an application is a sensory overload for Western visi- of this idea to the construction of a tors accustomed to well-regulated moduli space for vector bundles on streets and middle class societies. In curves. Unknown to me, Seshadri and the 60s, India still retained many of the Narasimhan had found a construction trappings of a medieval country. We of this same moduli space using uni- used to wait for cows as safe escorts tary representations of the fundamen- in order to cross the traffic-filled Brit- tal group of the (complex) curve. We ish boulevards of Mumbai and naked had converged, from totally different sadhus were still seen walking in the directions, on the same definition for city with their begging bowls. Repairs to the exterior of our apartment build- “stable” vector bundles, those that can ing were made from bamboo scaf- be included in one Hausdorff moduli folding, rising twenty stories and held space. This felt quite wonderful, both together only with string! A village from the mathematical perspective, to of fishermen, their fires, animals and see that one definition could arise in tents, subsisted in the very middle of such different contexts, and from the the downtown area. Gold smuggling geographical perspective, that ideas supported the “District of Song”, the worked-on half way across the world neighboring slum. In the midst of all – in Boston and Mumbai – could of this, Homi Bhabha, like Kublai Khan, turn out to be so close. Naturally, we had built his Xanadu: the Tata Institute wanted to work together and push of Fundamental Research (TIFR), an these ideas further. Seshadri was first ultra-modern paradise for scientists. to come to Harvard and, while he was Clean, air conditioned, efficiently run, there, said – “would you ever consider this was where all this wonderful math coming to India?” was being done.

51 in the pool of the Breach Candy Club alongside the Arabian Sea. Our whole family became very close friends with the Seshadris as well as the Ramanans and Narasimhans. We travelled north to Gujarat, Delhi, Agra and Nepal. Back home, I had been friends since college with Rohit Parikh, and his Gujarati family received us with great hospitality, both in Bombay and in his hometown of Palanpur. We traveled south to Tamil Nadu. There, in the town of Chingleput, we visited Seshadri’s family home. I was astonished when his father appeared, straight from court, in a wig and proceeded to put my Harvard education to shame quoting his favorite lines from Wordsworth. In back row, from left: M. S. Narasimhan, C. S. Seshadri, his wife Sundari and my wife Erika; in front row, from left: Seshadri’s nephew Narasi, his son, also Narasi, I used to commute to the Institute Steven and Peter (at the Gateway of India, 1967). by ‘BEST’ buses for a fraction of a cent My family spent the academic year each way. On the same buses were 1967/68 at the TIFR. My wife Erika and women workers with their babies, I rented an apartment in a tall new brightly clad in sarees, and chatting apartment building near the Colaba animatedly, who carried concrete on post office. Scribes could be found in their heads all day under the broiling front of the post office, reading and sun, for the new Navy construction writing letters for anyone in need of adjacent to TIFR. The idea that pov- their services. We had two sons then, erty crushes the spirit is obviously not Steven and Peter, who went to the always correct. At the end of the ride, Bombay International School where TIFR appeared like a mirage with the they learned some Hindi (they loved Arabian Sea as a backdrop, so chilled to say the numbers “ek, do, teen, char, by air conditioning that I always kept paanch” after which they performed a sweater in my office. Bhabha himself the action that their English ears heard had just died in a tragic airplane crash in the last word) as well as correct but his legacy was this amazing insti- English spelling – a vanishing skill in tute, filled with wonderful classic and the US, then and now. We enjoyed the contemporary Indian art and a brass afterglow of the British Raj, swimming and wood-paneled elevator, kept

Erika with Peter and Steven near Colaba Post Office. From left: Musili, Seshadri, Narasimhan, Raghunathan and myselft a TIFR (1967).

52 polished like the brass door handle in from his 2-year stay in India in the the Gilbert and Sullivan song. Bhabha 1930s. Bombay was still, in principle, a had started not only TIFR but its asso- dry state but we as Western residents, ciated nuclear research center BARC had obtained permits stating that we and, I assume, was centrally involved would die if we did not receive our in every aspect of India’s atomic pro- daily alcoholic “medicine”. So we were gramme. A friend of Nehru and of the able to host a proper cocktail party in industrialist Tata, Bhabha's Institute our apartment. put the newly-independent India on the international mathematical and We came again many times for short scientific circuit. After his death, he be- visits as well as for another academic came the God whose decisions were year in 1978/1979. Each time we came, never questioned. In the math depart- Bombay seemed more crowded and ment one felt equally, the presence of the British Raj receded further into the K. Chandrasekharan, who started with past. In 1978/1979, I lectured on Theta a core of Jesuit educated south Indian Functions, with equally superb note- The intellectual stimulation was Brahmins and created and shaped this takers. And this time, we were with intense: it felt as though half the department on a world-class model our third son Jeremy and our adopted before he left for Switzerland. Indian daughter Suchitra. Seshadri world's algebraic geometers were was Suchitra’s godfather and, with there [at the Tata Institute] and I lectured on Abelian Varieties. The our many stays in Bombay, it hardly the other half came through for a intellectual stimulation was intense: seemed that we had taken her off to visit. It felt as though half the world’s a foreign land. algebraic geometers were there and the other half came through for a visit. I know that these years abroad gave There were always large numbers of all my children a truly international brilliant graduate students and willing, outlook. Jeremy and Suchitra attended highly-skilled note-takers who fed the John Cannon Cathedral School, the manuscript to a legion of typists, also mastering the art of spelling. transforming with ancient Remington At this time, TIFR had built much of typewriters my often confused and its housing colony and we stayed in garbled lectures into a polished set Ramanan’s apartment in Bhaskara of lecture notes – almost overnight. since they were on leave. The shady, It was a highly interactive place: We paved areas underneath the buildings talked continuously over lunch in were a roller skating paradise where the cafeteria, tea and coffee breaks, all the children of the Institute played walks in the lush gardens amid the together after school. We had hired astonishing hoopoes and brilliantly colored parrots. This article is too small space to mention all the interactions I had but I do want to mention specifically, the pleasure I had talking with the brilliant Ramanujam who passed away tragically at an early age. He added many marvelous insights to these notes, which TIFR subsequently published (originally as an imprint of Oxford University Press, who, apparently still thinking that India was a British colony, could never find it in their catalogs when I inquired). While I was there, TIFR held one of its quadrennial international conferences, this one on algebraic geometry. The state of the field was laid out. Both Weil and Grothendieck were there, Weil having just come from a stay with the President of India, an old friend Me lecturing at the 1972 Colloquium. From left: Me, Jeremy, Suchitra and Erika in Bhaskara (1979).

53 a woman from the “District of Songs” India’s mathematical universe has now who made lentils and Buffalo milk expanded by orders of magnitude. curd every day and wondered at our Adding to the older mathematical suc- french toast, the “American khana” cess stories of TIFR and ISI (the equally my children relished along with the illustrious Indian Statistical Institute Indian fare. We spent Christmas at the built by the great statistician Maha- Krishnarajasagar Palace, built by the lanobis), there are numerous other Maharaja of Mysore and now a hotel flourishing centers of mathematics overlooking the water garden he had in India. Many of the universities now created along with the dam which have first class math departments, irrigated his lands. The monsoon found the Indian Institutes of Technology us in Goa watching hippies in bikinis are world renowned and the Indian bathe next to Hindu matrons in full Institute of Science in Bangalore, the sarees taking a salt water cure. After Institute of Mathematical Sciences this year, Erika wrote many poems in Chennai and, most recently, the about India that were published in her Chennai Mathematical Institute (CMI) ALEX TAYLOR’S CUP book The Karma Bazaar. (One of her are all very exciting places. poems is reproduced here). My measuring cup In the winter of 2008, Jenifer and I is a tin mug, found Erika passed away in 1988 and sub- visited Chennai Mathematical Insti- in a heap of broken forks sequently, I have come to India several tute. This remarkable Institute is the and bottles, in the Maine woods: times with my second wife, Jenifer, an creation of Seshadri. It is a unique Old Alex Taylor’s, who owned the land, artist. For her, the breathtaking thing blend of an American style liberal and used to feed the truants about India is its art, folk art as well as arts college with traditional Indian who came to swim. Tossing the dirty plates professional and its vivid, uninhibited guru one-on-one teaching, adding under his bed till they were needed again. use of color: a splash of red on a rock physics, computer science, history and and there is the presence of Ganesh. music to its maths curriculum. Only He’d have liked his cup Below are two of her photographs – in India could an intellectual with no to travel so far, one showing folk art in Rajasthan and business or management experience, and I took it with me to India the other, a Siva festival in Khajuraho. who spends all his spare time singing classical south Indian music, have though its bottom is round and wobbly. been the catalyst for such a unique My Indian ayah can’t believe that this educational experiment. My visit was dented old thing is American. She drinks especially exciting because it coincided her tea from it, thinking it with an intense seminar on the History the humblest cup in my house. of Indian Mathematics and Astronomy. In the last 5 years, I have become very I tell her I don’t care interested in a variety of topics in what she drinks from, just so the history of Math, and Chennai is my measuring cup isn’t half full of tea a center for Indian historical studies. when I need it. She has no use Folk art in Rajasthan. The extraordinarily rich Indian history for measuring cups. She cooks has been a revelation to me: From the by eye and handful, the way occurrence of ‘Pythagoras’s theorem’ her mother taught her. in India c. 800 BCE through the disco- very of the second order difference And every day she boils equation for sines and cosines in her strong, sweet, milky tea, the 5th century CE, to working out DQGVLWVRQP\NLWFKHQŞRRUWRGULQN the power series for sine, cosine and from Alex Taylor’s cup. It has become arctangent in the 14th century, it is hers, and when I leave an amazing story. Raised to see math she shall have it. in the lineage, Greece to Baghdad I think old Alex would enjoy to Renaissance Europe, it is clear to the thought of his measuring cup me now that a rich development in its place in Moti’s shack occurred in India in parallel, with besides the dabbas and kathoris fragmentary exchanges between the and the thalis and the great black two, benefiting both sides. There is a grinding stone. Siva festival in Khajuraho. major revival of Sanskrit scholarship

54 Raised to see math in the lineage, Greece to Baghdad to Renaissance Europe, it is clear to me now that a rich development How ancient Indian math survived – a palm-leaf The old and the new India, seen from the road by the mathematical manuscript. Chennai Math Institute. occurred in India in parallel, with fragmentary exchanges between the in India which is re-examining and It is wonderful that, thanks to the two, benefiting both the sides. deepening our understanding of the untiring efforts of Raghunathan and country’s heritage. countless others, India is now hosting the 2010 International Congress of I look forward to more visits in the Mathematicians. future: India has been a huge part of my life and has made it richer and more exciting in countless ways. Email: [email protected]

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55 Rajendra Bhatia Indian Statistical Institute New Delhi

Burrowings of a Bookworm Rajendra Bhatia

ONE OF my first mathematics text books, Joseph Edwards, Integral Calculus by High School Geometry by Hall and Benjamin Williamson, Higher Algebra Stevens, written in the early 20th cen- by S. Barnard and J. M. Child, Infinite tury for British schools, was widely used Series by Thomas Bromwich, . . . ). Some in India in the 1960s. Many students of my generation remember it as much for its neat presentation of Euclid as for the quaint phrases “pupil”, “rider”, “shew that”, and QED. Not everyone knew what this abbreviation really meant, and “Quite Easily Done” was one of the popular guesses.

This was followed by Higher Algebra by Hall and Knight, and then by Plane Trigonometry and Elements of Coordinate Geometry by S. L. Loney. This latter writer seems to have been a kind of Serge Lang of his times and his books on Statics, Hydrostatics, Particle Dynamics and Rigid Dynamics were the staple fare across India for the bachelor’s and master’s courses.

Students going to Indian universities in the 50s and early 60s read mainly British books (Differential Calculus by

56 of the English publishers of these books had branches in India. The English Language Book Society (ELBS) with grants from the British government brought out several books in various subjects at very low prices. G. H. Hardy's Pure Mathematics with a price tag of seven rupees was one of them. (In 1964 the British Council in India celebrated the quadricentennial of Shakespeare’s birth by selling a hard bound ELBS edition of his Complete Works for five rupees. I still have my copy).

Sometime in the 60s American text books began to replace British ones. With subsidies from the US govern- ment an Indo-American Text Book Pro- gram took shape. Publishers like Van Nostrand, John Wiley and Prentice Hall The front page and the first page of the two-page Contents from the first edition of Vaidyanathaswamy's book, brought out Indian editions of several published by the Indian Mathematical Society in 1947. (Courtesy: Tata Institute of Fundamental Research, Mumbai). books that could be afforded by stu- dents. P. R. Halmos’ Measure Theory was available for ten rupees (at that time ones I own are Higher Algebra by less than two dollars) and J. L. Kelley's A. Kurosh, Combinatorial Mathematics General Topology for fifteen rupees. for Recreation by N. Vilenkin and Another major publisher McGraw-Hill Functions of a Complex Variable by offered international student editions A. Sveshnikov and A. Tikhonov. at about twenty rupees. Thus, many Indian students learnt their analysis Learning from alien books can from W. Rudin, complex analysis from sometimes create strange difficulties. L. Ahlfors, topology from G. F. Simmons, The American physics text in my probability from J. V. Uspensky and college described a cyclotron tube W. Feller, and algebra from I. N. as being “doughnut shaped”. None Herstein. In 1980 copies of Notes on of the nuts I had known till then Differential Geometry by N. J. Hicks matched the rest of the description, were freely available on Bombay foot- and I was bewildered. More serious paths for two rupees. was the problem faced by a medical student whose British text book In the midst of these, there were a informed him that a certain disease few Indian books with a very wide could be diagnosed by observing a readership. Differential Calculus and particular body fluid turn the “colour Integral Calculus by Shantinarayan of a port wine stain on a white table were very popular, as were the calculus cloth”. Perhaps mathematics creates books by Gorakh Prasad. Indian fewer problems of this kind once a authors of calculus books seemed to reasonable proficiency in the foreign believe that readers older than sixteen language has been attained. years can imagine ladders, water tanks, or cars without being shown pictures It was only as a graduate student that of them. So their books weighed I learnt from Kelley’s Topology that much less than the two kilograms an Indian, R. Vaidyanathaswamy, had standard for American books. The written an outstanding book on the Mir Publishers from the Soviet Union subject in 1947. The book is based on offered many excellent books at very the author’s lectures at the University low prices. Somehow these did not of Madras. So he must have taught catch on as text books. Among the the subject for a few years before that.

57 Kelley gives the city of publication as In addition to this “in house” program- Madras but does not name the publi- me, some mathematicians from the sher (the Indian Mathematical Society). Tata Institute published other books. George Mackey in the Mathematical A particularly famous one is M. S. Reviews praised Vaidyanathaswamy's Raghunathan’s Discrete Subgroups of book for its originality and for the large Lie Groups. Another is Analysis on Real collection of exercises. In 1960 this and Complex Manifolds by Raghavan book was published by Chelsea, and Narasimhan. Chandrasekharan himself in 1998 a Dover edition came out. It remained an indefatigable writer is puzzling why this did not become after he left India in 1965, publishing a standard text in India. Another several books on number theory with pioneer in introducing modern topics Springer Verlag. away from the traditional British curriculum was Father C. J. Racine. C. R. Rao, the distinguished author According to one of his distinguished of Advanced Statistical Methods in students M. S. Narasimhan, Racine Biometric Research (Wiley, 1952) and was teaching “Modern Algebra” to Linear Statistical Inference (Wiley, 1965) undergraduates at Loyola College, was the Director of the Research Madras in 1950. He recommended van and Training School at the Indian der Waerden’s Modern Algebra whose Statistical Institute in Calcutta. The English translation had just appeared phenomenal activity at this school in in 1949. Narasimhan says that the the 1950s and 1960s led to a lecture English translation had errors. Racine note series, out of which came later, Learning from alien books can several well-known books. These sometimes create strange difficul- compared it with the original German edition he had and gave corrected include V. S. Varadarajan’s Geometry ties. The American physics text statements. He also gave his own notes of Quantum Theory, and Lie Groups, Lie Algebras and Their Representations, in my college described a cyclotron that appeared in 1956 as Introduction K. R. Parthasarathy’s Probability to Abstract Algebra published by tube as being "doughnut shaped". Measures on Metric Spaces, and C. S. Viswanathan. None of the nuts I had known R. Rao and S. K. Mitra’s Generalized Inverse of Matrices. All of these have till then matched the rest of the In 1949, K. Chandrasekharan published become classics in their fields. ISI’s description, and I was bewil- with S. Bochner the famous Fourier lecture note series too became Transforms that became the standard dered. defunct in the 1970s. Sporadically, reference for the subject. Soon after- books with high impact continue to wards, he returned from Princeton to appear from the Institute. One such lead the School of Mathematics at the title is K. R. Parthasarathy’s Introduction Tata Institute in Bombay. He initiated to Quantum Stochastic Calculus. a vigorous publication programme. The Lectures on Mathematics series In the 1970s Springer Verlag had notes of courses given by famous considerably expanded its publica- visiting mathematicians, often written tion programme and acquired by young students. One of the earliest Van Nostrand Company. In India, lecture note series, it includes authors it established a subsidiary Narosa like F. Bruhat, J. L. Koszul, J. L. Lions, Publishing House. It published L. Schwartz, C. L. Siegel and K. Yosida. “International Student Editions” of Another series Studies in Mathematics books like Yosida’s Functional Analysis, had more formal books. D. Mumford’s Halmos’ A Hilbert Space Problem Abelian Varieties from this series has Book, and Conway’s Functions of One been reprinted recently. In addition to Complex Variable all priced around these, a small number of Mathematical forty rupees. Another company Pamphlets presented instructional Tata McGraw Hill published Walter material for college and university Rudin’s Real and Complex Analysis teachers. This ambitious programme and Functional Analysis, while began losing steam after 1980, and is Wiley Eastern, Prentice Hall of India now almost at a stand-still. and Macmillan India also actively

58 reprinted some of their books. Some by K. B. Athreya and S. N. Lahiri, Indian authors were tapped in by Semisimple Groups and Riemannian these companies. The most notable Symmetric Space by A. Borel, and publication from this period is Noncommutative Geometry, Quantum K. R. Parthasarathy’s Introduction to Fields and Motives by A. Connes and Probability and Measure (Macmillan M. Marcolli; all new titles. Two of the India, 1977). Another is B. V. Limaye’s reprinted books are Characteristic Functional Analysis (Wiley Eastern, Classes by J. W. Milnor and J. D. 1981). Not all of these companies Stasheff, andMatrix Computations by were able to sustain their activities G. H. Golub and C. F. Van Loan. and most of these programmes lost direction. Recently, Springer India Another Indian publisher, Universities has entered the scene in a big way Press in Hyderabad has been publish- and brought out reprints of several ing a lot of mathematics books in low textbooks at affordable prices. cost paperback editions. Many of these are reprints of popular expository A fully Indian company, Hindustan books like Trigonometric Delights by Book Agency began a series Texts and E. Maor and Gamma by J. Havil. Some Readings in Mathematics in 1993 and are collections of articles that appeared has published sixty books since then, in Resonance, an Indian journal of most of them graduate level texts. science education, and some are new This is a different venture from the books at the college and school level. others mentioned earlier: most of the books are new titles and a majority of It seems unlikely that in the next few them are by Indian writers. To some years a school child in England, Canada extent, the series has reversed the or Kenya will learn geometry from an “book drain” as many of its titles have Indian book, as I did from Hall and been later brought out by major Stevens. However, a university student western publishers. As a sampler of the anywhere may well be learning her variety of this series I could mention harmonic analysis or probability theory Ergodic Theory and Spectral Theory of from an Indian book. Dynamical Systems by M. G. Nadkarni, Measure Theory and Probability Theory Email: [email protected]

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59 R. Ramaswamy Jawaharlal Nehru University New Delhi

Women in Mathematics: The Indian Experience R . Ramaswamy

‘LILAVATI’ [1] is a mathematical treatise Beautiful and dear Lilavati, whose of the twelfth century, composed by eyes are like a fawn’s! Tell me the mathematician and astronomer numbers resulting from one hundred Bhaskaracharya (1114–1185) who was and thirty-five taken into twelve, if also a teacher of repute. The name, ‘Lilavati’ which literally means ‘playful’ thou be skilled in multiplication by (and which is indicative of its contents) whole or by parts... is perhaps the first distinctively femi- nine title given to a scientific book! Since the text contains topics as di- According to legend, Lilavati was the verse as interest computation, arith- name of the author’s daughter and metical and geometrical progressions, the composition was intended to in- plane geometry, solid geometry, struct (or perhaps, console) her after a method of solving indeterminate a mishap due to which her proposed equations, and combinations, it ap- wedding could not take place. pears that at that time, in parts of the Indian subcontinent, women also Like other ancient Indian texts ‘Lilavati’ received instruction in mathematics. is composed in verse. Here, there are verses that are mathematical pos- The Indian Academy of Sciences, ers, many of which are directly ad- Bangalore has recently brought out dressed to the young Lilavati, whose a book of about one hundred essays feminine attributes are charmingly written by Indian women in science. recounted. Whether as supposed, this Titled ‘Lilavati’s Daughters: The Women is his daughter or one of his students Scientists of India’ [2], this collection of or even, just the author's poetic imag- first-hand accounts of what it means ery is not clear, but here is an example to become, and more importantly, to [1]– remain a practising scientist in India,

60 includes several mathematicians. The present article is, in part, loosely based on the essays contributed by women mathematicians; the author had the privilege of being associated with the project as one of the editors of the book.

There are, of course, more women in mathematics in the country than ever before, with around a hundred pro- fessional mathematicians in various academic departments. The number of women teachers of mathematics at the undergraduate level is much larger, and when it comes to enrol- ment, women students account for about 30% mathematics majors in most universities. However, there is the ever-present question of a “leaky pipeline”, with the numbers decreas- ing with advancing stages of a career in mathematics.

Pankajam and Padmavally were two of the earliest known modern women Historically, a major hurdle that many mathematicians, both students of girls have had to cross (and this is one R. Vaidyanathaswamy in Madras. The that, thankfully, is much less common former, active in the 1930s and 1940s at present) lay in being admitted to and the latter, in the 1950s have sever- schools in the first place. The number- al research publications to their credit. theorist, Rajinder Hans–Gill, born in There seems to have been something pre-Independence India recounts, in the air in Madras! Professor Bhama “The opportunities for education for Srinivasan who is now at the Univer- girls were non-existent in villages: sity of Illinois at Chicago, studied at there were no schools for girls, and Madras University where she also girls were not allowed to join schools taught briefly. She was influenced by for boys. So the first few years of my life Fr. Racine, the Jesuit mathematician were spent studying at home, longing who did so much to modernize the to go to school. [. . .] My uncle Narsher teaching of mathematics in India be- Singh [. . .] reluctantly agreed to let me tween the 1950s and the 1970s (see stay with him and study at a school for e.g., [3]). boys, where I could only go posing as a boy. This was a secret between our The mathematicians who figure in family and the headmaster. My uncle ‘Lilavati’s Daughters’ are, perhaps, was totally against the education of atypical, but there is a thread of com- women and it was only after many mon experience that runs through requests from me and my father that many of their life stories. These include he decided upon this course. I enjoyed considerable family support, parental wearing a turban and going to school encouragement, strong role models with my brother!” . . . and yet the choice of a career in mathematics remains a difficult one. While this is an unusual solution There are many barriers, so one qual- in many ways, it is undeniable that ity that all these women share in good throughout India, there are strong measure is perseverance. social pressures from both within the

61 family and outside it, for women to algebraic number theorist Khanduja stay away from mathematics. Such who studied in an all girls high school, career choices have therefore been the Arya Kanya Maha Vidyalaya with made against a background of rela- some extraordinary women teachers: tively low social support. However, be- “My interest in Mathematics was due ing able to sustain a career, especially to an excellent college teacher Miss one that has attained a level of aca- Gulshan Arora.” demic success has required tenacity, strong mentoring and considerable The representation of women in ap- family support. plied mathematics is somewhat larger compared to pure mathematics. This In an earlier generation – and to some is in part due to the larger number of extent even now – the following positions in applied mathematics if experience, articulated by Mangala one counts industry, defence and Narlikar is typical: “. . . even while I was other laboratories and teaching and studying for the M.A., my family was research departments in institutes of pressing me to get married. Soon after technology. Renuka Ravindran who a girl was twenty, she was expected worked in the area of nonlinear waves to get married and raise a family. Just and non-Newtonian fluids, and retired after I was promoted to be a research from the IISc, Bangalore after serving associate at the School of Mathematics as Dean, says “I had a passionate fond- [at the Tata Institute of Fundamental ness for mathematics. Here was an Research, Mumbai], I got a good area where the proofs were precise proposal and I decided to accept it. and unambiguous. There was clar- I had accepted the philosophy of my ity and precision at every step. What parents, namely, that the first priority could be more enchanting!” Indira for a young lady should be the family Narayanaswamy used a mathemat- and her spare time can be used for any ics education to join the Aeronautical study or hobby. [. . .] Fortunately, my Development Agency, to design and husband respected my wish to study develop a supersonic fighter Light mathematics although he too did not Combat Aircraft. wish me to prioritize career over the family.” Her choice was a part-time When asked “Why the choice of math- career, teaching when possible, and ematics?” many of the responses show number theory when possible. that chance plays no small role. The right inputs, the fortunate choices Many of the essayists have pointed out at the right times, and indeed, the the importance of teachers – particu- roads not taken. Mythily Ramaswamy larly women teachers of mathematics is like many of the women who are at the school and college level – who included in the book, an “accidental” recognised talent, provided encour- mathematician who found her voca- agement and often gave support tion through a set of fortunate circum- in many respects. The experience of stances (which also involved exclusion Riddhi Shah, gold medalist of Gujarat from other more traditional career University and recipient of the Young choices owing to a set of bureaucratic Scientist Award of the Indian National rules!). And several respond simply to Science Academy, who works in Lie the ineffable beauty of mathemat- groups and probabilities, ergodic ics. The algebraist Parimala who now theory and dynamical systems, is teaches at Emory University after a full characteristic: “Throughout my un- career at the TIFR, Mumbai, and her dergraduate studies, Shanti Prasanna student Sujatha Ramadorai are both inspired and encouraged me greatly. recipients of the Bhatnagar award, one She had faith in my abilities, took a lot of the highest forms of recognition in of interest in my well being and essen- the country. As it happens, they are tially treated me like her daughter. To also the only women mathematicians this day she is my friend and adviser.” to have been awarded this prize, and Similar sentiments are echoed by the both acknowledge the allure of the

62 aesthetics and the immense and aus- There are indications that the change is tere beauty of the subject. Their work underway: more than ever before, the has also earned them a considerable academic environment has become international reputation and accolades more inclusive, and more aware of the – Parimala is a recipient of the TWAS complex factors that enable women prize of the Academy of Sciences of in India to claim this new world of the Developing World, while Sujatha science. For instance, at a recent con- Ramadorai, who was also a member of ference held at the Jawaharlal Nehru India’s National Knowledge Commis- University in October 2009 on Advanc- sion, is a recipient of the Ramanujan es in Mathematics with a special Focus Prize of the International Center for on Women in Mathematics, of the one Theoretical Physics, Trieste. hundred and forty participants about two-thirds of the participants were At the present time there is much active women mathematicians. And of debate and discussion on issues of the 20 or so lectures at the meeting, gender, representation, and parti- again about two-thirds of these were As the country emerges as a cipation at various levels, not just in delivered by women mathematicians player on the global scene, it mathematics but in all disciplines and working in India as well as some work- professions. As the country emerges ing outside India. The numbers speak is also changing in a profound as a player on the global scene, it for themselves. and fundamental way. And is also changing in a profound and this change requires both patient fundamental way. And this change References understanding as well as sensitive requires both patient understand- handling. ing as well as sensitive handling. As [1] 'Bhaskaracharya’s Lilavati'. Colebroo- Ramadorai writes, “I am very conscious ke’s translation, with notes by H. C. that in India, women have multiple Banerji, (Asian Educational Services, contextual roles to play, and am also New Delhi, 1993). There are sev- eral other translations, some recent, constantly struck by the fact that including some digital versions. women do it with dexterity and ease, across sections of society! For women, [2] 'Lilavati’s Daughters: The women a scientific career perhaps offers more scientists of India', R. Godbole and flexibility in combining a career with a R. Ramaswamy, (eds) (Indian Academy of Sciences, Bangalore, 2008). family life. Scientific policies could be shaped towards making them sensi- [3] Narasimhan R. 'The Coming of Age of tive to the problems of women. I truly Mathematics in India', in Miscellanea feel that there is a whole new world in Mathematica (Springer-Verlag, Berlin, science waiting to be discovered and 1991) pp. 235–258. claimed by women.” Email: [email protected]

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63 Narendra Luther Former Administrator, Author Expert on Hyderabad

Hyderabad – A Procession of Cities Narendra Luther

HYDERABAD IS to cities what Taj Mahal is young prince Mohammad Quli, who travelers and chroniclers since the to buildings – a monument to love. was also a poet – indeed, the first poet 17th century onwards for its beauty, Both were named to commemorate in Urdu to have a published anthology charming layout and its vast green beloveds – Taj for Mumtaz; Bhagnagar to his credit – had fallen in love with cover. for Bhagmati. There is a major contrast, Bhagmati, a beautiful singer and dan- though. While the celebrated mauso- cer, and the city was originally named In the southwest of the Charminar is leum symbolizes a pining for a depart- after her – ‘Bhagnagar’. the Mecca Masjid. The construction of ed beloved, the city is a celebration of this mosque was started by the found- a union with a beloved. The creation Charminar1 was the city centre, and er’s son-in-law and successor, Mohd. of this city also embodies, albeit indi- the first building to be completed. It Qutb Shah in 1617. When Aurangzeb rectly, the integration of royalty with is one of the most magnificent monu- conquered Golkonda in 1687, the commoners, as also of one commu- ments in India. It was built with plas- mosque was still incomplete. When nity with another. ter and stone. It has a perfect square the emperor was approached for funds base – each side measuring 18.26 for its completion, he initially rejected Mohammad Quli Qutb Shah, the meters. Each of the four minarets rises the request, but later relented and the fifth ruler of the Qutb Shahi dynasty to a height of 48.7 meters from the mosque was thus completed over 70 of Golkonda (r. 1580–1611) issued a ground, and consists of four storeys. years after the start of its construc- decree to build a new city which The minarets contain 146 steps lead- tion. It is the second largest mosque in should be “a replica of heaven and un- ing to the top where can be found a India, after the Jama Masjid of Delhi, paralleled in the world”. Accordingly, small mosque that can accommodate and can accommodate 10,000 wor- the beautiful city of Hyderabad was 45 people. shippers at a time. founded in 1591. The architectural plan of the city incorporated many features Bhagnagar drew praise, consistent- The Golkonda sultanate had a flour- of the mythical Islamic heaven. The ly, from a large number of foreign ishing diamond cutting and polish-

1 Indeed, the monument, the snapshot of which adorns the cover of this Intelligencer! – Editor.

64 century, the Nizam had surpassed the Mughals, by ruling a vast dominion of about 510,000 square kilometers. Seven Nizams ruled Hyderabad for over two centuries.

The second Nizam entered into the 'Treaty of Subsidiary Alliance' with the East India Company in 1798. This led to the founding of the cantonment of Secunderabad which grew into the twin city of Hyderabad. In contrast to the medieval city of Hyderabad which bore a Muslim imprint, Secunderabad grew in the image of an English town. Its mall had numerous European shops which gave it a distinctively western identity. Secunderabad was marked by the presence of Christians, Anglo- The Mecca Masjid. Indians and Parsis, and acquired a cos- mopolitan character. When in 1939 it ing industry. The famed Koh-e-Noor into collectors' items. No one visit- was proposed to retrocede the civilian diamond was mined at Kolluru in ing Hyderabad can go back without part of Secunderabad to the Nizam, Golkonda. In 1656 it was presented buying a string of pearls. The city is also Indians living there protested against to the Mughal emperor Shah Jahan known for its handloom products. it. However, the retrocession did take by the renegade prime minister of place in 1945. The Hussain Sagar Lake Golkonda, Mir Jumla who sought Shah Just behind the Mecca Masjid is the built in 1562, separates the two cities Jahan’s intercession against his master, Khilwat Mahal complex comprising while the bund joins them. The Lake is Sultan Abdullah. The diamond, weigh- several palaces. The best known is embellished with the 18-meter high, ing 787 carats, changed numerous the group of four palaces called Chow 320-tonne monolithic statue of the hands before it finally came to adorn Mahalla built by the Nizam Salabat Buddha on an island in the middle. the crown of the British monarch in Jung in 1750. This was where the The statue sank into the lake while mid 19th century. Now Hyderabad is accession ceremony of the Nizams being transported for installation and known for fine pearls, thanks to skilled used to be held, and the governors- lay there for two years, before it was workers who drill and polish the crude general given a formal reception. The retrieved and installed on the island, pearls imported from China and Japan palaces lay in a neglected state for in 1992. long, till Princess Esra, the first wife of Mukarram Jah, took up its renova- The sixth Nizam ascended the throne tion in 2002, turning it into a tourist in 1869 when he was a mere toddler. attraction. He was the first Nizam to be educated

Churnings at Hyderabad

The Golkonda sultanate was con- quered by the Mughal emperor Aurangzeb in 1687 and became a part of the Deccan province of the Mugh- als. The title 'Nizam' was a shortened version of Nizam-ul-Mulk meaning Administrator of the Realm. The dynasty was founded by Mir Qamar-ud-Din, a viceroy of the Deccan under the Mughal emperors. After Aurangzeb's death in 1707, the Mughal Empire crumbled and in 1724 the viceroy in Hyderabad, Asaf Jah, virtually became Koh-e-Noor – the British Crown Jewel. independent. By the middle of 18th Buddha standing tall in the Hussain Sagar Lake.

65 confirm his view he pursuaded the Nizam to set up a Chloroform Com- mission in 1888. After conducting tests on various animals, the Commis- sion upheld Lawrie’s theory. Lancet, the renowned journal of the British Medical Association observed that the conclusion was based on inadequate evidence. Thereupon another Com- mission was set up in 1889 to which the Lancet was invited to send its nominee. The travel and hospitality ex- penses of the nominee, Dr. T. L Brunton were borne by the Nizam. The second Commission confirmed the findings of the first Commission. The editor of the Lancet praised the generoity of the Nizam in the interest of medicine. The sixth Nizam's wardrobe – largest in the world. Incidentally, subsequently the findings were proved wrong - but that is how along modern lines under a British developments were concentrated scientific progress takes place. tutor. He grew up to be such a dandy in the area between the Jubilee Hills that he never wore the same dress and the old Bombay Road towards the A Nobel Laureate twice. He had, therefore, the largest northwest of the city. In keeping with wardrobe in the world. It consisted of the new trend, he gave the new conur- The city also has a claim to scientific a hall 70 meters long with two storeys bation the hybrid name of Cyberabad. work leading to the coveted Nobel of almirahs to keep his dresses and A 'Hi-Tec City' was set up in 1998. The prize! In 1895 Major Ronald Ross, an other accessories. ‘International Institute of Information officer of the Indian Medical Service, Technology’ followed suit. Bill Gates was posted at Secunderabad. Con- Hyderabad was integrated with India set up the first office of Microsoft out- vinced personally that mosquitoes in 1948. In 1956 when the states side the U.S. in Hyderabad. The world- were carriers of the malaria parasites, were reorganized on linguistic basis, class 'Indian School of Business' was Hyderabad became the capital of the also established in the city. enlarged State of Andhra Pradesh. The first phase of industrialization of the The Movers and Shakers city started with the establishment of a number of public sector under- Hyderabad is known globally not takings by the Government of India. only for its fabled Nizams, but also for Over the years so many high-class certain scientific events which took training institutions were established place in its feudal society. Here are a in the city that it came to be known few stories of events and people that as the training capital of India. After brought renown to the city. the liberalization of the economy in 1991, Hyderabad not only became an The Chloroform Commissions important centre of activity for many foreign multinational companies, but Right from its first use in 1847 there also saw the rise of multinationals in were two theories about the risk different fields from within India, such involved in the administration of as Satyam, Dr. Reddy’s Labs, and GVK. chloroform. The Scottish view was that it constituted a risk to respira- Hyderabad entered a new phase of tion while the English held that it development after Chandrababu posed a risk to the heart. Surgeon- Naidu became the chief minister of Major Edward Lawrie, who was the the State in 1995. He initiated various principal of the Hyderabad Medical new programmes ushering the state School and physician to the Nizam into a 'knowledge society'. The new subscribed to the Scottish theory. To Ronald Ross – Nobel Laureate.

66 he carried out extensive research dur- States, Italy, Spain, England, and Japan, Jung had a vast collection of family ing his tenure in the city to confirm and was by any reckoning India’s great- heirlooms, antiques and gifts coming this hypothesis. A ramshackle building est photographer. His photographs are down from generations. He started which still stands near the old Begum- treasured all over the world. arranging them for display in his an- pet airport once served as Ross’s labo- cestral residence, the Dewan Deodhi. ratory. Ross would hatch mosquitoes Reconstruction of the City With 43,000 art objects, 47,000 books, from eggs, and would then feed them 9,000 paintings and numerous manu- on malarious persons. Subsequently Hyderabad suffered its worst flood in scripts, it became the largest one-man he would dissect them to look for para- 1908. Nearly 19,000 houses collapsed collection in the world. In 1961 the site gametes. Ross worked passionate- and about 80,000 persons, represent- museum was declared an institution ly, using only crude microscopes at his ing a quarter of the population were of national importance and moved disposal, and under extreme hot and rendered homeless. The renowned into a new building. It is a must-see humid conditions. After a great deal of engineer, M. Visweswaraya drew up place for every visitor. research, Ross established in 1897 that plans to prevent the recurrence of Some Superlatives the female of the Anopheles variety floods and for the development of of mosquitoes was the carrier of the Hyderabad. Based on his recom- Today Hyderabad can boast of some mendations, the seventh Nizam, Mir malaria parasite. This made it possible of the biggest and best features: to control transmission of malaria. For Osman Ali Khan undertook large-scale his monumental discovery Ross was construction of public buildings. By z The IMAX theatre with the world’s awarded the Nobel Prize for Medicine 1930s Hyderabad had become one largest screen, and the only true 3D in 1902 and was knighted in 1911. of the most beautiful cities in India. screen in India, is in Hyderabad. Urdu poets, who thronged to it from A Doyen of Photography all over India in search of patron- z Hitex, in Cyberabad is India’s big- age, praised it as an Uroos-ul-balad – gest and most modern exhibition Born in U. P. in 1846, Deen Dayal quali- 'a bride amongst cities'. centre. Close-by from there is the fied as a draughtsman and joined ‘Hyderabad International Conven- The Salar Jung Museum the service of the princely state of In- tion Centre’ which is the biggest dore. He showed extraordinary skill in and best of its kind in Asia. Yousuf Ali Khan, Salar Jung III was ap- photography and was patronized by pointed the Dewan (Prime Minister) z The Ramoji Film City is the world's the British Resident to Indore who in- of Hyderabad in 1912 by the seventh biggest film studio complex. It troduced him to the Viceroy. In 1885, Nizam. He was however dismissed provides total facilities including he came to Hyderabad with a letter in 1914, and this drove him into a 5-star accommodation. Its slogan of introduction from the Viceroy to depression. The Residency physician, is ‘bring in a script and take out a the Nizam, who appointed him as the Dr. Hunt suggested to him a diver- film’. Many Hollywood films have State Photographer. The Nizam was sion to get over the depression. Salar been shot here. so impressed by his work that he con- ferred the title of Raja on him. Deen Dayal set up a studio in Secunderabad. It employed about 50 persons includ- ing some Europeans; the staff also in- cluded English women appointed for taking photos of orthodox ladies who would not be photographed by men. In 1887, Raja Deen Dayal received the Royal Warrant of Appointment as pho- tographer to Queen Victoria.

The Archaeological Survey of India ac- quired 250 of his photographs. Various departments of the Nizam’s govern- ment, like Archaeology, Public Works, Forest, Revenue, Registration and oth- ers utilized his services. He thus con- tributed greatly to public causes. Deen Dayal won 12 international awards from various countries including the United 'Mephistopheles and Matilda' in the Salar Jung Museum.

67 z The country's largest and best ani- mation and art work studios are located in Hyderabad. z With a large number of modern corporate hospitals, Hyderabad is also leading in Health Tourism. The famous camp for the fish cure for asthma is held at the onset of the monsoon season and thousands of people flock from all over the country and even from abroad to take the medicine, the secret of which is held by one family. z Hyderabad has the highest tele- density connectivity in the country. To improve the physical connectiv- ity a new state-of-the-art airport Standing firm – for 2.5 billion years. (Asia’s biggest) was completed in 2008. An 11.6 km long fly-over – Asia’s longest – connects the air- port to the city.

The 'Fatal' Attraction

The landscape of Hyderabad is domi- nated by 2.5 billion-years old gneissic rock formations with fantastic shapes, and beautiful lakes. Except for a brief summer, the climate is generally equable. The city also abounds in his- toric monuments and heritage sites. It is a meeting point of the north and south of India. Bhagnagar, Hyderabad, Secunderabad, Cyberabad constitute a procession of cities under the brand of Hyderabad. To journey from the old city to Cyberabad is to travel from the A Chowki Dinner. medieval times, all the way into the future. choicest menu – the most famous be- to come back – probably to settle Hyderabad has long enjoyed a just ing Biryani. The menu is indeed very down permanently. You can avoid the reputation for its cosmopolitan cul- wide. The number of sweet dishes bench, but can’t escape drinking the ture. Its inhabitants are gracious, hos- (desserts) alone is put at 76. One rare water of the city. I believe that for every pitable people, notable for their social dish is Nimish. It is made out of the ten people visiting Hyderabad twelve graces and manners. They have a self- fluff of boiling milk and later kept in an settle down here. I know arithmetically deprecating sense of humour and earthen shallow dish overnight under it does not make sense. But the point speak Deccani, the older localized the open sky to absorb dew. is that the ten people settling down form of Urdu with a characteristic convince at least two of their friends accent. The cuisine of Hyderabad Wherever you may be from, you are or relatives to follow suit. is meat-based and is wide and var- bound to feel at home in Hyderabad. ied. The Chowki dinner of Hyderabad They say that if you sit on the stone is famous. Eight persons sit around bench at the Mecca Masjid or drink the a low table and are served with the water of Osman Sagar, you are bound Email: [email protected]

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68 University of Hyderabad memoranda of understanding have been signed, can come to the Univer- sity of Hyderabad and take courses in their specializations that will be accredited at their parent institution, and simultaneously learn about India in all its diverse aspects (history, socio- logy, the arts, economics).

Spread over an impressive 2,000+ acre campus, the University of Hyderabad is located away from the hustle and bustle of the city, offering its faculty and students an ideal tranquil environment conducive to academic excellence. The University works closely with several leading Science Complex institutions in the country, such as the Tata Institute of Fundamental Established in 1974 as a Central sciences, humanities, social sciences, University (an institution of the management studies and has one Government of India), the University of the finest schools of performing of Hyderabad is a premier institution arts in the country. The faculty is of postgraduate teaching and research highly qualified and internationally in India. It offers degree courses in a recognised for significant research diverse range of subjects, including contributions in frontier areas. Over a mathematics, physics, chemistry, life short span of time, the University has emerged as a centre for excellence in teaching, research and innovation.

The University endeavours to harness knowledge for development. Aiming Main Gate at broad-based education, the Uni- versity offers several multi-disciplin- ary courses. It has also a very unique Research, which indeed has plans to ‘Study in India Program’ under which establish new research facilities on a students from several universities in 200-acre plot within the campus of Main Administrative Building North America and Europe, with which the University of Hyderabad.

69 Hyderabad in August Situated in the Deccan Plateau, with an average elevation of about 500 meters above sea level, Hyderabad has a typical tropical climate. For most parts of the year the weather remains fairly moderate.

For the month of August the average minimum and maximum temperatures at Hyderabad are recorded to be 23°C and 31°C degrees, respectively, with a day average of 26°C. Relative humidity in the month of August has been on an average around 68%. Wearing light, cool cotton clothes, and slippers or open sandals is recommended. August is part of the monsoon period and sporadic light rain may be expected, so please carry an umbrella (and a small hand towel) on your outings.

70 Sight-seeing in Hyderabad the landscape of the region. At the western end of the roof of the Charminar is a beautiful mosque – the oldest in Hyderabad; the rest of the roof was earlier used as a court. The panoramic view of Hyderabad from here is spectacular. Around this grand monument are colourful bazaars, age-old in essence, giving one a sense of having gone back in history.

The Salar Jung Museum in Hyderabad, one of the oldest museums in the city, has an impressive collection of artefacts Charminar: Landmark of the city

The Charminar is the most famous landmark of the city, not to be missed by any visitor. Bearing the signature style of Islamic architecture, the Charminar is a structure with a square base with four towers in the four corners, each side measuring about 20 metres. Every side opens into a plaza through giant arches, which overlook four major thoroughfares. Each arch is 11 metres wide and rises 20 metres to the pinnacle from the plinth. The four minarets of the Charminar, which rise 48.7 meters from the ground, dominate Salar Jung Museum

71 modern art, wooden carvings, Would have liked to visit ivory carvings, jade carvings, Rajasthan, but don’t have the metal-ware, manuscripts, arms time? Not to worry, for Hyderabad & armor among others. Carpets, has its own Rajasthani niche at glass, furniture, lacquer and other Dhola-Ri-Dhani. At Dhola-Ri-Dhani, items collected from Persia, Arabia, the rustic ambience and charm Syria, and Egypt are on display of an ethnic Rajasthani village is here. Porcelain, bronze, enamel, recreated in an authentic manner. lacquer ware, embroidery and It offers plays, folk dances, puppet paintings, from China, Japan, Tibet, shows, camel rides and a variety of programmes for the whole Nepal and Thailand may be seen family. It also has a handicrafts under one roof at the Salar Jung market where you could pick Museum. The European chamber some souvenirs and objects of of artefact showcases oil and domestic utility. Rajasthani cuisine water paintings from England, Statue of 'Veiled Rebecca' in Salar Jung Museum gives you the taste of the northern France, Italy and Germany. A well- state of Rajasthan in every sense. maintained library is also housed Apart from the countryside within the museum. The statue setting, the plays and other from India and also from many of ‘Veiled Rebecca’, crafted knives entertainment programmes give countries all around the world. of Mughal Emperor Jehangir you a feel of the contemporary life To do justice to this museum, an and Queen Noor Jehan, famous in Rajasthan. The place also has a entire day is needed. European paintings like ‘Venice’, resort with lodging facilities and are other attractions of the accommodation for those inclined The Indian art on display includes museum. A huge clock in which to stay there; a swimming pool, an exotic collection of stone a tiny soldier comes out to strike a lake with boating facilities and sculptures, bronze images, painted the gong is very popular among an amphitheatre are some of the textiles, miniature paintings, visitors to this museum. other features of this exotic locale.

'Mephistopheles and Matilda' in Salar Jung Museum Dhola-Ri-Dhani: Rajasthani village in Hyderabad

72 Tips for your Taste-buds Nihari, a slow simmered, thick soup-like dish with lamb shanks in a spicy gravy which is eaten with a tandoor-baked bread called Kulcha or Naan is a fabled Hyderabadi breakfast. Hyderabad is also famous for its vegetarian dishes, the prominent mouthwatering ones being ‘bagaara baingan’ (a brinjal dish), ‘dahi vada’ (fried lentil balls in spicy, creamy curd) and ‘mirch- ka-sabu’ (hot Chillies, immersed in creamy gravy).

But a word of caution may be in order: the food here can be very spicy, hot and pungent, and even for those inured, it may be advisable to ask for a Hyderabadi Biryani: Delight for food lovers toned down version. Another trick could be to have one (or Hyderabad is truly a food-lover’s a Hyderabadi home an even more more) of the delicious desserts paradise. Hyderabadi cuisine memorable event is the hospitality ready at hand when you venture has been known down the ages and courtesy extended to guests. into the realms of Hyderabadi for its rich and aromatic flavors. This has to, indeed, be experienced cuisine. Some popular desserts of Traditional utensils made of to be believed. this area are ‘double-ka-meetha’ copper, brass, and earthen pots (bread pudding), ‘Gajar-ka-halwa’ are used for cooking. Exotic Hyderabadi Biryani, one of Hydera- (carrot sweet dish) and ‘Qubani-ka- herbs and spices and slow bad’s most popular dishes, consists meetha’ (apricot pudding). Wash cooking on fires for hours impart of flavored rice with meat or veg- down your meal with Iranian ‘chai’ a unique taste and texture to etables. The ‘kebabs’ such as the or tea – a hot drink with a distinct each dish. What makes a meal at ‘shammi kebab’ are also very popular. flavor.

73 Shopper's Section Some of the popular varieties of pearl are the rice pearl and basra pearl. The basra pearl is much sought after for its unmatched pearl color, luster etc. Ask for Kondapalli toys, bidri metal craft (silver inlaid on metal-copper and zinc). Bidri is an ancient art and in modern times can be found in forms like vases, ashtrays, ornaments, nameplates etc. gun metal craft, stone carvings, wood carvings, brassware. Pembarthi sheet metal art besides Nirmal art and Kalamkari art are not to be missed.

For those into fabrics, the overwhelming variety makes overweight baggage a real possibility. Apart from the handloom fabrics (cotton and silk fabrics used for clothing), there is Kalamkari, the pen-painted fabric from Machilipatnam which is used for clothing and wall decorations. Pearls and Pearl Jewellery Himroo is a special fabric used mainly for shawls and furnishing. The city is a shopper’s delight and Some of the items to look for in has much to offer in antiques, Hyderabad include Hyderabadi Lambadi mirror-work can be seen handicrafts, pearls and semi- lacquered bangles, and exquisite on blouses and skirts that are precious stones, jewellery, jewellery. Hyderabad is famous embroidered with mirrors and apparels and leather goods. for its refined and cultured pearls. very small, flat metallic disks.

74 Charminar Shops Bangle Bazaar

Besides these, Hyderabad has Dharmavaram, Pochampalli and shirt and flared trousers) and even sarees to please every woman in are known by these names. sandals with embroidery work are the world. The variety of prints available all over the city. Prices and materials for sarees is amazing A specialty of Hyderabad is zardozi depend mainly on the amount and though it can be confusing to (silk thread and cord) embroidery and quality of work put in. the uninitiated, it is nonetheless work. An amazing collection of a feast for the eyes. The most embroidered sarees, lacchas (long Abids, Basheer Bagh, Nampally, popular sarees are brought in from tops paired with long skirts and Begum Bazar, Laad Bazar and MG nearby towns like Venkatagiri, dupatta), ghagra and choli (skirt Road are the main shopping areas Siddipet, Gadwal, Naryanpet and and blouse), salwar kameez (long in Hyderabad city. These places feature some of the best stores and showrooms in the city.

Laad Bazar is famous for pearls, semiprecious stones, glass beads and colorful bangles and other jewellery. Charminar is a traditional place for silversmiths, pearl sellers and bangle sellers.

Shilparamam, an arts and craft village, offers exquisite traditional handlooms, textiles and carpets. The souvenir and handicrafts centre at Ravindra Bharati and Mahtaab Restaurant are other Shilparamam: An art and crafts village places to visit.

75 Entertainment in Hyderabad

and those interested in local culture, multiplexes, art galleries, various dance and music academies are sure to be a refreshing experience.

The Birla Planetarium in Hyderabad embodies educational entertainment and is much enjoyed by both children as well as adults. The Planetarium (timings between 10.30 am and 3.00 pm) offers daily sky shows (with commentaries in Telugu and English) delving into the enigmatic mysteries of the Universe. The Planetarium also Treasure Island boasts of the Dinosorium – a new wing that houses the fossil of the The city of Hyderabad offers a Apart from these parks, 160 million years old ‘Kotasaurus bouquet of entertainment options Hyderabad also has a number Yamanpalliensis’, along with a to choose from. Treasure Island, of pubs, discotheques, clubs, collection of smaller fossils of or TI, as it is popularly known, is snooker joints, bowling alleys and dinosaur eggs, marine shells a hot favorite as the Ocean Park, go-karting racetracks to keep you and fossilized tree trunks. The an amusement park that offers all entertained. For those who prefer Planetarium remains closed on the the thrills of being at the ocean. less adventurous engagements last Thursday of every month. The Ocean Park has a pool where waves emerge just like at an ocean. In addition, the park offers some really exciting joy rides that are bound to set your adrenalin rushing! A must-visit for those with children or those who wish to rediscover the child in them. Ocean Park Birla Planetarium

76 Taking a Break rides are also offered at some lakes.

There are a number of gardens and parks to choose from for relaxation. There are three big parks near the Hussain Sagar Lake – Indira Park in the east, Sanjeevaiah Park in the north and Lumbini Park in the south. The best time to visit the Lumbini Park is in the evening where one can enjoy the musical fountains. The Sanjeevaiah Park is home to some of the rare species of roses. The Public Gardens is a beautiful Buddha Stutue in Hussain Sagar Lake sprawling green garden where one can relax peacefully while To relax and just chill out, visit facilities, some adventure sports the children play to their hearts Hyderabad’s beautiful lakes, like para sailing and water scooter content. garden and parks. The Gandhipet Lake in Hyderabad, also called Osman Sagar, is spread over an area of 46 square kilometers and the lake is surrounded by lush greenery. At the Hussain Sagar Lake, the towering Buddha statue erected in the middle of the glistening waters is a breathtaking sight. In addition, nearby there are 33 statues of different celebrities. The Shamirpet Lake is also visited by deer, creating a delightful scenery. Apart from boating Boating pleasure at Indira Park

77 Suggestions for Sojourns Built in the 16th century, the Golconda fort is perched on a granite hill 120m high with a boundary wall covering a range of 10 kilometers. The fort has 4 smaller forts within it and 8 gates (darwazas). The main gate is called Fateh Darwaza. The gate has a unique in-built acoustic system which was used for security purposes. The ‘Light and Sound’ show brings to life the history and heydays of the Golconda Fort.

Golconda Fort

There are several places to visit around Hyderabad, some can be managed within a day while others require more time and planning.

Golconda Fort, Hyderabad

A majestic imposing monument, the Golconda Fort, lies on the western outskirts of the city, and beckons the visitor with its great cultural heritage of five centuries. Golconda was famous for its diamond mines and the world- famous 'Kohinoor' diamond is believed to have been mined here. Fateh Darwaza

78 centres, complexes storing props and period costumes, recreation joints, communication resources and a bewildering array of equipment and gadgetry for the film-maker. The wondrous mix of landscapes spanning continents and eras has to be seen to be believed. Small wonder that even film makers from Hollywood keep flocking here.

Mir Alam Tank, Hyderabad

Visit the Nehru Zoological Park and the Mir Alam Tank, an artificial lake that is about 16 kilometers Golconda Fort: Light and Sound Show

Ramoji Film City, Hyderabad floors, exotic outdoor locations, high-tech laboratories, state-of- Located about 35 kilometers the-art technology and the lush from the city and spread over green landscape and scenery. 2000 acres of land, the Ramoji Film City, is acknowledged by the Buses ply at intervals of five to ten Guinness Book of World Records minutes from the main gate of the as one of the largest film studios City to take the tourist on a guided in the world. The Ramoji Film City tour of the vast spread of exotic is not only a major film-making gardens, outdoor sets, studios, facility but is also a major tourist state-of-the-art technology labs, Lions at Nehru Zoological Park attraction. It boasts of 50 studio digital film facilities, hospitality from Hyderabad. The unique feature of this majestic Tank is that it has 21 in-built small masonry dams, semi-circular in shape. The peaceful and soothing ambience of this lake along with a visit to the Zoo will make you feel at one with nature.

Mahavir Harina Vanasthali, Hyderabad

Mahavir Harina Vanasthali National Park is situated in Vanasthalipuram, which is around 15 kilometers from the city. The park is particularly lush and beautiful during the monsoons when flowers of different colors blossom all around. This Park is home to the endangered black buck deer. Ramoji Film City The other species that can be

79 seen here include cheetahs, wild Shilparamam boars, monitor lizards, mongooses, porcupines and a vast variety of An arts, crafts & cultural village, snakes. In addition, several species Shilparamam, at Hyderabad, is of birds like partridges, quails, spread over 65 acres of prime peacocks, doves, pond herons, land in Madhapur, Hi-tech City. egrets, kingfishers, cormorants, Shilparamam showcases India’s kites, vultures, eagles can be seen rich and diverse cultural heritage here. among idyllic surroundings. Shilparamam displays ethnic art, The national park provides a crafts, skills and dances of rural tour around the park in vans for India and provides a platform for Nagarjuna Sagar Dam interested tourists. The park is the multifarious talents, which open on all days, except Mondays, otherwise would have remained from 9 am to 5 pm. hidden from public view.

Tiger at Srisailam Sanctuary

HiTech City

Boating on River Krishna in Vijaywada

The places to visit in and around Hyderabad are too numerous to be detailed here, but a special mention needs to be made of the Lakshminarasimha shrine at Yadagirigutta, which is about 69 km from Hyderabad and the Nagarjuna Sagar Dam on the Krishna River, 150 km from Hyderabad. The Ethipothala waterfalls, Srisailam Wildlife Reserve, boat ride on the Krishna River are once-in-a-lifetime Yadagirigutta Temple experiences.

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