 Srinivasa Ramanujan was a largely self-taught genius who emerged from an orthodox Hindu family to become one of the most influential mathematicians of the 20th century. Although he had almost no formal mathematical training, his remarkable intuition led him to make extraordinary contributions to several areas of mathematics, including, in particular, number theory. His work, grounded in the very ethos of the soil, much like this banyan tree, developed an enigmatic foliage that led others to sprout new roots and enlarge the canopy of contemporary mathematics. Ramanujan's mathematical explorations were abruptly cut short by his untimely death at the young age of 32. Now, almost 100 years later, we look at his life, his work, the legacy he left behind, and also meet some of the mathematicians influenced by his work, both here in India, as well as around the world. Ingenious. That's who a master. What he started in 100 years back is what we are doing today. Essentially, he was a self-trained man. Well, I was really fascinated by the story of Ramanujan when I was young. I think very influential in, I think, number theory. Maybe Ramanujan was very good at finding the tips of icebergs in some sense. In mathematics, we have somebody who is God figure, and that is Ramanujan. Srinivas Ramanujan was born on the 22nd of December, 1887, in a small district town of southern India. To commemorate his birth centenary, we made a docudrama film with enactments in 1987, more than 25 years ago. Today, walking in the footsteps of the genius, we visited many places associated with his life with a group of mathematicians, Ken Ono, E Raghuram, K Srinivas Rao, and TV Venkateshwara. It's in this small town of Eurot that Srinivas Ramanujan was born. Little Ramanujan grew up in the temple town of Kumbakonam, located on the banks of river Kaveri. During his childhood, Tennai village primary schools dotted the rural landscape of Tamil-speaking areas. Collective recitations set to metrical verses helped children develop a keen mnemonic ability. This oral tradition might have encouraged little Ramanujan's flair for logical thinking and deep reflection. In the Thinne schools, the unique function of the pedagogy is to cultivate a memory, which scholars would call it recollective memory, which is you remember and you memorize in order to be able to recollect. Here, writing is only an aid to memory and the particular ways in which the memory was trained was in terms of, you know, like you take the particular case or the pony like I'm the learning of the numbers. So there is a head monitor who recites the number and then the student is supposed to recite it after him and while reciting, which is to vocalize what he hears, and then there's a heap of sand in front of him and then he draws the graphic symbol of the number, which is the number notation and recites loudly so that you have the qualities of vocalization and visualization and memorization happening at the single continuum. Particularly what recollective memory trains you is to strong visual sense of the numbers in relation to each other. So the Kangayam Primary School that he went to, we know for sure that the Thinne schools continue to exist till about the 1920s. This familiar world of creating and cultivating and intimacy with the world of numbers was definitely part of the orientation of the Thinne schools. It was in this town high school of Kumbakonam that the magical world of numbers, trigonometry and geometry opened up for the ever curious little Ramanujan. That was an intelligent question you asked. You like some, don't you? Yes. You know, sir, I want to learn lots. I want to find out what the highest within mathematics is. What? What did you say? What do I have to do in mathematics test? What we learned from the early mathematical development of Ramanujan is the importance of problem solving at the primary school level. The Hungarian mathematician and educator George Polia observed that mathematics develops critical thinking and a love for expressing ideas precisely. In my own experience, early exposure to the beauty of abstract mathematical reasoning and access to math and puzzle books helped me develop a passion for mathematics. At the age of 12, Ramanujan borrowed from older students Loni's textbook of plain trigonometry and completely mastered its contents. This book went far beyond standard curriculum and introduced rudiments of calculus too. But the book that really transformed Ramanujan's life is Gar's synopsis of elementary results in pure mathematics borrowed from the local college library. He engrossed himself in its methods of analysis and demonstrations and proved the propositions and formulae contained therein all by himself. Excellent teachers like Seshu Iyer was very famous. So you got an opportunity for a good higher education in Kumbakonam. I mean, of course, other than Madras. You know, this is where he must have sat down and worked through car synopsis and Loni's trigonometry. Yes, and just imagine. This is where he worked on his big slate before he recorded his findings in those famous notebooks. Right, he used to write on the slate and then flip over, write again, erase, and then write. That was here. That was here. And what we call as the first notebook got written here. I mean, a large part of it is including that things on magic squares and things of that kind. In 1904, at the age of 16, based on his outstanding performance in school, Ramanujan won a scholarship to the government college in Kumbakonam. By this time, he had already started producing original research with not much encouragement from his professors and in isolation from the mathematical community of his time, he worked on problems in algebra, trigonometry, and calculus, studied infinite series, and played with prime numbers. A raw genius, he independently rediscovered several existing results, as well as made new discoveries. He said to have drawn inspiration from his family deity, the goddess Lakshmi Namagiri. However, being totally immersed in mathematics, he neglected all other subjects, and the government college did not renew his scholarship. So in 1906, he moved to Patehappa College in Madras. But there too, he concentrated exclusively on mathematics and did not pass the necessary examinations to enter the University of Madras. At the age of 14, Ramanujan wrote down his mathematical findings in three notebooks, which have become famous now as Ramanujan's freight notebooks. These notebooks contain a treasure of over 4,000 theorems, corollaries, and other results. Proofs formed the backbone of modern mathematics. But Ramanujan, inspired by Carr's book, had developed a highly unconventional style of working by finding patterns and numbers and formulae and relying on intuition and induction, rather than on proof and deduction. We are sitting in the University of Madras Library. Somehow, you know, I didn't know this until I got started on this project, this movie project, that these famous notebooks are housed here at the University of Madras Library. I'm glad that you discovered that fact. So these are the famous notebooks of Ramanujan. The very ones, we're holding them. Can you believe it? We're holding history in our hands here. Unbelievable. There are thousands of formulas in these notebooks. Look at the handwriting. Look at this. Chapter two. It's incredible. When we were filming in the University of Madras Library, the one defining moment, in fact, I will say this is one of the great moments of my life, was just holding Ramanujan's notebooks. You know, the very pages on which the great man wrote his mathematics. After a lot of hard work, he would come down to this final formulas which he thought were important. And that is what he would record in these notebooks. And what can I say? This just touches something very deep down in my psyche. It touches some kind of the intellectual fabric of our internal thoughts. It's just, yeah, just a great moment. Ramanujan made many fundamental contributions to the field of number theory. Number theory is a branch of mathematics that is primarily devoted to the study of whole numbers, such as 0, 1, 2, 3, et cetera. In particular, number theorists study what are called prime numbers, which serve as building blocks for whole numbers. A prime number is a number that is not divisible by any other number other than one and itself. For example, 2, 3, 5, and 7 are prime numbers. But 9 is not a prime number, because 9 can be written as 3 times 3. And so 3 divides 9 and so on. An intense Ramanujan, constantly toiling with equations, struck a rich purple patch with hypergeometric series, elliptic functions, Q series, continued fractions, consuming all his youthful energies. This productive period was briefly interrupted in 1909, when, as per the prevailing customs, Ramanujan's parents arranged his marriage to Janaki, a girl barely nine years old. Soon, household responsibilities compelled him to leave Gumbakonam and look for a steady job in Madras, the colonial center of British administration. For some time, he tutored students in arithmetic and somehow eaked out a frugal livelihood. Finally, he got a clerical job, largely due to the chairman, Sir Francis Spring, of the Madras Port Trust, getting thoroughly convinced of Ramanujan's mathematical ingenuity. There was a community of people in India who sort of also supported Ramanujan during his initial years. One is people like, for example, Mudaliar, who was a teacher in the Pachapa College, you know what I mean? Even after Ramanujan had to leave the college and for unemployed for quite some time, I mean, Hardee's books were introduced by Mudaliar to Ramanujan. The second is a collector, Ramchandra Rao, who paid him from his own personal salary, a small amount as a stipend, so that Ramanujan cannot work on maths while he was no longer a student. Third is people like Naranayur, you know, who got him a job in the Port Trust, who sort of entertained him every day in his house so that they can discuss maths. And he continued to help Ramanujan even after Ramanujan went to Cambridge. See, you know, people like Professor Ramaswamy, actually he was a government servant, he was one of the founders of Indian mathematical society. People like him encouraged Ramanujan to write. I mean, till then Ramanujan was sort of keeping a journal, you know, his notebooks, I mean, he was just keeping it to himself, maybe showing to a few people. They said, no, that's not the way, for example, science is done in, you know, modern world. So they encouraged him to publish. So some of his earlier papers came in India Journal for Mathematical Society, you know, I mean, his work on probabilistic number theory or elliptical function or continued fractions. They all have a lot of application even today. I mean, it's an important contribution. In 1957, the Tata Institute of Fundamental Research in Mumbai brought out a facsimile edition of Ramanujan's notebooks, which had hitherto been lying dormant at the University of Madras. For the first time, Ramanujan's startling results were brought to the attention of mathematicians around the world. Subsequently, in 2012, the Tata Institute published a second collector's edition of Ramanujan's notebooks. I'm here to talk with the director about this publication. So what motivated the Institute to bring out a second edition of Ramanujan's notebooks? Yes. Well, you know, 2012 marked the 125th birth anniversary of Srinivasa Ramanujan. It was an occasion to celebrate his life and his work, which has had profound impact in mathematics. TIFR organized a number of events to mark the occasion. One of the things we did was to produce the second edition of his notebooks, beautiful and very, very important set of notebooks of Ramanujan. And one reason to do that was that the earlier edition was now many years old. It was not in print anymore, not easy to find. So that was a good motivation. The other, I think, was that technology is better now. So we could do things we couldn't do, I think, in the first edition. For example, this one is colored. There are beautiful, you know, the original handwriting is also authentically in the colored ink in which Ramanujan wrote extremely neat handwriting. But also, you know, he wrote very carefully in different colored ink. Another reason for the second edition is that Professor Bruce Burnt, who's a well-known mathematician and has spent a good part of his life working on the notebooks. In fact, has contributed a preface to it. So that adds to the overall understanding we get of Ramanujan and his work. So those were two of the reasons. But really, the real reason is that more people can get access to these books now and hopefully enjoy them and get a true appreciation of this genius. For more than 50 years, people got to see this publication. And when this went on out of print, Tata Institute wanted to replant this book. When we saw the notebooks, we were first really very excited. While there was excitement, we were also feeling sad because it was laminated. But then this has to be preserved. Any material is laminated. It will deteriorate very soon. So it is important to capture the notebook in its original form, at least to whatever possible maximum levels. In fact, this factor pushed us to bring this book in color. 124th year is as good as any to celebrate the memory of a great mind. And the government declared the birthday of Ramanujan as National Mathematics Day. Mathematicians are remembered not only for their theorems, but also for their insightful conjectures or intelligent guesses they believe to be true. One example is Bertrand's postulate. The French mathematician Joseph Bertrand will forever be remembered for the conjecture he made in 1845 when he was just 23. Bertrand observed that between the number 3 and 6, there is a prime number, 5. Similarly, between the number 10 and its double 20, there are four primes, 11, 13, 17, and 19. Generalizing this observation, Bertrand postulated that between every number n and its double 2n, there is at least one prime number. And Bertrand checked that his postulate was true for all numbers less than 3 million. However, this still does not constitute a proof. For example, it does not show that there is a prime between 5 trillion and 10 trillion. It was the Russian mathematician Pafnuti Chebyshev, who five years later gained immortality by proving this result. And the postulate was promoted to the status of a theorem. However, Chebyshev's proof of the result was long and complicated. In 1919, Ramanujan provided an elegant two-page proof of this theorem in an article that appeared in the Journal of the Indian Mathematical Society. His proof gave rise to the notion of what are now called Ramanujan primes. Interestingly, 13 years later, unaware of Ramanujan's proof, the Hungarian mathematician Paul Erdosz, then just 18, gave another proof. This proof, although similar to Ramanujan's, contained a new idea that allowed Erdosz to generalize Bertrand's postulate in an entirely new way. Just as Bertrand's postulate inspired many mathematicians, including Ramanujan, Ramanujan's conjectures have, in turn, inspired other mathematicians to enlarge the edifice of mathematics brick by brick. I beg to introduce myself as a clerk in the Metra's Port Trist. I am now about 25 years of age and striking out a new path for myself. I have made a special investigation of divergent series. The results are termed by the local mathematicians who are startling, but they are not able to understand me in my higher flights. If you are convinced that there is anything of value, I would like to have my theorems published to Professor G.H. Hardy, Trinity. These theorems of his must be true, because no one would have the imagination to invent them. He must be a mathematician of the very highest order. Ramanujan's letter to Hardy, dated 16 January, 1913, is one of the most famous letters in the history of mathematics. It marks the first recognition outside India of Ramanujan's remarkable talent. In the late 1920s, after Ramanujan's death, this letter and a subsequent one he wrote to Hardy spawned numerous papers by several mathematicians. If you study the 1913 letter, the first letter of Ramanujan, there's a formula for the Fourier coefficient of a certain modular function. And he gives a rather remarkable formula, the first formula being some first term of a general series. And that statement itself tells you that he was aware of the circle method before he went to England. He was very convinced that the kind of results that he was writing were true. He was not postulating them as conjectures in the sense of modern mathematics. He was not educated into the philosophy of mathematics, which equated it into proving theorems only. He understood mathematics as arriving at interesting, beautiful results, which need to be justified. And he had worked out the justification for it himself. And he was prepared to sort of discuss the justification, which others could later on put them together. I think Klein, the great mathematician, had said 100 years ago, even in the early 20th century, that a great conjecture is much better than hundreds of sort of ordinary proofs. There are many problems in number theory, where one is interested in knowing whether every integer can be written as a sum of integers of a special type, important examples being Goldwasser conjectures and Waring's problem. And it was a method which was developed by Hardy and Ramanujan called circle method. It's a method which is normally used to solve such problems. And one can even see the genesis of the method in the very first letter of Ramanujan to Hardy, even before he met Hardy. And this is a method, one which he has enabled to get partial results on Goldwasser conjecture and a complete solution of Waring's problem. And the final result of Waring's problem was done somewhere in the mid-80s by myself and two French mathematicians, Ramak Dussoy and Frank Dress. It is here at the Trinity College that professors G. H. Hardy, and to a lesser extent, J. E. Littlewood, mentored Ramanujan only to be continually amazed by his remarkable originality. Later generations of mathematicians were to proclaim his conjectures as unexpected, elegant, and deep. But the most remarkable aspect of Ramanujan's encounter with the West was that, for the first time, and in his own right, he could pursue the professional career of a creative mathematician. When he arrived in England, he collaborated with Hardy and they developed the famous paper on the partition function, which was an asymptotic formula. But as Adley Selberg points out in his, I guess, his centenary lecture on Ramanujan that was delivered here at the Tata Institute in 1987, he says that if you look at Ramanujan's letter and see the first term that he had written, that coincides with the first term of Rada Marker's formula, which was discovered in the 1930s. Rada Marker discovered an explicit formula for the partition function, whereas Hardy and Ramanujan discovered only an asymptotic formula. Now Selberg, I think, semi-humorously points out that Hardy was to blame for this debacle because Ramanujan was going in the right direction of trying to get an exact formula and it was Hardy, he says, who steered him in the wrong direction. She wants to say wrong direction. Very interesting. T upon D N N minus 1 upon 24, he's right. But what's made him so certain? The function here is so many forms. To have chosen the right one is nothing less than an extraordinary stroke of genius. The First World War saw Cambridge University almost bereft of students, but the wonderful collaboration of Messers Ramanujan and Hardy during that time stands highlighted for its seminal contribution to the number theory. Our Indian research student wrote over 30 papers, seven of them jointly with Professor Hardy and ultimately he was delighted to be finally bestowed with the University of Cambridge degree. These papers shaped the course of 20th century mathematics in the most unimaginable way during his lifetime. This paper called On Certain Errorist Medical Functions. This is just great stuff. Great mathematics. It has led to hundreds and thousands of pages of modern number theory. He studied, he did lots of explicit calculations and he studied a certain function which we now call the Ramanujan Delta function and there are certain numbers, certain Fourier coefficients which show up in this function. And he literally explicitly computed these numbers and then he saw a lot of patterns. And here you again get a glimpse of Ramanujan's genius. Through these patterns he made some guesses that this must, how it must be, you know, this is how it, the general pattern, the general statement, this is how it should be. He computed the first I think 200 numbers. And then mathematicians took decades to prove Ramanujan's conjectures. The one of his conjectures was settled rather quickly by Mordel. And then later Hecker, this German mathematician, Eric Hecker put it into a conceptual framework. Later, Andrei Wei studied certain zeta functions and Wei had his conjectures. This is in the 40s, 50s. And then in the 70s, Delien was proving the Wei conjectures and as a corollary, Ramanujan's original conjecture followed. I mean, it's important. He's identified certain very important ideas and important directions in which to work. The Ramanujan conjecture, for example, is something you see all over the mathematical literature now. At the time he made it, it was just an isolated guess or a conjecture. That conjecture to prove, that to make that sort of transition from the trivial estimate to the guess Ramanujan made, it took 40, 50 years to prove. Somehow got proved at the right time when the mathematics was ready for it. I mean, a lot of other things had to happen. So, I mean, he had somehow sense, intuition for what is important. Often these sort of small things Ramanujan discovered to be just the tip of the iceberg. Now as mathematics is developing and we have a more general framework, then you see that a lot of Ramanujan's original work fits into a much bigger picture. And a lot of times that happens in mathematics that you don't fully understand a problem until you generalize it to a much more complicated problem and you establish all the necessary tools to be able to finally tackle. You see this progression of thought right from the early 20th century through Hacker's work in the 30s, through Andre Wei, through Deline in the 70s, and finally into the Langlands program which is a very huge, very fertile area of research in modern mathematics. It's a grand unifying principle and if you really want to get into the Langlands program you have to understand Ramanujan's original paper and this progression of thoughts. So it's just, to me, this is the grandeur of mathematics and Ramanujan's paper is at the heart of it. The Hardy-Ramanujan theorem concerning prime factors of a natural number being roughly log log n. That particular paper gave rise to a new branch of mathematics called probabilistic number theory and it was developed with full force by Airdish and Katz in the 20th century and now there's also a modular connection in which you can tie it to the theory of modular forms and deduce new theorems about prime devices of Fourier coefficients of modular forms using that particular approach. So the legacy is very broad, very vast and certainly there's a lot more work for us to do. The gold-damped English climate, wartime deprivation and a strict adherence to vegetarian diet all conspired to make Ramanujan seriously ill. He was admitted to several sanatoriums suffering from tuberculosis. It's good for you, Mr. Ramanujan. Ace it all up. Ramanujan, I just took a taxi cab. Number 1729. Seems rather a dull number to me. What do you think? 1729. One thousand and one. It is the smallest number, expressible as a sum of two cubes in two different ways. Guess which? Who? I? One cube plus 12 cube, nine cube plus 10 cube. Very simple. Is it indeed? Well, I suppose you wouldn't tell me the equivalent solutions to the fourth power. I see no obvious example. But I suppose the first such number must be a very large one. I'm going back now. Home. With the same disease inside me. You know you've been elected in FRS. You are a success. They want you here. You must fight your TB, you must. Back in India, with scant concern for a serious illness, Ramanujan exhausted all his energies in formulating a new set of theta functions which have now found applications in other areas of mathematics, today known as mock modular forms. It took mathematicians more than 80 years to unravel what Ramanujan meant by these enigmatic equations. A little over a year after returning to India in the festering heat of summer, an ailing Ramanujan lapsed into coma and passed away soon after in this very house. He was young, only 32 years old. Ramanujan recorded the discoveries that he made in the last year of his life on a collection of loosely ordered sheets of paper which were dispatched to Hadi after Ramanujan's death. But several years later, this collection was misplaced and so it came to be called the Lost Notebook. This collection was rediscovered only in 1976 when the mathematician George Andrews stumbled upon it in a box in the Wren Library of Trinity College, Cambridge. There were several things that were in Ramanujan's handwriting. One was the last letter that he'd actually written to G.H. Hardy in the last year of his life in 1920. And the other was a manuscript of about 100 sheets of paper, some written on both sides. And so seeing there, the Matheta functions meant that I had one of the mathematical treasures of a long time because this was Ramanujan when while he was physically ill and in fact dying, intellectually he was at the height of his powers. And so this promised to be a manuscript of great significance and that's what it's turned out to be. Very recently in the last five to 10 years there's a much better understanding of these mock modular forms, which Ramanujan wrote down many of these and discussed these and maybe for some 40, 50 years or more than that, it was not entirely clear what their significance was in the sense that they were interesting objects, but how do they relate to other objects in the theory of automotive forms and in mathematics as a whole? And that has only maybe begun to be understood in the last five to 10 years. So because of new insights, new methods, they find that these things are actually related to other things. This is what I was referring to about recent work. There's now a theorem, this is a theorem that I've proven with Amanda Folsom at Yale and Rob Rhodes at Stanford. And the theorem is about all even order roots of unity. What's surprising to us is that in the course of proving this theorem, we were required to understand the relationship between many different objects that Ramanujan inspired over the decades. It took us a lot of work to come to the realization that a theorem like this might be true. We have the benefit of 90 years of work of Dyson and Andrews and Garvin and so many others involving objects that Ramanujan didn't even would have never anticipated existed. And only once we understood them independently as their own theories were we able to assemble a theorem like this, which Ramanujan somehow had the intuition to predict that realization combined with the dozens of papers that people had written on the Mach data functions over the last 80 years led to a real explosion of mathematics. People are now applying this theory of Mach data functions in physics, number theory and combinatorics, just so many different areas. I have to say I certainly fancy that Ramanujan would have proven this theorem. Maybe even proved it the day after he wrote his last letter. This is the kind of formula that he would find. Some of his work is recently understood and this is fresh. So this is important also today. So this is not only for some mathematics from the beginning of the 20th century. No, this is fresh, this is livid. It is influential. A subject of great interest in my field of string theory has to do with the study of black holes. Now we know from the work of Beckenstein and Hawking that black holes have an entropy. And important question is how do we understand the entropy of this black hole from a more fundamental theory? This is the question string theory addresses. The answer low and behold is provided by the study of modular forms, a subject to which Ramanujan contributed a great deal. Ramanujan developed this subject because of his interest in number theory. So here we have a remarkable example of a subject developed by a mathematician because of their love for numbers, finding a central role in our understanding of string theory and in our quest to understand the fundamental laws of nature. There's another very interesting connection here. Ramanujan inspired the great Indian astrophysicist Subramaniam Chandrasekhar, who played a very important role in the understanding of black holes and their various implications. And now it turns out subsequently the study of black holes connects back to the work of Ramanujan. So in many ways, all the great truths in science and mathematics ultimately come together and are connected. Of course, on one hand, the legacy Ramanujan left for us is the body of work which he did and that remains an enigma and mathematicians are even now trying to decipher and wonder how he thought of this. I don't know why, but Ramanujan is his life story. It gives enthusiasm to the younger generation and at least a small fraction of them get attracted to mathematics. We get a lot of students who are inspired by the name of Ramanujan. They take up mathematics. They're happy that they're doing mathematics and they want to become like Ramanujan. Of course, in his story of mathematics, there are many, many, many great mathematicians, but he's really some kind of very exceptional. He's really unique. The role model, the idea that someone from such a background could rise to such great heights was a source of inspiration. And I think it was not only an inspiration to people like myself, but I think it was certainly an inspiration to many, many people. Subramanian Chandasekhar, for example, is on record as saying that when he was just a 12-year-old boy when Ramanujan passed away and his mother had been reading the newspaper article indicating that Ramanujan had gone to England and achieved greatness in mathematics and Subramanian Chandasekhar was inspired by this example. And I think that's the way it works. Ramanujan has left a huge legacy. There are, for example, many things named after him. There's the Ramanujan conjecture. There's the Ramanujan delta function, which existed before him. But Ramanujan kind of made it his own by his extensive study of it. And the Ramanujan tau function, the partition function, which again Ramanujan sort of made a great impact on. This is the Ramanujan gallery, which tells the life and work of Srinivasa Ramanujan to inspire children and adults about the natural geniuses' contributions to mathematics. The pi pavilion, which you see here, has been created mainly because one of the papers of Ramanujan is entitled on modular equations and approximations to pi. In this paper, he has given 19 infinite series representations for one by pi. And this was done by him some time ago in 1930, 1914 or so. Six decades later, in 1974, two brothers called Borewine brothers in Canada picked up this particular formula and evaluated pi up to 17 million digits. Of course, today pi has been calculated up to 51 billion digits. And in fact, there is a contribution which has been made from India by a Konpoor IIT student, which shows that any digit of pi can be calculated without reference to any other digit of pi. Indian children, for Indian students, the story is even more special. It's more inspiring because it shows that there are roots in our own culture, which there are deep roots of mathematical thinking, mathematical ideas, some of which Ramanujan was exposed to as a young person. And it gives a sense of, I think, of genuine pride and a sense of confidence that we can do this and it's worth doing this and it's worthwhile. I think this is a really great story. When Mr. P.K. Srinivasan and I jointly set up this museum, by the, that, we brought children here, we brought teachers here, we connected workshops for teachers here, I told them, I inspired them about Ramanujan and his works. Now, seeing this museum, though it is small, with all these pictures and these original letters written by him, we definitely feel that most students will take up higher mathematics when they grow up. As we know, Ramanujan was a pure mathematician who was interested in numbers for their own intrinsic beauty. But, surprising as it may be, his work still has an impact on our daily lives. For example, Ramanujan's work tells us something about how to build transportation networks. Consider a road network between different cities or between different centers within a city. Google uses these networks to recommend alternate routes. As I'm sure you've experienced, these road networks are very prone to failures. Sometimes intersections become overly congested or they're simply blocked off. So, one wants a network with many alternate routes. Specifically, even when several nodes are congested, one would like to be able to have high connectivity between different locations in the city. Building a robust network then corresponds to constructing a graph with a large spectral gap. In 1988, three mathematicians, Lubotsky, Philips and Sarnak, constructed a class of graphs that achieved the maximum possible spectral gap for a given sparsity. These graphs came to be known as Ramanujan graphs because their construction used a 1916 conjecture of the Indian mathematician. This conjecture was later proved by the Belgian mathematician Pierre Deline, who was awarded the Fields Medal, the most prestigious prize in mathematics. A second application of Ramanujan's work is important for modern web applications such as Facebook. In particular, Ramanujan-type graphs arise in the design of algorithms for detecting communities or groups of people with shared interests in such social networks. The study of such randomized algorithms is a part of the field of probability theory, which is my own domain of research. Thus, we see that Ramanujan's work from the early 1900s continues to have a profound impact on both fundamental and applied mathematical research to this day. You know, Ramanujan's discoveries are not like discoveries with proofs. It's discovery through intuition, okay? So he conjectured, you know, the existence of this mock modular forms. These are some very, very special modular forms which are being, whose properties are being unearthed, even in mathematics, only now. There's a work by string theorists and mathematicians together in trying to understand some new properties of some, you know, black holes, tiny black holes, and in which this is all married together and it's a, you know, wonderful unification. And both music and mathematics are about patterns and about stringing together patterns in a way to tell a complete story. Just like when you're improvising on tabla or in any improvisational form of music, you follow your intuition. It's not like you're using some formula or some algorithm I'm going to proceed in this way in order to make my musical piece. Mathematics should be done the same way and Ramanujan exemplifies that way of doing mathematics. As someone has said, mathematics is like a big symphony in which various, you hear various themes, they come together, they interview with each other, they sometimes dissonance is set up, right? Various things happen, but Ramanujan, for example, the themes Ramanujan set up through his work, they continue to be heard in mathematics, intertwined with various new ideas which also come up and they gather complexity, strength, interest, beauty. Mathematicians around the world find his body of work intriguing. Today, nearly a hundred years after his death, we find mathematics has slowly developed sophisticated tools which are unraveling some of the secrets hidden behind his intuitive suppositions. Over the years, the mythical tree in Ramanujan's legendary garden has opened up new vistas of possibilities. New branches of mathematics have sprouted and proliferated and found application in varied fields of different sciences, implausible in his own times. His fertile mind envisioned a world where science and mathematics converge to uncover the fundamental principles governing our universe. This then is the legacy of a self-drained mathematician who straddled the world of numbers in early 20th century and continues to inspire generations of mathematicians even today. Mathematics is like a beautiful jazz orchestra in which the creative melodies of different mathematicians harmoniously blend to form beautiful compositions. Ramanujan can be seen as one of the great conductors whose conjectures have spurred mathematical developments all over the world. Researchers continue to study properties of various mathematical constructs bearing his name. Thus, the legacy of Ramanujan is still very much felt to this day.