The Genius of Srinivasa Ramanujan | Vigyan Prasar | IISER Pune

Oh [Music] [Music] namaste in the miles of mathematics the inspirational story of Srinivas Ramanujan is unparalleled his letter to gh Hardy barely a few months before his death had hinted at the mysterious Mach theta functions now over 90 years later the last prophecy of Ramanujan is slowly revealing its tantalizing secrets uncovering fertile realms to the very edge of the mathematical world q series harmonic mass forms quantum modular forms automatic forms and many more leading to new and spectacular breakthroughs in the field of number theory in mathematics the genius of Ramanujan preoccupies generations of mathematicians even to this day [Music] you [Music] beautiful beautiful functions so unlike Rogers Mach theta functions that's what I will call them [Music] [Music] [Music] 25 years ago I made this film probably the first dramatized documentary produced in India to mark the Sentinel celebration of the great mathematician the Enigma of shinhwa's Ramanujan has entered the passage of time Ramanujan continues to be an enigma even today to unravel the mysteries of his mathematics is an infinite task yet we are not trying to do the impossible we are trying to present some facets of his life and work with a fond hope that it might excite the curiosity of some young student who would then go on to understand the depth and beauty of Ramanujan mathematics [Applause] it's in this small town of erode that shinhwa's Ramanujan was bought it Ramanujan did you say Ramanujan yeah you know I mean if you look at the horoscope which was prepared when he was born it is written as Ramanujam but the English and recited to Ramanujan the world knows him as Ramanujan but we know him as Ramanuja that's right although he was born here in erode but Kumbakonam created a much bigger role in his life this is the house where cumulative model the mother of Ramanujan was this is the parental home of Ramanujan and this is where he was born this is now house is named as Jyoti Nilayam and that seems to be only in the year 1951 so was this the was the original house broken down and rebuilt in 1951 somebody else bought it and rebuilt it reconstruct the original must have been like the house look at this what is this like 200 years old yeah the last remnants of a bygone era really Hey look at this the house is sunk way below the road after many years of probably many resurfacing of the road yeah look at that look at those tiles [Music] hey I'm on a girl without a Deena mind I'm very commendable design for that I know that I Monica know better than walk if we go back to the last letter of Ramanujan it's very clear now and in fact it's a shape inside too early he starts this I have discovered mock data functions unlike the fall state of functions studied by Rodgers and we'll talk about more about the raj for managing identities and how they relate to this when a letter says unlike the false native function studies by Rodgers they enter into mathematics as beautifully as the ordinary data functions the Ramanujan seems to be disappointed in the false data functions is somehow unnatural I first heard the story of Srinivasan Ramanujan when I was a teenager living in Baltimore Maryland in the 80s a letter came from India addressed to my father was a letter from Janaki Amal Ramanujan 's widow and she was thanking my father a gift he'd made was a contribution to the making of a bust of Ramanujan I didn't know who Ramanujan was it asked who was Ramanujan getting we are a way to go bottom is a journey of discovery in the spirit of pilgrimage visiting places associated with Ramanujan here Ken Ramanujan had a good sense of humor when Tangier was suggested to him he was sick he said Tanya what I don't want to go there it reminds me of Fanshawe Gugu he was planning on the word Tanya war a place where I died it is what it means this documentary is an attempt to imagine the cultural and mental landscape of Ramanujan to bring to light is intangible presence the depth of his creativity can only be allowed to to make evident his place in the history of mathematical sort this is the Crimean bungalow where Ramanujan stayed for a couple of months before he shifted to go Metro and unfortunately died eNOS go Maitreya is registered demócrata functions right yeah bye-bye mark yeah that's the question well the Mach theta functions I've I've spent years thinking about these functions and what you just asked is the main question so for many of us working in number theory the mathematics and Ramanujan z-- last letter the very last letter he wrote to Hardy has been a mystery to us his letter was written in 1920 and until very recently the question you just asked why Mach is exactly the question we would ask except we didn't know what the Mach theta functions were and it turns out a few years ago there was a great thesis by a dutch mathematician named by the name of sanders weygers who finally figured out what these Mach data functions are very technical I object but it turns out that with this understanding we've been able to approve lots of wonderful things in math math math physics and probability theory it's been it's been incredible useful today in current a madman oh absolutely I like to think of Ramanujan last prophecy as as you know a treasure trove of math people are even using them to study black holes now called black hole and you see this discovery that the Mach beta functions the discovery the Swagger's made is quite a historic event for mathematics there have now been many applications in mathematical physics combinatorics and number theory it's really an incredible sequence of events I'd like to think of it as this our understanding which is only the result of the last 10 years of work is the realization of perhaps Ramanujan last prophecy that's amazing it really is amazing so could Ramanujan have foreseen the mathematics that's come out of Mach theta functions today oh I don't think so most of the applications that I've discussed involve objects that weren't even defined until long after his death is this is all brand-new stuff but there's a related question many of us who work in number theory we feel like we've been playing catch-up every few years we figure out a little bit more about the jury of Ramanujan we've learned about Dyson's rank function and then we learned about the crank function and now we're beginning to learn about the mock data functions and some real and some real you know concrete rigorous sense and what we've now learned in its 2012 this letter was written in nineteen twenty we've learned that all of these objects are interrelated interrelated in ways that we're only beginning to understand now despite the fact we've been working on these objects for decades so I'd like to think of the mock beta functions the crank function the type that the rank function has beautiful instruments musical instruments that Ramanujan gave us hints of and now we see that they all fit together beautifully one harmonious orchestra that that's the result of officer Kennedy you know you had a colleague go to high school you are the excellent teachers like Aishwarya was very famous so you got an opportunity for a good higher education in Pomona I mean of course other than met us we have reached Robin geum's house you know this is where he must have sat down and worked through car synopsis and learn his trigonometry yes and just imagine this is where he worked in this big sways before recorded his findings and those famous nope okay so right on the slide and then flip over right again it is and then right here and what we call as the first notebook got written here I mean either large part of it is written here including that things aren't magic squares and things of that kind and you know his because of him Carson offices and even more so maybe we'll miss trigonometry so famous even and I was a college student raised to ceremonies trigonometric 70 or 80 years after Ramanujan yeah perhaps so without him nobody wouldn't even remember cars yeah Carson obsessed [Music] [Music] seventy bit of cold [Music] from time immemorial the pursuit of divine found reflection in all creative endeavors Ramanujan's mother was proficient in reciting verses from the DVR abundant little Ramanujan grew up immersed in the traditional milieu surrounding him for games and mathematical puzzles reflect the culture of those bygone days he was a gifted one marked by the civilizational ethos of that time his forays into the labyrinth of numbers with a way to experience beauty the grace of divinity yeah this is a new town high school which was formed in 1881 and Ramanujam joined here in 1898 in what was called as a first form which has six standard in those days the school discovered ramen isms you know intelligence and Ramon Jim discovered his mathematics in look at the layers of history in this place you know this place reminds me so much of my childhood is that right yeah almost I mean I think there's some similarity amongst all these government high schools the classrooms the benches the desks on the board you know maybe there will be a class monitor was written attendance how many were present yeah well this word all started for managing this is where it all started look at his little fellows these are managed Enza so then he famous stories about Ramanujan is a young mathematician yeah I mean in fact one of the stories where he was discovered was you know the 0x0 story zero no yeah so his teacher was asking well he posed this problem to the classroom in a thousand bananas are distributed amongst a thousand students then each would get one banana and then little fellow Ramanujan put his hand up and said sir if no bananas are distributed amongst those students would still get what haha silly everyone get one amazing so I guess maybe we already see at a young age he understood the importance of understanding variables and maybe producing something from seemingly nothing yeah all these early glimmers of his genius were already visible here yes I agree [Music] bishop okay [Music] Ramanujan was a student here in Kumbakonam in the department college he should have been studying all these other subjects but it's right what did you do i sat down and you just focus on mathematics and you work through car synopsis ah the crazy book with over 6000 theorems and formulas yeah you know what I've heard somewhere not 6,000 so the numbering is not linear it goes at every chapter there's a jump in numbering aha it's some 4800 whatever it is doesn't matter but that was his mathematical training well but surely we are synopsis was written in such unique way that Ramanujan presumably emulated the format right when he recorded his findings in his own personal notebooks right yeah do you have a favorite my favorite well you know there must have been a whole lot of CDs that he rediscovered Euler ah I know the special values of the Riemann zeta function and he rediscovered lightness or module 1 minus 1/3 is 1/5 of the dot is pi by 4 yeah you must have done all of that when he was a student here that's amazing absolutely it really feels nostalgic to be here after 25 long years the enigma Xinhua Ramanujan and now the genius of Xinhua Ramanujan in the very formative years it travels the mathematical canvas he pulled with distinguished luminaries his total absorption with numbers took him away from the prescribed curriculum and rendered him unfit or college education Ramanujan blossomed outside the university system he was already in his old realm discovering theorems that paved me to recognition the work that he did after he returned to India is although much more mature and more difficult pertains to his earlier work and not so much to his work in England I think that Ramanujan was creating a lot of new mathematics and this creation of mathematics really lifted his spirit rather than cast him down so he knew his collaborations with Hardy would be long remembered these were absolutely great papers but after he returned then he went back to q-series and it's just I mean they're just an enormous number of beautiful results on the on cue series in the lost notebook and so this is a continuation of his work and a much deeper study of his earlier work on cue series so cue series partitions mock data functions those sort of are the main three topics in the lost notebook modern development sir in Europe in mathematics passed him by even in the big city of Madras Presidency in a solitary pursuit he was far beyond the comprehension of local intellectuals [Music] later in Cambridge to his conjectures transcended the understanding of more than a generation of professional mathematicians Ramanujan had no interest in any subject other than mathematics the best teachers could not attract his attention but still his concentration was always in mathematics mathematics mathematics mathematics that's what it was when Ramanujan stayed in the Victoria hostel when he came to Madras yes I seemed to have a tray of throw of all kinds of books indeed related to his all our projects yeah few in mathematics right but history physiology theologies my literature literature terrific only if he had spent a little more time reading all these books but remember this interest was always in mathematics whatever other people might say I do he could not be distracted from mathematics it was madness twice that was probably the reason where he failed in the other subjects he was doing higher mathematics creative mathematics that is what he was a good [Laughter] representing a persona is an exercise in futility and yet is attempted again and again sometimes to open a window into the interior of the being sometimes too may not there are no true images only constructed ones enacting Ramanujan is far from being him it is an act of speculation to envision his presence to lay bare the depth-first enigma his gestures of arisia constitute folklore they continue to occupy a place in our collective memory he's alive looking into our eyes questioning our modes of understanding so these are these famous notebooks of Ramanujan the very original notebooks as incredible someone you know I didn't know this until I got started on this project this movie project that this famous normal cell house year with the University of Madras liability can't believe him I'm holding them to thousands of formulas in these notebooks look at the handwriting it's impeccable perfectly preserved and because I mean the pages are weathered but you can see we're holding history we had a voting involving history so here you can even see some of his very first works on the magic squares Magic's close and he has in our modern language this is a special value of the Riemann zeta-function yes Rita 2 n equals PI to the 2n Bernoulli be 2n factorial powers of 2 this is either he's rediscovered Euler and at some point of time he do that this was too precious and he wanted a copy you know he rewrote notebook one just didn't copy done he reported is like an revising and still here in the second notebook look at this page this page is a sea of beautiful numbers formula 38 triple I you find the continued fraction expansion for the Rogers Ramanujan function nestled among so many other seemingly unrelated formulas so here he is computing some of these class invariants in fact with no indication of how he computed these numbers at all that they're just listed as if is this was a tax code table yeah right here I have the third notebook this is the last of the notebook it turns out that most of the pages are blank but what I do really enjoy about this notebook is how it ends the very last page of this notebook contains nothing but gems and here on the very last page it begins with approximations 2pi here's an approximation to pi this accurate to fifteen decimals this one has 18 decimals and then he ends the page with erased the pi times the square root of 58 this is a gigantic number i would want to compute e to the PI times clear route 58 that he had no calculator there's no computers he did it by hand yeah I don't you know that's that's Ramanujan for you he's an enigma anything I don't know how to answer that question you know all these universities were not formed for doing research or undertaking higher education with the research orientation in India but they were formed for essentially certificate in and examining bodies so the nationalist Indians knew that sensor technology is very essential for a country like India thought that they should create a space for doing research so they are to form associations outside the framework of colonial universities it's in a similar fashion that in nineteen not seven some of the people got together and formed what is called the Indian mathematical society this society played a very important role in shaping pramana you know it's only after coming to Chennai Rama Rajyam got introduced to these people and the society and also Indian Journal of mathematics so now Ramanujan could publish in the journal so that a larger mathematician can look at it and give comments he can also be part of the mathematical community secondly this society also gave me much hope to learn new things to get access to new materials new areas of maths which i think is a very important part in Raman geum's life hey de Lauer take a look at this this is Ramanujan's very first published paper but this is on some properties of Bernoulli's numbers right and this original journal is just falling apart the papers that I don't even want to touch it so brittle done so many calculations it's calculated the numerator the denominator you know what to me is the best thing about this paper the fact that it is about bonad anomalous numbers went back two centuries before Ramanujan right and they're still relevant today they're still relevant today yeah hey isn't this some famous theorem crimes dividing the denominator of Bernoulli numbers mean is this is knocked out Claussen dry functional Claussen proved in 1840 I believe I think Hardy said something about this he says that the Ramanujan rediscovered the famous theorem of haunched out at a time of his life when yet hardly formed any definite concept of truth mmm-hmm so this means I wanted to just sat down calculated a whole lot of bonito numbers saw the prime factorizations and then he just saw the theorem that's exactly what it looks like total mystery I don't know how he did it [Music] the rule formula Toni equation million your barium is the Kuna carnival ah ha ha [Music] [Laughter] retiree good so screen what's wrong you have written some paper on Gauss Ramanujan and hypergeometric series can you tell me something about it yeah what you have shown is the most general result which is known as the 7s6 summation theorem and on this page in p8 is what is known as the gauss summation theorem what is beautiful about this satyr is that he starts with the most general result and this is the most general summation theorem which is known in literature today this is entry number 8 here which you can see is the GAO summation theorem and the beauty of it is that this is the one which he had seen in cars synopsis as an entry this particular on this Gauss emission theorem so the moment he saw that it should have come to him as a flash that this is the most general result how you cannot answer that is why you can call him as an enigma he says oh I am blue whatever word you like which is extraordinary you can say that everything which came to him came from within God said and Ramanujan wrote down and that's why I was fascinated to write this article on Gauss Ramanujan and hypergeometric series whatever Gauss and others had discovered from 1812 till the Ramanujan Stein the Ramanujan was able to reconstruct all that was discovered in Europe single-handed with only the hint of Gauss summation theorems in car synopsis isn't it remarkable it's amazing you spoke about something following beautifully what is beautiful in mathematics mean if you see how is it created it's not necessarily by a rash logical or systematic approach you know any one of us who's tried a problem knows that often you go off in the wrong direction first then you cross it out and you come back and you try many things until finally you see the right combination right arrangement of things that the logic just flows beautifully and brings the conclusion that you want and when you when you catch that that's an aesthetic experience you say aha kind of experience and you feel it and and the deepest mathematics is such that when others read what you wrote they can recreate that feeling of pleasure that feeling that s aesthetic experience can be recreated and so they can appreciate it it's a lot like music I think the nature of the human mind in trying to manipulate mathematical ideas is somewhat similar to the nature of the musician trying to compose a symphony or a new concerto or new composition in that both the rules of music have to be followed and the the aesthetic and harmonious component has to be kind of followed and the same applies in mathematics too and I think we're driven by by that kind of aesthetic dimension of mathematics sir orders of infinity agree buckler hard even the P often exact for melancholic earlier Allah in kita the exact formula Oh even the number of primes below a given number n PF n that's excellent I don't think you can do any better than that you have many more formally in this notebook and you have two notebooks so remember I was asking you to try to professor hardy Hardy is a very reputed mathematician he is well known in Europe and to write a letter to him I better introduce myself as a clerk in the Madras port trust I am now off about twenty-five years of age striking out a new path for myself I have made a special investigation into divergent series the results of which are termed by local mathematicians as 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 eternity College Cambridge is what went into his first famous letter to Hardy I see and there is so how do you had a hard time deciding you know whether this is genius or crying and then if there was a wonderful way for which me for it made the decision he said this has to be a genius because if not no one would have the imagination to you know come up with this kind of comment and then Ramanujan comes here and he gets in a few years a degree from Cambridge one of the top universities but he was a sense of how Hardy and Ramanujan worked on a day to day basis I'm just guessing but I suspect that Romano Ian would would sort of tell Hardy his remarkable calculations or formula and that Hardy being a very professional and someone who knew the subject very well would probably then go away and try prove them you see and then what he couldn't do it he'd go back to revenue you know he might realized I'm a new unit made a slight a slight mistake or something like I suspect it was more in these terms I'm sure that that Rama knew Yin did a lot to inspire Hardy in many ways and now it is it is a remarkable relationship I I can't think of any other example of course in terms of collaboration but I'm sure that that revenue Yin was the somehow the a very exceptional collaboration for him both mathematically and and personally really as the darkness of all that was happening in the first world war descended on England and and Cambridge me everyone usually terrible casualties and former Cambridge students were being killed with everyone else that I suspect in this gloomy atmosphere it probably was a ray of light of Hardy to have suddenly something new and and in inspiring and so on I suspect that's exactly no no I think this is probably an important important factor here [Music] so tell me do you have a favorite Ramanujan work yes of course I love this body of mathematics that has come out of his one paper it's called on arithmetic all functions in our favourites - yeah yeah this is where he studies the Ramanujan Delta function which is built up in terms of rama's and Tao function and the Delta function is the first prototypical example of all the forms is lead automatic forms and there's a huge body of mathematics if you want to understand that you got to understand this the delta function you know Rama did lots of calculations with the stall function and then he conjectured that based on his calculations that tau is multiplicative which led to hecka operators and hacker theory he made this guess about what is the size of towson and this sort of had later out this way conjectures and arithmetic geometry which Deline proved in the seventies and telling got a Fields Medal for this it's mind-boggling to see all these areas of mathematics which are opened out from this particular picture of Ramanujan it was incredible this this paper contains the Delta function which is the prototype for so many theories right and I guess in this way Ramanujan anticipated what would become the evolution of modern math right right it's like this this gigantic iceberg and it's tip of the iceberg and on top of this tip is the delta function you just got to understand the delta function and then you know and to understand this the rest of it this all comes out of a paper innocently called uncertain arithmetic functions the multitudes of images relocates the memory of shrinivas Ramanujan into a consciousness nikons fashioned in the hope of deferring his absence for all time to come his presence stands still in remembrance inspiring in our gaze the passport photo which was the best photographed of a head of Ramanujan Paul Granlund who is a sculptor and residents agreed to do a bust and it was awe-inspiring we were very very pleased with what it was copy was sent to mrs. Ramanujan we met her in 1987 at the centenary and when we visited her home she said when the bust arrived it was as if his spirit had returned and she garland it every day and that felt so nice because this was something that she could appreciate directly from the mathematical community [Music] Oh tell me about that you know that the map of the sky and I wanted in his pawn his horoscope people use this to predict that was his future Rommel is his mother komal optimal apparently knew his Oscar very well I believe somewhere in triplicate and one of the houses that he lived oh this is the janakiramaiah Trinity College I think he was elected shortly after he was elected FRS quite a place but a collection and what this is this must be two pages from one of the unpublished manuscripts of Ramanujan on the partition in the towel functions for the modulus Levin here's the way to ten Eisenstein series this was his notation Q times higher this is there's so much on this on the on these two pages this is the typical this is typical the kind of calculations that Ramanujan did to arrive at conferences for coefficients of modular forms and lives algebra those equations and so presumably if were to turn the page or a few pages later you'll use this formula as it's a very nice page I'm impressed with all of these original letters Wow look at this what this is this is one of the first letters that hardly wrote Ramanujan 1913 dear mister Ramanujan this is real Wow and so here he is talking about Ramanujan discussion on the distribution of primes in this newsletter and section for elliptic functions only this and your results concerning the continued fractions of the type that's great Raj Ramanujan continued fraction this is precious I've only read this in books it's the original it's amazing it's great they have this museum so I discovered the lost notebook in May of 1976 and it was a complete surprise to me because my original reason for going to Europe was to attend a conference in France and it was necessary for me to spend three weeks in Europe rather than just one because of the way the pricing of airline tickets worked so I wanted to have some appropriate academic activities besides going to the conference so when I went to the Trinity College library and asked to see these papers they came out in a couple of boxes and in these boxes besides the work of Watson there were several things that were in Ramanujan handwriting one was the last letter that he had actually written to G H Hardy in the last year of his life in 1920 and the other was a manuscript of about a hundred sheets of paper some written on both sides fortunately for me I knew what the Mach theta functions were because I had written my PhD thesis on what Ramanujan described in the letter that he wrote just before he died and looking through this document I saw here are the Mach theta functions so this must be the work from the last year of his life wowthis Ramanujan bedroom this is the famous window where he is to sit for us in the porch right here we can look out onto Saren copani Street and temples at the end of the block this is a very famous place he's to sit here and look right here at this window I'm speechless I don't know what to say [Music] India has produced many talented mathematicians in recent years a number of whom have come to Cambridge and attained high Academical distinction they will be the first to recognize that mr. Ramana John's work is of a different category he has justified abundantly all the hopes that were based upon his work in India and shown that he possesses powers as remarkable in their way as those of any living mathematician gh Hardy June 1916 the rudimentary ideas of the circle method were probably known to Ramanujan even before he left for England after arriving in England hardy and Ramanujan applied the method to this particular problem with the partition function the circle method is like a raga it's the underlying technique or underlying tool in new discoveries in mathematics and how you apply the circle method and how you improvise with it using the technique of the method is a kind of a determinant of the genius of the individual 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 gold was conjectures and bearings problem and it was a method which was developed by Hardy and Rama Rajyam called circle method is a method which is normally used to solve such problems and on and even see the genesis of the method in the very first letter or from origin to Hardy even before he met Hardy and this is the method some one which has enabled to get partial results on world was conjunction and a complete solution of variance problem and today what I saw is the profession of narsimha there are occasions from Ramanujan used to say that when he was being asked how did you get this result how did you get the result and so on he will just say I got it in a dream and any amount of his going to help so he would have just pull them by saying that he got it in a dream it's a lot narsimha dictating the equation to him and he would see that drift result in an equation and they wrote it down so can dedicate it you know beauty is very important in mathematics oh absolutely I didn't hardly say somewhere that beauty is the first test and that there's no permanent place in this world for ugly mathematics oh absolutely love that code you know theoretical math goes back to certainly if it goes back to the great Greeks people like Pythagoras you it was important to the Greeks it's very important to them and as you know they invented solid geometry plane geometry and they beautiful they built beautiful buildings and is there something about Ramanujan smack maddox connecting up to the Greeks oh absolutely actually we can just talk about the golden number the golden ratio will glory more beautiful than the golden ratio let me show you this so the golden ratio is a simple looking number it's one plus the square root of 5/2 and lots of famous architectural structures and and lots of pieces of artworks that are considered to be beautiful have been related to the properties of this single number do you think this this rectangle just length by brick is the golden ratio maybe not maybe not maybe but it's a beautiful board anyway so what's very beautiful mathematically in terms of a formula about the golden ratio is it can be expressed as a fraction what we call a continued fraction a continued fraction is a fraction that descends forever if you can imagine that being the case but this is very cumbersome it's very cumbersome but it's still quite beautiful this is a fraction that involves nothing but the number ones and if it went on forever you would find that this becomes very very close to and as a limit equals the number one plus root 5 over 2 isn't there some other short form notation of this we write 1 plus 1 and something like maybe it's sticking that one here yes I prefer this because it's easier for me to personally view it as a fraction but absolutely that would be the notation that Ramanujan used what's amazing is that Ramanujan Ramanujan is quite well known for his continued fraction he was an absolute monster yeah some people say that maybe as much as 20 percent of his notebooks involved formulas involving continued fractions so what I like about Ramanujan is that he would start with and I think this is what he did he would start with the golden ratio which has this beautiful continued fraction and try to understand a function say in a variable Q where everywhere the number 1 is replaced by Q being like U to the one v divided by 1 plus and now here where all the Q's come into play okay what Ramanujan did is he actually developed a theory of what we call class and variance which if you'd like that zone to do it's very deep this is stuff of the latter part of the 20th century in mathematics you could think of the roger's Ramanujan continued fraction as being like a factory that produces beautiful numbers these so-called algebraic numbers which the first happens to be the golden ratio what could be more beautiful than building a beautiful theory from the stuff of the ancient Greeks this is amazing and all that is linked to Ramanujan and continued fractions and starts here came up with a lot of astounding results in number theory do you think religion had some role to play in this we often hear of stories but I think probably religion in the in the commonplace understanding played no role I think that Ramanujan had tremendous faith in himself faith in his inability and in his in his abilities to make discoveries and I think he just pushed his you know inquiry passion into as far as it could reach and so in the in that sense he trained himself and and when you train yourself systematically and then you can rise above the training rise about the formal system of education then you enter into the realm of inspiration and no creativity there was no limit to his uncovering vast secrets you know even on his deathbed for example he was discovering the new theory of Mark theta functions which I think you may have heard of and that is again he found somehow in his illness he found refuge in the world of mathematical thought and I think from that sense and that is true religion but here we are underneath this beautiful temple that's nice man yep I find this rock you know it's amazing it's all whalemen it's it's beautiful it is nomicon you know this is the color of comical can you know what color literally means stone this is the abode of goddess namagiri the consort of [Applause] [Music] and then kind of amazed I'm on a dump let me down with it well the places like that and when I would bail them out [Music] when we anomaly tire [Music] sometimes I feel this is just it goes this way to explain this kind of divine intuition right do you have any good example of his Rahman in his intuition power to I I spent so many years thinking about a problem of Oilers partition numbers okay these are very simple numbers and how many ways can you add up numbers to get numbers like say four turns out there's five ways okay but amazingly for ten turns out there's 42 ways and by 20 there's 627 by 50 there's over 200,000 of these know what going mm yeah Ramanujan had this incredible intuition that allowed him to basically figure out the size of these numbers and on top of that as a number theorists were interested in their divisibility properties and he's the first that shed light on how to think about the divisibility properties of these numbers you know when Ramanujan was invited to Cambridge by Hardy he didn't want to go because he felt that he would lose his cusp crossing the Seas and so he came to namaka here and say stayed in in this mandapa with the nine a year for three nights it's in the second night I think he got a dream and the the goddess appeared in his dream and gave him an indication that he's blessed that he can leave for and he woke up nyah-nyah immediately and and told him the novel goddess has given her permission that's the sequel that we want you to do this being here in this yeah [Music] well what's striking for me is here we are at the famous namah Giri temple where the story of Ramanujan at least for the Western world it started here without the dream sequence here there would have been no Ramanujan for all of us and that leads me to feel that I have to stress that Ramanujan is still very relevant today it's a great legacy of course it's a great story that we've told the legi legacy is amazing legacy that theoretical math is unquestionable but what's important is the Ramanujan is important to us in many invisible ways he gave birth to you know parts of number theory which are used in cryptography the construction of you know networks a lot of graph theory depends on his ideas and so we're managing graphs and you know like I said he is relevant today she's not just a theoretical physics and math but he's relevant to all of us but in beautiful invisible ways [Music] 19:20 the April 26 about 10:00 a.m. in the morning rahman geum passed away it was a very sad moment and it was even saddest because there were hardly anybody for his cremation and the pro hits who were supposed to come to perform the last rites and refused to come and the family was in a great dilemma the last Rite has to be performed before the Sun doe and there was no Pro hits why because the Prophet said that he has cross the Seas and hence he is no longer a Brahman and we won't perform his last rites few prophets finally agreed to come after us a lot of persuasion and finally the cremation was hell it was same Rama Rajyam who one year ago in 1919 when he came back to India the people had gathered to felicitate him the people who are willing to rub shoulders with him but during his last right there was hardly anybody go Metro bungalow has been demolished at the altar of commercial development some years after this earlier film was shot here [Music] [Applause] Ramanujan spacian for the beauty of numbers is so intense that he endured pain by working on his last offering the mock theta functions [Applause] he was only 32 years old when he died the sequence of pillars evoke the symbolic form of Lord Narasimha a great Russian mathematician Schaaf ravitch once observed that a superficial glance at mathematics may seem to give an impression that it's a result of separate individual efforts scattered across continents and ages however the inner logic of its development brings to mind a work of singular intellect developing its thought systematically and consistently using a variety of individual efforts only as a mate it resembles an orchestra performing a symphony conceived by a single person this is no doubt true and yet the music reaches a crescendo in the hands of only a few luminaries once his luminary was undoubtedly sweet once Tremonti in this cosmic symphony of mathematics he played a major role [Applause] I'm extremely sorry for not writing you a single letter up to now I've discovered very interesting functions recently which I call Mach theta functions unlike the false data functions partially studied by Rodgers they enter into mathematics as beautifully as the ordinary theta functions the functions in this letter are pieces of what we call harmonic mass forms that realization combined with the dozens of papers at people that written on the mock theta functions over the last 80 years led to a real explosion of mathematics the letter begins with a discussion of asymptotic sneer roots of unity for what he refers to as oil Arian theta functions we call these modular forms special kinds of modular form did he use this phrase mortola forms no when he uses the word ordinary theta function that's what he means he means modular form so the one question that he raises is this can there be an oil aryan series that somehow pretends to be a modular form but isn't a modular form and hence the word mock math took many steps forward because of his letter even without understanding what he wrote in this letter now I think finally in 2012 we have that understanding and the picture is it's incredible it's breathtaking yeah [Music] that I don't that I don't that I don't that I [Music] none that I and number and I'm like I know that a I don't that I I don't I don't that I don't that I know I know I don't I don't that I don't that I I know they're numb that I don't that I know and that I am that I am [Music] I don't know I know I know I know die I know I don't I don't I don't I don't I don't I don't I don't [Music]

Channel: IISER Pune

Views: 46,846

Rating: 4.9281664 out of 5

Keywords: IISER Pune, Science Media Centre, SMC, Science Communication, https://sites.google.com/acads.iiserpune.ac.in/smc/home, Vigyan Prasar

Id: SYBOOjhAMsM

Length: 60min 22sec (3622 seconds)

Published: Tue Sep 19 2017