As part of the National Mathematics Day 2014 on 22nd December, Gonit Sora (https://gonitsora.com) in association with Sciensation Media (Pune) and Gyanome.org organised a mini web conference wherein various eminent mathematicians from India and abroad shared their perspectives on mathematics. For Gonit Sora, Manjil Saikia had a conversation with Prof. Ken Ono which we reproduce here verbatim (except for correction of slight mistakes and eliminating speech pauses, etc). The conversation was transcribed by Kritashri Sukanya and Ananya Guha.
Prof. Ken Ono is the Asa Griggs Chandler Professor of Mathematics at Emory Univesity in the USA. He is a well known number theorist and is recognised through out the world for his work related to the mathematics of Ramanujan. He has held numerous positions earlier and also serves on the editorial boards of numerous reputed journals specialising in number theory. Recently his work on mock theta functions was selected by Discover magazine as the second best scientific work of the year (2014).
Manjil: Thank you very much Sir for accepting our invitation.
Ken: Oh you are welcome.
Manjil: It’s a great pleasure to have you speak with us. Sir, we will have a very short conversation with you regarding your work related to Ramanujan’s Mathematics. First of all we would like to know how you got interested into working on areas influenced by Ramanujan.
Ken: Actually it’s a very long story. I first learned about Ramanujan when I was in high school in the early 1980’s. My father who was a Math Professor, he just retired from Johns Hopkins, he was one of the many mathematicians who made a gift, he contributed to this bust that Paul Granlund made of Ramanujan and I first learned about Ramanujan because of that story and well to make a long story short I never forget about Ramanujan, I watch the Nova special and before I knew it I was a graduate student in Los Angeles in early 1990’s and my advisor who just passed away, Basil Gordon was really quite a distinguished number theorist and he told me about the various Rogers-Ramanujan’s identities and before I knew it I wanted to work in areas of Math influenced by Ramanujan. So it was quite accidental but you know I believe in fate.
Manjil: Was your father a big influence in your early career and your love for Mathematics?
Ken: Oh absolutely. If my father had been anything other than a Mathematician I am sure I would have been doing something else. But as a young boy we had traveled all over the world for conferences and I just fell in love with the lifestyle. My father would spend afternoons in his study with his notepads thinking and a couple of times we have traveled to certain places talking about things that he was passionate about. Well when I started college my plan was to become a Doctor but I did not do well in Organic Chemistry so that was the end of that and so I switched to Mathematics. So my father played a very central role largely because he was following his passion and work was not work for him, it was a love.
Manjil: What kind of Mathematics do you do? Can you give a simple example for the school students or the college students.
Ken: Yes, one of the things that I really like about Ramanujan’s work and the kind of number theory that I work on is very simple to describe. Often the proofs involve complicated objects but the point is usually very clear. One of my favourite functions is the function p(n), the partition function. It is a simple count so if you work as students or anyone, in how many ways can you add up numbers to get numbers you would be talking about partition of integers and so one of the functions I am most interested in is p(n) and it acts in how many ways can you add up numbers to get 4 when we don’t take into accounts orders. It turns out that there are 5ways. In how many ways can you add up numbers to get 6? It turns out that there is 11 and these numbers the p(n), they grow at an incredible rate and Ramanujan was very famous for discovering in the early 20th century some divisibility properties. It turns out that every 5th partition number starting at 4 is a multiple of 5 and some of my recent works about 10 yrs. ago, I generalized that to number i other than 5. Ramanujan together with G. H. Hardy, were the first persons to really come up with a very good theorem about the size of these numbers and how rapidly they grow and two years ago with my collaborator Jan Brunier, we visited the Hardy-Ramanujan formula and found a different kind of formula and so my relationship to Ramanujan involves lots of things: Mock Theta functions, Modular forms and other types of objects but perhaps my favourite one just is related to adding and counting, his work in the partition functions.
Manjil: You have been a major exponent of the partition function. In recent years you have given us many outstanding results. For example the work that you have mentioned with Jan Brunier. Can you briefly describe what your work in that area is?
Ken: Oh yes. As I said that the partition function grows at a very rapid rate and Hardy and Ramanujan determined the growth of p(n) as n goes to infinity. So, what we like to do is find a formula. A formula wherein you find the partition number p(n). What Hardy and Ramanujan arrived at was an asymptotic one which gives with very reasonable accuracy these numbers. They were later perfected to get so called the perfect formula for the partition numbers where p(n) is expressed as an infinite sum, a convergent infinite sum and so that was the first kind of exact formula. But what we derived is a different kind of formula that comes from the theory of harmonic mass forms. We proved that the partition numbers p(n) are sums of algebraic numbers. So, for every n it turns out that there is a roughly square root of n algebraic numbers with a special value of a function that we find which is a special kind of Maass form. There is a universal single object called Maass form which helps us in getting our result.
Manjil: I had the pleasure of listening to you in the Ramanujan’s 125th Conference in New Delhi and there you mentioned that Ramanujan’s work has applications in physics, string theory and so on. Can you tell us in which areas in work is influenced apart from Mathematics?
Ken: Well sure. There is work of people like Sameer Murty, Atish Dabkolar and Don Zagier who show that we can have what are called multi-centered black holes using the mock Jacobi Forms. The first example of this forms are Ramanujan’s Mock Theta Functions, formulas of Ramanujan in his last letter to Hardy. Using much work in the last year and a half in this kind of Mathematics, we are now, infact, we have just finished a paper which I wrote with John Duncan and Mike Griffin which is available on the arXiv. It is about the subject of moonshine and in it we explain how the asymptotic properties of Mock Modular forms are related to three dimensional quantum gravity theory and so it turns out that the Mathematics of Ramanujan’s last letter has lot to do with quantum gravity, representation theory of the sporadic groups like the monster, there is a so called Umbral Moonshine conjecture that we have just proved and this is all related to the very stringy kind of Mathematics.
Manjil: What do you think was the impact of Ramanujan’s days in India towards his Mathematics? Would he had been a different mathematician if he would have got a different education?
Ken: Oh! That’s a great question. I am so glad that you raised that question because right now in the United States, we subject our students to many tests many examinations and sometimes we think that we are missing the point. While Ramanujan was a two times college dropout, he was brilliant and so I think if Ramanujan were to be born today, he would be in danger of losing his genius. We would be subjecting him to ordinary course work, ordinary routines of examination and test course and his credit was that his parents continued to support him and that’s something that we in the US can understand: how you could continue to believe in your child when you do not understand what they are doing and all the external evidence from examinations suggest that the student isn’t doing well. So, there is something about the makeup of his family and I give his parents a tremendous credit. I think his strong belief in himself played a very important role. Whatever influence was there in him from the environment that he grew up, I am sure it had a very important role.
I am a consultant on the movie project “The Man who knew Infinity”. This were Dev Patel who will play Ramanujan and Jeremy Irons will play G. H. Hardy. Before we started shooting the film, we had hours of discussion with your very problems and you see when you watch the movie, a beautiful scene in England where Ramanujan has a spiritual moment. He is sitting before a column design and he has his discussion with Hardy that “you know my mathematics is somehow divine origin” and there is a big conflict internal and external where Ramanujan has to know how to explain his ideas to Hardy not just mathematically but also culturally and personally and these are the things I think would be very difficult to replicate today so to answer to your question I wish if I had either a Time Machine, I wished to back in time and totally exhibit what the circumstances were in his home because they had been quite special and I think we have to give his parents quite a bit of credit for allowing him to pursue his Mathematics when certainly by today’s standards it won’t make any sense certainly as a parent I know it won’t make any sense to me.
Manjil: Since you mentioned the spiritual incident that you say will there in the movie, so there has been instances in various print articles or people habe this sense of belief that Ramanujan was a very spiritual man. Infact one of his very famous quotes is that “An equation had no meaning to me unless expresses a thought of God”. So according to you how much is it true?
Ken: Well, this is a very difficult question to answer because I have never met Ramanujan. You know, I know some people believe that there is no truth on the divine interventions of his works. My answer to that is when people ask me personally how I arrive at my findings….thats a hard moment…thats still magical…I can…if you are to ask me if my work were in some sense spiritual….I would have to say YES. I don’t know. What it is about…you know thinking about a problem for many days or hours and then finally having a solution come to you…popping to your brain…!! How can that not be somehow a spiritual…or special moment. Whether that is related to something religious or divine. NO. I think to each is own but certainly I think every mathematician, every theoretician, everyone who loves the art, everyone who loves the music certainly has to agree that there is something absolutely spiritual beyond human component to appreciating these arts and sciences.
Manjil: According to you what is the single most important lesson that Ramanujan’s life teaches us?
Ken: Ramanujan’s life teaches us?
Ken: Well this is what I get from Ramanujan’s life. Ramanujan first of all overcame many great obstacles. He become a world leader and teacher of western science, western mathematics. That’s just great distances geographically. He overcame great cultural barriers those days. I can’t even begin to understand. And all the time he was basically alone in England. And had very few close friends. We learn from that’s a triumph full human spirit. This was a very strong individual, very self-confident individual. And we can all learn from people with that exhibit that kind of strength.
Manjil: In present day India we do not see so many brilliant mathematicians like Ramanujan. There are many well known mathematicians in India but there is still no one in that class. So what do you think is lacking?
Ken: Well…I would put it this way… We don’t see many brilliant mathematicians like that at all ever. So I would like to place Ramanujan in group today consist of people like Terry Tao, Jean Perre Serre, some of the great people like this come along very infrequently. And when they do come along you just have to appreciate and celebrate them for their accomplishments and achievements that they provide to the rest of us. In terms of training young students today, well it is true you have to learn a body of work before you can make contributions. And it is harder now certainly that in Ramanujan’s days. But like I said one thing… one of the things we should learn is that…obviously he was very confident scientist and one who didn’t follow trends. He was a trend setter. And may be we need to invite our students to be a little bit more daring. Often our best PhD students and Post docs. work on problems which are 90% near completion when they start. If you are working towards the conjecture there may be a large number of experts working around round the problem. Waddling away and refining the problem so that you can really understand it. And then conjectures are proven. That’s important, or maybe we want to inspire some people to take some much more brave approach to mathematics. Start at least investing some efforts in terms of attacking those problems where looks like you are running into a brick wall. We need some people to run into the brick walls if we are to ever break through those walls. Now that’s kind of a gamble. Certainly we need people like that and Ramanujan was someone certainly like that.
Manjil: Can you describe to us a typical day of your work?
Ken: A typical day…! (laughs) Well not very different from yours. I live in Atlantam Georgia well known for having terrible traffic. So it takes me one hour to commute to work. I come in to my office or some days I lecture. On Mondays and Wednesdays I have either one or two classes. And the rest of my day I spent at my desk or my black board which is actually a glass board, talking to my PhD students and my Post Docs. And apart for that sometimes I travel. I travel quite a bit. Well my typical day is quite relaxed. I also exercise and do triathlon so I find time to swim, bike and run. And I have two children. My daughter is now a college student so I don’t see very much of her. My son is in 10th grade in high school. So we have two more year with him. And yes, my days are fairly simple I think.
Manjil: What advice would you give to your doctorate or post doctorate students?
Ken: I take graduate advice very seriously. There is no single speech that I give to my students. I love my students and have many of them. And they are very important to me. So what I like to do with my students on an individual basis is first to figure out what their skills are. There are many ways in which you can be a successful mathematician. You could be someone who invents an area. You could be someone who could be continually learning and because you have ended up learning so much, you could adapt ideas and transport them or import them into other questions. There are other students who are great problem solvers. They just need to be presented with problems and they can solve them. So on and so forth. One of the firsts things I do, is I try to understand very clearly what individual student strengths are. I tell them what I think their strengths are and we discuss what would be good sorts of question for research plans. And by the time they are third year graduate students we usually come to very clear agreement on what kind of projects we will work on. Hopefully they already have had some papers that are published. And the last 2 or 3 years the graduate scholars is about professional planning. What do you have to prove and what do you have to establish to get to the next level, where your thesis makes you someone who is in demand. And so what do I tell my students… I see them every day. Infact right before we were in chat with you, we were in lunch. And these are the kind of things that we talk about.
Manjil: So according to you how should one student choose his problems, the one that he wants to work on?
Ken: It really depends. So I have had students who ended up working in a Langlands Program. People who are big picture kind of mathematicians and there it takes great patience. You know those students who read for 1 to 2 years without even attempting a problem because in those subjects it takes much more, much greater investment in time before you have the right to really think about a problem. But I also have other students who are wizzes of solving problems. And so for them they might write a thesis in which they might solve 6 to 7 problems that were proposed by number theorists recently. I guess the most important thing is to recognize your skill. How patient are you? Are you a big picture mathematician or a problem solver? You like learning something new every day maybe you would be the kind of student that just learns so much that they place themselves in a position that they can solve problems. So this is very much a case by case analysis.
Manjil: What are some of the most outstanding unsolved problems as of today in mathematics?
Ken: Oh..! Let’s see. Well I have very high hopes on what’s called “The Twin Prime Problem “. We now know that there are bounded gaps between primes. 2 years ago we learnt that there are bounded gaps around 70 million due to the great work of Tony Zhang. Now the bound is down to 246 through the work of Polymath 8b, the crowd sourcing project headed by Terry Tao and there had been so much great work on primes in the last year and a half. I think in the coming 4-5 years we should expect more great era.
In terms of other areas I think umbral moonshine and some string theory in connection to mock theta functions we begin to see some very very big theorems. So from the very classical problems in analytic number theory to the most modern stringy type mathematics. I think we will be a big problem solve in very near future.
Manjil: You mentioned the polymath project. How efficient do you think this kind of massively open mathematical research is?
Ken: Ah..! That’s a great question. There are certainly many math problems that cannot be…that are not well suited for a polymath project. The Twin Prime Project, polymath 8b was perfect for this because the method that was invented to obtain bounded gaps in prime is a combination of many many different kinds estimates and many different kinds of inequalities where individuals could be experts on one particular component and by optimizing each of the components you arrive at this great theorem. So I think problems that require like that can benefit from croud sourcing you know probably could be problems like this. Problems where theorems are obtained by assembling many many different lemmas and propositions, where the end result is only good as the weakest proposition. So the ideas; this aren’t big picture ideas, the big picture paper probably come first but in the final analysis to get the most optimal theorem, questions like this again problems where lots of little lemmas and propositions come together in a very precise way, where the result is only as good as the weakest one. May be these are probably the kinds of problems best suited for polymath projects.
Manjil: What is your favorite mathematical result?
Ken: Well I think the proof of Fermat’s Last Theorem has to be the highlight of my mathematical career. I still like talking about it as if it was last year. That was a great mathematical moment.
Manjil: So any previous question you mentioned about the upcoming movie on Ramanujan? Can you tell us a little more about that movie?
Ken: It’s going to be great. Everybody should watch it. This movie is based on the famous biography by Robert Kanigel, the book “THE MAN WHO KNEW INFINITY”. And Matthew Brown he is the screen writer and also the director for the film, has adapted the book and has written a beautiful screenplay. Filming for the movie is now finished. I spend much of the summer in London at the PINEWOOD STUDIOs, working with the actors and the art department on this movie. And I think it’s going to be fantastic. Jeremy Irons is a great job playing G. H. Hardy, Dev Patel is the wonderful Ramanujan. There is footage actually filmed at Trinity College in Cambridge. This is the first time filming for a major movie has ever been permitted in Cambridge at Trinity College. I am very happy for that. I have already seen the trailer, it’s gonna be fantastic. I hope you watch it.
Manjil: So you have had a major role in the movie. You have been credited in the IMDB listing too. So how was the experience? Have you enjoyed it?
Ken: I had a time of my life. Actually I am still working with the film. I knew nothing about making movies before this project. AndI have to say it’s kind of amazing, what this film makers do. So days on are very busy. It starts very early in the morning as a very tight schedule. Art department is fantastic. We had one artist. Her name was Liz. Her one job was to master Ramanujan’s hand writing. And she would reproduce 10 copies of Ramanujan’s first letter to Hardy. The attention and details is that specific. And so I had a great time. And I think it will be great movie.
Manjil: Since you mentioned Ramanujan’s first letter to Hardy. So I have a question I wanted to ask to someone for a long time. If you get such a letter from some unknown mathematician would you read it? And would you comment on it?
Ken: Very good question. I do get letters occasionally from people claiming to prove something like the Riemann hypothesis. Because of my experience with Ramanujan, I do take a look at this letters. I probably shouldn’t admit this but even false conjecture very false proof of Fermat’s Last Theorem or the Riemann hypothesis. I will still get them refereed. I would insist somebody please read these papers. Even though with absolute certainty I know we will have a mistake because they are from amateurs. I do insist somebody read them and then submit it to me. Please don’t circulate that, I don’t want to get many more of this. But you are right. Because of what I know about Ramanujan, I think probably I am in the more of careful people. I take it as a responsibility and as a part of editorial boards of various journals to make sure each paper is evaluated.
Manjil: How do you see Ramanujan’s work in the coming year, say 25 years from now. How do you see his work?
Ken: Oh.. that’s a great question..! Well Ramanuja’s work is ubiquitous. People talk about Ramanujan–Petersson conjecture and analytic number theory. I am sure 25 years from now people would still be talking about that. I think the circle method that gives rise to formula for the partition numbers. People would still talk about that because we are using that in connection with math physics. My prediction is that those would probably two main areas in mathematics that will still be using Ramanujan’s work 25 years from now. The mock theta function will certainly be important in 25 years. May be there are some others that is quite important but that’s a quite a bit of work.
Manjil: To someone who is a young student who want to do something in mathematics. Who want to pursue a career in mathematics, what sort of books would you recommend they should read?
Ken: I think that depend on the age. Once the student has finished high school algebra and trigonometry, I still recommend the great book by Hardy and Wright ‘Introduction to Number Theory’. This days there are also high sort of combinatorics books.. These subjects are quite fashionable now like additive combinatorics. And I think it’s important for a young student to have some exciting books like that in addition to the standard course, the standard abstract algebra, and the standard analysis books. I would recommend a nice diversified cross-section of books.
Manjil: For someone who is from developing country like India. It gets very difficult to them to get access to the recent material, the recent research material or internet connectivity in the remote areas in India. At least from the part of India that I come from, it’s very difficult to access internet or to go get books from good libraries. So in that case how should a student use the resources that are already available to cultivate his own progress in mathematics.
Ken: Ok that’s very difficult for me. Because I live in a situation where I have internet in my pockets… well cellphones. And I have access to great libraries. Some of the books I have described are quite old. The book by Hardy and Wright has been around for decades. I would hope that a local school library would have those books. So for a student having access to most recent most cutting edge mathematics probably isn’t so vital. Because at the young age you want to be inspired and you don’t have to read about the latest work to be inspired by progress in mathematics. So some of the books that I have described like Hardy and Wright is a classic and I would hope the student could find those books.
Then the next best thing is try to find a local mentor. A local college or a local math teacher because the more people you knows who have access to these, the internet and books the opportunities the student are going to end up getting. And so I would hope that there is such facilities and such opportunities in place even in rural India. So I’ll be in Koumbakunam for 10 days and even there you can go to government college and find books. I think we have Internet at the hotel may be limited but it is true that I hope that the Internet becomes readily available in India. And I hope that happens in the near future.
Manjil: The final question. So which is your most favorite Ramanujan result?
Ken: My favorite Ramanujan result…I don’t know that I could pick one favorite out of all. I would pick top three.
One would be his study of congruences for the partition function and his Tau functions. Thats given birth to so much beautiful mathematics. Even it inspired Serre when he developed the theory of Galois representation which was assembled by Deligne in his Fields medal work.
Secondly I probably would say his development of the circle method with Hardy. This is one of the crowning achievements of classic arithmetic number theory of its day. People still use it.
And thirdly, is his work on mock theta functions. If Ramanujan’s last letter to Hardy had never been received then I am not sure what I would be doing right now. From the last 5 or 6 years I have been studying those works and developing it in a more modern perspective and the applications. There are so many applications. We are actually writing a book. I am writing a book with Amanda Folsonm of Yale and Kathrin Bringmann of Colonge. I think its going to be 600 pages and its going to be on mock modular forms and applications.
So my three favorite contributions are Ramanujan theory of congruences for p(n) and tau(n), two his work on the circle method with Hardy and three his imagination in terms of seeing that there is a theory of mock theta functions. All fantastic works.
Manjil: Thank you very much Prof. Ono for you time.
Ken: Thank you Manjil. My pleasure!