I know a little bit, Dr Ono, about how you have developed an interest in Ramanujan, but you have talked about it before in a similar Hangout we had about a year ago, and I’m sure you have talked about it elsewhere. Your adviser, Basil Gordon, actually sparked your interest. Can you start by telling us a little bit about his study of Ramanujan and how that sparked yours?
I have to say that my father was a math professor. He was a professor at Johns Hopkins for many years, and at first glance it might not come as a surprise that I ended up myself becoming a mathematician.
I credit my PhD adviser, Basil Gordon, for really being the person who sparked my love for and my passion for mathematics. Before graduate school, I could do well on examinations, I could memorize material, I was a robotic student.
But my PhD adviser was a genius. He’s not as well known as I think he should be. He’s the one who sparked my interest. We would meet every week at his house in Los Angeles, only a few blocks from the beach; it’s a very beautiful place. We would work for two or three hours each Saturday.
What was very ironic, or I should say unusual, about Basil Gordon is that he was a Renaissance man. He was an accomplished writer. He was an accomplished musician, and it was his habit to start our weekly meetings with a poetry reading or he would play the piano, maybe a Chopin nocturne, and his point was that mathematics is hard, we are confused all the time. And it’s not about solving equations, it’s looking for beauty.
And if we weren’t in the proper frame of mind, how could we produce mathematics? That was, I found, very surprising. No teacher or mentor before had ever tried to make me see mathematics that way.
What he did was, he used Ramanujan as a role model, as someone who succeeded not because of ordinary school training, and not because of ordinary coursework, but as a man who saw beauty in formulae. Ramanujan was an inspiration for Gordon.
Of course, I knew about Ramanujan long before I worked with Gordon, but it was during the year of 1991, when Robert Kanigel’s book, The Man Who Knew Infinity came out, Basil Gordon and I, we went to the bookstore, we bought copies, and we read the book together.
Basil is no longer alive, he regretfully passed away two years ago. I don’t want to cry, but I will tell you Basil Gordon was an amazing man, and he helped me find myself. I hope that I could be to my PhD students, you know, 10% of what Basil Gordon was to me. He was that important. Ramanujan was one of his great heroes.
I have always been fascinated, Dr Ono, by how Ramanujan found so many different formula for pi in particular. Did he have a particular interest in pi, or did pi just pop up as he was studying other infinite series and so on? What’s your idea about that?
Well, it’s a excellent question. I think that Ramanujan, if he were around today, would be very surprised that we celebrate pi day all over the world on 14 March. I think he would find that strange. I guess the short answer to your question is that pi would have certainly been an important number to Ramanujan, but it wasn’t important because it’s this crazy figure that he was trying to pursue for many different reasons. Rather it’s that pi appears in so many different areas of number theory as a critical constant.
Perhaps it’s a better testament to the taste that he had in mathematics that a lot of the formulae he produced ended up involving pi. Hypergeometric functions dating to the early works of (German mathematician Carl Friedrich) Gauss involve objects called elliptic integrals. These are related to volumes, and whenever you are computing a volume of an area of things that have arcs, you expect pi to appear.
Also, Ramanujan was a world expert on continued fractions and he found very beautiful expressions for pi. I gave a lecture yesterday at the International Centre for Theoretical Science (Bengaluru), and I made a point of it to say that to most of us, pi is this impossibly difficult number. There are some people who have memorized a hundred digits of pi, 3.14 a hundred digits, but at some point you have to say, pi is 3.14 and eventually nobody knows, right? There’s no repeating pattern.
The thing is, Ramanujan, when he wrote down pi in terms of continued fractions, it’s very easy to see the pattern. In fact, Ramanujan’s formulas for pi are perhaps some of the best examples of beautiful, symmetric formulas where most people would think that there is no such beauty apparent.
Ramanujan was very good at finding these formulas, and these formulas are of very different categories. Square root of two and numbers like the golden ratio are a little bit simpler than, say, E and pi, but Ramanujan’s formulas, he found equally beautiful formulas for all of them.
I didn’t know that. I didn’t know that he had found formula for other irrationals, too, but actually, that’s the core of what I want to get at today: can you give us an idea of how Ramanujan came up with these formulae at all? What is the process in his head? After all, these are not the kind of things that somebody like me is going to be just thinking about or playing around with numbers, even though I have an interest in numbers. How did this young man in rural Tamil Nadu come up with these ideas? What was going through his head?
Well, to answer your question, Dilip, I wish I knew what was going on in his head. If I had a time machine, if there was one person in my life I would like to go back in time and meet, it would be Ramanujan.
I can’t offer you a rock-solid answer, but let me do my best. The easiest examples to start with would be the golden ratio. The golden ratio is this number phi, it’s expressed as one plus the square root of five over two. That’s a little bit complicated. Some people express the golden ratio as the limits of the successive terms in the Fibonacci sequence.
The representation I liked most is the representation by a continued fraction. A continued fraction that expresses the golden ratio requires only knowing the number one. It’s one plus one divided by one plus one divided by, and you go on forever.
That’s a beautiful expression, you could teach this to bright high school students. All you need to know is the quadratic formula to realize that that is the golden ratio. Very bright students will quickly discover that the expression for the golden ratio as a continued fraction is shared by lots of quadratic irrationals.
Ramanujan certainly knew this fact... Instead of being content with knowing this kind of theory, he decided to throw in a parameter. “Why don’t I consider a continued fraction, not in terms of numbers, but in terms of variables?" Where, if you choose the variables to be simple numbers like one, you would happen to get back to golden ratio. He then asked, “Well, if you did this, what kind of numbers would you get if you chose numbers other than Q? What that open up a whole new world for other kinds of expressions?"
Well, Ramanujan did this, and in his very first letter to (English mathematician G.H.) Hardy, Ramanujan ended with three formulas, and these three formulas are called the Rogers-Ramanujan continued fraction, and these are the formulas that stunned Hardy. They astonished him.
These are the formulae that Hardy responded by saying, “They must be true, and Ramanujan must be a genius, because no one would possibly have the imagination to invent that."
How did Ramanujan end up producing that? That’s a great question. All I can tell you for sure is that he did and I’m so thankful that he did, because it’s a gift to the world of mathematics. There’s been over 7,000 papers that mathematicians have written over the last hundred years based on those two or three lines.
What do you think it must have been like to be Ramanujan? To think like him? To walk in his shoes? To talk like him? In fact, I’m asking you this because of your relationship with the film. You are a mathematical consultant to the film. Is this the kind of thing that you are telling the film crew about, or maybe Dev Patel, the actor?
Well, I did my best, having never met Ramanujan. There’s no footage of Ramanujan. In fact, there are only two surviving photographs, so there’s very little that we can go on.
On the set, as a math consultant, my job was to teach Dev how to communicate like mathematicians, and to do my best to try to infer what their personalities would have been like. Which was actually quite easy to do, because Hardy, of course, was the great British chaired professor. It’s easy for us to get him. He wrote important books, he was all about step-by-step, concrete formulation of mathematics.
Ramanujan was something completely different. How did he manage to do his work? We don’t know. It was certainly not by any procedure we would understand, and so in advising the producer of the director, it was very easy to say, “I can’t explain this to you. It’s a mystery to me. Try to imagine a professor trying to make sense out of someone who has brilliant ideas which are clearly right, and help them figure out how to bring those ideas to the surface, so the rest of the world of mathematicians could understand that."
How does the film compare to other similar films... say to Good Will Hunting?
Good Will Hunting is really not a film about mathematics, it’s a film about a young man who is struggling with his own identity; he was brilliant and he was struggling to embrace his brilliance. Ramanujan, in some ways, was obviously very brilliant, but very different in that he embraced his identity as the producer of mathematics.
First of all, the main theme, the way we described Ramanujan’s a character, a true, living human being, is very different from what one found in Good Will Hunting. I think our film will be one of the first films where the director took great pains in explaining how a mathematician does his work without compromising the human side of the story.
Let’s make it quite clear, the idea of Ramanujan is an idea that resonates with all teachers in the world everywhere. It’s the idea that greatness and talent is often found in the most unforgiving of circumstances, and teachers have the moral responsibility to try to recognize that, and then take the next step, and nurture these talented students, despite whatever circumstances.
Despite the financial circumstances. Despite their religious beliefs. Despite their countries of origin, and all that. That, I think, is truly the idea of Ramanujan.
Hardy perhaps was the only living person who could have recognized Ramanujan’s talents, and that’s what the story is about, how these two men from very different countries, very different cultures figured out how to work together and produce great mathematics.
Why do you say Hardy was the only living mathematician of that time who could have done this?
I often hear other mathematicians sort of complain that it’s a pity that Ramanujan had not sent his letter to Professor X in Germany or someone else, so on and so forth. It’s easy to try to imagine how mathematics would have developed differently had Ramanujan been discovered by someone else.
Say Henri Poincaré, the French mathematician who lived around the same time; what if Ramanujan had sent his letters there?
I actually write about this in my book, in the epilogue of my book. This is a question that I feel quite strongly about. My point is that Hardy was perhaps the only true person who could have recognized Ramanujan’s talent. Perhaps (British mathematician John Edensor) Littlewood would have been suitable.
Almost everything in Ramanujan’s notes involves analysis, classical analysis and formulae that relate to classical number theoretic constructions. Ramanujan didn’t know anything about any of the other standard courses in mathematics, and so, well, here’s a better test.
If we were to type up Ramanujan’s notes and distribute his notes to professors at major universities all over the world, you will discover that even today, in the 21st century, most mathematicians would not know what to make out of his formulas. It takes a very special person to see through pages and pages of formulas with no words to see their value.
If I could just conclude with one brief statement. It’s that Ramanujan transformed Hardy. Ramanujan transformed this generation of Japanese mathematicians. I applaud you and everyone who carries on his tradition, because it’s not just the mathematics.
Maybe for me it’s mathematics, but it’s really the idea of Ramanujan. Young people, even old people, middle-aged people, we all need our heroes, and when they are true stories, that’s when you have to take notes and pay attention, because we all have something to learn from that.
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