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Shelby Herbert:

Where does scientific inspiration come from? If given the opportunity to pursue it unhindered, where might it lead us? In a world where the scientific method and empirical data dominate the field, the idea that a complex mathematical equation could be delivered divinely in a dream feels at odds, yet it is in this space between science and the creative realm that the prodigy mathematician, Srinivasa Ramanujan, existed.

Michael Blane:

Born in 1887, Ramanujan was a self-trained two-time college dropout from south India. When he died at just 32 years old, he left behind at least three notebooks filled with mathematical equations, he described as divinely inspired. Mathematicians have been trying to figure out these equations ever since. The pursuit has led to solutions of ancient mathematical mysteries, breakthroughs in modern physics, and ideas that help power the internet. Ramanujan's story has been the source of inspiration for countless mathematicians and scientists, including world-renowned number theorist and our guest today, Dr. Ken Ono.

This is the Discover Science podcast, an offshoot of a public lecture series by the same name, where we speak with the world's leading scientists, researchers, and educators about important subjects that influence our world. I'm Michael Blane, a Ph.D. student in pure math here at the University of Nevada, Reno.

Shelby Herbert:

And I'm Shelby Herbert, a graduate student of journalism. I report for the Hitchcock Project for Visualizing Science, and I'm interested in making science accessible and exciting to people outside STEM fields. We are thrilled to be speaking today with Dr. Ono, who is an expert on the life and work of Srinivasa Ramanujan. He's written a book, given several public lectures, and produced and served as the mathematical consultant for the film, The Man Who Knew Infinity, which is about Ramanujan's life and work. Welcome to our show, Dr. Ono.

Dr. Ken Ono:

I'm glad to be here, Shelby.

Michael Blane:

You talked about how Ramanujan has influenced you, not only in your personal life but also in your career. Can you speak to what that influence has looked like, and how important he's been to you personally?

Dr. Ken Ono:

Yeah. There are so many layers to that question. Ramanujan mattered to me first when I was in high school, when I learned that he was a two-time college dropout. And as a high school student, I was told by my parents that it was all about the pursuit of high test scores and straight As. Let's make no mistake, good grades and good test scores are important. But as a high school kid, I really needed to understand that in the long run, the quality of your achievements and the quality of your character matter more. That's the first time Ramanujan mattered to me. Of course, in terms of his mathematics, I've been studying his mathematics my entire life. He left behind three notebooks that I am one of dozens of mathematicians around the world, trying to make sense of, even 100 years after his death.

Shelby Herbert:

Walking this back a little bit. Can you introduce us to Ramanujan and just his general impact on the field of mathematics and science?

Dr. Ken Ono:

Sure. Ramanujan, he was an actual living human being. He was born in the 19th century in India at a time when India was part of the British empire. It was just a colony. He was born a Brahman, which means that he came from a very devout religious family. And as a young boy, he believed that his family goddess would give him visions of mathematical formulas that he would record in notebooks. Fast forward 100 years into the future, we're still trying to figure out the meaning of those formulas. There are many dozens of mathematical structures that bear his name, and so, quite amazing that that's a true story.

Shelby Herbert:

You mentioned Ramanujan famously believed that his knowledge was revealed to him by his family goddess. What's the strangest way a proof has come to you?

Dr. Ken Ono:

Okay. I have an answer for this question. In my first tenure track job, I was a professor at Penn State, and as a young professor, I had perhaps the worst office in the building. This office was on the top floor of McAllister Hall. It was so awful that I was unable to open the window, because it had decades and many layers of paint keeping the window shut. But to answer your question, I had this yellow couch that I bought for $20 that I would sit on when I would do my work on mathematics. But this office was so terrible that it sat below the angled roof line of the building. So, you couldn't stand up unless you were less than five feet tall. I would have to duck to sit on this couch properly.

            Perhaps the best theorem I've ever proven in my career, the idea came to me as a flash insight when I was sitting on this chair, and I was so excited by what I'd realized that I stood up straight away and I banged my head on the ceiling and I knocked myself out. I came to maybe 10 minutes later, I was bleeding from the head. I had to run into the bathroom. To make a long story short, people came in to look after me, thinking that something terrible had happened without recognizing it was perhaps the most awesome experience of my life.

Shelby Herbert:

So, it was literally like a flash of light, and it was revealed to you.

Dr. Ken Ono:

Yeah, and together with a major bump. In fact, this might sound weird to you, but if you were to put your finger right here in my forehead and follow the line of my skull... You might even see it. You see that? Yeah. That's one of my best theorems. Do you see it? Yeah.

Michael Blane:

You see it?

Dr. Ken Ono:

Yeah. It hurts so much

Michael Blane:

In your writings about Ramanujan and how he's inspired your work, you talk about the individual problems that he left behind trying to solve those, but also trying to have an idea of the theory that he had in his mind maybe that he didn't share. Where are we now with that? What do you think the philosophy behind Ramanujan's work that you've been working on?

Dr. Ken Ono:

I get this question a lot and usually stutter, like I am now, trying to answer that question. I'm like you. I would love to know what was going on in Ramanujan's mind as he was scribbling down his formulas. Maybe he didn't himself know what he had in mind. Maybe there really is something to this Hindu goddess. But what I can tell you is, this happens many times a year and I've been studying Ramanujan since the late 1980s, I've gone back over his notebooks, gone over the same pages that I've looked at many, many, many, many times. I've often realized that, from a different perspective, I misunderstood what Ramanujan had in mind. This happened so often that I wish, like you, I had an answer to that question, because it'd be so much more efficient than having to spend decades literally confused, thinking you understood a formula only to recognize you only saw a small glimpse of what Ramanujan probably had in mind.

Shelby Herbert:

How many others, like Ramanujan, do you think exist out there today?

Dr. Ken Ono:

That's an excellent question. First of all, let's make something very clear. Ramanujan is not the product of an ordinary school system. He is one of those unusual minds. Those names that we all know about, think Einstein, think Newton. I want you to now think Ramanujan, he's one of those very few minds that come along, one of those very creative, brilliant minds that power generations of scientists after they have done their work. How many of them are out there? I don't know. It would be sort of asking how many Michael Jordans are there on planet Earth. We think they're out there, but they're so rare. What extraordinary confluence of events has to happen before they're revealed.

So to answer your question, I think they're out there, and we're searching for them. I run a program called the Spirit of Ramanujan, where we give fellowships, scholarships to people who wouldn't otherwise have opportunities. Based on what we've done in our little program for the last few years, I do know that there are extraordinary people out there that we need to discover and lift up and help them see the light.

Michael Blane:

We were speaking about other people out there. I wonder, as a professor, someone who advises young mathematicians, what advice do you have for being flexible and being a good advisor for different styles, or maybe people who didn't have the rigorous training in their youth?

Dr. Ken Ono:

That's a very profound question. One of the things that we learned from Ramanujan is the importance and the responsibility that mentors have for training their students. No two students are the same. I can tell you, in my many years as a teacher, going back to the early 1990s, the students that you remember the most are the ones that you felt like you didn't connect with. That loss. And what we have in Ramanujan is someone who could have so easily been lost to society.

And so, you are right. As a university professor, as a parent, as a colleague, you have to recognize that there is no one single path that leads to success. Certainly, there are some broad brushstrokes that seem to be common, but the true outliers are often so singular in their characteristics that a good mentor really has to ignore what has been done before.

Michael Blane:

A lot of modern math is a bit inaccessible, and maybe that's not so true for number theory, which oftentimes starts in simple questions or things that are easier to understand. I'm thinking about your recent talk about jellyfish swarms.

Dr. Ken Ono:

Did you see that?

Michael Blane:

I did.

Dr. Ken Ono:

Okay. Well, I love that. Thank you.

Michael Blane:

Yeah. It was an awesome problem that started with something so simple that we could teach to second graders, and then, leads to really modern math.

Dr. Ken Ono:

Yeah, yeah, yeah. I like that. Thank you. I'm glad you saw that. If my work has a trademark, or least, if I were to describe what I seek in the problems that I work on, it is that kind of coolness factor. For me, to really love a problem, really focus on it, it's important to me that the problem matter to someone. It doesn't have to matter to every mathematician. That's asking too much. But I won't work on a problem just because everyone says it's hard, hoping for some glory and solving a problem just because it's hard. If I can't explain why it matters, then it's not a problem that I think I want to work on. And so, you are right, in number theory, we deal with numbers. 1, 2, 3, 4, 5, you get it, but we don't have to seek or ask questions. We don't have to stray very hard from elementary to state problems to get to the point where we get questions that nobody knows the answer to. Yeah.

Shelby Herbert:

When you're working on a problem, do you see shapes? Do you see colors? Do you hear music?

Dr. Ken Ono:

That's really interesting. Yes. Yes. I don't know that I've ever really spoken about this. I'm not a professional musician, but I do understand that some tones represent colors to some musicians. And in mathematics, I think the same is true. The prime numbers certainly have a different feeling for me than some other numbers. At the end of the day, the intuition that I think we all work with benefits in this way. Yeah. So, as strange that may sound, I think I'm not unique in this, but I think most mathematicians will say, "Yeah, my favorite number is such and such, and this is why," and it might sound funny. Why is 24 better than 25, right? But depending on the questions that you think about, that could have a real meaning.

This is actually interesting, the elementary school joke. Why is... What is that joke?

Shelby Herbert:

Six afraid of seven.

Dr. Ken Ono:

Why is six mad at seven? Because seven ate nine, ha ha ha. But you don't have to venture very far from statements like that, where seven, eight, nine is actually replaced by a real mathematical statement. 24 is my favorite number.

Shelby Herbert:

I was going to ask.

Dr. Ken Ono:

Yeah, 24. Right. Maybe we don't want to get into why that is, but 24 seems to come up over and over and over again for many different reasons.

Shelby Herbert:

Including when you hit your head?

Dr. Ken Ono:

Oh my God. Yeah. Even when I... Including in my head. In that famous theorem, there's a formula where it has a denominator of 24, and that's actually really important. I'll give you one hint to why 24 is important. So, there's infinitely many prime numbers. The first two primes are two and three. Let's throw them out. They're just too small. But the primes after that 5, 7, 11, there's infinitely many of them. Square every prime number is starting at five and subtract one. If you square five, you get 25 subtract one. There you go, you got 24.

But check it out. What's the next prime? Seven. Square it, subtract one. 49 minus 1 is 48, that's a multiple of 24. Let's do it again. 11. 11 squared, 121. Subtract 1, 120. That's five copies of 24. 5 times 24. So if you take any prime, starting at five, any prime starting at five, multiply it with itself, subtract one, and you will have a multiple of 24. It's actually very easy to prove that, but the significance of that statement throughout mathematics is rather deep.

Shelby Herbert:

I got to ask, Michael, do you have a favorite number?

Michael Blane:

I've always said 19. I can't really give an answer why. I just like the feeling of that number.

Shelby Herbert:

I'm curious about Michael's question about the jellyfish swarm. You were saying that you only wanted to explore something if it held some kind of meaning. Could you talk a little bit about what you look for when you're trying to find meaning in an equation?

Dr. Ken Ono:

This paper that I wrote recently that Michael has seen, it's called jellyfish swarms. It has nothing to do with jellyfish. So don't think of jellyfish in an aquarium. It actually starts with a very elementary game that you could teach in elementary school that deals with averages. So if you take two numbers, add them together and divide by two, you get their average. It's called the arithmetic average. But that's not the only kind of average you could calculate in mathematics. You could multiply two numbers together and take their square root, and that would be a different kind of average. You might be asked, these averages are very different. The additive average and the multiplicative averages, they're generally very different numbers, but is there a way in which these types of averages intertwine or begin to know each other? And at first glance, it looks like they don't have the right to do that.

In this paper I wrote with some of my students shows how, in a certain perspective, there is repetitive structure, where these seemingly unrelated averages form structure when you looked at them through the right lens. We explain how the structures that they present look exactly like jellyfish, these bell heads with tentacles sticking out of them. It turns out that if you compile all the possible structures, it looks like a sea of jellyfish. And then, what we did in this paper is we explain the taxonomy, to borrow from biology, the taxonomy of all different kinds of jellyfish, including the kinds of spots that they're allowed to have. It's a beautiful mathematical picture.

Another thing I'd like to say, building on this, what makes a problem interesting? I think many professional mathematicians place a little bit too much emphasis on how hard the proof was. If you do an Ironman Triathlon, we get it. It was really, really hard, and we celebrate that. But let's make no mistake. Out of all the pursuits that one can do in athletics, in human performance, it doesn't have to be as hard as the Ironman Triathlon. And so, the same applies in mathematics. We were talking about how mathematics could be beautiful. And so for me, I think, the reputation I have is that I like to think about problems that are beautiful, that matter. And honestly, maybe I'm a little bit allergic to the super difficult hard problems, because there's so many problems out there that, maybe I'm a bit too afraid to try to tackle those that take years of work to solve. That's that's not who I am.

Shelby Herbert:

We were just talking about these jellyfish structures. You described that so poetically. Are there any other theories you've worked on that also kind of have some sort of call forward to a visual representation?

Dr. Ken Ono:

Yeah, yeah, yeah. I proved something called the umbral moonshine conjecture with collaborators a number of years ago. It appeared in the popular press, and the magazines actually paid a professional artist to render a visualization. You can probably find it online somewhere, but what it looks like is a planet that has these sources of light beaming away from it, where the beams of light reveal these little planets living below them. Without these beams of light, you wouldn't be able to see this. So yeah, the word umbral... If you don't know what the word umbral is, I didn't know it either before we did this work. But it's supposed to conjure visions of moonlight, and how out of the darkness, you can have moonlight lighting up whole regions of the universe.

So, this umbral moonshine... So, moonshine is supposed to be a nonsense. You speak moonshine, if what you're talking about is crazy. And the umbral part is where stuff like moonlight reveals structure that nobody would believe was actually true. So, umbral moonshine is this problem at the interface of mathematical physics and algebra, where we've learned a lot about certain groups using tools that really shouldn't be allowed.

Shelby Herbert:

So, we're kind of getting into the cosmos here. I read that some of Ramanujan's later theories are being used to explain the behavior of black holes.

Dr. Ken Ono:

Yeah, yeah, yeah. That's what this umbral moonshine is all about.

Shelby Herbert:

How is that possible?

Dr. Ken Ono:

Yeah. Well, how is it possible is the question that you would probably ask about almost any one of Ramanujan's formulas. Keeping in mind that he was an autodidact, coming up with formulas for objects that often wouldn't be defined until decades after his death. How is any of that possible? And in the case of studying multi-centered black holes, the formulas that we use, he wrote from his deathbed just weeks before he died. And when he died in 1920, nobody was talking about a black hole. Here we are in 2022, we've taken photographs of black holes. And we find that we need his formulas to do this string theory. How is any of that possible? So I can only answer your question by just repeating back to that same question.

Shelby Herbert:

Absolutely. If he had a longer life, I mean, what do you think? I mean, I've heard the dogma that like, "The best math you do is before 30." He died at 32, but how do you think the world would've changed if he had more time?

Dr. Ken Ono:

Wow. I get that question a lot. I honestly don't know how to answer it, because what Ramanujan was very good at was offering the scientists of his future, glimpses of the possible. But he died at 32, and I can't tell you what a glimpse of a possible would be when it took someone with his creative powers to discover them. So, whatever we've missed out on, we still don't know today.

Let me put it another way. To be a famous scientist, you either define your own theories and the scientists of your future realize the possibility of your theory, or you solve major open problems that stump people for decades, sometimes centuries. Ramanujan is a counter example. He was neither of those. He left behind notebooks. Just like many people study religious texts, as if all the answers are there. Except in the case of Ramanujan, the answers and the questions haven't even yet been written down. They're just parts of solutions to questions that maybe we haven't even asked yet.

And so, what have we missed to out on because Ramon died at such an early age? I don't know, because we haven't had many people with his powers that would be able to fill in what we've lost out on.

Shelby Herbert:

Absolutely. Other than the two theories we discussed earlier, are there any other ways his work still affects the body of mathematical research today?

Dr. Ken Ono:

Yeah. It turns out that many of the most important newsworthy theorems in my lifetime have revolved around Ramanujan's work. There is the proof for Fermat's last theorem, which made the front page of the New York Times. A math problem, the solution to a math problem, the day after its solution was announced, appeared on the front page of the New York Times. That would've never happened had Ramanujan not scribbled down some things called congruences for a function called tau. Major breakthroughs in black hole physics, as we've described. You can just Google it. It's shocking how often Ramanujan's name appears.

So, at the end of the day, had Ramanujan never lived, some of these major breakthroughs would still be open questions today. You can win the Nobel prize for conducting an experiment that confirms a prediction that Einstein already made 100 years ago. That is example of great science. Of course, you win the Nobel prize. But make no mistake. Some of the fundamental questions still to this day are, can you confirm a prediction of Einstein. So, I want you to think Ramanujan is, for us, something like that.

Michael Blane:

Although Ramanujan has a special place in your life story, I wonder if there are other mathematicians who inspire you, and maybe particular life stories you'd like to see portrayed.

Dr. Ken Ono:

In film?

Michael Blane:

Uh-huh (affirmative).

Shelby Herbert:

Very good question. So, two part question. Are there others who have inspired me? Of course. I think you can make a film about every human being. But in terms of an iconic scientific figure, absolutely. The first person that comes to mind is very recent. Her name is Vera Rubin. Vera Rubin is actually mother of one of my friends, Karl Rubin, who is a number theorist.

Vera Rubin discovered dark matter. She became a scientist at a time when women didn't go into physics. They weren't even admitted to graduate programs in physics. But she was a very strong woman, and she discovered most of what makes up the universe, dark matter. Someone should make a film of about her.

Emmy Noether was a famous mathematician, who lived in the early 20th century, whose ideas helped Einstein in her early work. You could definitely make a very interesting film about Emmy Noether. She lived at a very troubling time in our world's history, and her story should be told.

When you were producing this film and when you were the math consultant, did you encounter any challenges? I saw the movie. I thought you guys did such a great job at making these subjects engaging, making it look cool. What are some challenges you encountered trying to make that come across?

Dr. Ken Ono:

Yeah. I want to start by saying that I didn't originally start out as a producer on that project. This is, I consider, one of the most amazing things that ever happened to me in my life. I was brought on to the project to be a consultant for the art department, what formulas should be put on the blackboards, and so on and so forth. So, they invited me out to Pinewood studios to be part of that project.

It was actually really fascinating. That was the year that Harrison Ford broke his knee, when he fell out of the golf cart. We were there. And remember, I'm a math professor. And like Judi Dench, Harrison Ford, then Jeremy Irons, don't wake me up. I don't know how, honestly, any of that happened. But after about a week working on set, working with the art department, I got to know a lot of the people who were part of the film, and I think they found me interesting. They discovered that I knew a whole lot more about the story than maybe anybody else. Within a week, I was sitting in with the director and the actors, just the five of us, going over scenes. That's when I was elevated to the level of a producer.

So, you ask, what was challenging about that? So many things. The first thing that was challenging was... I'll never forget it was a Friday and I was working with Liz Colbert, who was our artist. She was fascinating. Her job was to master Ramanujan's handwriting. And she made Ramanujan's notebooks, copies of them, by hand. If you've seen the film, that notebook, she made the whole notebook, not just the pages that were open to. It was crazy what they do in Hollywood. But in the middle of one of our meetings, the director says, "We need to start rehearsals. Can you come in and help us?"

I walked into a room, and there's Jeremy Irons and Dev Patel and an Indian accent coach. We started reading the script, huddled around the table. I got to tell you, for the first 30 minutes, I sat 10 feet behind the table thinking I have no business here. I have no business here. And after that half hour, Jeremy Irons looks at me, "You're the mathematician, right? You are the most important person here today, because you have to teach us to know how to pretend to be mathematicians. Without you, we can't do this right."

Shelby Herbert:

He's just method, trying to embody that.

Dr. Ken Ono:

Oh my God. He even pulled the chair up. You got to pull the chair up. I even had to practice some of the line lines. "Now, you read this line. How am I supposed..." It was crazy. And then, where did we go from there? We're on set. I would give practice talks about what the lines would look like. I have to tell you, that was awesome. That they really cared what a mathematician sounds like. I can't tell you. Nobody can prepare a math professor for, "We're going on stage now." It's the Zurich International Film Festival. We are the opening film. After the standing ovation, I'm on stage with the actors. It was crazy. So, everything about that was frightening. I mean, even knowing how to get dressed for an event like that was frightening.

Getting back to this film, I want to emphasize that Hollywood made a film about mathematics. I know they've done that before. Think Good Will Hunting, or think The Theory of Everything, or A Beautiful Mind. But those were really Hollywood films. This was a small, independent film. This was really a film about mathematics. It was a film about two very different people who were, in their own right, extraordinary individuals. The last thought I want to leave you with is, what impact this film had on the actors? Jeremy Irons has played so many different kinds of characters.

So, what I want to leave you with is that this film was important to them. Jeremy had never played a Cambridge Don before, and he really wanted to do it. He was so excited by this film that he even offered the use of his own boat for the scenes at Cambridge. It was that important to him. Shortly after the conclusion of our theatrical run, he was invited to be a Chancellor of the University of Bath. And he said, "I can now honestly, genuinely take this, because I can say I've portrayed a Cambridge Don." So, it mattered to the actors.

For Dev, Dev was a 24-year-old boy. He's now quite the Hollywood star. But remember at that time, he was the boy from Slumdog Millionaire. And it was very important for him to take on a meaty role, one that mattered. It was really exciting for me to see how having the opportunity to work with someone like Jeremy Irons, who elevated every aspect of this film. It was really interesting to get to know Dev and see him do everything he can to learn from Jeremy, and I think it shows. So, I think this film meant a lot to both of these actors. I think it really meant a lot.

Michael Blane:

I think that the film contributed a lot to the popular perception of math and Ramanujan's memory. I think it's a beautiful thing. Like how your father contributed to the statue of Ramanujan, you're contributing to his memory in history as well.

Dr. Ken Ono:

Thank you. Yeah. Thank you very much.

Michael Blane:

Thank you for talking to us today.

Shelby Herbert:

One last question before we go, Dr. Ono, and I hope I say this right. What is your Erdős number?

Dr. Ken Ono:

My Erdős number? I think it's two.

Shelby Herbert:

Wow. Wow. That's incredible. What is my Erdős number now that I've spoken, now that I've collaborated with you?

Dr. Ken Ono:

We would have to write a paper together.

Shelby Herbert:

A podcast isn't a collaboration? I am devastated. Well, thank you so much. It was a pleasure. Yeah.

Dr. Ken Ono:

Great. All right. Thank you.

Shelby Herbert:

Thank you.