Episode 17 - Nalini Joshi

Join applied mathematician Nalini Joshi to learn about Mittag-Leffler's theorem, a fundamental result in complex analysis that tells us how to build meromorphic functions on the plane with any prescribed set of poles.

Evelyn Lamb: Hello and welcome to My Favorite Theorem. I’m your cohost Evelyn Lamb. I’m a freelance math and science writer in Salt Lake City, Utah. And this is your other cohost.

Kevin Knudson: Hi, I’m Kevin Knudson, professor of mathematics at the University of Florida. I’m looking forward to this because of the time zone issue here. This is taking place on two different days.

EL: Yes, yes, we are delighted to be joined by Nalini Joshi, who is joining us from tomorrow in Australia, which we’re getting a kick out of because we’re easily amused.

KK: That’s right.

EL: Hi, Nalini. Would you like to tell us a little bit about yourself?

Nalini Joshi: Sure. My name is Nalini Joshi. I’m a professor of applied mathematics at the University of Sydney. What else can I say except I’m broadcasting from the future? I was born in Burma and I moved to Australia as a child with my parents when they emigrated to Australia, and most of my education has been in Australia except for going to the U.S. to do a Ph.D., which I did at Princeton.

EL: Okay, so you’ve spent some time in both hemispheres. I guess in multiple times in your life.

NJ: Yeah.

EL: So when I was a little kid I had this idea that the world could never end because, you know, in the U.S., there’s always someone who’s a full day ahead, so I know that Thursday would have to happen because if it was Wednesday where I was, someone was already living in Thursday, so the world could never end.

NJ: That’s such a deep insight. That’s wonderful.

KK: That’s pretty good.

EL: Well…

KK: I was watching football when I was a kid.

NJ: I used to hang out at the back of the school library reading through all the old Scientific American magazines. If only they had columns like yours, Evelyn. Fantastic. I really, really wanted to work out what was happening in the universe, and so I thought about time travel and space travel a lot as a teenager.

EL: Oh. So did you start your career wanting to maybe go more into physics, or did you always know you wanted to be a mathematician?

NJ: No, I really wanted to become an astrophysicist, because I thought that was the way, surely, to understand space travel. I wanted to be an astronaut, actually. I went to an all-girls school for the first half of my school years, and I still remember going to see the careers counselor and telling her I wanted to be an astronaut. She looked at me and she said, you have to be more realistic, dear. There was no way that somebody like me could ever aspire to it. And nowadays it’s normal almost. People from all different countries around the world become astronauts. But at the time I had to think about something else, and I thought, okay, I’m going to become a scientist, explore things through my own mind, and that was one way I could explore the universe. So I wanted to do physics when I came to university. I studied at the University of Sydney as an undergraduate. When I got to first-year physics, I realized my other big problem, which is that I have no physical intuition. So I thought, I really needed to understand things from a really explicit, literal, logical, analytical point of view, and that’s how I came to know I must be more of a mathematician.

EL: Okay.

KK: I have the same problem. I was always going to be a math major, but I thought I might pick up a second major in physics, and then I walked into this junior-level relativity class, and I just couldn’t do it. I couldn’t wrap my head around it at all. I dropped it and took logic instead. I was much happier.

NJ: Yeah. Oh good.

EL: So we invited you on to find out what your favorite theorem is.

NJ: Yes. Well that was a very difficult thing to do. It was like choosing my favorite child, which I would never do. But I finally decided I would choose Mittag-Leffler’s theorem because that was something that really I was blown away by when I started reading more about complex analysis as a student. I mean, we all learnt the basics of complex analysis, which is beautiful in itself. But then when you went a little bit further, so I started reading, for example, the book by Lars Ahlfors, which I still have, called Complex Analysis.

KK: Still in use.

EL: That’s a great one.

NJ: Which was first I think published in 1953. I had the 1979 version. I saw that there were so many powerful things you could do with complex analysis. And the Mittag-Leffler theorem was one of the first ones that gave me that perspective. The main thing I loved about it is that you were taking what was a local, small piece of information, around, for example, poles of a function. So we’re talking about meromorphic functions here, that’s the subject of the theorem.

EL: Can we maybe set the stage a little bit? So what is a meromorphic function?

NJ: A meromorphic function is a function that’s analytic except at isolated points, which are poles. The worst singularities it has are poles.

EL: So these are places where the function explodes, but otherwise it’s very smooth and friendly.

KK: And it explodes in a controlled way, it’s like 1/zn for some finite n kind of thing.

NJ: Exactly. Right. An integer, positive n. When I try to explain this kind of thing to people who are not mathematicians, I say it’s like walking around in a landscape with volcanoes. Well-timed, well-controlled, well-spaced volcanoes. You’re walking in the landscape of just the Earth, say, walking around these places. There are well-defined pathways for you to move along by analytic continuation. You know ahead of time how strong the volcano’s eruption is going to be. You can observe it from a little distance away if you like because there is no danger because you can skirt all of these volcanoes.

KK: That’s a really good metaphor. I’m going to start using that. I teach complex variables in the summer. I’m going to start using that. That’s good.

NJ: So a meromorphic function, as I say, [cut ~7:29-7:31?] it’s a function that gives you a pathway and the elevation, the smoothness of your path in this landscape. And its poles are where the volcanoes are.

EL: So Mittag-Leffler’s theorem, then, is about controlling exactly where those poles are?

NJ: Not quite. It’s the other way around. If you give me information about locations of poles and how strong they are, the most singular part of that pole, then I can reconstruct a function that has poles exactly at those points and with exactly those strengths. That’s what the theorem tells you. And what you need is just a sequence of points and that information about the strength of the poles, and you need potentially an infinite number of these poles. There’s one other condition, that the sequence of these poles has a limit at infinity.

KK: Okay, so they don’t cluster, in other words.

NJ: Exactly. They don’t coalesce anywhere. They don’t have a limit point in the finite plane. Their limit point is at infinity.

EL: But there could be an infinite number of these poles if they’re isolated, on integer lattice points in the complex plane or something like that.

NJ: Right, for example.

KK: That’s pretty remarkable.

NJ: If you take your standard trigonometric functions, like the sine function or the cosine function, you know it has periodically spaced zeroes. You take the reciprocal of that function, then you’ve got periodically placed poles, and it’s a meromorphic function, and you can work out which trig function it is by knowing those poles. It’s powerful in the sense that you can reconstruct the function everywhere not just at the precise points which are poles. You can work out that function anywhere in between the poles by using this theorem.

KK: That’s really remarkable. That’s the surprising part, right?

NJ: Exactly.

KK: If you knew you had a finite number of poles, you could sort of imagine that you could kind of locally construct the function and glue it together, that wouldn’t be a problem. But the fact that you can do this for infinitely many is really pretty remarkable.

NJ: Right. It’s like going from local information that you might have in one little patch of time or one little patch of space and working out what happens everywhere in the universe by knowing those little local patches. It’s the local to global information I find so intriguing, so powerful. And then it struck me that this information is given in the form of a sum of those singular parts. So the function is reconstructed as a series, as an infinite sum of the singular parts of the information you’re given around each pole. That’s a very simple way of defining the function, just taking the sum of all these singular things.

KK: Right.

EL: Yeah, I love complex analysis. It’s just full of all of these things where you can take such a small amount of local information and suddenly know what has to be happening everywhere. It’s so wonderful.

NJ: Right, right. Those two elements, the local to global and the fact that you have information coming from a discrete set of points to give you continuous smooth information everywhere in between, those two elements, I realized much later, feature in a lot of the research that I do. So I was already primed to look for that kind of information in my later work.

EL: Yeah, so I was going to ask, I was wondering how this came up for you, maybe not the Mittag-Leffler theorem specifically, but using complex analysis in your work as an applied mathematician.

NJ: Right. So what I do is build toolboxes of methods. So I’m an applied mathematician in the sense that I want to make usable tools. So I study asymptotics of functions, I study how you define functions globally, functions that turn out to be useful in various mathematical physics contexts. I’m more of a theoretical applied mathematician, if you like, or I often say to people I’m actually a mathematician without an adjective.

KK: Right. Yeah.

NJ: You know that there is kind of a hierarchy of numbers in the number system. We start with the counting numbers, and we can add and subtract them. Subtraction leads you to negative integers. Multiplication and division leads you to rational numbers, and then solving polynomial equations leads you to algebraic numbers. Each time you’re building a higher being of a type of number. Beyond all of those are numbers like π and e, which are transcendental numbers, in the sense that they can’t be constructed in terms of a finite number of operations from these earlier known operations and earlier known objects.

So alongside that hierarchy of numbers there’s a hierarchy, a very, very closely related hierarchy of functions. So integers correspond to polynomials. Square roots and so on correspond to algebraic functions. And then there are transcendental functions, the exponential being one of them, exponential of x. So a lot of the transcendentality of functions is occupied by functions which are defined by differential equations.

I started off by studying differential equations and the corresponding functions that they define. So even when you’re looking at linear differential equations, you get very complicated transcendental functions, things like the exponential being one of them. So I study functions that are even more highly transcendental, in the sense that they solve nonlinear equations, and they are like π in the sense that these functions turn out to be universal models in many different contexts, particularly in random matrix theory where you might be, for example, trying to work out the statistics of how fundamental particles interact when you fire them around the huge loop of the CERN collider. You do that by looking at distributions of entries in infinitely large matrices where the entries are random variables. Now under certain symmetries, symmetry groups acting on, for example, you might have particles that have properties that allow these random matrices to be orthogonal matrices, or Hermitian matrices, or some other kind of matrices. So when you study these ensembles of matrices with these symmetry properties and you study properties like what’s their largest eigenvalue, then you get a probability distribution function which happens to be, by some miracle, one of those functions I’ve studied. There’s kind of a miraculous bridge there that nobody really knows why these happen. Then there’s another miraculous thing, which is that these models, using random matrices, happen to be valid not just for particle physics but if you’re studying bus arrival times in Cuernavaca, or aircraft boarding times, or when you study patient card sorting, all kinds of things are universally described by these models and therefore these functions. So first of all, these functions have this property: they’re locally defined by initial value problems given for the differential equation.

KK: Right.

NJ: But then they have these amazing properties which allow them to be globally defined in the complex plane. So even though we didn’t have the technology to describe these functions explicitly, not like I could say, take 1 over the sine function, that gives you a meromorphic function, whose formulae I could write down, whose picture I could draw, these functions are so transcendental that you can’t do that very easily, but I study their global properties that make them more predictable wherever you go in the complex plane. So the Mittag-Leffler theorem sort of sets up the baseline. I could just write them as the sum of their poles. And that’s just so powerful to me. There are so many facets of this. I could go on and on. There is another direction I wanted to insert into our conversation, which is that the next natural level when you go beyond things like trigonometric functions and their reciprocals is to take functions that are doubly periodic, so trigonometric functions have one period. If you take double periodicity in the complex plane, then you get elliptic functions, right? So these also have sums of their poles as an expression for them. Now take any one of these functions. They turn out to be functions that parametrize very nice curves, cubic curves, for example, in two dimensions. And so the whole picture shifts from an analytic one to an algebraic geometric one. There are two sides to the same function. You have meromorphic functions on one side, and differential equations, and on the other side you have algebraic functions and curves, and algebraic properties and geometric properties of these curves, and they give you information about the functions on the other side of that perspective. So that’s what I’ve been doing for the last ten years or so, trying to understand the converse side so I can get more information about those functions.

EL: Yeah, so using the algebraic world,

NJ: Exactly, the algebro-geometric world. This was a huge challenge at the beginning, because as I said, I was educated as an applied mathematician, and that means primarily the analytic point of view. But to try and marry that to the algebraic point of view is something that turned out to be a hurdle at the beginning, but once you get past that, it’s so freeing and so beautiful and so strikingly informative that I’m now saying to people, all applied mathematicians should be learning algebraic geometry.

KK: And I would say the converse is true. I think the algebraic geometers should probably learn some applied math, right?

NJ: True, that too. There’s so many different perspectives here. It all started for me with the Mittag-Leffler theorem.

EL: So something we like to do on this show is to ask our guest to pair their theorem with something: food, beverage, music, anything like that. So what have you chosen to pair your theorem with?

NJ: That was another difficult question, and I decided that I would concentrate on the discrete to continuous aspect of this, or volcanoes to landscapes if you like. As I said, I was born in Burma, and in Burma there are these amazing dishes called le thoke. I’ll send you a Wikipedia link so you can see the spelling and description. Not all of it is accurate, by the way, from what I remember, but anyway. Le thoke is a hand-mixed salad. “Le” is hand and “thoke” is mixture. In particular, the one that’s based on rice is one of my favorites. You take a series of different ingredients, so one is rice, another might be noodles, there have to be specific types, another is tamarind. Tamarind is a sour plant-based thing, which you make into a sauce. Another is fried onions, fried garlic. Then there’s roasted chickpea flour, or garbanzo flour.

KK: This sounds amazing.

NJ: Then another one is potatoes, boiled potatoes. Another one is coriander leaves. Each person might have their favorite suite of these many, many little dishes, which are all just independent ingredients. And you take each of them into a bigger bowl. You mix it with your hands. Add as much spices as you want: chili powder, salt, lemon juice, and what you’re doing is amalgamating and combining those discrete ingredients to create something that transcends the discrete. So you’re no longer tasting the distinct tamarind, or the distinct fried onion, or potatoes. You have something that’s a fusion, if you like, but the taste is totally different. You’ve created your meromorphic function, which is that taste in your mouth, by combining those discrete things, which each of them you wouldn’t eat separately.

KK: Sure. It’s not fair. It’s almost dinner time here, and I’m hungry.

NJ: I’m sorry!

EL: Are there any Burmese restaurants in Gainesville?

NJ: I don’t know. I think there’s one in San Francisco.

EL: Yes! I actually was just at a Burmese restaurant in San Francisco last month. I had this tea leaf salad that sounds like this.

NJ: Yeah, that’s a variation. Pickled tea leaves as an ingredient.

EL: Yeah, it was great.

NJ: I was also thinking about music. So there are these compositions by Philip Glass and Steve Reich which are basically percussive, independent sounds. Then when they interweave into those patterns you create these harmonies and music that transcends each of those particular percussive instruments, the strikes on the marimba and the xylophones and so on.

EL: Like Six Marimbas by Steve Reich?

NJ: Yeah.

EL: Another of our guests, her episode hasn’t aired yet, though it will by the time our listeners are hearing this, another of our guests chose Steve Reich to pair with her theorem.

KK: That’s right.

EL: One of the most popular musicians among mathematicians pairing their theorems with music.

NJ: Somebody should write a book about this.

KK: I’m sure. So my son is a college student. He’s studying music composition. He’s a percussionist. I need to get on him about this Steve Reich business. He must know.

EL: Yeah, he’s got to.

KK: This has been great fun, Nalini. I learned a lot about not just math, but I really knew nothing about Burmese food.

NJ: Right. I recommend it highly.

KK: Next time I’m there.

NJ: You said something about mentioning books?

EL: Yeah, yeah, if you have a website or book or anything you’d like to mention on here.

NJ: This is my book. I think it would be a bit too far away from the topic of this composition, but it has this idea of going from continuous to discrete.

EL: It’s called Discrete Systems and Integrability.

NJ: Yes.

EL: We’ll put a link to some information about that book, and we’ll also link to your website on the show notes so people can find you. You tweet some. I think we kind of met in the first place on Twitter.

NJ: That’s right, Exactly.

EL: We’ll put a link to that as well so people can follow you there.

NJ: Excellent. Thank you so much.

EL: Thank you so much for being here. I hope Friday is great. You can give us a preview while we’re still here.

KK: We’ll find out tomorrow, I guess.

NJ: Thank you for inviting me, and I’m sorry about the long delay. It’s been a very intense few years for me.

EL: Understandable. Well, we’re glad you could fit it in. Have a good day.

NJ: Thank you. Bye.

[outro]

Show notes (click here)