Evelyn Lamb: Hello and welcome to My Favorite Theorem, a math podcast where we ask mathematicians what their favorite theorem is. I’m one of your hosts, Evelyn Lamb. I’m a freelance math and science writer in Salt Lake City, Utah. And this is your other host.
EL: I’m all right. It’s fall here, or hopefully getting to be fall soon.
KK: Never heard of it.
EL: Yeah. Florida doesn’t have that so much. But yeah, things are going well here. We had a major plumbing emergency earlier this month that is now solved.
KK: My big news is that I’m now the chair of the math department here at the university.
EL: Oh yes, that’s right.
KK: So my volume of email has increased substantially, but it’s and exciting time. We’re hiring more people, and I’m really looking forward to this new phase of my career. So good times.
KK: But let’s talk about math.
EL: Yes, let’s talk about math. We’re very happy today to have Michèle Audin. Yeah, welcome, Michèle. Can you tell us a little bit about yourself?
Michèle Audin: Hello. I’m Michèle Audin. I used to be a mathematician. I’m retired now. But I was working on symplectic geometry, mainly, and I was interested also in the history of mathematics. More precisely, in the history of mathematicians.
EL: Yeah, and I came across you through, I was reading about Kovalevskaya, and I just loved your book about Kovalevskaya. It took me a little while to figure out what it was. It’s not a traditional biography. But I just loved it, and I was like, “I really want to talk to this person.” Yeah, I loved it.
MA: I wanted to write a book where there would be history and mathematics and literature also. Because she was a mathematician, but she was also a novelist. She wrote novels and things like that. I thought her mathematics were very beautiful. love her mathematics very much. But she was a very complete kind of person, so I wanted to have a book like that.
KK: So now I need to read this.
EL: Yeah. The English title is Remembering Sofya Kovalevskaya. Is that right?
KK: I’ll look this up.
EL: Yeah. So, what is your favorite theorem.
MA: My favorite theorem today is Stokes’ formula.
EL: Oh, great!
KK: Oh, Stokes’ theorem. Great.
EL: Can you tell our listeners a little bit about it?
MA: Okay, so, it’s a theorem, okay. Why I love this theorem: Usually when you are a mathematician, you are forced to face the question, what is it useful for? Usually I’ll try to explain that I’m doing very pure mathematics and maybe it will be useful someday, but I don’t know when and for what. And this theorem is quite the opposite in some sense. It just appeared at the beginning of the 19th century as a theorem on hydrodynamics and electrostatics, some things like that. It was very applied mathematics at the very beginning. The theorem became, after one century, became a very abstract thing, the basis of abstract mathematics, like algebraic topology and things like that. So this just inverts the movement of what we are thinking usually about applied and pure mathematics. So that’s the reason why I like this theorem. Also the fact that it has many different aspects. I mean, it’s a formula, but you have a lot of different ways to write it with integrals, so that’s nice. It’s like a character in a novel.
KK: Yeah, so the general version, of course, is that the integral of, what, d-omega over the manifold is the same as the integral of omega over the boundary. But that’s not how we teach it to students.
MA: Yeah, sure. That’s how it became at the very end of the story. But at the very beginning of the story, it was not like that. It was three integrals with a very complicated thing. It is equal to something with a different number of integrals. There are a lot of derivatives and integrals. It’s quite complicated. At the very end, it became something very abstract and very beautiful.
KK: So I don’t know that I know my history. When we teach this to calculus students anymore, we show them Green’s theorem, and there are two versions of Green’s theorem that we show them, even though they’re the same. Then we show them something we call Stokes’ theorem, which is about surface integrals and then the integral around the boundary. And then there’s Gauss’s divergence theorem, which relates a triple integral to a surface integral. The fact that Gauss’s name is attached to that is probably false, right? Did Gauss do it first?
MA: Gauss had this theorem about the flux—do you say flux?
MA: The flux of the electric—there are charges inside the surface, and you have the flux of the electric field. This was a theorem of Gauss at the very beginning. That was the first occurrence of the Stokes’ formula. Then there was this Ostrogradsky formula, which is related to water flowing from somewhere. So he just replaced the electric charges by water.
KK: Sort of the same difference, right? Electricity, water, whatever.
MA: Yes, it’s how you come to abstraction.
KK: That’s right.
MA: Then there was the Green theorem, then there is Stokes’ formula that Stokes never proved. There was this very beautiful history. And then in the 20th century, it became the basis for De Rahm theory. That’s very interesting, and moreover there were very interesting people working on that in the various countries in Europe. At the time mathematics were made in Europe, I’m sorry about that.
KK: Well, that’s how it was.
MA: And so there are many interesting mathematicians, many characters, different characters. So it's like it's like a novel. The main character is the formula, and the others are the mathematicians.
EL: Yeah. And so who are some of your favorite mathematicians from that story? Anyone that stands out to you?
MA: Okay, there are two of them: Ostrogradsky and Green. Do you know who was Green?
EL: I don't know about him as a person really.
MA: Yeah, really? Do you know, Kevin? No.
KK: No, I don't.
MA: Okay. So nobody knows, by the way. So he was he was just the son of a baker in Nottingham. And this baker became very rich and decided to buy a mill and then to put his son to be the miller. The son was Green. Nobody knows where he learned anything. He spent one year in primary school in Nottingham, and that’s it. And he was a member of some kind of, you know, there are books…it’s not a library, but morally it’s a library. Okay. And that’s it. And then appears a book, which is called, let me remember how it is called. It’s called An essay on the application of mathematical analysis to the theories of electricity and magnetism. And this appears in 1828.
EL: And this is just out of nowhere?
MA: Out of nowhere. And then the professors in Cambridge say, “Okay, it’s impossible. We have to bring here that guy.” So they take the miller from his mill and they put him in the University of Cambridge. So he was about, I don’t know, 30 or 40. and of course, it was not very convenient for the son of a baker to be a student with the sons of the gentlemen of England.
MA: Okay. So he didn't us stay there. He left, and then he died and nobody knew about that. There was this book, and that’s it.
KK: So he was he was 13 or 14 years old when he wrote this? [Ed. note: Kevin and Evelyn had misheard Dr. Audin. Green was about 35 when he wrote it. The joys of international video call reception!]
MA: Yeah. And then and then he died, and nobody knew except that—
MA: Wow. And then appears a guy called Thomson, Lord Kelvin later. This was a very young guy, and he decided to go to Paris to speak with a French mathematicians like Cauchy, Liouville. And then it was a very long trip, and he took with him a few books to read during the journey. And among these books was this Green book, and he was completely excited about that. And he arrived in Paris and decided to speak of this Green theorem and this work to everybody in Paris. There are letters and lots of documentation about that. And then this is how the Green formula appeared in the mathematics.
EL: Interesting! Yeah, I didn't know about that story at all. Thanks.
KK: It’s fascinating.
MA: Nobody knows. Yeah, that's very interesting.
KK: Isn’t what we know about Stokes’ theorem, wasn't it set as an exam problem at Cambridge?
MA: Yeah, exactly. So it began It began with a letter of Lord Kelvin to Stokes. They were very friendly together, on the same age and doing say mathematics and physics and they were very friendly together and and they were but they were not at the same at the same place of the world writing letters. And once Thomson, Kelvin, sent a letter to Stokes speaking of mathematics, and at the very end a postscript where he said: You know that this formula should be very interesting. And he writes something which is what we now know as the Stokes theorem.
And then the guy Stokes, he had to make a problem for an exam, and he gave this as an examination. You know, it was in Cambridge, they have to be very strong.
MA: And this is why it’s called the Stokes’ formula.
KK: Wow. Yeah ,I sort of knew that story. I didn't know exactly how it came to be. I knew somewhere in the back of my mind that it had been set as an exam problem.
MA: It’s written in a book of Maxwell.
EL: And so the second person you mentioned, I forget the name,
MA: Ostrogradsky. Well, I don’t know how to pronounce it in Russian, and even in English, but Ostrogradsky, something like that. So he was a student in mathematics in Ukraine at that time, which was Russia at that time, by the way. And he was passing his exams, and the among the examination topics there was religion. So he didn't go for that, so he was expelled from the university, and he decided to go to Paris. So it was in 1820, something like that. He went to Paris. He arrived there, and had no exams, and he knew nobody, and he made connections with a lot of people, especially with Cauchy, who was a was not a very nice guy, but he was very nice to Ostrogradsky.
And then he came back to Russia and he was the director of all the professors teaching mathematics in military schools in in Russia. So it was quite important. And he wrote books about differential calculus—what we call differential calculus in France but you call calculus in the U.S. He wrote a book like that, and for instance, because we were speaking of Kovalevskaya when she was a child, on the walls of her bedroom there were the sheets of the course of Ostrogradsky on the wall, and she read that when she was a little girl. She was very good in calculus.
This is another story, I’m sorry.
KK: No, this is the best part.
MA: And so, next question.
KK: Okay, so now I’ve got to know: What does one pair with Stokes’ theorem?
MA: Ah, a novel, of course.
EL: Of course!
KK: A novel. Which one?
MA: Okay, I wrote one, so I’m doing my own advertisement.
EL: Yeah, I was hoping we could talk about this. So yeah, tell us more about this novel.
MA: Okay, this is called Stokes’ Formula, a novel—La formule de Stokes, roman. I mean, the word “novel” is in the title. In this book I tell lots of stories about the mathematicians, but also about the about the formula itself, the theorem itself. How to say that? It’s not written like the historians of mathematics like, or want you to write. There are people speaking and dialogues and things like that. For instance, at the end of the book there is a first meeting of the Bourbaki mathematicians, the Boubaki group. They are in a restaurant, and they are having a small talk, like you have in a restaurant. There are six of them, and they order the food and they discuss mathematics. It looks like just small talk like that, but actually everything they say comes from the Bourbaki archives.
EL: Oh wow.
MA: Well, this is a way to write. And also this is a book. How to say that? I decided it would be very boring if the history of Stokes’ formula was told from a chronological point of view, so it doesn’t start at the beginning, and it does not end at the end of the story. All the chapters, the title is a date: first of January, second of January, and they are ordered according to the dates. So you have for instance, it starts with the first of January, and then you have first of February, and so on, until the end, which is in December, of course. But it’s not during the same year.
MA: Well, the first of January is in 1862, and the fifth of January is in 1857, and so on. I was very, very fortunate, I was very happy, that the very end of the story is in December because the first Bourbaki meeting was in December, and I wanted to have the end there. Okay, so there are different stories, and they are told on different dates, but not using the chronology. And also in the book I explain what the formula means. You are comparing things inside the volume, and what happens on the surface face of the volume. I tried to explain the mathematics.
Also, in every chapter there is a formula, a different formula. I think it’s very important to show that formulas can be beautiful. And some are more beautiful than others. And the reader can just skip the formula, but look at it and just points out that it's beautiful, even if I don't understand it completely.
There were different constraints I used to write the book, and one of them was to have a formula, exactly one formula in every chapter.
EL; Yeah, and one of the reasons we wanted to talk to you—not just that I read your book about Kovalevskaya and kind of fell in love with it—but also because since leaving math academia, you've been doing a lot more literature, including being part of the Oulipo group, right, in France?
MA: Yes. You want me to explain what it is?
EL: Yeah, I don't really know what it is, so it'd be great if you could tell us a little more about that.
KK: Okay. It's a group—for mathematicians, I should say it’s a set—of writers and a few mathematicians. It was founded in 1960 by Raymond Queneau and François Le Lionnais. The idea is to the idea is to find constraints to write some literary texts. For instance, the most famous may be the novel by George Perec, La Disparition. It was translated in English with the title A Void, which is a rather long novel which doesn’t use the letter e. In French, it is really very difficult.
MA: In English also, but in French even more.
EL: Oh, wow.
MA: Because you cannot use the feminine, for instance.
EL: Oh, right. That is kind of a problem.
MA: Okay, so some of the constraints have a mathematical background. For instance, this is not the case for La Disparition, but this is a case for for some other constraints, like I don't know, using permutations or graph theory to construct a text.
KK: I actually know a little about this. I taught a class in mathematics and literature a few years ago, and I did talk about Oulipo. We did some of these—there are these generators on the internet where you can, one rule is where you pick a number, say five, and you look at every noun and replace it by the one that is five entries later than that in the dictionary, for example. And there are websites that will, you feed it text, and it's a bit imperfect because it doesn't classify things as nouns properly sometimes, it's an interesting exercise. Or there was another one where—sonnets. So you would you would create sonnets. Sonnets have 14 lines, but you would do it sort of as an Exquisite Corpse, where you would write all these different lines for sonnets, and then you could remove them one at a time to get a really large number, I forget now however many you do so, yeah, things like that, right?
MA: Yeah, this is cent mille milliards, which is 10 to the 14.
KK: That’s right, yeah. So 10 different sonnets. But yeah, it’s really really interesting.
MA: The first example you gave then, which is called in French “X plus sept,” X plus seven, you do you start from a substantive, a noun, you take the seventh in a dictionary following it.
KK: That’s right.
MA: It depends on the dictionary you use, of course.
MA: So that's what they did at the beginning, but now they're all different.
EL: Yeah, it's a really neat creative exercise to try to do that kind of constraint writing.
MA: That forms a constraint, the calendar constraint I used to in this book, is based on books by Michelle Grangaud, who is a poet from the Oulipo also, and she wrote Calendars, which were books of poetry. That's where the idea comes from.
EL: Yeah, and I assume this, your novel has been translated into English?
MA: Not yet.
EL: Oh, okay.
MA: Somebody told me she would do it, and she started, and I have no news now. I don’t know if she were thinking of a published or not. If she can do something, I will be very grateful.
EL: Yeah, so it’s a good reason to brush up your French, then, to read this novel.
And where can people find—is there writing work of yours that people can find on a website or something that has it all together?
MA: Okay, there is a website of the Oulipo, first of all, oulipo.net or something like that. Very easy to find.
KK: We’ll find it.
MA: Also, I have a webpage myself, but what I write is usually on the Oulipo site. I have also a site, a history site. It’s about history but not about mathematics. It’s about the Paris Commune in 1871. It has nothing to do with mathematics, but this is one of the things I am working on.
EL: Okay. Yeah, we'll share that with with people so they can find out more of this stuff.
MA: Thank you.
KK: Alright, this has been great fun. I learned a lot today. This is this is the best part of doing this podcast, actually, that Evelyn and I really learn all kinds of cool stuff and talk to interesting people. So we really appreciate you to appreciate you taking the time to talk to us today, and thanks for persevering through the technical difficulties.
MA: Yes. So we are finished? Okay. Goodbye.