This transcript is provided as a courtesy and may contain errors.
EL: Welcome to My Favorite Theorem. I’m your host Evelyn Lamb. I’m a freelance math and science writer based in Paris for a few more days, but after that I’ll be based in Salt Lake City, Utah. And this is your other host.
KK: Hi, I’m Kevin Knudson, professor of mathematics at the University of Florida, where it’s raining. It’s been raining for a week. After a spring of no rain, now it’s raining. But that’s OK.
EL: I probably shouldn’t tell you that it’s absolutely gorgeous, sunny and 75 degrees in Paris right now.
KK: You really shouldn’t.
EL: OK, then I won’t. OK. Each episode we invite a mathematician on to find out about their favorite theorem. Today we’re very happy to welcome Eriko Hironaka onto the show. So would you like to tell us a little bit about yourself, Eriko?
EH: Yes, thank you, first of all, for having me on this show. It’s very flattering and exciting. I worked at Florida State University for almost twenty years. I was a professor there, and I recently moved to the American Mathematical Society. I’ve been working there for two years. One year, I guess, full time, so far. I work in the book program. I’m somebody who is a mathematician but is doing it from various angles.
EL: Yeah, I was really interested in having you on the podcast because I think that’s a cool perspective to have where you’ve been in the research world for a long time, but now you’re also seeing a broader view, maybe, of math, or kind of looking at it from a different angle than before. Do you mind telling us a little bit about what you do as a book person for the AMS?
EH: Yeah, what do I do? Actually I was thrown into this job, in a way. They said, OK, you’re going to work in the book program. Your job is basically to talk to people about books, and see if anybody wants to write a book, and if they do, you keep talking with them, and when they finally submit something, you prepare, maybe the real job part is to, once there’s a submission, start it through a review process, then also what’s kind of exciting is to convince the publishing group to actually publish the book. That part requires me to think about how this book fits into mathematics and mathematical literature and then also how much it’ll cost to produce the book and what’s involved in selling the book. Who is the audience and how can it be presented in the best possible way? I think of myself as sort of the connector between the author, who is thinking about the mathematics, and the publishers, who are thinking about. The AMS is a nonprofit, but to cover costs and make this a reasonable project.
EL: You see a lot of different aspects of this, then.
EH: Yeah, so I don’t know if I was more naive than most mathematicians, but I think most mathematicians don’t think beyond proving theorems and conveying and communicating their ideas to other people. Maybe they also think about what to write on their vita, and things like that. That kind of thing is very different. Right now I don’t really have a reason to keep up my vita in the same way that I used to. That was a big change for me.
EH: I still do mathematics, I still do research, give talks, and things like that. I still write papers. But that’s really become something just for me. Not for me, it’s for math, I guess. But it’s not for an institute.
KK: It’s not for the dean.
EH: It’s not for the dean. Exactly.
KK: That’s really liberating, I would think, right?
EH: It’s super liberating, actually. It’s really great.
KK: Very cool. I dream about that. One of these days.
EH: I feel like I’m supporting mathematics kind of from the background. Now I think about professors as being on the battlefield. They’re directly communicating with people, with students, with individuals. And working with the deans. Making their curriculum and program and everything work.
EL: So what have you chosen as your favorite theorem?
EH: OK, Well, I thought about that question. It’s very interesting. I’ve even asked other people, just to get their reaction. It’s a very interesting question, and I’m curious to know what other people have answered on your podcast. When I think of a theorem I think about not just the statement, but more the proof. I might think of proofs I like, theorems whose proofs I like, or I might think about how this theorem helped me because I really needed something. It’s actually kind of utilitarian, but a favorite theorem should be more like what made you feel great. I have to say for that, it’s a theorem of my own.
KK: Cool, great.
EH: So I have a favorite theorem, or the theorem that made me so excited, and it was the first theorem I ever proved. The reason it’s my favorite theorem is because of a mixture not just of feeling great that I’d proved the theorem but also feeling like it was a big turning point in my life. I felt like I had proved myself in a way. That’s this theorem, I think of it as a polynomial periodicity theorem, and what it says. Do you want me to say what the theorem is?
KK: Yeah, throw it out there, and we’ll unpack it.
EH: So the theorem in most generality, it says that if you have a finite CW complex, if you have a sort of nice space, in my case I was looking at quasiprojective varieties, but any kind of reasonably nice space, you can take a sequence of coverings, of regular coverings, corresponding to a choice of map from the fundamental group of the space to some, say, free abelian group, and the way you get the sequence of coverings is you take that map and compose it with the map from that free abelian group to the free abelian group tensored with Z mod n. So if everything is finitely generated, that gives you a surjective map from the fundamental group of your space to a finite abelian group. And now by the general theorem of covering spaces gives you a sequence of finite coverings of your space. And then if you have that space having a natural completion you can talk about natural branch coverings associated to those for each n. My theorem was what happens to the first Betti numbers of these things? The rank of the first homology of these coverings. I showed that this sequence actually has a pattern. In fact, there is a polynomial for every set base space and map from the fundamental group of the base space to a free abelian group, there is a polynomial with possibly periodically changing coefficients so that the first Betti number is that polynomial evaluated at n.
EH: So n is the degree of the covering. The Betti numbers are changing periodically, the polynomials are changing periodically, but it’s a pattern, it’s a nice pattern, and there’s a single polynomial telling you what all of these Betti numbers are doing.
EL: So what was the motivation behind this theorem?
EH: This problem of understanding the first Betti number of coverings comes from work of Zariski back in the early 1900s. His goal was to understand moduli of plane curves with various kinds of singularities. Simply put, what he did was he tried to distinguish curves by looking at topology, blending topology with algebraic geometry. This was kind of a new idea. This is not very well known about Zariski, but one of his innovations was bringing in topology to the study of algebraic geometry.
KK: That’s why it’s called the Zariski topology, right? I don’t know. One assumes.
EH: In a way. Not really!
KK: I’m not a historian. My bad.
EH: He brought geometry topology in. The Zariski topology is more of an algebraic definition. What he did was he was interested, for example, he showed that, what he was interested in when you’re talking about moduli of plane curves is whether or not you can get from one plane curve with prescribed singularities, say a sextic curve, a degree six curve, in C2, in the complex plane, with exactly six simple cusps. So six points in the plane can either lie on a conic or not. General position means it doesn’t lie on lines, doesn’t lie on a conic. But if the six points lie on a conic, it turns out you cannot move within the space of sextics with six cusps to a sextic with six cusps not lying on a conic.
EH: They’re two distinct families. You’d have to leave that family to get from one to another. You can’t deform them in the algebraic category. To prove this, he said, well, basically, even though the idea of fundamental groups and studying fundamental groups was really new still and was just starting to be considered a tool for knot theory, for example, that came a little bit later. But he said, you can tell they’re different because their topology is different. For example, take coverings. Take your curve, and say it’s given by the equation F(x,y)=0. So F(x,y) is a polynomial. Take the polynomial z^n=F(x,y). You get a surface in three-dimensional space, and now look at the first Betti number of that. So the first Betti number, the first homology, can kind of be described algebraically in terms of other things, divisors and things like that. You can think of it as a very algebraic invariant, but you can also think of it as a topological invariant. Forget algebra, forget complex analysis, forget everything. And he showed that if you take the sextics with six cusps and you took z^n=F(x,y), you get things with first Betti number nontrivial, and by the way, periodically changing with n. In fact, when 6 divides n, it’s nontrivial. It jumps. Every time 6 divides n, it jumps. Otherwise I can’t remember, I think it’s zero. But in the case that the cusps are in general position, the first Betti numbers are always zero.
EH: So that must mean that the topology is different. And if the topology is different, they can’t be algebraically equivalent. So that was the process of thinking, that topology can tell you something about algebraic geometry. And that kind of topology is what geometric topologists study now, fundamental groups, etc. But this was all a very new idea.
EL: So that’s kind of the environment that your theorem lives in, this intersection between topology and algebraic geometry.
EH: That’s right. So my theorem, Sarnak conjectured, I was working on fundamental groups of complements of plane curves, especially with multiple components, for my thesis. And Peter Sarnak was looking at certain problems coming from number theory that had to do with arithmetic subgroups of GL(n) and looking at what happens when you take your fields to be finite, and things like that. You get these finite fields. Somehow in his work, something coming from number theory, he wondered, hearing about what I was doing with fundamental groups and Alexander polynomials, which have to do with Betti numbers of coverings, he asked, “Can you show that the Betti numbers of coverings are periodic or polynomial periodic,” which is that other thing. I thought, OK, I’ll do this, and since I was already working topologically, I could get the topological part by looking at the unbranched coverings, and then I had to complete it. To understand the completion, the rest, there’s a difference between the Betti numbers of the unbranched coverings and the Betti numbers of the branched coverings, to understand that, I needed to understand intersections of curves on the surface, to sort of understand intersection theory of algebraic curves. And these have very special properties, nice properties coming from the fact that we’re talking about varieties.
EH: And I used that to complete the proof. It was a real blend of topology and algebraic geometry. That’s what made it really fun.
KK: That’s a lot of mathematics going in. And I love your confidence. Peter Sarnak said, “Hey, can you do this?” and you said, “Yeah, I can do this.”
EH: Right, well I was feeling pretty desperate. It was really a time of: Should I do math? Should I not do math? Do I belong here? And I thought, “OK, I’ll try this. If it works, maybe that’s a sign.”
EL: So what have you chosen to pair with this theorem?
EH: As it happens, after I proved this theorem and I showed it to Sarnak, I basically wrote a three-page outline of the proof. I showed it to him, and he looked at it carefully and said, “Yeah, this looks right.” Also, you know, you can feel it when you have it. Suddenly everything has become so clear. I was glowing with this and driving from Stanford to Berkeley, which is about an hour drive, and I usually took a nicer route through the hills to the west, so you can imagine driving with these vales and woods, and it was beautiful sunshine and everything, and the Firebird suite starts out very quiet, and it just perfectly represented what it feels like to prove a theorem. It starts really quiet, and then it gets really choppy and frenzied.
KK: And scary.
EH: Exactly, scary. The struggling bird, he’s anxious and frightened, really, really unsettling. And then there’s this real gentleness, feeling like it’s going to be OK, it’s going to be OK. But that also is a bit disturbing. There’s something about it that’s disturbing. So it keeps you listening, even though it’s very sweet and the themes are developed, it’s a very beautiful theme. Then there’s this bang and then it becomes really frenzied again, super frenzied, but excited. And then it becomes bolder and bolder. And then that melody comes in, and it starts to really come together. And it starts to feel like you’re running, like there’s a direction, and then finally it gets quiet again. There’s this serenity. And this time the serenity is real. All this stuff has built up to it, and that starts to build and the beautiful theme comes out in the end. It’s just this glorious wonder at the very end. It was like all my excitement was just exemplified in this piece of music.
EL: I love that picture of you driving through California, blasting Firebird.
EH: Yes, exactly.
EL: With this triumphant proof that you’d just done. That’s really a great picture.
KK: So my son just finished high school, and he wants to be a composer. He’s going to go to college and study composition. And I actually sort of credit that piece, Firebird suite, as one of the pieces that really motivated him to become a composer. That and Rhapsody in Blue.
EH: It really tells a story.
KK: Yes, it does. It’s really spectacular. So I think maybe a lot of our listeners don’t know that you have a rather famous father.
KK: Your father won the Fields medal for proving resolution of singularities in characteristic zero, right?
KK: What was that like?
EH: Yeah, so I had a really strange relationship with mathematics. Because I grew up with a mathematician father, I avoided math like the plague, you know. Partly because my father was a mathematician, and I thought that was kind of strange, that it didn’t fit in with the rest of the world that I knew. I grew up in the suburbs. It wasn’t a particularly intellectual background. For me, the challenge to my life was to figure out how to fit in, which I was failing at miserably. But I thought that was my challenge. Doing well in math was not the way to fit in in school. I would kind of deliberately add in mistakes to make sure that I didn’t get a good grade.
KK: Really? Wow.
EH: I would kick myself if I forgot and I would get a high grade and everybody would say, “How did she do that?” You know what I mean? I thought of math as this embarrassment, in a way, to tell the truth, strangely enough. But on the other hand, through my father and his friends and colleagues, I knew that mathematics also had this very beautiful side, and the people who did it were very happy people, it seemed. I saw that other side as well. And I think that was an advantage because I knew that math was really cool. It’s just that that wasn’t my thing. I didn’t want to do that. Also, my teachers were not very exciting. The math teachers seemed to make math as boring as possible. So I had this kind of split personality when it came to math, or split feeling about what math was.
EH: But then when I started to do math, I started somehow accidentally to do math in college, and I actually got much more attracted to it. It was after vaguely stumbling through calculus and things like that. So I never really learned calculus, but I started skipping through calculus, and I took more advanced classes, and it just really clicked, and I got hooked. I learned calculus in graduate school, as some people do, by teaching it.
KK: Well that’s when you really learn it anyway, that’s right.
EH: Some people have this impression that lots of mathematicians had the advantage of having access to mathematics from a young age, but I think it’s not obvious how that’s an advantage. In some cases it could be that they were nurtured in mathematics. I mean, I talk to my kids about mathematics, and it’s a fun thing we do together. But I don’t think that’s necessarily the case of people with mathematical parents. In my case it certainly wasn’t the case for me. But still it was an advantage because I knew that there was this thing called mathematics, and many people don’t know that.
EL: Yeah. And, like you said, you knew that mathematicians were happy with their work, and just even knowing that there’s still math to prove. That was something, when I started to do math, I didn’t really understand that there was still more math to do, it wasn’t just learning calculus really well. But going and finding and exploring these new things.
KK: I had that same experience. I remember when I was in high school, telling people I was going to go to graduate school and be a math professor, and they said, “Well, what do you do?” I said, “I don’t know, I guess you write another calculus book.” Which we certainly do not need, right?
EH: Or we need different kinds.
KK: So, I say that, but I’m actually writing one, so you know.
EH: Oh, are you?
KK: Just in my spare time, right? I have so much of it these days.
EH: I think there is a need for calculus books, it’s just maybe different kinds.
KK: Well now that I know someone at the publishing house at the AMS…
EH: Absolutely. I’m going to follow up on this.
KK: Oh, wow. Well this has been fun.
EL: Yeah, thank you so much.
EH: Well thank you for asking me. It gave me a chance to think about different things, and it’s been fun talking to people about, “What’s you’re favorite theorem?”
EL: Good math conversation starter.
EH: Yeah, absolutely.
KK: Thanks for joining us, Eko.
EH: Thank you.
KK: Thanks for listening to My Favorite Theorem, hosted by Kevin Knudson and Evelyn Lamb. The music you’re hearing is a piece called Fractalia, a percussion quartet performed by four high school students from Gainesville, Florida. They are Blake Crawford, Gus Knudson, Del Mitchell, and Bao-xian Lin. You can find more information about the mathematicians and theorems featured in this podcast, along with other delightful mathematical treats, at Kevin’s website, kpknudson.com, and Evelyn’s blog, Roots of Unity, on the Scientific American blog network. We love to hear from our listeners, so please drop us a line at firstname.lastname@example.org. Or you can find us on Facebook and Twitter. Kevin’s handle on Twitter is @niveknosdunk, and Evelyn’s is @evelynjlamb. The show itself also has a Twitter feed. The handle is @myfavethm. Join us next time to learn another fascinating piece of mathematics.