Kevin Knudson: Welcome to My Favorite Theorem, a podcast about math and so much more. I'm one of your hosts, Kevin Knudson, professor of mathematics at University of Florida. And here is your other host.

Evelyn Lamb: Hi, I'm Evelyn Lamb. I'm a math and science writer in Salt Lake City, Utah. So how are you, Kevin?

KK: I’m fine. It's it's stay at home time. You know, my wife and son are here and we're sheltered against the coronavirus, and we've not really had any fights or anything. It's been okay.

EL: That’s great!

KK: Yeah, we're pretty good at ignoring each other. So that's pretty good. How about you guys?

EL: Yeah, an essential skill. Oh, things are good. I was just texting with a friend today about how to do an Easter egg hunt for a cat. So I think everyone is staying, you know, really mentally alert right now.

KK: Yeah.

EL: She’s thinking about putting bonito flakes in the little eggs and putting them out in the yard.

KK: That’s a brilliant idea. I mean, we were walking the dog earlier, and I was lamenting how I just sort of feel like I'm drifting and not doing anything. But then, you know, I've cooked a lot, and I'm still working. It's just sort of weird. You know, it's just very.

EL: Yeah, time has no meaning.

KK: Yeah, it's it's been March for weeks, at least. I saw something on Twitter, Somebody said, “How is tomorrow finally March 30,000th?”

EL: Yeah.

KK: That’s exactly what it feels like. Anyway, today, we are pleased to welcome Susan D'Agostino to our show. Susan, why don't you introduce yourself?

Susan D’Agostino: Hi. Thanks so much for having me. I really appreciate being here. I’m a great fan of your show. So yeah, I'm Susan D’Agostino. I'm a writer and a mathematician. I have a forthcoming book, How to Free Your Inner Mathematician, which is coming out from Oxford University Press. Actually, it was just released in the UK last week and the US release will be in late May. And otherwise, I write for publications like Quanta, Scientific American, Financial Times, and others. And I'm currently working on an MA in science writing at Johns Hopkins University.

KK: Yeah, that's pretty cool. In fact, I pre-ordered your book. During the Joint Meetings, I think you tweeted out a discount code. So I took advantage of that.

SD: Yes. And actually, that discount code is still in effect, and it's on my website, which I'll mention later.

EL: Great. So you said you're at Hopkins, but you actually live in New Hampshire?

SD: Exactly. Yes. I'm just pursuing the program part-time, and it's a low-residency program. So I’m a full-time writer, and then just one class a semester. It creates community, and it's a great way to meet other mathematicians and scientists who are interested in writing about the subject for the general public.

EL: Nice. I went to Maine for the first time when I was living in Providence last semester and drove through New Hampshire, which I don't think is actually my first time in New Hampshire, but might have been. We did stop at one of the liquor stores there off the highway, which seems like a big thing in New Hampshire because I guess they don't have sales tax.

SD: No sales tax, no income tax, “Live Free or Die.” Yeah, and you probably test right around where I live because I live in New Hampshire has a very short seacoast, about 18 miles, depending on how you measure it. We live right on the seacoast.

EL: Oh yeah, we did pass right there. Wonderful. Yeah, the coast is very beautiful out there.

SD: I love it. Absolutely love it. I'm feeling very lucky because there's lots of room to oo outside these days. So, yeah, just taking walks every day.

EL: Wonderful.

KK: So you used to be a math professor, correct?

SD: Yes.

KK: And you just decided that wasn't for you anymore?

SD: Yeah, well, you know, life is short. There's a lot to do. And I love teaching. I had tenure and everything. And I did it for a decade. And then I thought, “You know, if I don't write the books I have in mind soon, then maybe they won't get done.” I've got my first one out already, only two years into this career pivot to writing, and I’m working on my next one. And I always had in mind, in fact, I have a PhD, but I also have an MFA. So I have a terminal degrees both in math and writing. And I always had one foot in the math world and one foot in the writing world, and I realized I didn't want to only live in one. So this is my effort to live fully in both worlds.

KK: That’s awesome.

EL: Yeah. Nice. So the big question we have now of course, is what is your favorite theorem?

SD: Okay, great. My favorite theorem is the Jordan curve theorem.

KK: Nice.

SD: Yeah. It’s a statement about simple closed curves in a 2-d space. So before I talk about what the Jordan curve theorem is, let's just make sure we're abundantly clear about what a simple closed curve is.

EL: Yes.

SD: So, a curve—you can think about it as just a line you might draw on a piece of paper. It has a start point, it has an end point. It could be straight, it could be bent, it could be wiggly, it could intersect itself or not. The starting point and the end point may be different or not. And because this is audio, I thought maybe we could think about capital letters in a very simple font like Helvetica, or Arial. So for example, the capital letter O is a is a curve. When you draw it, it has a start point and an end point that are the same. The capital letter C is also a curve. That one has a different starting and end point, but that's okay. It satisfies our definition. Capital letter P also. That one intersects itself in the middle, but it's still it's a curve.

Okay, so a simple curve is a curve that doesn't intersect itself along the way. It may or may not have the same starting and end point, but it won't intersect itself along the way. So capital letter O and capital letter C are both simple. But for example, the capital letter B is not simple, because if you were to start at the bottom, go up in a vertical line, draw that first upper loop and then the second upper loop, between the first and second upper bubbles of the B, you will hit that initial vertical line that you drew. So it's not simple because it touches itself along the way.

And a closed curve is a curve that starts and ends at the same point. So the letter O is closed, but the letter C is not because that one starts in one place ends in another.

KK: Right.

SD: Moving forward as we talk about the Jordan curve theorem, let's just keep in mind two great examples of simple closed curves: the letter O, and even the capital letter D. It's fine that that D has some angles, in the bottom left and upper left. So corners are fine, but it needs to start and end in the same place and doesn't intersect itself other than where it starts and ends.

Okay, so the Jordan curves theorem tells us that every simple closed curve in the plane separates the plane into an inside and an outside. So a plane, you might just think of as a piece of paper, you know, an 8 1/2 by 11 piece of paper, let's draw the letter O on it. And when you draw that letter O, you are separating that piece of paper, the surface, into a region that you might call inside the letter O and another region that you might call outside the letter O. And the second part of the Jordan curve theorem tells you that the boundary between this inside and that outside formed by this letter O is actually the curve itself. So if you're standing inside the O, and you want to get to the outside of the O, you've got across that letter O, which is the curve.

Okay, so that doesn’t sound very profound.

KK: It’s obvious. It’s just completely obvious.

EL: Any of us who are big doodlers—like, when I was a kid, at church, I was always doodling inside the letters in the church bulletin. That’s the thing. I know that there's an inside and outside to the letter O.

SD: You do. Yes. And you could ask your kid brother, kid, sister, whoever. Anyone—you probably didn't need a big mathematical theorem to assure you of this somewhat obvious statement when it comes to the letter O. Okay, so, I do want to tell you why I think it's really interesting beyond this fact that it seems obvious. But before I do, I just want to make two quick notes. And one is that you really do need the simple part, and you really do need the closed part of the theorem because, for example, if you think about a non-closed curve, like the letter C, and you're standing on the piece of paper around that letter C, maybe even inside, like where the C is surrounding you, it actually doesn't separate the piece of paper into an inside and an outside. And then you also need the non-simple part because if you think about the letter P, which is not simple because it intersects itself, if you think about the segment of the P that's not the loop, so the vertical bottom part of that P, that is part of the curve, the letter P, and that piece of the curve doesn't separate—so even though that P seems to have a little bit of a bubble up there, in the in the loop of the P, the bottom part of the P is part of the curve, and it's not the boundary between the inside, what you might consider the inside of the P, and the outside of the P. So you really do need the simple part and the closed part.

KK: Right, right.

SD: Okay, so the reason I think it's interesting, in spite of the fact that it seems obvious, is because it actually isn't very obvious. And it's not obvious when you talk about what mathematicians love to call pathological curves.

KK: Yeah. Okay. No, I know, I know, the theorem I just wanted to shrug my shoulders and say, “Oh, look, it's just a special case of Alexander duality.” Right? And so surely it works. But yeah, okay.

SD: And there are other poorly-behaved curves, or misbehaved curves, like another curve you might think about is the Koch snowflake. So one way of thinking about the Koch snowflake is—again, I'm going to wave my hands a little bit here because we're in audio and I can't draw you a picture—but if you think about the outline of a snowflake, and there's a prescribed way to draw the Koch snowflake, but I'm going to simplify it a little bit. Imagine the outline of a snowflake, so not the inside or the outside of the snowflake, just the outline of it. And on a Koch snowflake, that snowflake is going to have jagged edges. It's going to zig and zag as it goes along the outline of the snowflake. The Koch snowflake actually has an infinitely jagged curve, line, to draw it. So it's not that it has 1000 zigs and zags or 1 million or even 1 billion. It has an infinite number of zigs and zags going back and forth. So you know, it's a little bit easier to imagine the— what could loosely be defined as the inside of the Koch snowflake, and the outside of the Koch snowflake when you imagine one being drawn on a piece of paper. You know, right in the heart of the very dead center of that Koch snowflake, you could probably feel pretty confident saying, “Hey, I'm inside the Koch snowflake.” And then far outside, you could be confident saying, “I'm outside of the snowflake.” But if you think about yourself right up against the edge of this Koch snowflake. And put yourself right there. Then as you think about this boundary of the Koch snowflake, the boundary is supposed to be what separates the inside from the outside, but if you're right up close to that boundary, and in the process of drawing an infinite number of constructions to get the ultimate Koch snowflake. You continue zigging and zagging, you add more zigs and zags every time. Then even in the steps that it takes you to get to your drawing of the Koch snowflake, at some point, it might seem like “Hey, I'm inside. Oh wait, now they zigged and zagged and I’m outside. Oh, wait, they zigged and zagged some more. Now I'm inside again.” So it seems like even in the finite steps that you need to take to draw that Koch snowflake, to imagine what the it is in its infinite world, it seems like that boundary is not really clear. So again, another place where it makes you stop and say, “Wait a minute, maybe the Jordan curve theorem is not as obvious as it first looked.”

KK: Right. Why do you love this theorem so much?

SD: Yeah, so I love it. It actually it kind of goes along with your question of what do you pair it well with? So maybe I'll just jump ahead to what's sugar. Yeah. So, um, because even in my book and in the chapter that in which I discuss the Jordan curve theorem, I actually paired it with a poem. And the poem is by a New Hampshire native, Robert Frost, who actually went to Dartmouth, which is where I got my doctorate. And one of my favorite poems by Frost is called “The Road Not Taken.” And in the beginning of the poem, he's standing in front of this fork in the road, essentially, and he's looking at both options, realizing, “Okay, I've got to go left or I've got to go right.” You know, he starts off:

Two roads diverged in a yellow wood, And sorry I could not travel both And be one traveler, long I stood And looked down one as far as I could To where it bent in the undergrowth;

So he's standing here and he's saying, “Well, which path should I take?” And he notices one that he calls you know, “it was grassy and wanted wear” and had no leaves—what was what was the line—“in leaves no step had trodden black.” And he ultimately comes to the conclusion that he's going to take the past path less traveled. You know, at the very end of the poem, he says, “Two roads diverged in a wood and I—/ I took the one less traveled by,/ And that has made all the difference.” And it strikes me that what Frost is telling us, and what the Jordan curve theorem is telling us, is take the paths that are more unusual, that aren't well trodden, that people don't always look at first, that aren't as obvious or as paved for us. Maybe it's a path that's going to make you question whether you're inside or outside. Or maybe it’s going to have what feels like this amorphous boundary that you can't quite put your finger on. I guess it reminds me that sometimes making a non-traditional choice in life, or looking at pathological objects in math, is actually something very engaging to do, and can can make a life a little bit more interesting.

You know, when I first heard about this theorem, I had the same reaction that most everybody else does: Okay, so I can just draw a curve—you know, you say a curve and you think, “Oh, I can just draw a curve.” I'm just going to do a squiggle on a piece of paper. And as long as I make it simple and closed, then it might be the letter O or it might be some blob that doesn't intersect, but at least starts and ends where it ends where it started. You know, I remember thinking, wait, why does this theorem get its own name? Why isn’t it just lemma 113.7?

EL: An observation.

KK: Clearly.

SD: Why did it get its own name? A I remember asking, and a lot of people, at first everybody was happy to recite the theorem and and say what it was and laugh at how obvious it was, but then later, I kept searching and searching, and then finally I ended up discovering that in fact, it wasn't as obvious, but in order to appreciate how it’s not that obvious, you needed to look at the paths not taken, the more unusual lines and curves.

EL: Yeah, so this is a theorem that, of course, I I feel like I've known for a long time, not just in the “it's obvious” sense, but in the sense that it's been stated in classes that I took—and feel entirely unconfident about knowing anything about it's proof, at least in the general case. I feel like the the difference between how much I have used it and relied on it and what I actually understand of how to prove it is very large.

SD: Yeah, honestly I can say the same thing. My background is in coding theory, definitely not topology. And honestly, I never saw topology as my strength. It was always something that I was in awe of, but also found extremely challenging or less intuitive to me. But I had looked at the proof long ago. I haven't looked at them deeply recently. There are a number of different approaches. But yeah, I feel the same, that even—the statement sounds simple and it's not, and to my understanding, the proofs are also non trivial.

KK: Yeah. I mean, I was sort of being glib earlier and saying it's just a special case of Alexander duality, like that's easy to prove.

EL: Yeah. Right.

KK: I mean, I was teaching topology this this semester, and I was proving Poincaré duality, which is a similar sort of thing, and it's highly non-trivial. I mean, you break it into a bunch of steps, and it sort of magically pops out of it. And I think that's kind of the case here. It's like, you break it into enough discrete steps where each thing seems okay. But in the end, it is a lot of heavy machinery. And like even for Poincaré duality, in the end you use Zorn’s lemma I mean, there's some kind of choice going on. I think when when Jordan—actually, did Jordan even state this theorem? Or is this one of those things where where Jordan gets the credit, but it wasn't really him?

SD: Actually, I don’t know, and now I need to know that answer.

EL: I think he did.

KK: Did he?

EL: Yeah, not to toot my own horn but I’m, gonna anyway, the calendar that I published this year, the page-a-day calendar, still available for purchase, I think Camille Jordan’s birthday is pretty early. It's sometime in January, so I've actually even read this not too long ago. And I think he did publish it and did have a proof of it. And there's an interesting article, I believe by Thomas Hales, about his about Jordan’s proof of the Jordan curve theorem, I guess maybe to some extent defending from the claim some people have that that he never had a rigorous proof of it. I did read that for doing the calendar, but it was over a year ago at this point and I don't quite remember. But yeah, you can find a reference to it on my calendar. I will also include that in the show notes.

KK: And also the same Jordan of Jordan canonical form.

SD: Right.

KK: Pretty serious contributions there from one person.

SD: Absolutely.

KK: Yeah. All right. I actually like this pairing a lot.

EL: Yeah.

KK: And and since you live in New Hampshire, it's perfect.

SD: Yes. I have a number of New Hampshire references in my book because I just feel like I wanted to humanize math to the extent that I could, while still tackling pretty substantial ideas. But any time I had an invitation to bring in something from left field that was actually meaningful to me, I just went for it.

EL: Yeah.

SD: I’m sure Evelyn, too, it sounds like you're up on all of the mathematicians’ birthdays at this point because of your calendar.

EL: I know a few of them now. More than I did two years ago.

SD: Right.

KK: So it was like to give our guests a chance to plug anything. You’ve already plugged your book. Any other places we can find you online?

SD: Yeah, well, lately, I've been writing for Quanta magazine, which has been very exciting. And in fact, I have a few math articles already out this year. And I have a very special one—I can't tell you the topic. I'm not supposed to—it should be coming out April 15. And I'm very excited about that article that I believe is going to be on April 15, assuming everything is fine with the publication schedule, given the pandemic. But yeah, listeners can find links to my articles on my website, which is just susandagostino.com. And you can find information about my books and my articles and what I'm up to there. vKK: Cool. Well, thanks so much for joining us, Susan. This was a good one.

EL: Yeah, lovely to chat.

SD: Great. Well, thank you so much. And you know, I love the show, and really, it was my honor to be here. Thank you.

KK: Thanks.

On this episode of My Favorite Theorem, we talked with mathematician and science writer Susan D'Agostino. Here are some links you might find interesting as you listen.

D'Agostino's website
How to Free Your Inner Mathematician, her new book (find a discount code on her website)
Evelyn's article about the Koch snowflake
Thomas Hales' article about Camille Jordan's proof of the Jordan curve theorem (pdf)

Evelyn's page-a-day math calendar

The article D'Agostino was excited about towards the end of the podcast was this interview with Donald Knuth