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Evelyn Lamb: Hello and welcome to My Favorite Theorem, the math podcasts with no quiz at the end. I'm your host Evelyn Lamb. I'm a freelance math and science writer in Salt Lake City, Utah. And this is your other host.



Kevin Knudson: I’m Kevin Knudson, professor of mathematics at the University of Florida. How's it going?



EL: All right. I got to take an overnight Amtrak trip last weekend, my first time, so that was pretty fun. Went from Salt Lake to Sacramento and got to see lots of beautiful Nevada and California landscapes on the way.



KK: Yeah, I did an overnight Amtrak once and it was less fun. It was from Jackson, Mississippi to Chicago. And — which, I mean, it's, you know, it's all night, right? So you don't really see anything. And it's remarkable how many times have to pull over for the freight trains, right?



EL: Yeah.



KK: This is how American rail is really different from European rail. You're at the mercy of all the freight, but that's okay. Anyway, yeah.



EL: I guess, today, living on the only portion of Amtrak's corridor for which they actually own the tracks, is our guest, Juliette Bruce. At least I hope I'm correct, that that's where you're living. Otherwise, that was a weird introduction. So please tell us a little bit about yourself.



Juliette Bruce: Thank you so much for the introduction. I'm Juliette Bruce, as you said, and I am a postdoc at Brown University. So in fact, I am in the northeast along the Acela Express corridor. In fact, I've never taken that Amtrak corridor, I've only taken the very slow ones that you were talking about, but I hope to take it soon.



EL: Yes. Find yourself someplace to go between New York, DC, Boston, I guess to Boston, you don't really need the Acela. It's already pretty close.



KK: You can walk to Boston.



EL: If you're really dedicated.



JB: It’s a pretty far walk.



EL: Yes. So I guess this isn't the train cast. This is a math podcast. So, so yeah. What are your mathematical interests at Brown?



JB: Yeah, so my area of math is kind of in the intersection of algebraic geometry and commutative algebra, which is all about studying the interaction between this algebra, coming from kind of the symbolic equations we get when we write down systems of polynomial equations, and the kind of geometry we can look at when we study the zero set of those equations. So we can look at the simultaneous solutions to the system of polynomials, and that's some lovely geometric object. And alternatively, we can look at these symbols we write on our paper, and somehow, in some point in math, we learned that we can do lovely things, like finding the roots of a quadratic polynomial by graphing them on our graphing calculator pictorially, or we learn we can use symbols and write down things like the quadratic formula, and magically they give the same answer. A lot of my research is sometimes a generalization of this fact that there's two different ways to study the solutions to a system of polynomial equations.



EL: Right. I must admit, I'm pretty naive about algebraic geometry, but there is this kind of magic in it, which is — you know, like, in, what, seventh or eighth grade or something, you start learning to graph the zeros of polynomials. Maybe you might not use that exact language for it, but you start to understand that you can intersect two different polynomial equations and find these intersection points and stuff like that. And yet, this is also like cutting edge math, you know, just add a few variables, or bump up the powers of the the numbers that you're using. And suddenly, this is stuff that, you know, people are getting PhDs in. I's kind of kind of cool,



KK: Right? Or work over a finite field, whatever those are. Yeah.



JB: I mean, I always find it fascinating with just how many different areas algebraic geometry has touched in mathematics and in the world. It seems to start from such a lovely and beautiful, simple idea that we learn in, you know, middle school or high school, and just kind of grows exponentially. And it turns out, it's actually a very deep idea that maybe we don't always appreciate when we first see it. I know I certainly did not.



EL: Yeah.



KK: All right.



EL: So then what is your favorite theorem?



JB: So my favorite theorem, or the theorem I want to talk about today, I know it as Petri’s theorem. I know some people know it as the Babbage-Enriques-Noether-Petri theorem. I'm not sure exactly on the correct attribution here, so I'll stick with Petri’s thereom and apologize to Babbage, Noether, and Enriques, who maybe want the appropriate attribution here. And this is a theorem from classical algebraic geometry, which means from the 19th century, and it's about understanding the interaction between thinking about systems of solutions of polynomial equations abstractly, and how we can realize that abstract solution set concretely as solutions to an honest-to-God set of polynomial equations that we could write down and describing what those polynomials might look like.



KK: Okay.



JB: And so the statement of the theorem, I'll state the theorem, and then we'll walk through it, maybe. And you can ask questions, because I know when it’s stated, it's a little bit of a mouthful and a little scary, is that if I have a curve that is non-hyperelliptic, and I embed it via the canonical embedding, then the image of the canonical embedding is cut out by quadratics unless the curve is trigonal, meaning it admits a three-to-one map to the Riemann sphere, or it's a curve in the plane of degree five. So that's the statement of the theorem. That's a mouthful, I know, to get through.



KK: Yeah, sure. Right.



EL: That’s interesting. So, you know, as I already confessed, this is outside of maybe my, my mathematical comfort zone a little bit. And how, how should I think about these exceptions? Like how exceptional are the exceptions? Is it, like, a lot of things? Or just a couple of little things that and otherwise, everything falls under this umbrella?



JB: Yeah, so that's a fabulous question. And so I gave — there are two exceptions to this theorem, right? If a curve admits a three-to-one map to Riemann sphere, so there's a map that goes to the Riemann sphere, that kind of every preimage has three points, it kind of looks like a sheet wrapped up three times around the sphere. Or it's a very specific curve in the plane of degree five. And so these exceptions, there's an infinite number of them. But it turns out if you think about them correctly, it's kind of a small proportion, or it's not most curves that will satisfy this. So this is somehow saying, with these few exceptions aside, we can actually understand the image of what's called the canonical bundle. So maybe I should say, what is actually going on here. It's something a little deep. So kind of the starting point of algebraic geometry is that we want — I said, we want to study the solution sets of polynomial equations. Well, it turns out that that's how the field started. But pretty quickly, people realized, well, this is some geometric space, it's a set of points. And instead of looking at the solution set to a particular set of polynomial equations, we can kind of abstract this away and forget the polynomial equations together and just think about what possible sets of solutions could I have, and think about that kind of abstractly in the ether. There's no polynomial in sight, we can just say, oh, you know, this is a solution set to some system of polynomial equations. We don't know which. And it's a lovely theorem that, you know, if we're talking about curves, it turns out algebraic geometers have this very weird convention that curves would look to people like us, like a two-dimensional surface. This is because I like to work over the complex numbers. So my polynomials have solutions and the complex plane is two-dimensional. So we have this weird terminology. So abstractly, a curve, if it's smooth and has to satisfy some other conditions, just looks like a closed surface, possibly with some holes in it. So we'd have a genus g surface. So if you've seen a doughnut, or a torus, that's just an algebraic curve of genus one. And if you seen a sphere, that's just an abstract algebraic curve of genus zero. And the beauty of these is that somehow, if we take these objects, we can realize them in space, we can put them into some large, complex space, or some large projective space, and once we've done that, you can ask, well, I know there is some set of polynomials that cut the space out, we have this algebraic variety. It's a system where we know it's by definition, a solution set to some polynomials. And you could ask what polynomials actually cut it out under this realization in space. And often, there are many different realizations. So for example, you could look at the parabola, a very simple example. We can look at the parabola, x2−y=0. This gives us the normal parabola going through the origin. It’s realized in space. But we can also abstractly think about just kind of the parabola floating around, no coordinate system at all. And we could also realize that same parabola in space by just, you know, shifting it up or down the y-axis and moving it around, and the polynomials that cut it out when I start moving it around, we learn, are different, right? We learn how to do transforms, we knew somehow, like (x−1)2−y=0 gives a different solution set, but it looks the same in the plane, just moved around. So you could ask, when I put my abstract curv