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 curves in space, what are the polynomials that actually cut this thing out? And so those are kind of the input to the theorem, is these abstract curves. And we put them into space. And what are they cut out by? So that's kind of the input. And the theorem is answering that question, what are they cut out by? What are they defined by?

KK: Right. So are you assuming you're starting with a plane curve? Or?

JB: No, so this curve doesn't have to be in the plane, although it's kind of just this abstract notion of a curve, so it’s somehow, just in general, a kind of smooth looking surface that's compact and has g holes, so maybe like a 2-holed torus or a 3-holed torus. It's kind of some very weird donut-looking shapes, essentially, is what the curve goes in, what is the input of this theorem?

KK: Right. And you mentioned something called the canonical embedding. So that might require a little terminology.

JB: Exactly. So what do I mean by the canonical embedding? Defining it exactly is complicated. And it's not something I would want to try to do on this podcast.

EL: Especially with audio.

JB: Especially with audio. But instead, let me just kind of give this notion. I said, you know, if we're looking at perhaps standard parabolas in the plane, there's a lot of different ways we could put it in the plane. We could put it through the origin, we could put it so like the vertex is at (1,1) or (2,1), or we could do all these things, and there isn't, doesn't seem to necessarily be a natural best choice for how we put a parabola in the plane.

EL: Right. It feels very arbitrary.

JB: It’s very arbitrary. And when we change our arbitrary choice, we change the set of polynomials that define the parabola in the plane. It turns out that when we're kind of working in a slightly more abstract setting, where instead of looking at parabolas in the plane, but we're looking at these two-dimensional surfaces, which are what algebraic geometers will think of as curves, because we're looking at the complex set of points, there’s an almost canonical way to put them into some kind of space. And that's called the canonical embedding. It kind of arises by looking at ways you can kind of differentiate on your surface. It comes from looking at what are known as differentials on your surface. And I won't say anything more than that, other than to say somehow, it's this beautiful fact that was developed by people in the 19th century that there exists such a thing that allows you to transport these abstract curves into different spaces in a way that has beautiful properties. And somehow, it's a great tool for studying curves.

EL: Not quite sure if this is the right question asked, but you know, you have this input to this theorem, and then it tells you something about like, you know, what polynomials can be your solutions? How specific is it? Like, would it output something that, you know, we would have recognized as a polynomial in seventh grade? Or does it output something that maybe has a little more technical machinery behind it?

JB: This is an absolutely fantastic question. This is a fantastic question. So, right, as you're saying, the input is I input this abstract surface abstract Riemann surface of genus g that satisfies some properties and the output of the theorem and saying if it doesn't satisfy, if it doesn't fall into these two exceptional collections, which are relatively small when it comes to lists of exceptions, then we know that the defining equations are degree two. And you might ask, well, does the proof actually give — like what are the polynomials? Can you actually write them down? And in part, the version of Petri’s theorem I know, in fact, gives you those polynomials in some sense. There are some choices that have to be made, and those choices kind of arise from some technicalities about defining exactly what is the canonical embedding. There are some choices there. But once you've made those choices, Petri’s theorem actually comes down and says we can write down an honest set of degree two polynomials in a lot of variables now. The number of variables is the number of holes on my surface, is the genus of my surface. So it's a lot of variables. But we can write down an honest set of equations that cut this out. And this is this beautiful thing that takes an extremely abstract thing, you know, this curve that's abstract sitting in our mind, and realize it in space, and it outputs a list of polynomial equations.

EL: Okay, wow.

KK: So okay, so now I'm thinking about elliptic curves in particular. So you mentioned there's just one variable. So that's a torus. Right? You can you can actually write this with one variable?

JB: Yes. Yeah. So if you're looking at elliptic curves,

KK: But I always think of elliptic curves as being, like, y2=x3 plus some change, right? That's not degree two, is it?

JB: That’s not degree two. Right. And you're calling me on a on a little technical point that I swept away in the beginning, which is that I said — when I said theorems carefully, I said that if our curve is non-hyperelliptic, right, and it happens, that elliptic curves will kind of not be in a case where the canonical embedding is, in fact, an embedding. Somehow you can talk about what that map might be, and for an elliptic curve, that map would take your elliptic curve and map it all to a single point.

KK: Yeah, that's a bummer.

JB: And that's a bummer. So sadly, elliptic curves, it doesn't quite work. So it is this interesting issue where if we're looking at abstract curves of small genus, so like elliptic curves are doughnuts of genus one, so there's one hole, or if we're looking at curves of genus two, so there's donuts with two holes, this theorem doesn't really apply, because those curves are extra special. And in fact, that's some of the beauty of things like elliptic curves. But once we're looking at more higher genuses, like genus three or four, and so on, you start to see very interesting things. So for example, if you take a genus three curve, and it satisfies this property being non-hyperelliptic, whatever that means, you can realize this curve in the projective plane, which is kind of a three-dimensional object cut out by some polynomials of degree four.

KK: So I mean, is this this love at first sight? Like, did you, you know, as a student read this in, is this in Hartshorne somewhere, or is it somewhere else and just fall in love?

JB: Yeah, so this is a great question. And it's actually as far as I'm aware of, not in Hartshorne, which is kind of a weird thing, because Hartshorne is notoriously quite comprehensive.

KK: Sure.

JB: But all the players are in Hartshorne. And in fact, the lead-up to kind of this theorem is in Hartshorne. And so the build up to this is to get to this theorem, you spend a lot of time in this fairly hard textbook, and you get to the end, and all of a sudden, they say, let's look at curves. And you think, wow, that's pretty simple. I've spent a year and a half, two years of my life learning all this complicated machinery and now you're going to tell me we're doing the simplest case. And you start looking at them and you see these beautiful things where all of a sudden, you built this machinery that lets you compute these equations in specific cases, so say small genus. And later on, you can read this amazing theorem of Petri and see that there's actually a full argument there, of how you can write down these polynomials, and it's kind of this beautiful synergy of all the things you learned coming together in one.

EL: I mean, I guess you've kind of answered this a little already, but what do you think draws you to this theorem so much that makes you love it?

JB: Yeah. So what draws me to this theorem, I think, is a number of different things. So (a) is this beautiful combination, or culmination, of learning so much, so many of these kind of complicated tools that don't seem closely related to the spirit of algebraic geometry, which is again, studying polynomial equations and their solution sets. But also, its has this amazingly surprising thing, which is that somehow if I take this abstract curve, and I take some realization of it in space and I ask for the equations that cut it out, those equations should really depend on how I put it into space. You know, if I put them into space a different way, I'll get different equations. And what this theorem is saying is that since these exceptions to this theorem don't depend on how I put it into space, those things only depend on the actual curve itself in its abstract form, that somehow, there's this beautiful thing that sometimes when you put things into space in the correct way, and you look at their defining equation, that's telling you something very, very special about not just that particular realization of your curve, but kind of the abstract, ethereal curve that lives kind of off in our imagination.

KK: So part two, we like our guests to pair their theorem with something. So what pairs well with Petri’s theorem?

JB: Yeah, so the thing that I thought of when I was thinking what pairs well with Petri’s theorem, is I was thinking about how, when I first started to see the, you know, glimpses of this theorem, it was Chapter Four of Hartshorne, so partway through this huge book that I had spent, you know, multiple years trying to get to that point. And all of a sudden, you get there and you see this beautiful vista of mathematics, these beautiful examples, that use all the tools you’ve got, and there was no easier path there. And it made me think of another one of my favorite hobbies, which is mountain climbing, or mountaineering. And how, you know, oftentimes you have to slog through these very tedious, long, hard, difficult and exhausting things, exhausting work, not always the most enjoyable — sometimes you're tired, sometimes your legs hurt, sometimes you're just kind of looking around and saying, wow, this is kind of a dusty desert, this isn't very pretty. But all of a sudden, if you put in that work, you get to this vista, and you see kind of the beauty of the world around you. And to me you get to this theorem, and you see the beauty of algebraic geometry, and kind of the essence of why you did what you did why you put in that work.

EL: Yeah, so what are some of your favorite vistas or mountains that you've gotten to climb?

JB: Yeah, that's a great question. So I have lived in California with my partner for a while and so a lot of the things I've liked to climb and do are kind of in the Sierra Nevada range or the Cascades, and not all of these ones I've actually fully summited, but I think things like looking off from Mount Shasta in Northern California has a beautiful view where you see the changing from how it's beautiful green forest around the mountain where it's snowy to this dry browner desert as you move off into kind of Northern California. There's some beautiful vistas out near Lake Tahoe and you kind of climb these peaks and get to the top and all of a sudden you can see this absolutely gorgeous lake spread out in this beautiful forest with peaks kind of circling it as a rim.

EL: That does sound amazing. As you can see, but our listeners can I've got my Zoom background is from a hike I did recently where — I must say this hike is basically beautiful the entire time. So there, there wasn't so much slogging, but you know, it was covered in aspens and the evergreens and things and just, you know, you do sometimes come around this curve, and suddenly you can see the Salt Lake valley, below where before it had just been trees, and there is something really special about that.

KK: Yeah, absolutely. I'm an East Coast kid. So Appalachians, which is still of course very beautiful, but different vibes. All right. So we'd like to give our guests a chance to plug anything they're doing where can we find you on on the intertubes?

JB: That’s a great question. I guess I would say I have a professional website. Google my name, you'll find it. Of course I’m also on Twitter for who knows how long.

EL: Limited time offer.

JB: A limited time offer, potentially, also under my name. And you know, otherwise I'm happy to respond to emails or things like that. And I'll just plug as a final thing, you can't reach me this way, but I am the president of Spectra, the association for LGBTQ+ mathematicians. So I'm also very heavily involved and happy to talk, and, you know, look at that sort of work. So if you're interested in LGBTQ+ mathematicians, I'd plug looking up Spectra and the work we've been doing there.

EL: Yeah. And did they just kind of recently, sort of — I got the feeling maybe it was a little more of an amorphous organization, and now it's sort of coalescing into something that has has a little more structure. I've tried to make an algebraic geometry analogy here and it’s just not working. But yeah, this is relatively recent, right? So are you the first president of Spectra?

JB: Yeah, so that's absolutely right. Spectra has kind of a long and amorphous history, coming from kind of a lot of grass roots activism in the ‘90s, through the 2000s. And it's existed in some form, at least with a website for, you know, a number of years. But in the last few years, we've really been trying to grow and formalize and expand our reach and the ability of support we're able to give LGBTQ+ mathematicians. Part of this includes kind of creating a formal board structure, and we did that over a number of years, going to effect this last year, and I was lucky enough to be chosen by the previous board members to be the inaugural president for this year. So I've been lucky to kind of take the reins and guide the organization through its first kind of formal year this year, although building upon all the amazing work a number of extremely dedicated and thoughtful people have done many years previously.

EL: Yeah, and I think it really has been maybe a lifeline, or really as a place that, you know — young LGBTQ mathematicians have maybe sometimes felt isolated where they are, and able to be like, is there anyone else like me? And, like, of course, there are a lot of people like you. And it's been a place that people can find, and I think that's really special.

JB: Yeah, that's exactly the goal. One of the goals we have is trying to make sure people see other visible LGBTQ mathematicians and see people they might be able to aspire to and reach out to or seek advice from or support from. So that's been one of our goals of formalizing and trying to increase our presence.

EL: Well, that's great. Yeah, check out Spectra. And yeah, send Juliette an email, you know, about anything related to that.

KK: Or algebraic geometry, or mountaineering.

JB: Or mountaineering. Yeah.

KK: All right. Well, this has been great fun. Thanks for joining us, Juliette.

JB: Yeah, thank you so much for having me. It's been a pleasure. I've really enjoyed listening to your podcast prior to this. So I really appreciate the opportunity to tell you some hopefully coherent words about my favorite theorem.

EL: Yes, thank you. I just love all the different perspectives we get by talking with so many different people here.

KK: Yeah, that's the best part. All right. Take care.

[outro]

On this episode, we were happy to talk with Juliette Bruce, a mathematician at Brown University, about Petri's theorem. Here are some links you might enjoy as you listen to the episode.
Her website and Twitter profile
The canonical bundle and Petri's theorem on Wikipedia
Robin Hartshorne's (in)famous Algebraic Geometry textbook
Spectra, the association for LGBTQ+ mathematicians