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Evelyn Lamb: Hello and welcome to my favorite theorem, the math podcast with no quiz at the end. I'm Evelyn Lamb, a freelance writer in Salt Lake City, Utah, where it is gorgeous spring weather, perfect weather to be sitting in my basement talking to people on Zoom. This is your other host.


Kevin Knudson: I’m Kevin Knudson, professor of mathematics at the University of Florida. I don't know, Evelyn, I saw the pictures on Instagram over the weekend and it looked cold in Utah. You wrote that you rode a century, right?



EL: Metric century.




KK: Okay. Metric.




EL: Just in case — you know, I don’t want people to think I'm quite that hardcore. Yeah, at least at this point in the season. Yeah, I hadn't managed to ride more than about 25 miles since last fall because weather, travel, just things conspiring against me. The week before I was like, I really need to get 30 or 40 miles in on Monday. And then it was, like, 20 mile an hour winds and sleet and I was like, well, I guess I'm just going into this cold, but it was fine. It was actually gorgeous weather. It was a little chilly at the start, but better than being too hot.




KK: Well, you know, the muscle memory takes over, right? So you can do — I mean, 62 miles isn't that much more than 25, really, once you have the legs, so congratulations.




EL: Yes, thank you. Well, we are delighted today to be joined by Kate Stange, who is in the Mountain Time Zone, something that I always feel thrilled about because I'm constantly converting time zones when I'm talking to people, and finally, someone I didn't have to do that for! So Kate, other than being in the Mountain Time Zone, what would you like to tell us about yourself?




Kate Stange: Oh, geez. Um, well, I'm also a cyclist. And so I'm jealous hearing about your rides.




EL: Wonderful!




KS: Here in Colorado we have we have this ride called the The Buff Classic. And so it has a 100 mile option where they close Boulder Canyon so that you can bike up the canyon without any cars.




EL: Oh, wow.




KS: Then you bike along the peak to peak highway. It's just wonderful.




EL: Yeah.




KK: Yeah. That sounds great.




EL: So you're at CU Boulder?




KS: Yes. And it's run from the campus. It starts right outside the math department.




EL: Oh, perfect. Yeah, just drop your stuff in your office and hop on and ride it?




KS: Yeah.




EL: Yeah, great. Well, we are thrilled to have you here today. And I guess we can just dive right in. What you're on what do you like to tell us about?




KS: My favorite theorem, at least for today, is the bijection between quadratic forms and ideal classes.




KK: That’s a lot of words.




EL: Yeah, and I'm so excited to hear about this, because I am honestly a little nervous about both quadratic forms and ideal classes, and a little embarrassed about being nervous about quadratic forms, not so much with ideal classes. So yeah, can you tell us a little bit about what that all means?




KS: Yeah, sure. So quadratic forms is probably what sort of comes first in the story, at least sort of the way that the mathematics tells it, and also probably the historical way. And so a quadratic form is just a polynomial with an x2, a y2 and an xy. So it's like 3x2 + 7xy − y2. So that's quadratic form. And, as number theorists, one of the things that we're most interested in studying is what are the integer solutions to polynomial equations? And so first you start with linear equations. And there's a wonderful story to do with Euclidean algorithm and stuff there. And then you move on to quadratic. And really, these are sort of some of the first equations that you would start studying next, I guess. And so they go back to the classical days of number theory, Gauss and Euler and everybody. And, yeah, so they come in, what happens is that they come together in families. So different quadratic forms, you can actually just do a change of variables. And it'll look different, but it won't really be too different, particularly if you're interested in what numbers it can represent when you put in integers. So say I take x2 + y2, which is the simplest one, if I put in various different integers to that I get various different integers out. And then if I do a change of variables on that, just a little change of variables — like maybe I change x to x + y, but I leave y alone — the formula will look different after I do that change of variables, but as I put in all integers and look at all the stuff I get out, those two sets, the in and out, they're going to look the same. And so we kind of want to mod out, we want to ignore that difference. So I'm really thinking of equivalence classes of quadratic forms. So that's the first object.




EL: And that change of variables is kind of the only equivalence class thing that happens with them?




KS: Yeah. Yeah. Because they could really behave differently between the different classes.




KK: And you only allow a linear change of variables, right?




KS: Yes, exactly. Yes. Thank you.




EL: Yeah. Okay. So now, ideal classes.




KS: Now ideal classes. So this is an interesting one, to describe where it comes from, I think. So there's sort of — if you think about the history of math, I would say there are sort of two versions, there are sort of two histories of math at the same time. There's one, which is sort of the human history, which is fascinating and human and quirky. And then there's sort of the way that the math would like to unfold to human understanding the way that as any human coming to it, they might discover the pieces of the mathematics. And I don't know too much about the details of the human history. But in terms of how you might discover this, if you're just looking at the integers, you are interested in how they behave, you discover things like prime numbers, you've got addition, you've got multiplication, you've got powers, you might ask how these things interact. And at some point, maybe when complex numbers are discovered, you think about whether there are possibly other collections, or other number systems, other collections of numbers in which you could do the same kind of thing. And so one of the first examples of this would be what's called the Gaussian integers, which is where you take complex numbers, I don't know whether I should dive into complex numbers, but you take complex numbers with integer coefficients.




EL: So that means things like 1 + 2i or something. So the i and the 1 both have integers in front of them.




KK: Right.




KS: Yeah, exactly. And so this is a collection of integers, kind of, right? And we ask things like, okay, are there prime numbers? And so it turns out that there are in that system, there are Gaussian prime, so, like, 1 + i is a prime number. And so you kind of start to develop this whole theory that you have for the integers. But what you find is that in some of these systems, you lose unique factorization. So we love unique factorization in the integers, right? Every integer, up to reordering the factors and maybe putting on a minus sign, you have always a unique factorization into prime numbers. And in the Gaussian integers, that's true. But in some of these other systems, you lose that. And so what people tried to do is to try to fix it. And it turns out, the way to fix it is to add in what were I think, originally called ideal numbers. They were thought of as numbers that should be in the system that weren't in the system. And what they actually were were collections of numbers. They were sets of numbers instead of individual numbers. And the idea here is that, say, you were to take — like in the integers, if you took the number two, you could replace that idea, that idea of two-ness with just the collection of even integers. And so that's an ideal now. Instead of a number, it’s an ideal, and it's really carrying the same information. But now it's a subset. And so by moving from individual elements of the ring, of the collection of numbers, you move to subsets of them. Now you have more things, and so now you can recover unique factorization in that world. So those are ideals.




EL: Yeah. And so the Gaussian integers do have unique factorization.




KS: They do. Yeah.




EL: So this — actually, I kind of forgot, but recently, this came up in something I was writing where I wanted the example to be the Gaussian integers so bad because it doesn't have any square roots in it. But then it didn't work because it isn't true for that. I was trying to show how unique factorization could fail, but I didn't want to have to use square roots. But as far as I know, you can't do that. So then I fixed it by putting a square root of negative five in there and hoping that people would be patient with me about it. But yeah.




KK: So that's the example of one where you don't get unique factorization, right? So you take the integers and you join the square root of minus five. That's one example.




KS: That’s one of them. Yeah.




EL: And then it's like two and three are no longer primes.




KS: So if you multiply (1+ √ −5) × (1− √ −5)




KK: You get six. Yeah.




KS: You get six, which is also two times three. And those are two different prime factorizations of six.




KK: Right.




EL: Yeah, but it's so fun that you can do that, and I like your way of putting it where regardless of how these ideas actually formed, you can as a human, looking at some of these basic pieces like p