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Kevin Knudson: Welcome to My Favorite Theorem, a math podcast with no quiz at the end. I'm Kevin Knudson, professor of mathematics at the University of Florida, and I'm joined today by my fabulous co-host.



Evelyn Lamb: Hi, I'm Evelyn Lamb, a freelance math and science writer in Salt Lake City where we are preparing for another snowstorm this week after we had one last week, which is great because we are so low on water right now and we need every bit of precipitation. So even though I'm from Texas, and I don't naturally love shoveling snow or being below 50 degrees, I am thrilled that we're supposed to get snow tonight.




KK: So when I lived in Michigan — you know, I grew up in North Carolina, so snow was a thing, but we didn't shovel it. We just sort of lived with it — and I had a neighbor across the street who was in his 70s. And he had a snowblower and he let me use it and I thought this is amazing. So if you and John haven't invested in a snowblower yet, you know, maybe it's time.




EL: But we'll see. Climate change means that we might have to do less and less snow shoveling.




KK: Well, it's true. Actually, I remember growing up, you know skiing was a thing in North Carolina and I think you might still be able to but like the natural snow ski resorts, kind of they have to manufacture all their snow now. It's, it's things have changed even in my lifetime, but it's not real. As we're told, it's not real. Sorry to editorialize. Anyway, let's talk math. Today, we are pleased to welcome Courtney Gibbons. Why don't you introduce yourself?




Courtney Gibbons: Hi, I'm Courtney Gibbons. I will see far more snow up here in Clinton, New York, then either of you, I think.




EL: Definitely.




CG: I just sent in my plowing contract for the year. So that's awesome. Make sure that I don't have to shovel or snowblow my own driveway, which is not long. I'm a professor. I'm an associate professor of mathematics at Hamilton College up here in beautiful Clinton, New York. There is a Hamilton, New York, but that's where Colgate is. So don't get them confused.




KK: Oh, right.




CG: Yeah. It’s weird. It's a strange thing.




EL: Yeah. At least it’s not the whole Indiana University of Pennsylvania thing because that is not okay.




CG: Yeah, no, no. I think there was an incident on one of our campuses where, like, Albany sent some kind of emergency response squad to the wrong Hamilton. But they're only 20 minutes away, so it was a quick thing. Yeah.




KK: All right. Well, welcome. So, I mean, well, maybe we just get into it.




EL: Well, I will say that we have talked to Courtney before on the podcast, although extremely briefly, when we, I guess this must have been the joint meetings that 2019 a decade ago. (Hahaha.)




KK: Yeah.




EL: We had people give us, you know, little, like, minute or two sound bites of their favorite theorems. And she did talk about a theorem, although I understand it's not the theorem that she's going to talk about today, which, I mean, you don't have one and only theorem in your life? Come on, Courtney.




CG: I am a lover of many theorems. I think back then I mentioned Hilbert’s Nullstellensatz, which is the beautiful zero point theorem that links roots of polynomials to factors of polynomials over algebraically closed fields. It's beautiful. It's a really nice theorem. I initially thought I was going to talk about Hilbert’s syzygy theorem today, which by the way, syzygy, excellent hangman word.




EL: Yeah.




CG: Unless you're playing with people. You've used that word on before, in which case, their first guess will be Y. But I decided today I wanted to talk about Emmy Noether’s isomorphism theorems, in part because they're usually just called the isomorphism theorems. And Emmy Noether’s attribution gets lost somehow. So I wanted to talk about those today because I'm a huge Noether fan. I mean, I'm also a huge Hilbert fan. You kind of have to be a big fan of both. But these theorems are super cool. They're theorems you could see in your first course in abstract algebra, and that's actually where I first saw them. I'm a commutative algebraist and I do a lot of homological algebra. So I love arrows. I love kernels. I love cokernels. I love images. I love anything you can set up in an exact sequence, and I think this was my first exposure to a theorem that was best explained with a diagram. And I remember at that moment being like, “This is what I want to do! I want to draw these arrows.” And I'm lucky because I got to grow up to do what I want to do.




EL: That’s kind of funny because I loved abstract algebra when I took it in undergrad, and I think as it got more to, like, you know, having all these kernels and cokernels and arrows, that was when I was like, “I just can't do this,” and ended up more in geometry and topology. So, you know, different, different things for different people. That's fine. So yeah, let's get into it. So what are these theorems?




CG: Excellent, well, they are often numbered. I grabbed a couple books off my shelf, and it wasn't consistent, but Rotman and Dummit and Foote, kind of numbered them the same way. So the first one, which is usually the first one that you see, it's true for rings and groups and modules. I most often use it for modules, but I'll state it for groups. And it says that if you've got a homomorphism F from a group G to a group H, then the kernel of that homomorphism is a normal subgroup of your group G. Or if you're working with rings, it's an ideal of your ring, or you know, a submodule of your module. And you can mod out by it. So you take G mod the kernel, and it's going to be isomorphic to the image of your homomorphism. And so if you've got a surjection from G to H, and you are like, “I kind of want to build something isomorphic to H, but built out of the parts of G,” you're like, “Cool, I can just take the kernel and mod out by that and look at the group of cosets of of that normal subgroup.” And you've got this — you're done! You've built this cool isomorphism. And I advertise it to my students as, like, a work-saving thing. Because usually to build an isomorphism, you've got to show one-to-one/injective and onto/surjective. And this is like, well, take the thing you want to be isomorphic to, try to imagine it as a better group, a nicer group, mod the kernel of something, and build that homomorphism. Make it surjective and then you get one-to-one for free from this theorem. So I love this theorem. It's a really, it's a nice theorem. I actually use it. I don't reference it, but I think when you do the first step of finding a free resolution, which is which is what I do, it’s like my bread and butter, I love doing this. If you're calculating it by hand, you take a module, and you surject onto it with a free module. You look at the kernel of that thing, and then you build a map whose image is that kernel. And the big deal here is that your module M is isomorphic to the cokernel of that image map, which is the same thing as what you get from the isomorphism theorem. It's that first free module mod the kernel. So this gives you a nice presentation for a module. You can do this in certain nice cases. And I always sort of give a little thanks to Emmy when I start building free resolutions. I'm like, “I know that this is your theorem in disguise.”




EL: Yeah.




KK: I don’t think I actually knew that attribution, that Emmy Noether was the first person to explicitly notice these. But you know, she's the one who figured out that, you know, homology is a group, right?




CG: Yeah, exactly. So she was thinking about rings and groups. And, you know, a lot of the terminology is thanks to her and Hilbert. When you think about integral domains, I think that was what Hilbert initially called rings. I dug this up at one point for my students, they're like, where don't come from?




EL: That makes sense, right? Because I think of a ring as something that's like the integers.




CG: Yeah, it is really. Yeah. And like when people started generalizing to super bonkers weird examples, like, you know, the ring of quaternions and stuff, you're like, okay, so everything isn't quite like the integers. But we've got these integral domains, integral being the “like the integers” adjective and domain, I think of like, where stuff lives. The stuff that's like the integers lives here.




EL: Yes.




CG: Which is nice. But yeah, these these are, these are attributed to me. And it really bugs me to see them without her name attached since we have Hilbert’s syzygy theorem and Hilbert’s Nullstellensatz and it's like, but what about Emmy?




KK: Yeah, she she, she doesn't get the recognition she deserves. I mean, everybody knows like, she's like a mathematics mathematician. But yeah, she she definitely doesn't get credit a lot of the times.




EL: Yeah, I will say I am so, so tired of the headline, “The most important mathematician you've never heard of,” and I read this, and I know that I know more mathematicians than your average person. But like, if it's Emmy Noether, I’m just like, come on. You can't say you’ve never heard of her.




KK: Even the physicists.




EL: A lot of people haven’t heard of her. But yeah.




CG: But it's also interesting the way she's talked about. Because at the time, of course, it was difficult to be a woman in math even though she somehow was able to be a professor, although unpaid. But you know, you look at the way people described her and