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## Episode 73 - Courtney Gibbons

## Manage episode 317465358 series 1516226

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 there's that one guy, I forget, who was like, “I can testify that she was a mathematician, but I can't testify to the fact that she was a woman.” You know. And it's like she was accepted because she was so unfeminine. And it was not threatening to the status quo of men doing math to have this, in their words, coarse, rough, simple soul wearing men’s shoes, blah, blah, blah, great heft, among them, because it's like, well, she's basically a man. So that always bothered me, too. I'm not a particularly feminine person, but occasionally, I do feel like I get a little bit of a brush off because they're like, “Oh, you're so cute.” I'm not cute! I'm a big strong mathematician!

EL: Yeah.

CG: That’s okay. I'm aging and rapidly getting less cute.

EL: You’ll become invisible soon.

CG: Yeah, I know. I'm really I'm excited for that. Yeah, I think I'll enjoy that better than the like, “Oh, aren't you a cute little mathematician?”

EL: Yeah. All right. So…

KK: So there's a first isomorphism theorem. So there must be a second.

CG: There’s a second, there's a third, there's a fourth, there's probably a fifth, then at some point, you just start calling them nth. And I mean, I think I've only seen four. I'm only going to talk about the first three, because I actually don't really know what the fourth one says. There's no quiz at the end. But there's homework. The fourth, look up the fourth. Okay, so the second isomorphism theorem is kind of a special case of the first one. You can prove it using the first one. It says that if you've got a group G, and you've got a normal subgroup, and you've got — we’ll call it N for normal — and you have another subgroup doesn't have to be normal, so we'll just call it S for subgroup, or Su or whatever. This says a couple of things. It says that the product of S and N is a group (and you form the product the way you would expect; you just multiply pairwise elements together, things from S with things from N). The intersection S intersect N is a normal subgroup of S, which is a really nice exercise. And you've got an isomorphism between the product SN mod N, and the quotient group S mod S intersect N. So this is a way of sort of saying like you've got this one group, maybe you don't understand SN really well, and you're working with SN mod N. Instead, replace it with a group you might understand better, which was your original group S mod S intersect N, which is a nice, normal subgroup. So it gives you some opportunity to work with a nicer group to understand the one you don't know. And the proof of this is you build a surjection from S to SN mod N: you’re going to just sort of send S to its coset representative over there and argue that it's onto. And then you calculate that the kernel is S intersect N and then you're like, boom, first isomorphism theorem, take it away. Which is, which is really cool.

KK: It’s kind of a corollary.

CG: Yeah, it's kind of a corollary, but the diagram is super cool. It looks like a diamond because you've got S and S intersect N and then SN and and then you've got all these things all over, the arrows. I love the arrows! And you can put the group G at the top and you put the identity at the bottom, and then you've got like a nice lattice. So I think it even is sometimes maybe called the lattice theorem. But that may. Yeah, I think so. But I, I wouldn't bet your final exam on that if you need to say something.

EL: Yeah. Well, and lattices are a different thing also. So I would be confused if I called it the lattice theorem. I’d think it was about like lattices in the plane or something.

CG: Right. Yeah. I think the idea is it looks like like a lattice. And most algebraists are like, “Oh, cool. We can call that a lattice.”

KK: It is a lattice though, right? The set of subgroups is a lattice. I mean, it’s a partially ordered set with meets and joins and blah, blah, blah.

CG: It is. Yeah, and what's funny is that the person I learned abstract algebra from in undergrad studied lattice-ordered stuff. I forget exactly. It was algebraic stuff. Now I'm of course blanking on it. That was Marlow Anderson, and I am the retirement replacement for Bob Redfield at Hamilton, who studied lattice-ordered subgroups. So I inhabit this space where lattices surround me. It’s kind of exciting.

KK: Yeah. But you're right. This one's a little more obscure. I can't remember ever using this in life except for, like doing homework?

CG: Yeah, I don't recognize a place in my life where I've used this one. If I have I was ignorant of the fact that that was what was making things work, which is not unusual. This is why I need to have collaborators. So they can be like, “Ah, Gibbons.” Hope only gets you so far.

KK: Right? All right. All right, number three.

CG: So number three, I think this one is low-key my favorite even though again, this is not one — actually no, I did use this. We used this in my modern algebra class this semester. This says let's say you've got a group G, and you've got a normal subgroup K in G. And you've got a normal subgroup N in K, so you're normal all the way up. This says that if you take G mod N mod K mod N, which kind of looks like — if you think of it as fractions, like if you're taking 1/2 then dividing by 3/2, this basically says you can “cancel” the denominators, and you're going to get G mod K. Which is really nice, because if you think about what G mod N mod K mod N is, G mod N is a group of cosets and K mod N is a subgroup of cosets. And now you're making cosets out of cosets. And that's a level of abstraction that you don't really want to work with directly, and so having this isomorphism that's like, that’s the same thing as G mod K, just work over there, that’s it's a really powerful thing. And in my class, we're doing a bunch of Galois stuff. Like, our first course in algebra follows that the story of why the quintic isn't solvable by radicals, basically, why there's no nice quintic formula that looks like the quadratic formula. And when we get into the solvable groups thing, and you're building these chains of subgroups, and you want to do stuff with them, we got on this sort of sidetrack of like, well, what if you wanted to mod out by one of the modded out things? Yeah, there’s an isomorphism theorem. This is so cool.

EL: So yeah, I remember seeing this and possibly having it like a homework question or something in abstract algebra, probably in grad school. And I remember feeling like I was getting away with something. When you could just like, like, cancel the denominators, which is not what this actually is.

CG: It’s one of those things where like, I'd be like, sure, that should work. And that's the hope I'm talking about, like, you’ve got to check the details. But again, this one is one you prove with the first isomorphism theorem. You set up your surjection from G mod N to G mod K, and you show all the stuff that's going to go to zero there was the stuff in K mod N. So this is another nice one, but it just looks so cool. You're like, “I can cancel the denominators. Yes, life is good.” You know, some people worry that mathematics is fundamentally flawed, and we don't know where the error is. And I'm like, it can't be. Look, this worked out. We can cancel denominators. That’s got to be on solid ground! It worked out exactly how we think it should.

EL: I love that. That's a great attitude to go through life with.

CG: W could worry about the fundamental flaws, or we could enjoy the wins where denominators cancel. I'd say we've done pretty well for ourselves. We've built this in a sensible way.

KK: I sort of feel like even if we do find the place where it's bad, it doesn't mean that bridges are going to fall down. Right? I mean, most of what we’ve built it seems okay. I mean, yeah, yeah.

EL: It will probably only break something like what Kameryn talked about with us last month.

KK: Some weird set theory.

EL: Yeah, who cares about that?

CG: Let it burn.

EL: Just kidding, Kameryn.

CG: No, I mean, isn’t there, I don't follow this very much. But I thought there was a modern push to sort of replace the foundations of math, the set theory stuff with category theory instead.

EL: Homotopy?

CG: Yeah, homotopy type theory, which in my mind is just like, you know, an evil cousin of category theory. But I, I don't know. I like homotopy. But I don't know what the homotopy type theory is.

EL: I know. I’m afraid to say it.

CG: It is a little scary to say.

EL: Yeah, I have not been able to figure out the correspondence between what I think of as a homotopy, which is like, “Look, you could drag the little loop around on the doughnut,” and homotopy type theory. Don't tell anyone.

CG: Well, and as someone who studies homology, I'm like, why would we do homotopies? They're kind of yucky. Like homology makes it nicer, right? Homology is the abelian version.

KK: It’s abelian, right.

CG: I mean, I guess, you know, if you actually want to describe stuff in the real world, he can't just live in commutative land, but who wants to live in the real world? That’s what the physicists do.

EL: Yeah, that's why we do math, to not have to do that.

CG: Exactly. Exactly. The real world is full of snow. And climate change deniers.

KK: Sure. Yeah. So it seems like you've had a long love for these theorems, right? This goes way back, right?

CG: This goes way back. For me, I was not actually intending to be a math major. I failed seventh grade math. My elementary school report cards were like, “She's very creative. Not so great at math.” “Asks too many questions,” I think was one of them. My first existential crisis was like, what happens if we keep adding forever, and we run out of names for the numbers, and I was like, I can't go to school. They asked me to add as high as I can, and I got to 100. And I was like, I'm going to stop here because at some point, I know what to do, but I don't know what to say. And that was pretty stressful. For me, I hadn't worked out like the time calculation, like how long it would take me to get to the point where I didn't know the names of the numbers anymore. But that was at a time where like, most of my classmates thought 31 was the biggest number because we only ever counted on the calendar.

So I was really surprised in college. I mean, I was also a college dropout who went back. I was surprised in college to get caught up in math and actually find that I could ask these questions. And my professors weren't like, “Oh, my god, shut up.” They were like, “What a cool question!” And they would actually talk to me, and I think it was the relationships more than the math that drew me in at first. By the time I transferred I was starting in multivariable calculus, having finished my first year elsewhere. And I was like, it's cool. We're gonna parameterize the way a leaf falls from a tree, mathematical poetry, all that, but like, it didn't set my soul on fire. And then I take differential equations, which really didn't set my soul on fire, although it was fun. I had fun time in it. But when I got to algebra, I felt like the language I was trying to speak my whole life was finally available to me. It was all about relationships and these like, very picky, not like picky in the sense of real analysis, but you could talk about really subtle differences in things. Like the difference between an ideal and a subring to me, was this really compelling thing, because they're so similar, but they're just a little bit different. And that little difference makes such a huge difference when you start talking about quotient objects and cosets and things like that. I really loved it. And then when I saw the arrows come out, I was like, I'm hooked. I want to build the isomorphism theorem of friendship. This is the way I want to describe everything in my world. And I just got worse from there. I had a brief flirtation with topology in grad school, and then I realized that you could be a commutative homological algebra person and steal all the cool diagrams, like the Meyer-Vietoris sequence, and all of that, and you could do with polynomials! And, and I was like, all right, that's, that's for me. That's what I'm gonna do.

KK: Very cool. I mean, I'm a pretty algebraic topologist myself. I mean, not not so much anymore. But earlier in my career, I was doing topology, but I was doing, like, homology of groups. So there's some geometry there, but not really, right. You’re really just thinking about algebra all day long.

CG: Yeah, I love that. I love that the tools of homology. I was tempted a little bit by geometric group theory, because it's so pretty and so fun. And I think some of the best, most fun conversations I had in grad school before starting my research work were with my friends who were taking that class too. We could just draw something on the board and argue about it for hours. And that’s one of the things I love most. I think I'm a mathematician — some people are mathematicians because they want to uncover these deep, beautiful, abstract truths. And I'm a mathematician because I like talking to people about math. So all my research work, my job is really like, if I didn't do research, I wouldn't have math to talk to people about, so I've got to keep doing the research and ideally doing with my friends, so we can talk about math. I love talking to my students about math. I love talking to you guys about math. I love talking my family about math, and they're like, “Oh my god. Please don't explain again what an algebraic geophysicist does.” Which is at one point what my mom decided. I said, “I do commutative algebra, which is close to algebraic geometry.” And in her mind, it became algebraic geophysicist.

KK: That sounds cool though.

CG: Which, that sounds like a cool gig. Yeah, like I would do it if I knew what it was. Yeah, like homology of rocks. Yeah, yeah.

EL: Figure out when the next earthquake is gonna be using commutative diagrams.

CG: Yeah. Maybe that's the the new topological data analysis stuff where you're doing persistent homology.

KK: That’s right. Well, you can use that to try to understand the structure of various things like you're looking for cavities inside of these. Yeah. So I mean, there are people who've done these analyses. It's real.

CG: All right. Maybe that's my mid-career pivot. Yeah. Algebraic geophysicist. Okay, joint appointment in the geology department. I get to go on their cool trips where they go hiking. I’m in.

KK: Right. Okay, so part two of this podcast. So well, we don't have a part three. Maybe there is a part three? Anyway, we’ve got three things here. We ask our guests to pair their theorem with something. So what have you chosen to pair the isomorphism theorems with?

CG: Well, I decided the best way to do this would be a clickbait-y Buzzfeed listicle kind of thing. So I'm going to start with theorem number three. Theorem number three was the theorem where you are cancelling denominators. I think it's a really fun and satisfying theorem, and so what I recommend you pair this with is a small batch craft beverage, alcoholic or non, of your choosing because when you use this, you should indulge all your senses and just be overwhelmed with joy that this works out and you could just cross out the denominators. For me, that would be a nice local, unfiltered wheat beer, just sit back, cross those little denominators out and just be like, “Yes, life is good. We have built a good thing.”

EL: That sounds great.

KK: Actually, one of the things I might be looking forward to the most about the joint meetings being in Seattle. [Transcriber’s note: Whoops! The 2021 Joint Mathematics Meetings were canceled/postponed after all.] And Evelyn knows this. My favorite distillery happens to be in Bainbridge Island. And so I think I'm going to make a little side trip to procure their excellent whiskey that I cannot get here. I can only get it there. So this is good. This is a good pairing.

CG: Oh, excellent. Are you taking orders?

KK: Well, I don't know. For you? I guess I could ship you something.

CG: Yeah. I’m not going to make it in person this year. [Narrator: She wasn’t the only one.] But I do love a good whiskey.

KK: Their whiskey is spectacular.

CG: We’re going to have to have an offline conversation about this and compare tasting notes. All right. Theorem two. It's about multiplication and intersections. And what goes better with multiplication than bunnies?

EL: Okay.

CG: And did you know there's a huge intersection of math people with bunnies, especially on Twitter?

KK: Yes.

CG: I think you've got to pair theorem number two with bunnies, you've got to it. It's not a theorem you use super often, but if you do use it, you have to immediately go out and acquire a pet bunny and join the math bunny Twitter crowd. It's just, I don't think there's any way around that. I think that's one of the laws of the universe.

KK: Okay.

CG: I should apologize to the math bunny Twitter folks, because they can infer that this is not the best thing, right? Number one is always the best thing you can you can have. But I hope they'll forgive me when they hear that my pairing for theorem number one, the OG, the original isomorphism theorem, is friends. When you use this theorem, you should think about your friends, you should write a note to your friends, you should talk to your friends. I've got friends on the brain this week because I was just thinking about a couple of friends of mine who passed away pretty young one was in his late 30s and one was in his mid-40s. And you know, I thought about Emmy's life and how she had a pretty tumultuous life, right, trying to make it onto the faculty at a university in the first place and then being shipped out of the faculty to Bryn Mawr’s gain. She ended up at Bryn Mawr.

EL: Basically dodging Nazis.

CG: Yeah, dodging Nazis. We talked about “math is apolitical,” but is it? Bryn Mawr lucked out, I guess? Because they got — I mean, America in general lucked out because we got all these amazing mathematicians who were fleeing Nazi Germany. But that sort of seems to me, and I'm not a math historian, but it seems to me that that must sort of mark the point where mathematics in Germany really took a backseat and why English has become the predominant language of mathematics.

EL: Yeah, I mean, the American mathematicians, the great, the big names from the early 20th century studied in Germany. And now it does very much go the other way. There's even a quote about, like, the center of gravity of research crossed the Atlantic because of the Nazis.

CG: Yeah, so I mean, that must have been such an awful thing to experience, being expelled and the uprooting of all of that. And then she died fairly young. She died at 53. I think it was complications from surgery, a cancer surgery. And, you know, it just it makes me wonder what did she have planned? What were her mathematical plans? How was she about to revolutionize physics again? Her most famous theorem probably is the theorem that talks about conservation of stuff under different types of symmetries. Like when you do something, something is conserved, which is, I guess, super important to physicists. I've never used that one in my own work. Right. But it seems like 53, she must have had so much more stuff planned for herself mathematically, physics-y stuff, personal stuff.

EL: Well, and how would Bryn Mawr have been different? And how would American — you know, would she have stayed at Bryn Mawr? Would she have ended up at a different university? Because I think she was only in the US for a couple of years before that happened. Yeah, she barely even got to teach anyone. I mean, I guess it should really be reversed. People in the US barely got to learn from her. And yeah, it's really tragic when you read about it. I think it was maybe an ovarian cancer surgery or something.

CG: I think that's right.

EL: I'm trying to remember if she's one of the people who seemed to be recovering fine, and then just kind of dropped dead a couple days later, or something like that. I know that I’ve read about it. I don't know for sure.

CG: I read up this morning, actually, to make sure that I had her age right. And it seemed it seems like the internet consensus at least is it was a viral infection. So the surgery went fine. But she had incurred some infection after the surgery and just dropped dead. You think of Maryam Mirzakhani. Also someone who was taken really young with so much math on the horizon. But it's sort of hopeful, right? Like, you know, Emmy really didn't get the recognition she deserved. But Maryam Mirzakhani, of course, got at least some of the recognition she deserved, which is, I guess we're making some progress.

This weekend, my friend and collaborator Nick Baeth passed away from a really rapid pancreatic cancer. I think he got the diagnosis maybe six weeks before he died.

EL: Gosh.

KK: That’s awful.

CG: And I wish I had finished writing the note to him that I was saying to him that it was just such a — he and I are math siblings. He was my math “older brother,” although we were in grad school at different times. But we met through our advisor, Roger Wiegand, who — his students become basically family. I know all my “siblings,” pretty much now and we hang out and they're great. But that's how I started collaborating with Nick. And we started working on some semigroup stuff, which is a little outside of my wheelhouse, but related to some questions that I want to answer in my research life. And it was just so much fun to work with him. We'd started up a new collaboration, and we paused it, obviously, six weeks ago. But I thought I had enough time to tell him just what a wonderful surprise it was to find such a like-minded person to collaborate with, somebody who was super generous with ideas, super generous with coauthorship. It really felt like if you'd had a good conversation with Nick about something, he'd be like, “Do you want to be a coauthor? Can you prove this thing? Let’s actually keep doing it.” And I know that's not always the norm in math, and people are very protective of their ideas, not without reason. But it was just such a joy to work with somebody who actually felt like it was more fun to do math with people, and that was why you did math, than you have to go chasing down these big, outstanding results. Not that what he worked on wasn't important. I mean, he had a Fulbright in Austria. It was pretty cool. And I've just been thinking about that. And you know, this other, my graduate office-mate who died, like, a year and a half ago, who was also really young, and just thinking about for someone like me, who does math, to talk math with friends, I need the reminder that I should tell my friends that I enjoy talking math with them, and I enjoy being friends more often. And so for me every time I sit down and calculate a free resolution, and I do my little “Thanks, Emmy,” I'm going to also jot a note to somebody, just a short note like, “Hey, I was using Emmy Noether’s first isomorphism theorem today, and I thought I'd say hi.”

EL: That’s great. I love that.

KK: So you know, maybe there was a third part of this podcast. We always like to give our guests a chance to plug anything, if they like. Where can we find you on the internets? Or do you have something you really want to promote?

CG: Oh boy. You can find me on Twitter, I am addicted to Twitter. I am @virtualcourtney. And that is my online presence as opposed to my corporeal presence, which is at corporeal Courtney, although I don't know if that's on Twitter? Anyway, I tweet a lot there. My pinned tweet, there is a 10-minute pep talk that I recorded for my students during pandemic times. This is a pep talk I give pretty frequently to students who are maybe in their first math course where it's not just calculus 1-20. We're starting to develop proof techniques, or we're dealing with things that are more abstract than things they've thought about before. Or we're encountering the isomorphism theorems and they're like, “Oh my God, what do these arrows mean?” And they're having this crisis of confidence, like “am I cut out to be a mathematician?” And my little 10-minute pep talk goes into my own bumpy route to becoming a mathematician. I'm plugging it a little bit self-consciously. I put it as an email attachment to my students. And I had put it on YouTube and meant to make it unpublished but clicked the wrong button. And then suddenly it had been shared, and I was getting emails about it and thousands of views. And I was like, I guess I'll leave it up. So I'll plug that for anybody who's feeling a little bit, I don't know, shaky in their mathematician identity, or, you know, not sure that they're allowed to have a mathematician identity. I certainly feel like the last person who was “supposed” to become a mathematician. My initial plan was to be a French major. And I did take my graduate language exams in French. So it's good that I studied enough French to know the subjunctive. But yeah, so I'll plug that and then I'll say one more thing. I am working on writing an open-source, remixable, free WebWork-enabled algebra textbook that takes my predecessor Bob's idea of using Galois theory for the first course in modern algebra, and makes it into an active pedagogy kind of thing, sort of in the model of Active Calculus.

KK: Oh, cool.

EL: Oh, cool.

CG: So I am on sabbatical next academic year, and that's my main project. And if anybody wants to, test WebWork problems, or help out when I'm like, “Okay, why did I think I could write a textbook?” I'm happy to talk math with old friends, new friends, math book friends, WebWork friends, anybody who wants to help make that thing actually happen.

EL: Yeah, and selfishly keep me in the loop about it, I'd love to know about it when it comes out here. You're actually convincing me, and several of our guests are always like, working on me. Like, I should really learn algebra a little better and feel a little more comfortable with some of these ideas. CG: Sure. I’ll keep you in mind as a beta tester. Over the years, I've developed all these active learning worksheets that are the basis for the book. They look fun. My students are — well, they used to smile. Now they’re masked, so I just assumed they're smiling all the time. But they would smile and argue when they were working on them, which is what I had hoped would happen.

EL: Oh, that’s great. Yeah.

CG: You should rope some friends in to learn algebra with.

EL: Yeah!

CG: Because I think that's the most important thing for sticking with it is having pals.

KK: Well, good luck. You know, as somebody who, I’ve written a couple of books, you always reach this point where it's like, “Why am I doing this? What did I get myself into?” But once you push through that, it's fine.

CG: I’m in the planning stage where I have a big wiki essentially, like a big table with, like, here’s chapter one’s outline, here’s all the resources for everything in here and now, it's a matter of actually connecting the things. I mean, PreTeXt is its own special thing, learning learning PreTeXt. But yeah, I think it will be a good experience. I feel a little bad because you know, both my algebra Professor Marlow Anderson and my predecessor Bob Redfield have these great books. And I'm like, “No, I'm not going to use those. I’m going to write my own. Stick it to you guys.” But I think what I haven't seen is a mixture of the active pedagogy with the Galois theory as the thing that links rings, groups, and fields. So that's what I want to create and put out there.

KK: Cool. All right. Well, this has been a lot of fun, Courtney. Thanks for joining us today.

CG: Yeah, absolutely. I have sort of been like, “When are they going to ask me?” for a little while, not thinking like, maybe I could just reach out and be like, “Hey, ask me.” So I was really excited when you did. And I've loved listening to the podcast. And I appreciate you both for for making it happen. It’s awesome.

EL: Well thank you.

KK: Yeah. All right. Well, take care.

CG: Bye.

[outro]

On this episode of My Favorite Theorem, we were delighted to talk with Courtney Gibbons, a mathematician at Hamilton College, about Emmy Noether's isomorphism theorems. Below are some related links you might find useful.

Courtney Gibbons's website and Twitter account

The Wikipedia article on Noether's isomorphism theorems, which includes a helpful chart describing differences in labeling the theorems

An article about Emmy Noether by astrophysicist Katie Mack and her biography on the MacTutor History of Mathematics Archive

Evelyn's 2017 article in Undark about the effect of Nazism on German mathematics in the 1930s

Our episode with Kameryn Williams

Active Calculus, a free, open-source resource for teaching calculus

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## Manage episode 317465358 series 1516226

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 there's that one guy, I forget, who was like, “I can testify that she was a mathematician, but I can't testify to the fact that she was a woman.” You know. And it's like she was accepted because she was so unfeminine. And it was not threatening to the status quo of men doing math to have this, in their words, coarse, rough, simple soul wearing men’s shoes, blah, blah, blah, great heft, among them, because it's like, well, she's basically a man. So that always bothered me, too. I'm not a particularly feminine person, but occasionally, I do feel like I get a little bit of a brush off because they're like, “Oh, you're so cute.” I'm not cute! I'm a big strong mathematician!

EL: Yeah.

CG: That’s okay. I'm aging and rapidly getting less cute.

EL: You’ll become invisible soon.

CG: Yeah, I know. I'm really I'm excited for that. Yeah, I think I'll enjoy that better than the like, “Oh, aren't you a cute little mathematician?”

EL: Yeah. All right. So…

KK: So there's a first isomorphism theorem. So there must be a second.

CG: There’s a second, there's a third, there's a fourth, there's probably a fifth, then at some point, you just start calling them nth. And I mean, I think I've only seen four. I'm only going to talk about the first three, because I actually don't really know what the fourth one says. There's no quiz at the end. But there's homework. The fourth, look up the fourth. Okay, so the second isomorphism theorem is kind of a special case of the first one. You can prove it using the first one. It says that if you've got a group G, and you've got a normal subgroup, and you've got — we’ll call it N for normal — and you have another subgroup doesn't have to be normal, so we'll just call it S for subgroup, or Su or whatever. This says a couple of things. It says that the product of S and N is a group (and you form the product the way you would expect; you just multiply pairwise elements together, things from S with things from N). The intersection S intersect N is a normal subgroup of S, which is a really nice exercise. And you've got an isomorphism between the product SN mod N, and the quotient group S mod S intersect N. So this is a way of sort of saying like you've got this one group, maybe you don't understand SN really well, and you're working with SN mod N. Instead, replace it with a group you might understand better, which was your original group S mod S intersect N, which is a nice, normal subgroup. So it gives you some opportunity to work with a nicer group to understand the one you don't know. And the proof of this is you build a surjection from S to SN mod N: you’re going to just sort of send S to its coset representative over there and argue that it's onto. And then you calculate that the kernel is S intersect N and then you're like, boom, first isomorphism theorem, take it away. Which is, which is really cool.

KK: It’s kind of a corollary.

CG: Yeah, it's kind of a corollary, but the diagram is super cool. It looks like a diamond because you've got S and S intersect N and then SN and and then you've got all these things all over, the arrows. I love the arrows! And you can put the group G at the top and you put the identity at the bottom, and then you've got like a nice lattice. So I think it even is sometimes maybe called the lattice theorem. But that may. Yeah, I think so. But I, I wouldn't bet your final exam on that if you need to say something.

EL: Yeah. Well, and lattices are a different thing also. So I would be confused if I called it the lattice theorem. I’d think it was about like lattices in the plane or something.

CG: Right. Yeah. I think the idea is it looks like like a lattice. And most algebraists are like, “Oh, cool. We can call that a lattice.”

KK: It is a lattice though, right? The set of subgroups is a lattice. I mean, it’s a partially ordered set with meets and joins and blah, blah, blah.

CG: It is. Yeah, and what's funny is that the person I learned abstract algebra from in undergrad studied lattice-ordered stuff. I forget exactly. It was algebraic stuff. Now I'm of course blanking on it. That was Marlow Anderson, and I am the retirement replacement for Bob Redfield at Hamilton, who studied lattice-ordered subgroups. So I inhabit this space where lattices surround me. It’s kind of exciting.

KK: Yeah. But you're right. This one's a little more obscure. I can't remember ever using this in life except for, like doing homework?

CG: Yeah, I don't recognize a place in my life where I've used this one. If I have I was ignorant of the fact that that was what was making things work, which is not unusual. This is why I need to have collaborators. So they can be like, “Ah, Gibbons.” Hope only gets you so far.

KK: Right? All right. All right, number three.

CG: So number three, I think this one is low-key my favorite even though again, this is not one — actually no, I did use this. We used this in my modern algebra class this semester. This says let's say you've got a group G, and you've got a normal subgroup K in G. And you've got a normal subgroup N in K, so you're normal all the way up. This says that if you take G mod N mod K mod N, which kind of looks like — if you think of it as fractions, like if you're taking 1/2 then dividing by 3/2, this basically says you can “cancel” the denominators, and you're going to get G mod K. Which is really nice, because if you think about what G mod N mod K mod N is, G mod N is a group of cosets and K mod N is a subgroup of cosets. And now you're making cosets out of cosets. And that's a level of abstraction that you don't really want to work with directly, and so having this isomorphism that's like, that’s the same thing as G mod K, just work over there, that’s it's a really powerful thing. And in my class, we're doing a bunch of Galois stuff. Like, our first course in algebra follows that the story of why the quintic isn't solvable by radicals, basically, why there's no nice quintic formula that looks like the quadratic formula. And when we get into the solvable groups thing, and you're building these chains of subgroups, and you want to do stuff with them, we got on this sort of sidetrack of like, well, what if you wanted to mod out by one of the modded out things? Yeah, there’s an isomorphism theorem. This is so cool.

EL: So yeah, I remember seeing this and possibly having it like a homework question or something in abstract algebra, probably in grad school. And I remember feeling like I was getting away with something. When you could just like, like, cancel the denominators, which is not what this actually is.

CG: It’s one of those things where like, I'd be like, sure, that should work. And that's the hope I'm talking about, like, you’ve got to check the details. But again, this one is one you prove with the first isomorphism theorem. You set up your surjection from G mod N to G mod K, and you show all the stuff that's going to go to zero there was the stuff in K mod N. So this is another nice one, but it just looks so cool. You're like, “I can cancel the denominators. Yes, life is good.” You know, some people worry that mathematics is fundamentally flawed, and we don't know where the error is. And I'm like, it can't be. Look, this worked out. We can cancel denominators. That’s got to be on solid ground! It worked out exactly how we think it should.

EL: I love that. That's a great attitude to go through life with.

CG: W could worry about the fundamental flaws, or we could enjoy the wins where denominators cancel. I'd say we've done pretty well for ourselves. We've built this in a sensible way.

KK: I sort of feel like even if we do find the place where it's bad, it doesn't mean that bridges are going to fall down. Right? I mean, most of what we’ve built it seems okay. I mean, yeah, yeah.

EL: It will probably only break something like what Kameryn talked about with us last month.

KK: Some weird set theory.

EL: Yeah, who cares about that?

CG: Let it burn.

EL: Just kidding, Kameryn.

CG: No, I mean, isn’t there, I don't follow this very much. But I thought there was a modern push to sort of replace the foundations of math, the set theory stuff with category theory instead.

EL: Homotopy?

CG: Yeah, homotopy type theory, which in my mind is just like, you know, an evil cousin of category theory. But I, I don't know. I like homotopy. But I don't know what the homotopy type theory is.

EL: I know. I’m afraid to say it.

CG: It is a little scary to say.

EL: Yeah, I have not been able to figure out the correspondence between what I think of as a homotopy, which is like, “Look, you could drag the little loop around on the doughnut,” and homotopy type theory. Don't tell anyone.

CG: Well, and as someone who studies homology, I'm like, why would we do homotopies? They're kind of yucky. Like homology makes it nicer, right? Homology is the abelian version.

KK: It’s abelian, right.

CG: I mean, I guess, you know, if you actually want to describe stuff in the real world, he can't just live in commutative land, but who wants to live in the real world? That’s what the physicists do.

EL: Yeah, that's why we do math, to not have to do that.

CG: Exactly. Exactly. The real world is full of snow. And climate change deniers.

KK: Sure. Yeah. So it seems like you've had a long love for these theorems, right? This goes way back, right?

CG: This goes way back. For me, I was not actually intending to be a math major. I failed seventh grade math. My elementary school report cards were like, “She's very creative. Not so great at math.” “Asks too many questions,” I think was one of them. My first existential crisis was like, what happens if we keep adding forever, and we run out of names for the numbers, and I was like, I can't go to school. They asked me to add as high as I can, and I got to 100. And I was like, I'm going to stop here because at some point, I know what to do, but I don't know what to say. And that was pretty stressful. For me, I hadn't worked out like the time calculation, like how long it would take me to get to the point where I didn't know the names of the numbers anymore. But that was at a time where like, most of my classmates thought 31 was the biggest number because we only ever counted on the calendar.

So I was really surprised in college. I mean, I was also a college dropout who went back. I was surprised in college to get caught up in math and actually find that I could ask these questions. And my professors weren't like, “Oh, my god, shut up.” They were like, “What a cool question!” And they would actually talk to me, and I think it was the relationships more than the math that drew me in at first. By the time I transferred I was starting in multivariable calculus, having finished my first year elsewhere. And I was like, it's cool. We're gonna parameterize the way a leaf falls from a tree, mathematical poetry, all that, but like, it didn't set my soul on fire. And then I take differential equations, which really didn't set my soul on fire, although it was fun. I had fun time in it. But when I got to algebra, I felt like the language I was trying to speak my whole life was finally available to me. It was all about relationships and these like, very picky, not like picky in the sense of real analysis, but you could talk about really subtle differences in things. Like the difference between an ideal and a subring to me, was this really compelling thing, because they're so similar, but they're just a little bit different. And that little difference makes such a huge difference when you start talking about quotient objects and cosets and things like that. I really loved it. And then when I saw the arrows come out, I was like, I'm hooked. I want to build the isomorphism theorem of friendship. This is the way I want to describe everything in my world. And I just got worse from there. I had a brief flirtation with topology in grad school, and then I realized that you could be a commutative homological algebra person and steal all the cool diagrams, like the Meyer-Vietoris sequence, and all of that, and you could do with polynomials! And, and I was like, all right, that's, that's for me. That's what I'm gonna do.

KK: Very cool. I mean, I'm a pretty algebraic topologist myself. I mean, not not so much anymore. But earlier in my career, I was doing topology, but I was doing, like, homology of groups. So there's some geometry there, but not really, right. You’re really just thinking about algebra all day long.

CG: Yeah, I love that. I love that the tools of homology. I was tempted a little bit by geometric group theory, because it's so pretty and so fun. And I think some of the best, most fun conversations I had in grad school before starting my research work were with my friends who were taking that class too. We could just draw something on the board and argue about it for hours. And that’s one of the things I love most. I think I'm a mathematician — some people are mathematicians because they want to uncover these deep, beautiful, abstract truths. And I'm a mathematician because I like talking to people about math. So all my research work, my job is really like, if I didn't do research, I wouldn't have math to talk to people about, so I've got to keep doing the research and ideally doing with my friends, so we can talk about math. I love talking to my students about math. I love talking to you guys about math. I love talking my family about math, and they're like, “Oh my god. Please don't explain again what an algebraic geophysicist does.” Which is at one point what my mom decided. I said, “I do commutative algebra, which is close to algebraic geometry.” And in her mind, it became algebraic geophysicist.

KK: That sounds cool though.

CG: Which, that sounds like a cool gig. Yeah, like I would do it if I knew what it was. Yeah, like homology of rocks. Yeah, yeah.

EL: Figure out when the next earthquake is gonna be using commutative diagrams.

CG: Yeah. Maybe that's the the new topological data analysis stuff where you're doing persistent homology.

KK: That’s right. Well, you can use that to try to understand the structure of various things like you're looking for cavities inside of these. Yeah. So I mean, there are people who've done these analyses. It's real.

CG: All right. Maybe that's my mid-career pivot. Yeah. Algebraic geophysicist. Okay, joint appointment in the geology department. I get to go on their cool trips where they go hiking. I’m in.

KK: Right. Okay, so part two of this podcast. So well, we don't have a part three. Maybe there is a part three? Anyway, we’ve got three things here. We ask our guests to pair their theorem with something. So what have you chosen to pair the isomorphism theorems with?

CG: Well, I decided the best way to do this would be a clickbait-y Buzzfeed listicle kind of thing. So I'm going to start with theorem number three. Theorem number three was the theorem where you are cancelling denominators. I think it's a really fun and satisfying theorem, and so what I recommend you pair this with is a small batch craft beverage, alcoholic or non, of your choosing because when you use this, you should indulge all your senses and just be overwhelmed with joy that this works out and you could just cross out the denominators. For me, that would be a nice local, unfiltered wheat beer, just sit back, cross those little denominators out and just be like, “Yes, life is good. We have built a good thing.”

EL: That sounds great.

KK: Actually, one of the things I might be looking forward to the most about the joint meetings being in Seattle. [Transcriber’s note: Whoops! The 2021 Joint Mathematics Meetings were canceled/postponed after all.] And Evelyn knows this. My favorite distillery happens to be in Bainbridge Island. And so I think I'm going to make a little side trip to procure their excellent whiskey that I cannot get here. I can only get it there. So this is good. This is a good pairing.

CG: Oh, excellent. Are you taking orders?

KK: Well, I don't know. For you? I guess I could ship you something.

CG: Yeah. I’m not going to make it in person this year. [Narrator: She wasn’t the only one.] But I do love a good whiskey.

KK: Their whiskey is spectacular.

CG: We’re going to have to have an offline conversation about this and compare tasting notes. All right. Theorem two. It's about multiplication and intersections. And what goes better with multiplication than bunnies?

EL: Okay.

CG: And did you know there's a huge intersection of math people with bunnies, especially on Twitter?

KK: Yes.

CG: I think you've got to pair theorem number two with bunnies, you've got to it. It's not a theorem you use super often, but if you do use it, you have to immediately go out and acquire a pet bunny and join the math bunny Twitter crowd. It's just, I don't think there's any way around that. I think that's one of the laws of the universe.

KK: Okay.

CG: I should apologize to the math bunny Twitter folks, because they can infer that this is not the best thing, right? Number one is always the best thing you can you can have. But I hope they'll forgive me when they hear that my pairing for theorem number one, the OG, the original isomorphism theorem, is friends. When you use this theorem, you should think about your friends, you should write a note to your friends, you should talk to your friends. I've got friends on the brain this week because I was just thinking about a couple of friends of mine who passed away pretty young one was in his late 30s and one was in his mid-40s. And you know, I thought about Emmy's life and how she had a pretty tumultuous life, right, trying to make it onto the faculty at a university in the first place and then being shipped out of the faculty to Bryn Mawr’s gain. She ended up at Bryn Mawr.

EL: Basically dodging Nazis.

CG: Yeah, dodging Nazis. We talked about “math is apolitical,” but is it? Bryn Mawr lucked out, I guess? Because they got — I mean, America in general lucked out because we got all these amazing mathematicians who were fleeing Nazi Germany. But that sort of seems to me, and I'm not a math historian, but it seems to me that that must sort of mark the point where mathematics in Germany really took a backseat and why English has become the predominant language of mathematics.

EL: Yeah, I mean, the American mathematicians, the great, the big names from the early 20th century studied in Germany. And now it does very much go the other way. There's even a quote about, like, the center of gravity of research crossed the Atlantic because of the Nazis.

CG: Yeah, so I mean, that must have been such an awful thing to experience, being expelled and the uprooting of all of that. And then she died fairly young. She died at 53. I think it was complications from surgery, a cancer surgery. And, you know, it just it makes me wonder what did she have planned? What were her mathematical plans? How was she about to revolutionize physics again? Her most famous theorem probably is the theorem that talks about conservation of stuff under different types of symmetries. Like when you do something, something is conserved, which is, I guess, super important to physicists. I've never used that one in my own work. Right. But it seems like 53, she must have had so much more stuff planned for herself mathematically, physics-y stuff, personal stuff.

EL: Well, and how would Bryn Mawr have been different? And how would American — you know, would she have stayed at Bryn Mawr? Would she have ended up at a different university? Because I think she was only in the US for a couple of years before that happened. Yeah, she barely even got to teach anyone. I mean, I guess it should really be reversed. People in the US barely got to learn from her. And yeah, it's really tragic when you read about it. I think it was maybe an ovarian cancer surgery or something.

CG: I think that's right.

EL: I'm trying to remember if she's one of the people who seemed to be recovering fine, and then just kind of dropped dead a couple days later, or something like that. I know that I’ve read about it. I don't know for sure.

CG: I read up this morning, actually, to make sure that I had her age right. And it seemed it seems like the internet consensus at least is it was a viral infection. So the surgery went fine. But she had incurred some infection after the surgery and just dropped dead. You think of Maryam Mirzakhani. Also someone who was taken really young with so much math on the horizon. But it's sort of hopeful, right? Like, you know, Emmy really didn't get the recognition she deserved. But Maryam Mirzakhani, of course, got at least some of the recognition she deserved, which is, I guess we're making some progress.

This weekend, my friend and collaborator Nick Baeth passed away from a really rapid pancreatic cancer. I think he got the diagnosis maybe six weeks before he died.

EL: Gosh.

KK: That’s awful.

CG: And I wish I had finished writing the note to him that I was saying to him that it was just such a — he and I are math siblings. He was my math “older brother,” although we were in grad school at different times. But we met through our advisor, Roger Wiegand, who — his students become basically family. I know all my “siblings,” pretty much now and we hang out and they're great. But that's how I started collaborating with Nick. And we started working on some semigroup stuff, which is a little outside of my wheelhouse, but related to some questions that I want to answer in my research life. And it was just so much fun to work with him. We'd started up a new collaboration, and we paused it, obviously, six weeks ago. But I thought I had enough time to tell him just what a wonderful surprise it was to find such a like-minded person to collaborate with, somebody who was super generous with ideas, super generous with coauthorship. It really felt like if you'd had a good conversation with Nick about something, he'd be like, “Do you want to be a coauthor? Can you prove this thing? Let’s actually keep doing it.” And I know that's not always the norm in math, and people are very protective of their ideas, not without reason. But it was just such a joy to work with somebody who actually felt like it was more fun to do math with people, and that was why you did math, than you have to go chasing down these big, outstanding results. Not that what he worked on wasn't important. I mean, he had a Fulbright in Austria. It was pretty cool. And I've just been thinking about that. And you know, this other, my graduate office-mate who died, like, a year and a half ago, who was also really young, and just thinking about for someone like me, who does math, to talk math with friends, I need the reminder that I should tell my friends that I enjoy talking math with them, and I enjoy being friends more often. And so for me every time I sit down and calculate a free resolution, and I do my little “Thanks, Emmy,” I'm going to also jot a note to somebody, just a short note like, “Hey, I was using Emmy Noether’s first isomorphism theorem today, and I thought I'd say hi.”

EL: That’s great. I love that.

KK: So you know, maybe there was a third part of this podcast. We always like to give our guests a chance to plug anything, if they like. Where can we find you on the internets? Or do you have something you really want to promote?

CG: Oh boy. You can find me on Twitter, I am addicted to Twitter. I am @virtualcourtney. And that is my online presence as opposed to my corporeal presence, which is at corporeal Courtney, although I don't know if that's on Twitter? Anyway, I tweet a lot there. My pinned tweet, there is a 10-minute pep talk that I recorded for my students during pandemic times. This is a pep talk I give pretty frequently to students who are maybe in their first math course where it's not just calculus 1-20. We're starting to develop proof techniques, or we're dealing with things that are more abstract than things they've thought about before. Or we're encountering the isomorphism theorems and they're like, “Oh my God, what do these arrows mean?” And they're having this crisis of confidence, like “am I cut out to be a mathematician?” And my little 10-minute pep talk goes into my own bumpy route to becoming a mathematician. I'm plugging it a little bit self-consciously. I put it as an email attachment to my students. And I had put it on YouTube and meant to make it unpublished but clicked the wrong button. And then suddenly it had been shared, and I was getting emails about it and thousands of views. And I was like, I guess I'll leave it up. So I'll plug that for anybody who's feeling a little bit, I don't know, shaky in their mathematician identity, or, you know, not sure that they're allowed to have a mathematician identity. I certainly feel like the last person who was “supposed” to become a mathematician. My initial plan was to be a French major. And I did take my graduate language exams in French. So it's good that I studied enough French to know the subjunctive. But yeah, so I'll plug that and then I'll say one more thing. I am working on writing an open-source, remixable, free WebWork-enabled algebra textbook that takes my predecessor Bob's idea of using Galois theory for the first course in modern algebra, and makes it into an active pedagogy kind of thing, sort of in the model of Active Calculus.

KK: Oh, cool.

EL: Oh, cool.

CG: So I am on sabbatical next academic year, and that's my main project. And if anybody wants to, test WebWork problems, or help out when I'm like, “Okay, why did I think I could write a textbook?” I'm happy to talk math with old friends, new friends, math book friends, WebWork friends, anybody who wants to help make that thing actually happen.

EL: Yeah, and selfishly keep me in the loop about it, I'd love to know about it when it comes out here. You're actually convincing me, and several of our guests are always like, working on me. Like, I should really learn algebra a little better and feel a little more comfortable with some of these ideas. CG: Sure. I’ll keep you in mind as a beta tester. Over the years, I've developed all these active learning worksheets that are the basis for the book. They look fun. My students are — well, they used to smile. Now they’re masked, so I just assumed they're smiling all the time. But they would smile and argue when they were working on them, which is what I had hoped would happen.

EL: Oh, that’s great. Yeah.

CG: You should rope some friends in to learn algebra with.

EL: Yeah!

CG: Because I think that's the most important thing for sticking with it is having pals.

KK: Well, good luck. You know, as somebody who, I’ve written a couple of books, you always reach this point where it's like, “Why am I doing this? What did I get myself into?” But once you push through that, it's fine.

CG: I’m in the planning stage where I have a big wiki essentially, like a big table with, like, here’s chapter one’s outline, here’s all the resources for everything in here and now, it's a matter of actually connecting the things. I mean, PreTeXt is its own special thing, learning learning PreTeXt. But yeah, I think it will be a good experience. I feel a little bad because you know, both my algebra Professor Marlow Anderson and my predecessor Bob Redfield have these great books. And I'm like, “No, I'm not going to use those. I’m going to write my own. Stick it to you guys.” But I think what I haven't seen is a mixture of the active pedagogy with the Galois theory as the thing that links rings, groups, and fields. So that's what I want to create and put out there.

KK: Cool. All right. Well, this has been a lot of fun, Courtney. Thanks for joining us today.

CG: Yeah, absolutely. I have sort of been like, “When are they going to ask me?” for a little while, not thinking like, maybe I could just reach out and be like, “Hey, ask me.” So I was really excited when you did. And I've loved listening to the podcast. And I appreciate you both for for making it happen. It’s awesome.

EL: Well thank you.

KK: Yeah. All right. Well, take care.

CG: Bye.

[outro]

On this episode of My Favorite Theorem, we were delighted to talk with Courtney Gibbons, a mathematician at Hamilton College, about Emmy Noether's isomorphism theorems. Below are some related links you might find useful.

Courtney Gibbons's website and Twitter account

The Wikipedia article on Noether's isomorphism theorems, which includes a helpful chart describing differences in labeling the theorems

An article about Emmy Noether by astrophysicist Katie Mack and her biography on the MacTutor History of Mathematics Archive

Evelyn's 2017 article in Undark about the effect of Nazism on German mathematics in the 1930s

Our episode with Kameryn Williams

Active Calculus, a free, open-source resource for teaching calculus

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