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Kevin Knudson: Welcome to my favorite theorem, the math podcast with no quiz at the end. I'm Kevin Knudson, professor of mathematics at the University of Florida, and I am joined, as always, by my fabulous co-host.



Evelyn Lamb: Hi, I'm Evelyn Lamb, a freelance math and science writer in Salt Lake City, Utah, trying to remember how to do this. It's been a minute since we've recorded one of these. We kind of went dormant for the winter.



KK: Yeah, a little bit, a little bit. Yeah. But Punxsutawney Phil told us — I don’t, what did he say? Let's pretend he said six more weeks of winter.



EL: I think he usually does. I don’t know.



KK: I mean, objectively, there are always six more weeks of winter. Like, the calendar says so, right?



EL: Yeah.



KK: Anyway, yeah.



EL: And, you know, he probably is pretty good at seeing shadows if he's a prey animal because he'd be used to seeing, like, a bird coming overhead.



KK: That’s an interesting question.



EL: Do birds eat groundhogs?



KK: That’s what I was going to wonder. I mean, like, eagles, maybe, but groundhogs are pretty large, right? I mean,



EL: Yeah. What eats groundhogs?



KK: Well, that's something to investigate later.



EL: Yeah.



KK: So it is Pi Day, right?



EL: It is! Well…



KK: We’re actually, we're recording this on Pi Day. When our listeners hear this, it won't be, but we're recording.



EL: And, I always have to put in a plug for my calendar.



KK: That’s right.



EL: The AMS math page-a-day calendar on which Pi Day does not occur on this day.



KK: That’s right.



EL: There are other Pi days on this calendar, none of which is this day, my little joke here. So you can find that in the AMS bookstore.



KK: Right. Are you Team Pi or Team Tau?



EL: I’m Team whichever one works for the calculation that you’re doing. It’s not that big a deal.



KK: That’s right. That's right. Okay. All right. Enough of us, enough of our useless banter, although we did discuss what's the ratio of banter to actual talk, right, that there's, there's like a perfect ratio. But we are pleased today to welcome Karen Saxe. Karen, why don't you introduce yourself and let us know all about you?



Karen Saxe: Hi, there, everybody. So first of all, happy Pi Day. If listeners know who I am, I was a professor at Macalester College for about for over 25 years. And then about seven years ago came to work at the American Mathematical Society, where I am very happy to be the director of the Government Relations Office. So I work in DC with Congress and federal agencies. And could quite a bit about this. I'm also happy to be here because it's Women's History Month. And it will be appropriate that it is Pi Day when you hear what my favorite theorem is.



KK: Okay, good to know. So, I'm curious to know more about this government relations business. So I mean, I know that the AMS does a lot of work on Capitol Hill, but maybe some of our listeners don’t. Can you explain a little more about what your office does?



KS: Yeah, so we do a lot of things. So first of all, we communicate — I sort of view the work of our office as going two ways. One is to communicate to Congress why mathematics is important to almost everything they make decisions about, you know, our national security, health care, you know, modeling epidemics, thinking, like you’re in Florida, thinking about how to model severe weather and things they care about, and then why they should fund fundamental research in mathematics and all sciences. And then also you know, how they make decisions about education. So we tell Congress, we give them advice and feedback on our view about what they should do in those realms. And then on the sort of flip side, I tell the AMS community, the whole math community about what Congress is doing and what's happening at the agencies like the NSF, and Department of Defense and Department of Energy, that that they might care about things, things that would affect their lives. So that’s sort of it in a nutshell. I spend a lot of time on the hill. I just came this morning, I went to a briefing put on by the National Science Board, which is the presidentially-appointed board that oversees the NSF. And they put out a congressionally mandated report every few years on the state of, it's called the indicators report. I'm sure I found it more interesting than everybody else, but it's pretty fascinating. You know, it covers everything from publications around the world, like which countries are are putting out the most science publications, what the collaborator network looks like around the world, and that to sort of US demographic information about education, you know, who's getting undergraduate degrees? Who's getting two year degrees? Who's getting PhDs, that that sort of thing. It covers a lot, actually. Pretty interesting.



KK: Yeah, yeah. All that in like two hours, right, and then it's over.



KS: Yeah, all that in two hours. And then they give you the big report that you can. And I've got them sitting in front of me. But given that this is a podcast, showing things doesn't work.



KK: Well, we do it all the time.



KS: Here’s one of the reports I picked up this morning. Actually, one really, so they're, you know, they're one thing. And you might end up cutting this, but one thing that's sort of fascinating to me is they always list barriers for getting into STEM degrees. And you know, there are things listed, like college accessibility, things that — and even going back. So like, you know, school kids who say they don't have science teachers in their schools, they don't have math teachers, but they've added to this list. “I can't support my family on a graduate student stipend.” So this is something.



EL: Yeah.



KK: That’s real.



KS: And we are, we've endorsed a bill in Congress that would look that would help to improve the financial stability, I guess, you would say, or the ability to be a grad student or a postdoc. So it's looking at stipends, it's looking at benefits, you know, leave time, all that sort of stuff, making it a job that you can choose to take when you're 23, and have a family to support and could make a hell of a lot more money doing something else with a math undergraduate degree.



EL: Yeah, and not see it as something where it's like, you're kind of putting off real life for a little longer, which I think maybe in the past was more of the model, like, oh, yeah, you'll have a real career later. But you know, in your mid-20s, you'll just keep being a student and not have kids or, you know, things, you know, not have parents to support or things like that.



KS: Exactly.



KK: Yeah. Okay. That's, that's good to know. Thank you for all that hard work you do, Karen. So but this is a math podcast.



KS: Right.



KK: So what’s your favorite theorem?



KS: Okay, so first, I'm going to tell you about the three theorems that I didn't choose.



KK: Cool.



EL: Great.



KS: So — I'm sure everybody goes through this — and thinking about my research, it would probably have to be the Riesz-Thorin interpolation theorem, which basically tells you that if you've got a bounded linear operator on two Lp spaces, then it's bounded on every Lp space in between those two values of p, so I used that all the time when I did research on that sort of thing. Then, but I was primarily a teacher of undergraduates, and kind of my two favorite theorems to teach are always Liouville’s theorem and, and then the uncountability of the real numbers.



EL: Yeah.



KS: And Liouville, they’re the one that says, you know, that there's a bounded — if you have a bounded entire function function, it's got to be constant. And the result is so stunning, and it gives a great proof of the fundamental theorem of algebra, that every non-constant polynomial has a root. So I always love teaching that. And then of course, like, Cantor’s diagonal argument about the real numbers, nothing beats that proof in terms of like, cool proof, in my opinion.



EL: Yeah. All-time great.



KS: Yeah, all-time great, right. And I think it's been mentioned on your podcast before. But what I picked was this theorem that says that if you have a given fixed perimeter, then the circle maximizes the two-dimensional shape you can make, so the isoperimetric theorem.



EL: Nice! And as you said, very appropriate for Pi Day.



KS: Yeah, which, I hadn’t even thought about that, which is sort of also embarrassing. But until we started acknowledging Pi Day, I hadn't thought about that. So another way to say it, or the way you might see it in a textbook, is if you have a perimeter P and an area A, then P2−4πA is greater than or equal to 0, with equality if and only if you have a circle. So this theorem has a very long, fascinating history. Lots of great applications. And for all those reasons, I love it. I love history.



KK: Yeah.



KS: I love math.



KK: Yeah. Do you have a favorite proof of this theorem?



KS: I do, actually. Yeah. Well, I didn't know you'd ask that. So there are a lot of proofs. And the one that I like, and this comes from being an analyst probably, is in the early 1900s. Hurwitz gave a proof using Fourier series. I love that proof. And proofs are quite old, going back thousands of years to the Greeks. And then in 1995, Peter Lax actually gave a new short calculus-based pro