Evelyn Lamb: Hello and welcome to My Favorite Theorem, the podcast that was already quarantined. I’m one of your hosts, Evelyn Lamb. I am holed up in my house in Salt Lake City, Utah, where I'm a freelance writer. So, honestly, I have worked in my basement, you know, every day for the past five years, and that hasn't changed. This is your other host.

Kevin Knudson: Hi, I’m Kevin Knudson, professor of mathematics at the University of Florida, which is open for business…But you can't go to campus.

EL: Okay.

KK: Yeah, we moved all of our classes online two weeks ago, I'm just teaching a graduate course this term, so that's sort of easier for me. I feel bad for the people who have to actually lecture and figure out how to do this all at once. My faculty have actually been great. They really stepped up. And, remarkably, I've had very few complaints from students, and I'm the chai,r so you know, they would come to me. And it's just really not—I mean, everybody has really taken the whole thing in stride. A lot of anxiety out there, though, among our students. Really, this is a really challenging time for everybody. And I just encourage my faculty to, you know, be kind to their students and to themselves. So let’s shelter in place and get through this thing, right?

EL: Yup. Yeah, we had an earthquake a week and a half ago to just, like, shake things up, literally. So it's just like, oh, as if I pandemic sweeping through town was not enough. We'll just literally shake your house for a while.

KK: Yeah, well, you know, we can go outside. We have a Shelter in Place Order, but it's been 90 degrees every day for the last week. And so you know, I like to go bird watching, but my favorite bird watching spot is a city park, and it's closed. So I have to just kind of sit on my back porch and see what's up. Yeah. Oh, well,

EL: Well, yes, we're making it through it. And I hope—I mean by the time this is—we have a bit of a backlog in our past episodes, and so who even knows what's going to be happening when this is airing. [Editor’s note: We decided to publish this one out of order, so we actually recorded it pretty recently.] But whatever is happening, I know our guests will be very thrilled to be listening to Ruthi Hortsch! Hi, Ruthi. How are you today?

Ruthi Hortsch: Hey, I'm managing.

EL: Yeah.

RH: It’s a weird time.

EL: Definitely. So what do you do, and where are you?

RH: Yeah, so I'm in New York City right now, which is kind of right now the hotbed of lots of new infections. But I've been in my apartment for the last two and a half weeks and haven't really directly been experiencing that.

I work for an organization called Bridge to Enter Advanced Mathematics. So we're a education nonprofit. We work with low-income and historically marginalized youth. And we're trying to create a realistic pathway for them to become mathematicians, scientists, engineers, programmers.

We start working with students when they're in middle school and we try to figure out, like, what are the things you need to get you to a place where you'll have a successful STEM career? And so we do a lot of different things, but they all are to that purpose.

EL: Yeah, and I'm so glad that we have you on the show to talk about this. Because, yeah, I've been thinking like, we really need to get someone from BEAM on here because I think BEAM is just such a great program. My spouse, and I donate to it every year. I mean, obviously not every year, I don't even know how old it is. But you know, we've made that part of our yearly giving, and yeah, I just think it does great work. So, does that have programs in both New York and LA now?

RH: Yes. So we started in New York City in 2011. And a few years ago, we expanded to LA. So the LA programs are still pretty new. They're building up, kind of starting with students in the first year of contact, and then adding in programming for the older students as that first class gets older. So they now have eighth graders, and that's their oldest class, and they'll continue to add in the ninth grade and the 10th grade program, et cetera, as it goes on. The other kind of exciting thing is, last year, we got a grant from the Gates Foundation. And that grant was to partner with other local programs and other cities to help them build up programs that could do some of the same things we do. So it's not the same comprehensive, really intensive support that we give our students in New York City and LA. But assuming summer camps don't get canceled this summer because of corona, there are going to be day camps in Albuquerque and Memphis that are advised by us.

EL: Oh, that's so great. Yeah, because that's the one thing about it is that it is so localized and, of course, important places for it to be localized. But, you know, the more the, the wider, the better. So that's awesome. And what's your role there? What do you do?

RH: Yeah, I have a hard time answering this question. So I work in programs, which is like, I work on things that are directly affecting students. I run one of our summer camps in the summer. So I run a sleepaway camp at Union College, in which students learn proof-based mathematics for the first time. The students at the sleepaway camp are all rising eighth graders, and so they get to learn number theory and combinatorics and group theory. They also do some modeling and programming and stuff.

During the year I do some managing our other programs team, so supporting other staff. I also do all of our faculty hiring. So certainly we hire a lot of people just for the summer, and most of them are—so we hire college, university students, we hire grad students, we hire professors in various different roles. And I handle all of the, like, hiring people to teach math courses.

EL: Wow.

KK: That’s a lot. Are your programs sort of face to face, or are they online? Is it sort of a combination of stuff?

RH: Yeah, so our summer we run six in-person summer camps each summer. So there's two in upstate New York that are sleepaway, one in Southern California that's sleepaway, and then one day camp in LA and two day camps in New York City. And those are all in-person, face to face. And then during the school year, we also have Saturday classes, which is a mix of life skills and enrichment. And we also do in-person advising. So we have office hours where students can come ask us anything, and then also kind of more intensive. Like, how do you apply to college? How do you get into other summer programs or other STEM opportunities? So most of our programs are face to face. Right now, we've had to cancel a bunch of our year-round stuff. So we don't have Saturday classes right now. We are doing one class for the eighth graders virtually, because we really thought it was critical. And at the moment, we're hoping the summer programs will still run, but it's really hard to say what's going to be going on in two weeks.

KK: Yeah, well, fingers crossed.

EL: But as wonderful as it is to talk about BEAM, what we're dying to know is what is your favorite theorem?

RH: Yeah, so this was actually really fast for me to think of. My favorite theorem is Falting’s theorem. So Falting’s theorem is also actually known as the Mordell conjecture, because Mordell originally conjectured it in the same paper in which he proved Mordell’s theorem, I believe, or at least during the same process of research for him.

EL: Yeah, and so for longtime listeners, was it Mathilde Lalín who, that was her favorite theorem?

RH: Mm-hmm.

EL: Okay, that's right. So we're kind of dovetailing right in.

RH: Yeah. So Mordell’s theorem is about—so when you look at elliptic curves, they have a finitely-generated abelian group. And Mordell’s theorem is the theorem that proves that it actually is finitely-generated.

KK: Right.

RH: So when I say the finitely-generated part, it's actually only looking at the rational points on the curve. So we care about algebraic curves, kind of in general. And then we want to think about, like, how do different algebraic curves behave differently? And because I'm trained as a number theorist, I also specifically care about how many rational points are on that curve and how they behave. So this intersects also with algebraic geometry. And in some sense, this is a statement about how the arithmetic part of the curves—the rational points—interacts with the geometry of it.

So one thing that people care about a lot in geometry is the notion of a genus. This is one of the ways to classify things. And of course, when you're looking at visual shapes, one way of thinking about the genus is how many holes does it have? So if you're just looking at a shape that’s, like, a big sphere, there's no way of poking a hole through it without actually breaking it apart. And so that has genus zero because there are zero holes. But if you're looking at a doughnut, a torus, that has one hole because there's like one place where you can poke something through. And then you can generalize from there that having more holes is higher genus. And so that's kind of a wishy-washy way of looking at things, and a very visual way. There are ways to define that formally in the algebraic sense, but in the places where both definitions make sense, the definition is the same.

And so when you look at algebraic curves, we can ask ourselves, how do genus zero curves act differently than genus one curves, act differently than genus two curves, and does that tell us anything about the number of rational points? And so it turns out that with genus zero curves, genus zero curves are actually really just conic sections. So basically the nice lines that you study in like algebra in high school. And those have infinitely many rational points, right? So when I say rational point, you can kind of think of it as being like the points where the components have rational values.

And genus one curves are actually exactly elliptic curves. So in that case, that's when Mordell’s theorem kicks in and the rational points are this finally generated abelian group. And sometimes they have infinitely many rational points, and sometimes they don't, and it kind of depends on what this algebraic structure, this algebraic group structure, looks like. So that's the most complicated weird point. And for genus two or higher curves, it turns out to be true that there are only finitely many rational points on a genus two or higher curve. And that's the statement of Falting’s theorem.

EL: Okay, and so I, there's something that I, you know, you hear like genus two or higher. And I always wonder, is there a limit to how high the genus can be of these curves? Or, like, is there a maximum complexity that these curves can have?

RH: So no. And actually, there's a statement in algebraic geometry that makes it really easy-ish— you know, “ish”— to calculate the genus, which is called Riemann-Roch. And it gives you a relationship between the degree of the equation defining it and the genus. And essentially, the genus grows quadratically with the degree. There's an asterisk on everything I'm saying. It’s mostly true.

KK: It’s mostly true.

EL: So if I'm remembering correctly, Mordell’s—let’s see, Mordell’s conjecture, Falting’s theorem—was really important for proving Fermat’s last theorem. Is that correct?

RH: I don't think so, no. But all of these things are related to each other.

EL: Okay.

RH: A lot of the common definitions and theorems that play into all these things, they share a lot, but it's not directly, like, one thing implied the other.

EL: Okay, yeah.

RH: In particular, Fermat’s Last Theorem was reduced to a statement about elliptic curves, which is about genus one curves, while Falting’s theorem is really a statement about genus two or higher curves.

EL: Okay.

KK: So was this a love at first sight kind of theorem?

RH: I think no. I think part of the reason that I really started appreciating it was because I had a mentor in undergrad who was really excited about it. And I didn't really understand the full implications and the context, but I was like, “Okay, this mentor I have is really about it, so I'm going to be really about it.”

And we actually used Falting’s theorem as a black box for the REU project I was working on. So we assumed it was true and then used that to show other things. And then later on in grad school, I had a number of things that I was really interested in that Falting’s theorem was related to. One of the things that I think is really cool that's being researched right now is there’s a bunch of like, tropical geometry that is being studied. And this is, like, relating algebraic verbs to kind of more combinatorial objects. So you can actually translate these lcurves that have a more—I don't want to say analytic, but a smooth structure, and then turning them into a question about, like, counting more straight-edged structures instead.

One of the things about Falting’s proof of Falting’s theorem is that it's not, it doesn't actually give you a bound. So it tells you that there are only finitely many points, but it doesn't give you a constructive way of saying, like, what does it actually bounded by, the number of finite points? And using tropical geometry, people have been able to make statements about bounds in certain situations, which is really cool.

KK: Okay, I always like these tropical pictures, you know, because suddenly everything just looks almost like Voronoi diagrams in the plane, these piecewise linear things. So I guess the idea of genus probably still makes sense there in some way, once you define it properly. Right?

RH: Yeah. And there's a correspondence between, there’s a notion of a tropical curve, which still looks like one of those Voronoi diagrams. There’s an actual correspondence, this curve in classical algebraic geometry gives you this particular diagram.

EL: Nice. And so you say it was very easy to choose this theorem. So what's your, like, elevator sales pitch for this theorem? Keeping in mind that no one is going to be in an elevator with anyone else anytime soon. We're staying far apart, but you know.

RH: Yeah. So, I think it’s kind of amazing that geometry can tell you something about the arithmetic of a curve. I think this is what drew me to arithmetic algebraic geometry, that there is this kind of relationship. When you think, okay, arithmetic, geometry, those are totally different fields, people study them in totally different ways, but in fact, it turns out that the geometry of a curve can tell you information about the arithmetic. And that's just bizarre, and also very powerful in that you can make a statement about how many rational solutions there are to an equation using correspondence in geometry.

The REU project that I worked on actually is a statement that I think is really easy to understand. If you have a rational polynomial, that gives you a function from the rationals to the rationals, right?

And so you can ask yourself: how many-to-one is that function? How many points gets sent to the same point? And if you look at only rational points, our REU project showed that it can't be more than four-to-one off a finite number points.

So if you are willing to ignore some finite number of points, then no rational polynomial is ever more than four-to-one.

KK: Interesting.

RH: And that feels like a very powerful statement. And it's because we had this hammer of Falting’s theorem to just smash it in the middle.

KK: That’s really fascinating. So no matter how high the degree it's no more than four-to-one? I wouldn’t have guessed that.

RH: Off a finite number of points.

KK: Yeah, sure. Generically. Yeah. Right. Interesting.

RH: I think the real powerful thing there is that Falting’s theorem comes in.

KK: Yes.

RH: Oh, actually, higher degree means high complexity means high genus.

KK: Okay, cool. So another thing we like to do is ask our guests to pair their theorem with something. So what pairs well with Falting’s theorem?

RH: Yeah, so this is a maybe a little bit of a stretch, but I've been living in New York City for four years, and I love bagels. They’re definitely one of the best parts of living in New York City. I'm always two blocks away from a really good bagel. Traditionally, bagels are genus one, so it's actually not quite appropriate. You have to, I don't know, do the fancy cut to increase the genus—there’s a way to cut a bagel to get higher genus. But I still think since we're thinking about genuses, we're thinking about complexity of things.

EL: Yeah. Well, like, you cut the bagel in in half, you know, to get like the cream cheese surface, and then just stick them together and you've got a genus two. Put a little cream cheese on the side. You know?

RH: Yeah. I mean, if we're cutting holes we can cut as we want.

EL: That’s true. So, are you more—what do you put on the bagel? What kind of bagel, also, do you prefer?

RH: Ao I mostly like everything bagels.

EL: Of course. Yeah. Great bagel.

RH: There is a weird thing that goes on where some bagel shops put salt on their everything bagel and some don't. And I feel like the salt is important.

KK: Yeah. Agree.

EL: As long as it's not too much. Like just the right amount of salt is—

RH: Yeah. It’s definitely important.

KK: Well a salt bagel is a pretzel.

EL: Yes.

RH: And I don't actually eat cream cheese. So I do eat fish sometimes, but I generally don't eat dairy. And I so I usually get, like, tofu scallion spread. And the tofu spread that gets sold in the bagel shops here is actually really good.

KK: Well yeah, I'm not surprised. I can't get a decent bagel in Gainesville. I mean, there's a couple of bagel shops, but they're no good.

RH: Yeah. This is what you get for leaving New York City.

KK: Right, right.

EL: Yeah, it's funny, actually one of our quarantine projects we're thinking about is making bagels. I've made bagels one other time. But, yeah.

KK: It's kind of a nuisance. You know that. That boiling step is really—I mean, it's crucial, but it just takes so much time and space.

EL: Yeah, I mean, they were not nearly as good as a real bagel shop bagel, but fun to play with.

KK: Yeah. So what's everyone doing to keep themselves occupied? So far I've got a batch of sauerkraut fermenting. I just started a batch of limoncello that'll be ready in a month. I made scones. Maybe that’s it. Yeah. How about you guys?

RH: Well, I'm still trying to work 40 hours a week.

KK: Yeah, I'm doing that too.

RH: We're still trying to help our students respond to the crisis and helping support them both academically, but holistically also.

KK: Yeah, it's very stressful.

RH: And at the moment, we're still doing all of our prep work for the summer, which is a huge undertaking? But when I have free time, I've been cooking more. And I'm actually also working on writing a puzzle hunt.

EL Ooh, cool. Well if that happens, we'll include a link to that in the show notes—if it's the kind of thing that you can do out of a particular geographical place.

RH: Yeah, so the puzzle hunt I'm helping write is actually for Math Camp.

EL: Okay.

RH: So before I worked for BEAM I worked for Canada-USA Math Camp, and in theory, they're running a camp this summer, and one of the traditional events there is [the puzzle hunt]. I think the puzzle hunt often gets put up after the summer, but I’m not sure.

EL: Oh, cool. The last thing that I, or library book that I got out from the library—it was actually supposed to be due, like, the day after the library shut down here—was 660 Curries, which is an Indian cookbook that—we don’t really cook meat at home, but it's got, I don't know, maybe a hundred-page section of legume curries and a bunch of vegetable curries, so we've been kind of working through that. We made one last night that was great. It was a mixture of moong dal and masoor dal. Yeah, we’ve been eating a lot of curry, and it just makes my early-this-year plan of, like, “Oh, I want to make more dal, so I've got to go stock up on lentils and rice,” brilliant plan, really has made it a lot easier. So yeah.

RH: I love dal, and I don't feel like anybody around me ever likes dal as much as I do.

KK: This is a dal-lover convention right here. It's one of my favorite things to eat. Yeah.

EL: Oh, yeah. Well, I can recommend, if you get a chance to get 660 Curries, I don't remember if it's called mixed red and lentil dal with garlic and curry leaves, or something like that.

KK: Yeah, I'm actually making curry tonight, but chicken curry so we'll we'll see.

EL: Yeah, so other than that, just panicking most of the time. It’s been a big pastime for me.

RH: I’ve had to, like, ban myself from reading the news in the evening.

KK: Good call.

EL: That is very smart.

RH: I haven’t done a good job keeping to it.

EL: Yeah, I have not done a good job with my self-control with that. So, I’m really trying to do that. I'm hoping to do some sewing projects too, maybe making some masks that I can leave out for people in the neighborhood to take. Obviously not medical grade, but maybe make people feel a little better.

KK: So yeah, Ellen, my wife, started doing that yesterday. She made, you know, probably 15 of them yesterday real quick.

EL: Nice.

KK: I went to the store yesterday and you know—

EL: Hopefully it gives people a little peace of mind and maybe decreases droplet transmission.

KK: Let’s hope.

EL: I’ve refrained from armchair epidemiology, which I encourage everyone to do. So yeah, I hope everyone stays safe and tries to keep keep a good spirit and help the people in your lives. I hope our listeners can do that too. And I hope they find some enjoyment in thinking about math for a little while with us.

KK: So yeah, thanks for joining us, Ruthi. We really appreciate it.

EL: Yeah, everyone go find BEAM online if you want to learn more about that.

RH: Yeah. Follow us on social media.

EL: Yeah. So what are the handles for that?

RH: Yeah, I should have this memorized. You can find it on our website. They're all linked to on our website, beammath.org. If you're in New York or LA, we have trivia night, which is a puzzle-y, mathy trivia, usually in the fall, that you can buy tickets to. So I definitely recommend that. And otherwise, sign up for our newsletter, which you can also do on our website.

EL: And you're on Twitter also, right?

RH: Yes, I am. You do have to know how to spell my last name, though.

EL: Okay.

RH: Yeah, I'm @ruthihortsch.

EL: All right. And that's H-O-R-T-S-C-H?

RH: Good job!

EL: Yeah, it’s funny, I was actually in a Zoom spelling bee last night. So yeah, I got second place.

KK: Good for you.

EL: Got knocked out on diaphoresis.

KK: Diaphoresis. Wow. Yeah, that's pretty—okay, anyway. All right. Well, thanks for joining us and take care everyone.

RH: Right. Yeah, it was nice to meet you.

EL: Bye.

[outro]

On today's episode of My Favorite Theorem, we had the privilege to talk with Ruthi Hortsch, a program coordinator at Bridge to Enter Advanced Mathematics (BEAM), a math program for low-income and historically marginalized middle- and high-school students. Dr. Hortsch lives in New York City, which is currently being hit hard by covid-19. We love all our listeners and guests, and right now we are especially thinking about those in New York and other virus hot spots. You may be sick, you may be worried about loved ones, you may be suddenly parenting or caregiving in ways you hadn't expected. We wish you the best, and we hope you enjoy thinking about math for a little bit instead of the news cycle. Stay strong and healthy, friends!

As you listen to this episode, you may find these links helpful.
The Bridge to Enter Advanced Mathematics website, Twitter, Facebook, and Instagram pages.
Ruthi Hortsch on Twitter
Faltings’s theorem, Dr. Hortsch's favorite theorem
Our episode with Matilde Lalín, whose favorite theorem was the closely-related Mordell's theorem.
660 Curries

Canada/USA Mathcamp

Tropical Geometry wikipedia page