Evelyn Lamb: Hello and welcome to my favorite theorem. Math podcast. I'm one of your hosts Evelyn Lamb. I'm a freelance math and science writer in Salt Lake City, Utah. And this is your other host.

Kevin Knudson: Hi. I’m Kevin Knudson, professor of mathematics at the University of Florida. How's it going?

EL: All right, it is a bright sunny winter day today, so I really like—I mean, I'm from Texas originally, so I'm not big on winter in general, but if winter has to exist, sunny winter is better than cloudy winter.

KK: Sure, sunny winter is great. I mean, it's a sunny winter day in Florida, too, which today means it is currently, according to my watch, 81 degrees.

EL: Oh, great. Yeah.

KK: Sorry to rub it in.

EL: Fantastic. It is a bit cooler than that here.

KK: I’d imagine so.

EL: So yeah. Anything new with you?

KK: No, no. Well, actually so so my I might be going to visit my son in a couple of weeks because he's studying music composition, right? And the the orchestra at his at his university is going to play one of his pieces, and so kind of excited about that.

EL: Very exciting! Yeah, that's awesome.

KK: Yeah, but that's about it. Otherwise, you know, just dealing with downed to trees in the neighborhood. Not in our yard, luckily, but yeah, stuff like that. That's it.

EL: Yeah. Well, we are very happy today to have Rebecca Garcia as a guest. Hi, Rebecca. How are you?

Rebecca Garcia: Hi, Evelyn. Håfa ådai, I should say, håfa ådai, Evelyn, and håfa ådai, Kevin. Thanks for having me on the program.

EL: Okay, and what—håfa ådai, did you say?

RG: Yeah, that's right. That's how we, that's our greeting in Chamorro.

EL: Okay, so you are originally from Guam, and is Chamorro the name of a language or the name of a group of people, or I guess, both?

RG: It’s both actually. Yes. That's right. And so Chamorro is the native language in the island. But people there speak English mostly, and as far as I'm able to tell I think I'm the first Chamorro PhD in pure mathematics.

EL: Well, you’re definitely the first Chamorro guest on our show. I think the first Pacific Island guest also.

KK: I think that's correct. Yeah.

EL: So yeah, how did you—so you currently are not in Guam. You actually live in Texas, right?

RG: I do. I'm a professor at Sam Houston State University, which is in Huntsville, Texas, north of Houston. And I'm also one of five co-directors of the MSRI undergraduate program.

EL: Oh, nice. That seems like it is a great program. So how did you how how did you get from Guam to Huntsville?

RG: Oh my goodness. Wow. That is a that is a long, long journey.

KK: Literally.

RG: I started out as a as a undergraduate at Loyola Marymount University, and I had the thought of becoming a medical doctor. And so I thought we were supposed to do some, you know, life science or you know, chemistry or biology or something along those lines. And so I started out as one of those majors and had to take calculus and fell in love with calculus and the professors in the math department. And I was drawn to mathematics. And that's how I ended up on the mathematics side. And one of the things that I learned in my undergraduate career was these really crazy math facts about the rational numbers. And so that's one of the things that interested me in mathematics, was just the different types of infinities the concept of countable, uncountable, those sorts of things.

EL: Yeah, those those seem to be the kinds of facts that draw a lot of people into this rich world of creativity and math that you might not initially think of as related to math when you're going through school. So I think this brings us to your favorite theorem, or at least the favorite theorem you want to talk about today.

KK: Sounds like it.

EL: Yeah, so what’s that?

RG: Yeah. So it’s more, I would say, more of a fun fact of mathematics that the rationals first of all are countable, meaning they are in one-to-one correspondence with the natural numbers. And so you can kind of, you know, label them, there's a first one and a second one in some way, not necessarily in the obvious way. But then, at the same time, they are dense in the real numbers. So that to me, just blows my mind, that between any two real numbers, there's a rational number.

EL: And yeah, so you can't like take a little chunk of the real line and miss all the rational numbers.

RG: That’s right.

KK: Right.

RG: That to me just blows my mind. Because—and then you just sort of start, you know, your brain just starts messing with you, you know, between zero and one there are infinitely many rational numbers and yet they're still countable. And it just, it just starts to mess with your mind a little bit. Right?

EL: Yeah. Well, and we were we were talking about this a little bit before and it's this weird thing. Like, yeah, there's, like a countable is like a smallness thing. And yet dense is like, they're, you know, they fill up the whole interval this way. I mean, it is really weird. So where did you first encounter this?

RG: This was in a class in real analysis. And, yeah, so that's where I started to…I thought I was going to be a functional analyst. I thought I was that's what I wanted to do this. I love real analysis. That didn't happen either. But it was in that class where we were talking about just these strange facts, like the Cantor set: that set is a subset of the reals that is uncountable and yet it’s sparse.

KK: Totally disconnected, as the topologists say.

RG: Totally disconnected. There you go. Yeah. Right. And so then all these weird things are happening. And you're just in this world where you thought you understood the real line, and then they throw these things at you like, the reals are dense. I mean, the rationals are dense in the reals, you have these weird uncountable sets that are totally disconnected. What's going on? Yeah, so that's where I started to hear about all these weird things happening.

KK: Right. So one of two things happens when people learn these things, right? It either blows their minds so much they can't keep going. Or it intrigues them so much that you want to learn more. But not be an analyst. Right?

RG: [laughing] That’s right. At some point I fell in love with computational algebraic geometry and these Gröbner bases, and how you can really get your hands on some of these things and their applications to combinatorics. So I ended up, I had an algebraist’s heart, but I was exposed to some really good analysts early in my career. And so I was very confused. But I've always, I stay true to my algebraic heart and follow that mostly.

EL: And so is this a fact that you get to teach to your students now ever?

RG: So no, this is not, but I do like to talk about the the different infinities and things along those lines. And I like to, before class I come in early, and I'll have a little chat with them about just the fact that—you know, they they don't understand that math is not “done.” So, there's still so much to do. And they have no idea that, you know, there's what, what is research like? What does that mean? And so I talk about open questions. And I bring some of that in the beginning of class. And these concepts that had also drawn me in, about the different kinds of infinities and these weird concepts about the rationals being dense and, you know, just things like that. I do get to talk about it, but it's not in a class that I would teach the material on.

EL: Yeah, just going back to this idea that you've got the rationals that are dense, so it's this, like, measure zero small set, but it's like everywhere. And then you've got the Cantor set, which is uncountable and sparse. It's like, we've got these various ways of measuring these sets. And you think that they line up in some natural way. And yet they don't. It's just like, you know, the density is measuring a different type of property of the numbers than the measure is.

RG: That’s exactly.

EL: And actually, I guess countability is a different thing. Also, I mean, it's, yeah, it's so weird. And it's hard to keep all these things straight. My husband does a lot of analysis and like has, yeah, all of these, like, what kinds of sets are what.

RG: And what properties they have. And yeah, I don’t have that completely straight.

KK: This is why I’m a topologist.

EL: But I mean, topology is like,

KK: Oh, it's weird too.

EL: It’s secretly analysis.

KK: Well…

EL: Analysis wishes it was topology, maybe.

KK: So my old undergraduate advisor—who passed away last summer, and I was really sad about that—but he always he always referred to topology as analysis done right.

EL: Shots fired.

KK: Which is cheap, of course, right? Because you prove all this stuff in topology Oh, the image of a connected set is connected. Yeah, that's easy now go off to the real line and prove that the connected sets are the intervals. That's the hard part. Right? So yeah, he's being disingenuous, but it was. It's a good line. Right.

RG: Right.

EL: So you said that you ran into this, was this an undergraduate class where you first saw these notions of countability and everything?

RG: Right, it was an undergraduate class where I ran into those notions and I was a junior, well, I guess it was in my second semester as a junior, where we were talking about these strange sets. And that's when I had also thought about going on to graduate school and wanting to do mathematics for the rest of my life. I mean, I was a major by then, of course, but I just didn't know what I was going to do. But it wasn't until then, when I learned about, well, this is this could be a career for you. This may be something you like to do. And of course, this was many, many years ago. And nowadays, you can do so much more with mathematics, obviously. I mean, we know that we can do so much more, I should say. We've always been able to do so much more. We just haven't been able to share that with our students so much. We never really spent the time to let them know there's so many careers and mathematics that one can do. But anyway, at that, at that time I was I was drawn into really thinking about becoming a mathematician, and that was one of the experiences that that made me think that there's so much more to this than than I originally thought.

EL: Yeah, well, I talk to a lot of people, you know, in my job writing and doing podcasts and stuff about math, and there's so many people who don't realize that, like, math research is a career you can do.

RG: Right.

EL: And the more we can share these kinds of “aha” moments and insights, the better and, you know, just show like, well, you can use, you know, kind of the logic and the rules of the game to like, find out these really surprising aspects of numbers.

RG: Right. And I think also, one of the experiences that I've had as an undergrad that really just sort of sealed the deal—I’m going to go into mathematics—was doing an undergraduate research program as a student. Well wasn't really at the time an undergraduate research program, it was just another summer program. This is many years ago, almost before all of that. And I had the chance to spend a summer just thinking about mathematics at a higher level with a cohort of other students who were like-minded as well, you know. And it was really—it was it was like, “Oh, I can do this for the rest of my life? Like how amazing is that?” And so, I was part of a summer program as an undergrad. And then when I was a graduate student, my lifelong mentor, Herbert Medina, was running a program in Puerto Rico and asked me to be a TA while I was a grad student. And so these were some of the things that led me to do what I do now, working with undergraduates, doing research and mathematics.

EL: And so that ties in to the MSRI program that you are part of, right?

RG: Right.

EL: I guess it I've seen it written like MSRI-UP. So I guess that's undergraduate program?

RG: Yes. Undergraduate Program. That's right. Yeah. Well, that that's sort of like, a different stage that I'm at now. But yeah, before that, I started my own undergraduate research program together with colleagues in Hawaii, at the University of Hawaii at Hilo. And we ran an undergraduate research program called PURE Math, and that was Pacific Undergraduate Research Experience in Mathematics. And we ran that for five years. And then, and then I ended up moving into the co-director role at MSRI-UP.

EL: Nice.

KK: That’s a great program.

EL: Yeah. So the other thing we like to ask our guests to do, is to pair their theorem with something. You know, just like the right wine can enhance that meal, you know, what would you recommend enjoying the density of the rationals with?

RG: Well, I did think about this a bit. And one of the things that I think, you know, you think the rationals are dense but they really shouldn't be? So, I think of foods that are dense, but they really shouldn't be, and one of those foods that comes to mind, especially being here in Texas, but also being married to a mathematician who is from Mexico, is tamales. So tamales really should not be dense. They should be fluffy and sumptuous, but here in Texas, you find really dense the most, unfortunately. But it It was strange to also discover that growing up in Guam, we also have our own version of tamales, and a lot of the foods are related in some way to foods from Mexico. So I feel like there's this huge rich connection between myself being from Guam, my husband being Mexican and there's just this strange richness that we share this culture, that I don't know, it just blows my mind too. So the same way that the rational is being dense in the reals blows my mind.

EL: All right, well, I have to ask more about this tamale like creation from in traditional Guam cuisine. What, is that wrapped in, like, banana leaves or something like that?

RG: It ought to be, and maybe traditionally it was. I think that nowadays it's not that way. They usually serve it in aluminum foil, and it's made—it's a mixture like tamales. So tamales in Mexico are made with corn, right?

KK: I was about to ask this. What are they made of in Guam?

RG: Yeah, yeah. And so in Guam we actually use, like, a rice product.

EL: Okay.

RG: It's ground up just like corn. And so instead of corn, we're using rice, and it's flavored in different ways.

KK: Interesting.

EL: All right. I have kind of in my mind because I'm more familiar with this like almost, is it kind of like a mochi texture? Because, I mean, that's a rice product, but maybe it's not maybe that's like more gelatinous than this would be.

RG: Yeah, I guess mochi is really pounded and yeah, so yeah, that's more chewy. I think that the tamal, well, you wouldn't say it like that, but the tamales in Guam are very soft and, gosh, I don't know how to describe it. But it's a very soft textured food.

KK: I would imagine the rice could be softer, and I mean, corn can get very dense, especially when you start to put lard in it and things like that.

RG: Yes.

KK: I mean, it’s delicious.

RG: It is delicious. And oh my, I can’t get enough tamales. Oh, well.

KK: Yeah, maybe you can.

RG: Yeah, I should learn.

EL: Yeah, well, nice. I unfortunately, we do have a couple restaurants in Salt Lake that are Pacific Island restaurants, but we have more people from Samoa and Tonga here. I don't know if we have a lot of people from Guam here. Yeah, there's actually like a surprising number of like, Samoans who live in Salt Lake. Who knew?

RG: Right.

EL: But yeah, it's it's because of like the history of Mormon missionaries.

KK: That’s what I was gonna say.

EL: Yeah, the world is very interesting, but yeah I don't know if I've seen this kind of food there. I will just have to, you know, if I'm ever in Huntsville I’ve got to get you to make me some of this. I’m just inviting myself over for dinner now. Hope you don't mind.

RG: That would be great. It would be wonderful to have you here.

EL: Is there anything else you'd like to share? We'd like to give our guests a chance to like, share, you know if they've got a website or blog or book or anything, but also if you want to share information about MSRI-UP, application information, anything like that for students? Anything you'd like to share?

RG: Oh, wow. That's a lot of stuff.

EL: Yeah, I know. I just rattled off a ton of things.

RG: Well, yes, I do have, I guess I would like to say for the undergraduate listeners in the audience, please consider applying to our MSRI-UP program, and just in general apply to a research program in the summer. These are paid opportunities for you to expand your mind and do some mathematics in a great environment, and so I highly recommend considering applying for that. And so this is the time right now of course by the time the listeners hear this, I’m sure it will be over, but consider doing some undergraduate research or using your summer wisely.

KK: I parked cars in the summer in college. I did.

EL: Well, you never know the connections that might happen though because I was talking to someone one time who basically his big break to get to go to grad school came because, like, somehow he was involved in like parking enforcement somewhere, and some math professor called in to complain about, like, getting a ticket, and one thing led to another and then he ended up in grad school. So really, you never know. Maybe that's not the ideal route to take. There are more direct routes, but yeah, there are many paths.

RG: Yes, there are. And there's also another, I guess another thing to flag would be, well, contributed to a book that Dr. Pamela Harris and others have put together on undergraduate research. So that just I guess that was just released. I'm not entirely sure now. I think it was accepted, and I don't know if if one is able to purchase it, but if you if you consider working with your students on undergraduate research, this is a great resource to use to get you going, I guess.

KK: Great.

EL: Oh, awesome. So this is like a resource for like faculty who want to work with undergraduates? Oh, that's great.

RG: Yes.

EL: We will find a link to that and put that in the show notes for people.

RG: That sounds good.

EL: Okay, great. Thanks so much for joining us.

KK: It’s been great.

RG: Thank you so much.

On this episode of My Favorite Theorem, we were happy to talk with Rebecca Garcia, a mathematician at Sam Houston State University, about the density of the rational numbers in the reals. Here are some links you might find helpful.

Her website
A biography of Garcia for SACNAS
MSRI-UP
A Project-Based Guide to Undergraduate Research in Mathematics, the book she mentioned contributing to