Kevin Knudson: 1-2-3
Kevin Knudson and Evelyn Lamb: Welcome to My Favorite Theorem!
KK: Okay, good.
EL: Yeah.
[Theme music]
KK: So we’re at the JMM.
EL: Yeah, we’re here at the Joint Math Meetings. They’re in Baltimore this year. The last time I was at the Joint Meetings in Baltimore I got really sick, but so far I seem to not be sick.
KK: That’s good. You’ve only been here a couple of days, though.
EL: Yeah. There’s still time.
KK: Yeah, so I’ve only been to the Joint Meetings one other time in my life, 20 years ago as a postdoc in Baltimore. I’ve just got a thing for Baltimore, I guess.
EL: Yeah, I guess so.
KK: So people may have seen this on Twitter. Fun fact: this is our first time meeting in person.
EL: Yeah.
KK: And you’re every bit as charming in real life as you are over video.
EL: And you’re taller than I expected because my first approximation of all humans is that they are my height, and you are not my height.
KK: But you’re not exceptionally short.
EL: No.
KK: You’re actually above average height, right?
EL: I’m about average for a woman, which makes me below average for humans.
KK: Well, if we’re going to the Netherlands, for example, I’m below average for the Netherlands.
EL: Yes.
KK: So I’m actually leaving today. I was only here for a couple of days. I was here for the department chairs workshop. You’re here through when?
EL: I’m leaving on Friday, tomorrow. Yeah, while we’ve been here we’ve been collecting flash favorite theorems where people have been telling us about their favorite theorem in a small amount of time. So yeah, we’re excited to share those with you.
KK: Yeah, this is going to be a good compilation. I’m going to try to get a couple more before I leave town. We’ll see what happens.
EL: Yeah. All right.
KK: Enjoy.
EL: I am here with Eric Sullivan. Can you tell us a little bit about yourself?
Eric Sullivan: Yeah, I'm an associate professor at Carroll College in Helena, Montana, lover of all things mathematics.
EL: And here with me in the Salt Lake City Airport, I assume catching a connecting flight to the Joint Math Meetings.
ES: You got it.
EL: All right, and what is your favorite theorem, or the favorite theorem you'd like to tell me about right now?
ES: Oh, I have many favorite theorems, but the one that's really coming to mind right now, especially since I'm teaching complex analysis this semester, are the Cauchy-Riemann equations.
EL: Very nice.
ES: Giving us a beautiful connection between analytic functions, and ultimately, harmonic functions. Really lovely. And it seems like a mystery to my students when they first see it, but it's beautiful math.
EL: Yeah, it is. They are kind of mysterious, even after you've seen them for a while. It's like, why does this balance so beautifully?
ES: Right? And the way you get there with the limit, so I'm just going to take the limit going one way, then I’ll take the limit going the other way and voila, out comes these beautiful partial differential equations.
EL: Yeah, very lovely. And I know I'm putting you on the spot. But do you have a pairing for this theorem?
ES: Ooh, a pairing? Oh boy, something was a very complex taste. Maybe chili.
EL: Okay.
ES: I’ll say chili because there's all sorts of flavors mixed in with chili, and complex analysis seems to mix all sorts of flavors together.
EL: All right, I like it. Well, thank you. This is the first lightning My Favorite Theorem I'm recording so far at the joing meetings, or even before, on the way, so yeah, thanks for joining me.
Courtney Gibbons: I'm Courtney Gibbons. I'm a professor at Hamilton College in upstate New York. And my favorite theorem is Hilbert’s Nullstellensatz, which translates to zero point theorem, but if you run it through Google Translate, it's actually quite beautiful. It's like the “empty star theorem” or something like that. It's very astronomical. And I love this theorem because it's one of those magical theorems that connect one area that I love, algebra, to another area that I don't really understand, but would like to love, geometry. And I find that in my classes, when I ask someone, “What's a parabola?” I have a handful of students who do some sort of interpretive dance. And I have a handful of students were like, “Oh, it's like y equals some x squared stuff.” And I'm like, “I'm with you.” I think of the equation. And some people think of the curve, the plot, and that's the geometric object, and the Nullstellensatz tells you how to take ideals and relate them to varieties. So it connects algebra and geometry. And it's just gorgeous, and the proof is gorgeous, and everything about it is wonderful, and David Hilbert was wonderful. And if I were going to pair it with something, I’d probably pair it with a trip to an observatory, so that you could go appreciate the beauty of the stars, and think about the wonderful connectedness of all of mathematics and the universe. And maybe you should have, like, a beer or something too.
EL: Why not?
CG: Yeah. Why not? Exactly.
EL: Good. Well, thank you. Absolutely.
KK: All right, JMM flash theorem time. Introduce yourself, please.
Shelley Kandola: Hi. My name is Shelly Kandola. I'm a grad student at the University of Minnesota.
KK: And it’s warmer here than where you are usually.
SK: Yeah, it's 15 degrees in Minnesota right now.
KK: That’s awful.
SK: Yeah.
KK: Well anyway, we’ve got to be quick here. What's your favorite theorem?
SK: The Banach-Tarski paradox.
KK: This is an amazing result that I still don't really understand and I can't wrap my head around.
SK: Yeah, you've got a solid sphere, a filled-in S2, and you can cut it into four pieces using rigid motions, and then put them back together and get two solid spheres that are the same size as the original.
KK: Well, theoretically, you can do this, right? This isn't something you can actually do, is it?
SK: Physically no, but with the power of group theory, yes.
KK: With the power of group theory.
SK: The free group on two generators.
KK: Why do you like this theorem so much?
SK: So I like it because it was the basis of my senior research project in college.
KK: It just seems so weird it was something you should think about?
SK: Yeah, it intrigued me. It's a paradox. And it's the first theorem I dove really deep into, and we found a way to generalize it to arbitrarily many dimensions with one tweak added.
KK: Cool. So what does one pair with the Banach-Tarski paradox?
SK: One of my favorite Futurama episodes. There's this one episode where there's a Banach-Tarski duplicator, and Bender jumps into the duplicator, and he makes two more, and he wants to build an army of himself.
KK: Sure.
SK: But every time he jumps in, the two copies that come out, are half the size of the original. He ends up with an army of nanobots. It contradicts the whole statement of the paradox that you're getting two things back that are the same size as the original.
KK: Although an army of Benders might be fun.
SK: Yeah, they certainly wreak havoc.
KK: Don’t we all have a little inner Bender?
SK: Oh yeah. He's powered by beer.
KK: Well, thanks for joining us. You gave a really good talk this morning.
SK: Thanks.
KK: Good luck.
SK: Thank you for having me.
KK: Sure.
David Plaxco: My name is David Plaxco. I'm a math education researcher at Clayton State University. And my favorite theorem is really more of an exercise, I think most people would think. It's proving that the set of all elements in a group that conjugate with a fixed element is a subgroup of the group. I'll tell you why. Because in my dissertation, that exercise was the linchpin in understanding how students can learn by proving.
EL: Okay.
DP: So I was working with a student. He had read ahead in the textbook and knew that not all groups are commutative, so you can't always commute any two elements you feel like. And he generalized this to thinking about inverses. He didn’t think that every inverse was necessarily two-sided, which in a group you are. Anyway, so he was trying to prove that that set was a subgroup and came to this impasse because he wanted to left cancel and right cancel with inverses and could only do them on one side. And then he started to question, like, maybe I'm just crazy, like maybe you can use the same inverse on both sides. And then he proved it himself using associativity. So he made, like, I call it John’s lemma, he came up with this kind of side proof to show that, well, if you're associative and you have a right inverse and a left inverse, then those have to be the same. And then he came back and was able to left and right cancel at free will any inverse, and then proved that it was a subgroup, so through his own proof activity, he was able to change his own conceptual understanding about what it means to be an inverse, like how groups work, all these things, and it gave him so much more power moving forward. So that's how that theorem became my favorite theorem because it gave me insight into how individuals can learn.
EL: Nice. And do you have a pairing for this theorem?
DP: My diploma. Because it helped me get it.
EL: That seems appropriate. Thanks.
DP: Thanks.
Terence Tsui: So I'm Terrance, and I'm currently a final year undergraduate studying in Oxford. My favorite theorem is actually a really elegant proof of Euler’s identity on the Riemann zeta function. We all know that the Riemann zeta function is defined in a way of the sum of 1/ks where k runs across all the natural numbers. But at the same time, Euler has given a really good other formulation: we say status is the same as the infinite product of 1-1/ps, where p runs across the primes. And then it's really interesting, because if you look at you see, on one hand, an infinite some, and on the other hand, you have an infinite product. And it’s very rare that we see that infinite sums and infinite products actually coincide. And they’re only there because it is a function that actually works on nearly every s larger than 1. And that means that this beautiful, elegant identity actually runs correct for infinitely many values. And the most interesting thing about this theorem is that the proof to it could be done probabilistically, where we consider some certain particular events, and we realize that the Riemann zeta series sum is actually equivalent to finding a certain intersection of infinitely many independent events. And first it is just an infinite product of certain events. And first we have the Riemann zeta function equalling a particular infinite product. And I think that is something that is really out of out of our imagination, because not only does it link two things—a sum to an infinite product, but at the same time, the way that it proves it comes from somewhere we could not even imagine, which is from probability. So if I need to pair this theorem with something, I would say it’s like a spider web, because you can see that there's very intricate connections and that things connect to each other, but in the most mysterious ways.
ELL: Cool. Well, thanks.
TT: Thank you.
Courtney Davis: So Hi, I'm Courtney Davis. I am an associate professor at Pepperdine University out in LA.
EL: Okay. And I hear that we have a favorite model, not a favorite theorem from you.
CD: Yes. So I'm a math biologist. So I'm going to say the obvious one, which is SIR modeling, because it is the entry way into getting to do this cool stuff. It’s the way that I get to show students how to write models. It's the first model I ever saw that had biology in it. And it's something that is ubiquitous and used widely. And so despite being the first thing everyone learns, it's still the first thing everyone learns. And that's what makes it interesting to me.
EL: Yeah. And and can you kind of just sum up in a couple sentences what this model is, what SIR means?
CD: Yeah. So SIR is you are modeling the spread of disease through a susceptible (S) population through infected and into recovered or immune, and you can change that up quite a lot. There are a lot of different ways to do it. It's not one fixed model. And it's all founded on the very simple premise that when two individuals run into each other in a population, that looks like multiplication. And so you can take multiplication, and with that build all the interactions that you really need, in order to capture what's actually happening in a population that at least is well mixed, so that you have a big room of people moving around about it, for instance.
EL: Okay. And I'm going to spring something on you, which is that usually we pair something with our theorem, or in this case model, so we have our guests, you know, choose a food, beverage, piece of art, or anything. Is there anything that you would suggest that pairs well with SIR?
CD: With an SIR model, I would say, a paint gun.
EL: Okay.
CD: I don't know that that's what you're looking for.
EL: That’s great.
CD: Simply because running around and doing pandemic games or other such things is also a common way to get data on college campuses so that you can introduce students, and they can parameterize their models by paint guns or water guns or something like that.
EL: Oh, cool. I like it. Thank you.
CD: Absolutely. Thank you.
Jenny Kenkel: I’m Jenny Kenkel. I'm a graduate student at the University of Utah. I study commutative algebra. My favorite theorem is this isomorphism between a particular local cohomology module and an injective module: The top local cohomology of a Gorenstein ring is isomorphic to the injective hull of its residue field. But I was thinking that maybe it would pair really well with like, a dark chocolate and a sharp cheddar, because these two things are isomorphic, and you would never expect that. But then they go really well together, just in the same way that I think a dark chocolate and a sharp cheddar seem kind of like a weird pairing, but then it's amazing. Also, they're both beautiful.
EL: Nice, thank you.
JK: Thank you.
Dan Daly: My name is Dan Daly. And I am the interim chair of the Department of Mathematics at Southeast Missouri State University.
KK: Southeast—is that in the boot?
DD: That is close to the boot heel. It's about two hours south of St. Louis.
KK: Okay. I'm a Cardinals fan. So I'm ready, we’ve got something here. So what's your favorite theorem?
DD: So my favorite theorem is actually the classification of finite simple groups.
KK: That’s a big theorem.
DD: That is a very big, big,
KK: Like 10,000 pages of theorem.
DD: At least
KK: Yeah. So what draws you to this? Is it your area?
DD: So I am interested in algebraic and combinatorics, and I am generally interested in all things related to permutations.
KK: Okay.
DD: And one of the things that drew me to this theorem is that it's such an amazing, collaborative effort and one of the landmarks of 20th century mathematics.
KK: Big deal. Yeah.
DD: And, you know, it just to me, it seems such a such an amazing result that we can classify these building blocks of finite groups.
KK: Right. So what does one pair this with?
DD: So I think since it's such a collaborative effort, I'm going to pair it with Louvre museum.
KK: The Louvre, okay.
DD: Because it's a collection of all of these different results that are paired together to create something that is really, truly one of a kind.
KK: I’ve never been. Have you?
DD: I have. It’s a wonderful place. Yeah. It’s a fabulous place. One of my favorite places.
KK: I’m going to wait until I can afford to rent it out like Beyonce and Jay Z.
DD: Yeah, right.
KK: All right, well thanks, Dan. Enjoy your time at the Joint Math Meetings.
DD: All right, thank you much.
Charlie Cunningham: My name's Charlie Cunningham. I'm visiting assistant professor at Haverford College. And my area of research originally is, or still is, geometric group theory. But the theorem that I want to talk about was a little bit closer to set theory, which is I want to talk about the existence of solutions to Cauchy’s functional equation.
EL: Okay. And what is Cauchy’s functional equation?
CC: So Cauchy’s functional equation is a really basic sort of thing you can ask about a function. It's asking, all right, you take the real numbers, and you ask is there—what are the functions from the real numbers to the real numbers where if you add two numbers together, and then apply the function, it's the same thing as applying the function to both of those numbers and then adding them together?
EL: Okay. So kind of like you're naive student and wanting to—how a function should behave.
CC: Yes. Right. So this would come up in a couple of places. So if you’ve taken linear algebra, that's the first axiom of a linear function. It doesn't ask about the scaling part. It's just the additive part. And if you've done group theory, it's a fancy way, is it's all the homomorphisms from the real numbers to themselves, an additive group. So the theorem basically, is that well, well, first of all, the question is, well, there are some obvious ones. There are all the functions where you just multiply by a fixed number, all the linear functions you’d know from linear algebra, like 2 times x, 3 times x, or π times x, any real number times x. So the question is, are there any others? Or are those the only functions that exists at all that satisfy this equation? And the theorem turns out that the answer depends on the fundamental axioms you take for mathematics.
EL: Wow. Okay.
CC: Right. So the answer is just to use a little bit of set theory, that if you are working in a set theory, which most mathematicians do, that has something called the axiom of choice in it, then the answer is no, there are lots and lots and lots of other functions that satisfy this equation, other than those obvious ones, but they're almost impossible to think about or write down. They're not continuous anywhere, they are not differentiable anywhere. They're not measurable, if anyone knows what that means. Their graph, if you tried to draw them, are dense in the entire plane, which means any little circle you draw on the plane intersects of the graph somewhere. They still pass the vertical line test. They’re still functions that are well-defined. And I really like this theorem. One reason is because it's a really great place for math students to learn that there isn't always one right answer in math. Sometimes the answer to very reasonably posed questions isn't true or false. It depends on the fundamental universe we’re working in. It depends on the what we all sit down and agree are the starting rules of our system. And it's a sort of question where you wouldn't realize that those sorts of considerations would come up. It also comes up—When I've asked linear algebra students, it's equivalent to the statement are both parts of the definition of a linear function actually necessary? We usually give them to you as two pieces: one, it satisfies this, and the other is scalars pull out. Do we actually need that second part? Can we prove that scalars pull out just from the first part? And this is the only way to prove the answer's no. It's a good exercise to try yourself to prove just from this axiom, that rational scalars pull out, any rational number has to pull out of that function. But real numbers, not necessarily. And these are the counter examples. So it's a good place at that level when you're first learning math, to realize that there are really subtle issues of what we really think truth means when we're beginning to have these conversations
EL: Nice. And what is your theorem pairing?
CC: My theorem pairing, I'm going to pair it with artichokes.
EL: Okay.
CC: I think that artichokes also had a bad rap for a lot of time, for a long time. You should also look at the artichoke war, if you've never heard of it, a great piece of history of New York City, and it took a long time for people to really understand that these prickly, weird looking vegetables can actually be delicious if approached from the right perspective.
EL: Nice. Well, thank you.
Ellie Dannenburg: So I'm Ellie Dannenberg, and I am visiting assistant professor at Pomona College in Claremont, California. And my favorite theorem is the Koebe-Andreev-Thurston circle packing theorem, which says that if you give me a triangulation of a surface, that I can find you exactly one circle packing where the vertices of your triangulation correspond to circles, and an edge between two vertices says that those circles are tangent.
EL: Okay, so this seems site kind of related to Voronoi things? Maybe I'm totally going in a wrong direction.
ED: So, I know that these are—so I don't think they're exactly related.
EL: Okay. Nevermind. Continue!
ED: Okay. But, right, it’s cool because the theorem says you can find a circle packing if I hand you a triangulation. But what is more exciting is you can only find one. So that's it.
EL: Oh, huh. Cool. All right. And do you have something that you would like to pair with this theorem?
ED: So I will pair this theorem with muhammara, which is this excellent Middle Eastern dip made from walnuts and red peppers and pomegranate molasses that is delicious and goes well with anything.
EL: Okay. Well, it's a good pairing. My husband makes a very good version. Yeah. Thank you.
ED: Thank you.
Manuel González Villa: This is Manuel González Villa. I'm a researcher in CIMAT [Centro de Investigación en Matemáticas] in Guanajuato, Mexico, and my favorite theorem is the Newton-Puiseux theorem. This is a generalization of implicit function theorem but for singular points of algebraic curves. That means you can parameterize a neighborhood of a singular point on an algebraic curve with a power series expansion, but with rational exponents, and the denominators of those exponents are bounded. The amazing thing about this theorem is that it’s very old. It comes back from Newton. But some people will still use it in research. I learned this theorem in Madrid where I made my PhD from a professor call Antonia Díaz-Cano. And also I learned with the topologist José María Montesinos to apply this theorem. It has some high-dimensional generalizations for some type of singularities, which are called quasi-ordinary.
The exponents—so you get a power series, so you get an infinite number of exponents. But there is a finite subset of those exponents which are the important ones, because they codify all the topology around the singular point of the algebraic curve. And this is why this theorem is very important. And the book I learned it from is Robert Walker’s Algebraic Curves. And if you want a more recent reference, I recommend you to look at Eduardo Casas-Alvero’s book on singularities of plane curves. Thank you very much.
EL: Okay.
EL: Yeah. So can you introduce yourself?
JoAnne Growney: My name is JoAnne Growney. I'm a retired math professor and a poet.
EL: And what is your favorite theorem?
JG: Well, the last talk I went to has had me debating about it. What I was prepared to say an hour ago was that it was the proof by contradiction that the real numbers are countable, and Cantor's diagonal proof. I like proofs by contradiction because I kind of like to think that way: on the one hand, and then the opposite. But I just returned from listening to a program on math and art. And I thought, wow, the Pythagorean theorem is something that I use every day. And maybe I'm being unfair to take something about infinity instead of something practical, but I like both of them.
EL: Okay, so we've got a tie there. And have you chosen something to pair with either of your theorems? We like to do, like, a wine and food pairing or, you know, but with theorems, you know, is there something that you think goes especially well, for example a poem, if you’ve got one.
JG: Well, actually, I was thinking of—the Pythagorean theorem, and it's probably a sound thing, made me think of a carrot.
EL: Okay.
JG: And oh, the theorem about infinity, it truly should make me think of a poem, but I don't have a pairing in mind.
EL: Okay. Well, thank you.
JG: Thank you.
Mikael Vejdemo-Johansson: I’m Michael Vejdemo-Johansson. I'm from the City University of New York.
KK: City University of New York. Which one?
MVJ: College of Staten Island and the Graduate Center.
KK: Excellent. All right, so we're sitting in an Afghan restaurant at the JMM. And what is your favorite theorem?
MVJ: My favorite theorem is the nerve lemma.
KK: Okay, so remind everyone what this is.
MVJ: So the nerve lemma says—well, it’s basically a family of theorems, but the original one as I understand it says that if you have a covering of a topological space where all the cover elements and all arbitrary intersections of cover elements are simple enough, then the intersection complex, the nerve complex of the covering that inserts a simplex for each nonlinear intersection is homotopy equivalent to the whole space.
KK: Right. This is extremely important in topology.
MVJ: It fuels most of topological data analysis one way or another.
KK: Absolutely. Very important theorem. So what pairs well, with the nerve lemma?
MVJ: I’m going to go with cotton candy.
KK: Cotton candy. Okay, why is that?
MVJ: Because the way that you end up collapsing a large and fluffy cloud of sugar into just thick, chewy fibers if you handle it right.
KK: That's right. Okay. Right. This pairing makes total sense to me. Of course, I’m a topologist, so that helps. Thanks for joining us, Mikael.
MVJ: Thank you for having me.
Michelle Manes: I’m Michelle Manes. I'm a professor at the University of Hawaii. And my favorite theorem is Sharkovskii’s theorem, which is sometimes called period three implies chaos. So the statement is very simple. You have a weird ordering of the natural numbers. So 3 is bigger than 5 is bigger than 7 is bigger than 9, etc, all the odd numbers. And then those are all bigger than 2 times 3 is bigger than 2 times 5 is bigger than 2 times 7, etc. And then down a row 4 times every odd number, and you get the idea. And then everything with an odd factor is bigger than every power of 2. And the powers of 2 are listed in decreasing order. So 23 is bigger than 22 is bigger than 2 is bigger than 1.
EL: Okay.
MM: So 1 is the smallest, 3 is the biggest, and you have this big weird array. And the statement says that if you have a continuous function on the real line, and it has a point of period n, for n somewhere in the Sharkovskii ordering, so put your finger down on n, it’s got a point of period everything less than n in that ordering. So in particular, if it has a point of period 3, it has points of every period, every integer. So I mean, I like the theorem, because the hypothesis is remarkable. The hypothesis is continuity. It's so minimal.
EL: Yeah.
MM: And you have this crazy ordering. And the conclusion is so strong. And the proof is just really lovely. It basically uses the intermediate value theorem and pretty pictures of folding the real line back on itself and things like that.
EL: Oh, cool.
MM: So yeah, it's my favorite theorem. Absolutely.
EL: Okay. And do you have something that you would suggest pairing with this theorem?
MM: So for me, because when I think of the theorem, I think of the proof of it, which involves this, like stretching and wrapping and stretching and wrapping, and an intermediate value theorem, it feels very kinetic to me. And so I feel like it pairs with one of these kind of moving sculptures that moves in the wind, where things sort of flow around.
EL: Oh, nice.
MM: Yeah, it feels like a kinetic theorem to me. So I'm going to start with the kinetic sculpture.
EL: Okay. Thank you.
MM: Thanks.
John Cobb: Hey there, I’m John Cobb, and I'm going to tell you my favorite theorem.
EL: Yeah. And where are you?
JC: I’m at College of Charleston applying for PhD programs right now.
EL: Okay.
JC: Okay. So I picked one I thought was really important, and I'm surprised it isn't on the podcast already. I have to say it's Gödel’s incompleteness theorems. Partly because for personal reasons. I'm in a logic class right now regarding the mechanics of the actual proof. But when I heard it, I was becoming aware of the power of mathematics, and hearing the power of math to talk about its own limitations, mathematics about mathematics, was something that really solidified my journey into math.
EL: And so what have you chosen to pair with your theorems?
JC: Yeah, I was unprepared for this question. So I’m making up on the spot.
EL: So you would say your your preparation was…incomplete?
JC: [laughing] I would say that! Man. I'll go with the crowd favorite pizza for no reason in particular.
EL: Well pizza is the best food and it's good with everything.
JC: Yeah.
EL: So that's a reason enough.
JC: Awesome. Well, thank you for the opportunity.
EL: Yeah, thanks.
Talia Fernós: My name is Talia Fernós, and I'm an associate professor at the University of North Carolina at Greensboro. My favorite theorem is Riemann’s rearrangement theorem. And basically, what it says is that if you have a conditionally convergent series, you can rearrange the terms in the series that the series converges to your favorite number.
EL: Oh, yeah. Okay, when you said the name of it earlier, I didn't remember, I didn't know that was the name of the theorem. But yes, that's a great theorem!
TF: Yeah. So the proof basically goes as follows. So if you do this with, for example, the series which is 1/n times -1 to, say, the n+1, so that looks like 1-1/2+1/3-1/4, and so on. So when you try to see why this is itself convergent, what you'll see is that you jump forward 1, then back a half, and then forward a third, back a fourth, so if you kind of draw this on the board, you get this spiral. And you see that it very quickly, kind of zooms in or spirals into whatever the limit is.
So now, this is conditionally convergent, because if you sum just 1/n, this diverges. And you can use the integral test to show that. So now, if you have a conditionally convergent series, you will have necessarily that it has infinitely many positive terms and infinitely many negative terms. And that each of those series independently also diverge. So when you want to show that a rearrangement is possible, so that it converges to your favorite number, what you're going to do is, let's say that you're trying to make this converge to 1, okay? So you're going to add up as many positive terms as necessary, until you overshoot 1, and then as many negative terms as necessary until you undershoot, and you continue in this way until you kind of have again, this spiraling effect into 1. And now the reason why this does converge is that the fact that it's conditionally convergent also tells you that the terms go to zero. So you can add sort of smaller and smaller things.
EL: Yeah, and you you don't run out of things to use.
TF: Right.
EL: Yeah. Cool. And what have you chosen to pair with this theorem?
F: For its spiraling behavior, escargot, which I don't eat.
EL: Yeah, I have eaten it. I don't seek it out necessarily. But it is very spiraly.
TF: Okay. What does it taste like?
EL: It tastes like butter and parsley.
TF: Okay. Whatever it’s cooked in.
EL: Basically. It's a little chewy. It's not unpleasant. I don't find a terribly unpleasant, but I don’t
TF: think it's a delicacy.
EL: Yeah. But I'm not very French. So I guess that's fair. Well, thanks.
TF: Sure.
This episode of my favorite theorem is a whirlwind of “flash favorite theorems” we recorded at the Joint Mathematics Meetings in Baltimore in January 2019. We had 16 guests, so we’ll keep this brief. Below is a list of our guests and their theorems with timestamps for each guest in case you want to skip around in the episode. We hope you enjoy this festival of theorem love as much as we enjoyed talking to all of these mathematicians!
1:58 Eric Sullivan from Carroll College in Montana loves the Cauchy-Riemann equations.
3:48 Courtney Gibbons from Hamilton College in New York loves Hilbert’s Nullstellensatz.
5:08 Shelley Kandola from the University of Minnesota loves the Banach-Tarski paradox.
7:20 David Plaxco from Clayton State University in Georgia loves a group theory exercise that helped him with his dissertation.
9:40 Terence Tsui from Oxford University in the UK loves a probabilistic proof of the equivalence of two forms of the Riemann zeta function.
12:14 Courtney Davis from Pepperdine University in California loves the SIR model.
14:25 Jenny Kenkel from the University of Utah loves the isomorphism between the top local cohomology of a Gorenstein ring and the injective hull of its residue field.
15:14 Dan Daly of Southeast Missouri State University loves the classification of finite simple groups.
16:42 Charlie Cunningham of Haverford College in Pennsylvania loves Cauchy’s functional equation.
20:38 Ellie Dannenberg of Pomona College in California loves the Koebe-Andreev-Thurston circle packing theorem.
22:15 Manuel González Villa of CIMAT (Centro de Investigación en Matemáticas) in Guanajuato, Mexico loves the Newton-Puiseux theorem.
24:08 JoAnne Growney, a retired math professor and current math poet, loves the Pythagorean theorem.
25:54 Mikael Vejdemo-Johansson of CUNY loves the nerve lemma.
27:27 Michelle Manes of the University of Hawaii loves Sharkovskii’s theorem.
29:38 John Cobb of the College of Charleston loves Gödel’s incompleteness theorems.
31:04 Talia Fernós of the University of North Carolina at Greensboro loves Riemann’s rearrangement theorem.
More description
Kevin Knudson: 1-2-3
Kevin Knudson and Evelyn Lamb: Welcome to My Favorite Theorem!
KK: Okay, good.
EL: Yeah.
[Theme music]
KK: So we’re at the JMM.
EL: Yeah, we’re here at the Joint Math Meetings. They’re in Baltimore this year. The last time I was at the Joint Meetings in Baltimore I got really sick, but so far I seem to not be sick.
KK: That’s good. You’ve only been here a couple of days, though.
EL: Yeah. There’s still time.
KK: Yeah, so I’ve only been to the Joint Meetings one other time in my life, 20 years ago as a postdoc in Baltimore. I’ve just got a thing for Baltimore, I guess.
EL: Yeah, I guess so.
KK: So people may have seen this on Twitter. Fun fact: this is our first time meeting in person.
EL: Yeah.
KK: And you’re every bit as charming in real life as you are over video.
EL: And you’re taller than I expected because my first approximation of all humans is that they are my height, and you are not my height.
KK: But you’re not exceptionally short.
EL: No.
KK: You’re actually above average height, right?
EL: I’m about average for a woman, which makes me below average for humans.
KK: Well, if we’re going to the Netherlands, for example, I’m below average for the Netherlands.
EL: Yes.
KK: So I’m actually leaving today. I was only here for a couple of days. I was here for the department chairs workshop. You’re here through when?
EL: I’m leaving on Friday, tomorrow. Yeah, while we’ve been here we’ve been collecting flash favorite theorems where people have been telling us about their favorite theorem in a small amount of time. So yeah, we’re excited to share those with you.
KK: Yeah, this is going to be a good compilation. I’m going to try to get a couple more before I leave town. We’ll see what happens.
EL: Yeah. All right.
KK: Enjoy.
EL: I am here with Eric Sullivan. Can you tell us a little bit about yourself?
Eric Sullivan: Yeah, I'm an associate professor at Carroll College in Helena, Montana, lover of all things mathematics.
EL: And here with me in the Salt Lake City Airport, I assume catching a connecting flight to the Joint Math Meetings.
ES: You got it.
EL: All right, and what is your favorite theorem, or the favorite theorem you'd like to tell me about right now?
ES: Oh, I have many favorite theorems, but the one that's really coming to mind right now, especially since I'm teaching complex analysis this semester, are the Cauchy-Riemann equations.
EL: Very nice.
ES: Giving us a beautiful connection between analytic functions, and ultimately, harmonic functions. Really lovely. And it seems like a mystery to my students when they first see it, but it's beautiful math.
EL: Yeah, it is. They are kind of mysterious, even after you've seen them for a while. It's like, why does this balance so beautifully?
ES: Right? And the way you get there with the limit, so I'm just going to take the limit going one way, then I’ll take the limit going the other way and voila, out comes these beautiful partial differential equations.
EL: Yeah, very lovely. And I know I'm putting you on the spot. But do you have a pairing for this theorem?
ES: Ooh, a pairing? Oh boy, something was a very complex taste. Maybe chili.
EL: Okay.
ES: I’ll say chili because there's all sorts of flavors mixed in with chili, and complex analysis seems to mix all sorts of flavors together.
EL: All right, I like it. Well, thank you. This is the first lightning My Favorite Theorem I'm recording so far at the joing meetings, or even before, on the way, so yeah, thanks for joining me.
Courtney Gibbons: I'm Courtney Gibbons. I'm a professor at Hamilton College in upstate New York. And my favorite theorem is Hilbert’s Nullstellensatz, which translates to zero point theorem, but if you run it through Google Translate, it's actually quite beautiful. It's like the “empty star theorem” or something like that. It's very astronomical. And I love this theorem because it's one of those magical theorems that connect one area that I love, algebra, to another area that I don't really understand, but would like to love, geometry. And I find that in my classes, when I ask someone, “What's a parabola?” I have a handful of students who do some sort of interpretive dance. And I have a handful of students were like, “Oh, it's like y equals some x squared stuff.” And I'm like, “I'm with you.” I think of the equation. And some people think of the curve, the plot, and that's the geometric object, and the Nullstellensatz tells you how to take ideals and relate them to varieties. So it connects algebra and geometry. And it's just gorgeous, and the proof is gorgeous, and everything about it is wonderful, and David Hilbert was wonderful. And if I were going to pair it with something, I’d probably pair it with a trip to an observatory, so that you could go appreciate the beauty of the stars, and think about the wonderful connectedness of all of mathematics and the universe. And maybe you should have, like, a beer or something too.
EL: Why not?
CG: Yeah. Why not? Exactly.
EL: Good. Well, thank you. Absolutely.
KK: All right, JMM flash theorem time. Introduce yourself, please.
Shelley Kandola: Hi. My name is Shelly Kandola. I'm a grad student at the University of Minnesota.
KK: And it’s warmer here than where you are usually.
SK: Yeah, it's 15 degrees in Minnesota right now.
KK: That’s awful.
SK: Yeah.
KK: Well anyway, we’ve got to be quick here. What's your favorite theorem?
SK: The Banach-Tarski paradox.
KK: This is an amazing result that I still don't really understand and I can't wrap my head around.
SK: Yeah, you've got a solid sphere, a filled-in S2, and you can cut it into four pieces using rigid motions, and then put them back together and get two solid spheres that are the same size as the original.
KK: Well, theoretically, you can do this, right? This isn't something you can actually do, is it?
SK: Physically no, but with the power of group theory, yes.
KK: With the power of group theory.
SK: The free group on two generators.
KK: Why do you like this theorem so much?
SK: So I like it because it was the basis of my senior research project in college.
KK: It just seems so weird it was something you should think about?
SK: Yeah, it intrigued me. It's a paradox. And it's the first theorem I dove really deep into, and we found a way to generalize it to arbitrarily many dimensions with one tweak added.
KK: Cool. So what does one pair with the Banach-Tarski paradox?
SK: One of my favorite Futurama episodes. There's this one episode where there's a Banach-Tarski duplicator, and Bender jumps into the duplicator, and he makes two more, and he wants to build an army of himself.
KK: Sure.
SK: But every time he jumps in, the two copies that come out, are half the size of the original. He ends up with an army of nanobots. It contradicts the whole statement of the paradox that you're getting two things back that are the same size as the original.
KK: Although an army of Benders might be fun.
SK: Yeah, they certainly wreak havoc.
KK: Don’t we all have a little inner Bender?
SK: Oh yeah. He's powered by beer.
KK: Well, thanks for joining us. You gave a really good talk this morning.
SK: Thanks.
KK: Good luck.
SK: Thank you for having me.
KK: Sure.
David Plaxco: My name is David Plaxco. I'm a math education researcher at Clayton State University. And my favorite theorem is really more of an exercise, I think most people would think. It's proving that the set of all elements in a group that conjugate with a fixed element is a subgroup of the group. I'll tell you why. Because in my dissertation, that exercise was the linchpin in understanding how students can learn by proving.
EL: Okay.
DP: So I was working with a student. He had read ahead in the textbook and knew that not all groups are commutative, so you can't always commute any two elements you feel like. And he generalized this to thinking about inverses. He didn’t think that every inverse was necessarily two-sided, which in a group you are. Anyway, so he was trying to prove that that set was a subgroup and came to this impasse because he wanted to left cancel and right cancel with inverses and could only do them on one side. And then he started to question, like, maybe I'm just crazy, like maybe you can use the same inverse on both sides. And then he proved it himself using associativity. So he made, like, I call it John’s lemma, he came up with this kind of side proof to show that, well, if you're associative and you have a right inverse and a left inverse, then those have to be the same. And then he came back and was able to left and right cancel at free will any inverse, and then proved that it was a subgroup, so through his own proof activity, he was able to change his own conceptual understanding about what it means to be an inverse, like how groups work, all these things, and it gave him so much more power moving forward. So that's how that theorem became my favorite theorem because it gave me insight into how individuals can learn.
EL: Nice. And do you have a pairing for this theorem?
DP: My diploma. Because it helped me get it.
EL: That seems appropriate. Thanks.
DP: Thanks.
Terence Tsui: So I'm Terrance, and I'm currently a final year undergraduate studying in Oxford. My favorite theorem is actually a really elegant proof of Euler’s identity on the Riemann zeta function. We all know that the Riemann zeta function is defined in a way of the sum of 1/ks where k runs across all the natural numbers. But at the same time, Euler has given a really good other formulation: we say status is the same as the infinite product of 1-1/ps, where p runs across the primes. And then it's really interesting, because if you look at you see, on one hand, an infinite some, and on the other hand, you have an infinite product. And it’s very rare that we see that infinite sums and infinite products actually coincide. And they’re only there because it is a function that actually works on nearly every s larger than 1. And that means that this beautiful, elegant identity actually runs correct for infinitely many values. And the most interesting thing about this theorem is that the proof to it could be done probabilistically, where we consider some certain particular events, and we realize that the Riemann zeta series sum is actually equivalent to finding a certain intersection of infinitely many independent events. And first it is just an infinite product of certain events. And first we have the Riemann zeta function equalling a particular infinite product. And I think that is something that is really out of out of our imagination, because not only does it link two things—a sum to an infinite product, but at the same time, the way that it proves it comes from somewhere we could not even imagine, which is from probability. So if I need to pair this theorem with something, I would say it’s like a spider web, because you can see that there's very intricate connections and that things connect to each other, but in the most mysterious ways.
ELL: Cool. Well, thanks.
TT: Thank you.
Courtney Davis: So Hi, I'm Courtney Davis. I am an associate professor at Pepperdine University out in LA.
EL: Okay. And I hear that we have a favorite model, not a favorite theorem from you.
CD: Yes. So I'm a math biologist. So I'm going to say the obvious one, which is SIR modeling, because it is the entry way into getting to do this cool stuff. It’s the way that I get to show students how to write models. It's the first model I ever saw that had biology in it. And it's something that is ubiquitous and used widely. And so despite being the first thing everyone learns, it's still the first thing everyone learns. And that's what makes it interesting to me.
EL: Yeah. And and can you kind of just sum up in a couple sentences what this model is, what SIR means?
CD: Yeah. So SIR is you are modeling the spread of disease through a susceptible (S) population through infected and into recovered or immune, and you can change that up quite a lot. There are a lot of different ways to do it. It's not one fixed model. And it's all founded on the very simple premise that when two individuals run into each other in a population, that looks like multiplication. And so you can take multiplication, and with that build all the interactions that you really need, in order to capture what's actually happening in a population that at least is well mixed, so that you have a big room of people moving around about it, for instance.
EL: Okay. And I'm going to spring something on you, which is that usually we pair something with our theorem, or in this case model, so we have our guests, you know, choose a food, beverage, piece of art, or anything. Is there anything that you would suggest that pairs well with SIR?
CD: With an SIR model, I would say, a paint gun.
EL: Okay.
CD: I don't know that that's what you're looking for.
EL: That’s great.
CD: Simply because running around and doing pandemic games or other such things is also a common way to get data on college campuses so that you can introduce students, and they can parameterize their models by paint guns or water guns or something like that.
EL: Oh, cool. I like it. Thank you.
CD: Absolutely. Thank you.
Jenny Kenkel: I’m Jenny Kenkel. I'm a graduate student at the University of Utah. I study commutative algebra. My favorite theorem is this isomorphism between a particular local cohomology module and an injective module: The top local cohomology of a Gorenstein ring is isomorphic to the injective hull of its residue field. But I was thinking that maybe it would pair really well with like, a dark chocolate and a sharp cheddar, because these two things are isomorphic, and you would never expect that. But then they go really well together, just in the same way that I think a dark chocolate and a sharp cheddar seem kind of like a weird pairing, but then it's amazing. Also, they're both beautiful.
EL: Nice, thank you.
JK: Thank you.
Dan Daly: My name is Dan Daly. And I am the interim chair of the Department of Mathematics at Southeast Missouri State University.
KK: Southeast—is that in the boot?
DD: That is close to the boot heel. It's about two hours south of St. Louis.
KK: Okay. I'm a Cardinals fan. So I'm ready, we’ve got something here. So what's your favorite theorem?
DD: So my favorite theorem is actually the classification of finite simple groups.
KK: That’s a big theorem.
DD: That is a very big, big,
KK: Like 10,000 pages of theorem.
DD: At least
KK: Yeah. So what draws you to this? Is it your area?
DD: So I am interested in algebraic and combinatorics, and I am generally interested in all things related to permutations.
KK: Okay.
DD: And one of the things that drew me to this theorem is that it's such an amazing, collaborative effort and one of the landmarks of 20th century mathematics.
KK: Big deal. Yeah.
DD: And, you know, it just to me, it seems such a such an amazing result that we can classify these building blocks of finite groups.
KK: Right. So what does one pair this with?
DD: So I think since it's such a collaborative effort, I'm going to pair it with Louvre museum.
KK: The Louvre, okay.
DD: Because it's a collection of all of these different results that are paired together to create something that is really, truly one of a kind.
KK: I’ve never been. Have you?
DD: I have. It’s a wonderful place. Yeah. It’s a fabulous place. One of my favorite places.
KK: I’m going to wait until I can afford to rent it out like Beyonce and Jay Z.
DD: Yeah, right.
KK: All right, well thanks, Dan. Enjoy your time at the Joint Math Meetings.
DD: All right, thank you much.
Charlie Cunningham: My name's Charlie Cunningham. I'm visiting assistant professor at Haverford College. And my area of research originally is, or still is, geometric group theory. But the theorem that I want to talk about was a little bit closer to set theory, which is I want to talk about the existence of solutions to Cauchy’s functional equation.
EL: Okay. And what is Cauchy’s functional equation?
CC: So Cauchy’s functional equation is a really basic sort of thing you can ask about a function. It's asking, all right, you take the real numbers, and you ask is there—what are the functions from the real numbers to the real numbers where if you add two numbers together, and then apply the function, it's the same thing as applying the function to both of those numbers and then adding them together?
EL: Okay. So kind of like you're naive student and wanting to—how a function should behave.
CC: Yes. Right. So this would come up in a couple of places. So if you’ve taken linear algebra, that's the first axiom of a linear function. It doesn't ask about the scaling part. It's just the additive part. And if you've done group theory, it's a fancy way, is it's all the homomorphisms from the real numbers to themselves, an additive group. So the theorem basically, is that well, well, first of all, the question is, well, there are some obvious ones. There are all the functions where you just multiply by a fixed number, all the linear functions you’d know from linear algebra, like 2 times x, 3 times x, or π times x, any real number times x. So the question is, are there any others? Or are those the only functions that exists at all that satisfy this equation? And the theorem turns out that the answer depends on the fundamental axioms you take for mathematics.
EL: Wow. Okay.
CC: Right. So the answer is just to use a little bit of set theory, that if you are working in a set theory, which most mathematicians do, that has something called the axiom of choice in it, then the answer is no, there are lots and lots and lots of other functions that satisfy this equation, other than those obvious ones, but they're almost impossible to think about or write down. They're not continuous anywhere, they are not differentiable anywhere. They're not measurable, if anyone knows what that means. Their graph, if you tried to draw them, are dense in the entire plane, which means any little circle you draw on the plane intersects of the graph somewhere. They still pass the vertical line test. They’re still functions that are well-defined. And I really like this theorem. One reason is because it's a really great place for math students to learn that there isn't always one right answer in math. Sometimes the answer to very reasonably posed questions isn't true or false. It depends on the fundamental universe we’re working in. It depends on the what we all sit down and agree are the starting rules of our system. And it's a sort of question where you wouldn't realize that those sorts of considerations would come up. It also comes up—When I've asked linear algebra students, it's equivalent to the statement are both parts of the definition of a linear function actually necessary? We usually give them to you as two pieces: one, it satisfies this, and the other is scalars pull out. Do we actually need that second part? Can we prove that scalars pull out just from the first part? And this is the only way to prove the answer's no. It's a good exercise to try yourself to prove just from this axiom, that rational scalars pull out, any rational number has to pull out of that function. But real numbers, not necessarily. And these are the counter examples. So it's a good place at that level when you're first learning math, to realize that there are really subtle issues of what we really think truth means when we're beginning to have these conversations
EL: Nice. And what is your theorem pairing?
CC: My theorem pairing, I'm going to pair it with artichokes.
EL: Okay.
CC: I think that artichokes also had a bad rap for a lot of time, for a long time. You should also look at the artichoke war, if you've never heard of it, a great piece of history of New York City, and it took a long time for people to really understand that these prickly, weird looking vegetables can actually be delicious if approached from the right perspective.
EL: Nice. Well, thank you.
Ellie Dannenburg: So I'm Ellie Dannenberg, and I am visiting assistant professor at Pomona College in Claremont, California. And my favorite theorem is the Koebe-Andreev-Thurston circle packing theorem, which says that if you give me a triangulation of a surface, that I can find you exactly one circle packing where the vertices of your triangulation correspond to circles, and an edge between two vertices says that those circles are tangent.
EL: Okay, so this seems site kind of related to Voronoi things? Maybe I'm totally going in a wrong direction.
ED: So, I know that these are—so I don't think they're exactly related.
EL: Okay. Nevermind. Continue!
ED: Okay. But, right, it’s cool because the theorem says you can find a circle packing if I hand you a triangulation. But what is more exciting is you can only find one. So that's it.
EL: Oh, huh. Cool. All right. And do you have something that you would like to pair with this theorem?
ED: So I will pair this theorem with muhammara, which is this excellent Middle Eastern dip made from walnuts and red peppers and pomegranate molasses that is delicious and goes well with anything.
EL: Okay. Well, it's a good pairing. My husband makes a very good version. Yeah. Thank you.
ED: Thank you.
Manuel González Villa: This is Manuel González Villa. I'm a researcher in CIMAT [Centro de Investigación en Matemáticas] in Guanajuato, Mexico, and my favorite theorem is the Newton-Puiseux theorem. This is a generalization of implicit function theorem but for singular points of algebraic curves. That means you can parameterize a neighborhood of a singular point on an algebraic curve with a power series expansion, but with rational exponents, and the denominators of those exponents are bounded. The amazing thing about this theorem is that it’s very old. It comes back from Newton. But some people will still use it in research. I learned this theorem in Madrid where I made my PhD from a professor call Antonia Díaz-Cano. And also I learned with the topologist José María Montesinos to apply this theorem. It has some high-dimensional generalizations for some type of singularities, which are called quasi-ordinary.
The exponents—so you get a power series, so you get an infinite number of exponents. But there is a finite subset of those exponents which are the important ones, because they codify all the topology around the singular point of the algebraic curve. And this is why this theorem is very important. And the book I learned it from is Robert Walker’s Algebraic Curves. And if you want a more recent reference, I recommend you to look at Eduardo Casas-Alvero’s book on singularities of plane curves. Thank you very much.
EL: Okay.
EL: Yeah. So can you introduce yourself?
JoAnne Growney: My name is JoAnne Growney. I'm a retired math professor and a poet.
EL: And what is your favorite theorem?
JG: Well, the last talk I went to has had me debating about it. What I was prepared to say an hour ago was that it was the proof by contradiction that the real numbers are countable, and Cantor's diagonal proof. I like proofs by contradiction because I kind of like to think that way: on the one hand, and then the opposite. But I just returned from listening to a program on math and art. And I thought, wow, the Pythagorean theorem is something that I use every day. And maybe I'm being unfair to take something about infinity instead of something practical, but I like both of them.
EL: Okay, so we've got a tie there. And have you chosen something to pair with either of your theorems? We like to do, like, a wine and food pairing or, you know, but with theorems, you know, is there something that you think goes especially well, for example a poem, if you’ve got one.
JG: Well, actually, I was thinking of—the Pythagorean theorem, and it's probably a sound thing, made me think of a carrot.
EL: Okay.
JG: And oh, the theorem about infinity, it truly should make me think of a poem, but I don't have a pairing in mind.
EL: Okay. Well, thank you.
JG: Thank you.
Mikael Vejdemo-Johansson: I’m Michael Vejdemo-Johansson. I'm from the City University of New York.
KK: City University of New York. Which one?
MVJ: College of Staten Island and the Graduate Center.
KK: Excellent. All right, so we're sitting in an Afghan restaurant at the JMM. And what is your favorite theorem?
MVJ: My favorite theorem is the nerve lemma.
KK: Okay, so remind everyone what this is.
MVJ: So the nerve lemma says—well, it’s basically a family of theorems, but the original one as I understand it says that if you have a covering of a topological space where all the cover elements and all arbitrary intersections of cover elements are simple enough, then the intersection complex, the nerve complex of the covering that inserts a simplex for each nonlinear intersection is homotopy equivalent to the whole space.
KK: Right. This is extremely important in topology.
MVJ: It fuels most of topological data analysis one way or another.
KK: Absolutely. Very important theorem. So what pairs well, with the nerve lemma?
MVJ: I’m going to go with cotton candy.
KK: Cotton candy. Okay, why is that?
MVJ: Because the way that you end up collapsing a large and fluffy cloud of sugar into just thick, chewy fibers if you handle it right.
KK: That's right. Okay. Right. This pairing makes total sense to me. Of course, I’m a topologist, so that helps. Thanks for joining us, Mikael.
MVJ: Thank you for having me.
Michelle Manes: I’m Michelle Manes. I'm a professor at the University of Hawaii. And my favorite theorem is Sharkovskii’s theorem, which is sometimes called period three implies chaos. So the statement is very simple. You have a weird ordering of the natural numbers. So 3 is bigger than 5 is bigger than 7 is bigger than 9, etc, all the odd numbers. And then those are all bigger than 2 times 3 is bigger than 2 times 5 is bigger than 2 times 7, etc. And then down a row 4 times every odd number, and you get the idea. And then everything with an odd factor is bigger than every power of 2. And the powers of 2 are listed in decreasing order. So 23 is bigger than 22 is bigger than 2 is bigger than 1.
EL: Okay.
MM: So 1 is the smallest, 3 is the biggest, and you have this big weird array. And the statement says that if you have a continuous function on the real line, and it has a point of period n, for n somewhere in the Sharkovskii ordering, so put your finger down on n, it’s got a point of period everything less than n in that ordering. So in particular, if it has a point of period 3, it has points of every period, every integer. So I mean, I like the theorem, because the hypothesis is remarkable. The hypothesis is continuity. It's so minimal.
EL: Yeah.
MM: And you have this crazy ordering. And the conclusion is so strong. And the proof is just really lovely. It basically uses the intermediate value theorem and pretty pictures of folding the real line back on itself and things like that.
EL: Oh, cool.
MM: So yeah, it's my favorite theorem. Absolutely.
EL: Okay. And do you have something that you would suggest pairing with this theorem?
MM: So for me, because when I think of the theorem, I think of the proof of it, which involves this, like stretching and wrapping and stretching and wrapping, and an intermediate value theorem, it feels very kinetic to me. And so I feel like it pairs with one of these kind of moving sculptures that moves in the wind, where things sort of flow around.
EL: Oh, nice.
MM: Yeah, it feels like a kinetic theorem to me. So I'm going to start with the kinetic sculpture.
EL: Okay. Thank you.
MM: Thanks.
John Cobb: Hey there, I’m John Cobb, and I'm going to tell you my favorite theorem.
EL: Yeah. And where are you?
JC: I’m at College of Charleston applying for PhD programs right now.
EL: Okay.
JC: Okay. So I picked one I thought was really important, and I'm surprised it isn't on the podcast already. I have to say it's Gödel’s incompleteness theorems. Partly because for personal reasons. I'm in a logic class right now regarding the mechanics of the actual proof. But when I heard it, I was becoming aware of the power of mathematics, and hearing the power of math to talk about its own limitations, mathematics about mathematics, was something that really solidified my journey into math.
EL: And so what have you chosen to pair with your theorems?
JC: Yeah, I was unprepared for this question. So I’m making up on the spot.
EL: So you would say your your preparation was…incomplete?
JC: [laughing] I would say that! Man. I'll go with the crowd favorite pizza for no reason in particular.
EL: Well pizza is the best food and it's good with everything.
JC: Yeah.
EL: So that's a reason enough.
JC: Awesome. Well, thank you for the opportunity.
EL: Yeah, thanks.
Talia Fernós: My name is Talia Fernós, and I'm an associate professor at the University of North Carolina at Greensboro. My favorite theorem is Riemann’s rearrangement theorem. And basically, what it says is that if you have a conditionally convergent series, you can rearrange the terms in the series that the series converges to your favorite number.
EL: Oh, yeah. Okay, when you said the name of it earlier, I didn't remember, I didn't know that was the name of the theorem. But yes, that's a great theorem!
TF: Yeah. So the proof basically goes as follows. So if you do this with, for example, the series which is 1/n times -1 to, say, the n+1, so that looks like 1-1/2+1/3-1/4, and so on. So when you try to see why this is itself convergent, what you'll see is that you jump forward 1, then back a half, and then forward a third, back a fourth, so if you kind of draw this on the board, you get this spiral. And you see that it very quickly, kind of zooms in or spirals into whatever the limit is.
So now, this is conditionally convergent, because if you sum just 1/n, this diverges. And you can use the integral test to show that. So now, if you have a conditionally convergent series, you will have necessarily that it has infinitely many positive terms and infinitely many negative terms. And that each of those series independently also diverge. So when you want to show that a rearrangement is possible, so that it converges to your favorite number, what you're going to do is, let's say that you're trying to make this converge to 1, okay? So you're going to add up as many positive terms as necessary, until you overshoot 1, and then as many negative terms as necessary until you undershoot, and you continue in this way until you kind of have again, this spiraling effect into 1. And now the reason why this does converge is that the fact that it's conditionally convergent also tells you that the terms go to zero. So you can add sort of smaller and smaller things.
EL: Yeah, and you you don't run out of things to use.
TF: Right.
EL: Yeah. Cool. And what have you chosen to pair with this theorem?
F: For its spiraling behavior, escargot, which I don't eat.
EL: Yeah, I have eaten it. I don't seek it out necessarily. But it is very spiraly.
TF: Okay. What does it taste like?
EL: It tastes like butter and parsley.
TF: Okay. Whatever it’s cooked in.
EL: Basically. It's a little chewy. It's not unpleasant. I don't find a terribly unpleasant, but I don’t
TF: think it's a delicacy.
EL: Yeah. But I'm not very French. So I guess that's fair. Well, thanks.
TF: Sure.
This episode of my favorite theorem is a whirlwind of “flash favorite theorems” we recorded at the Joint Mathematics Meetings in Baltimore in January 2019. We had 16 guests, so we’ll keep this brief. Below is a list of our guests and their theorems with timestamps for each guest in case you want to skip around in the episode. We hope you enjoy this festival of theorem love as much as we enjoyed talking to all of these mathematicians!
1:58 Eric Sullivan from Carroll College in Montana loves the Cauchy-Riemann equations.
3:48 Courtney Gibbons from Hamilton College in New York loves Hilbert’s Nullstellensatz.
5:08 Shelley Kandola from the University of Minnesota loves the Banach-Tarski paradox.
7:20 David Plaxco from Clayton State University in Georgia loves a group theory exercise that helped him with his dissertation.
9:40 Terence Tsui from Oxford University in the UK loves a probabilistic proof of the equivalence of two forms of the Riemann zeta function.
12:14 Courtney Davis from Pepperdine University in California loves the SIR model.
14:25 Jenny Kenkel from the University of Utah loves the isomorphism between the top local cohomology of a Gorenstein ring and the injective hull of its residue field.
15:14 Dan Daly of Southeast Missouri State University loves the classification of finite simple groups.
16:42 Charlie Cunningham of Haverford College in Pennsylvania loves Cauchy’s functional equation.
20:38 Ellie Dannenberg of Pomona College in California loves the Koebe-Andreev-Thurston circle packing theorem.
22:15 Manuel González Villa of CIMAT (Centro de Investigación en Matemáticas) in Guanajuato, Mexico loves the Newton-Puiseux theorem.
24:08 JoAnne Growney, a retired math professor and current math poet, loves the Pythagorean theorem.
25:54 Mikael Vejdemo-Johansson of CUNY loves the nerve lemma.
27:27 Michelle Manes of the University of Hawaii loves Sharkovskii’s theorem.
29:38 John Cobb of the College of Charleston loves Gödel’s incompleteness theorems.
31:04 Talia Fernós of the University of North Carolina at Greensboro loves Riemann’s rearrangement theorem.