Evelyn Lamb: Hello and welcome to My Favorite Theorem, a podcast where we ask mathematicians to tell us about their favorite theorems. I'm Evelyn Lamb. I'm one of your hosts. I am a freelance math and science writer in Salt Lake City, Utah. Here's your other host.
Kevin Knudson: Hi, I'm Kevin Knudson, professor of mathematics at the University of Florida. How's it going, Evelyn?
All right. I am making some bread right now, and it smells really great in my house. So slightly torturous because I won't be able to eat it for a while.
KK: Sure. So I make my own pizza dough. But I always stop at bread. I never that extra step. I don't. I don't know why. Are you making baguettes? Are you doing the whole...
EL: No. I do it in the bread machine.
EL: Because I'm not going to make, yeah, I'm not going to knead and shape a loaf. So that's the compromise.
KK: Oh. That that's the fun part. So I've had a sourdough starter for, the same one for at least three or four years now. I've kept this thing going. And I make my own pizza crust, but I'm just lazy with bread. I don't eat a lot of bread.
EL: So yeah. And joining us to talk about--on BreadCast today--I'm very delighted to welcome Cynthia Flores. So hi. Tell us a little bit about yourself.
Cynthia Flores. Oh, thanks for having me on the show. I'm so grateful to join you today, Kevin and Evelyn. Well, I'm an assistant professor of mathematics and applied physics at California State University Channel Islands in the Department of Math and Physics. The main campus is not located on the Channel Islands.
EL: Oh, that's a bummer.
CF: It's actually located in Camarillo, California. It's one hour south of Santa Barbara, one hour north of downtown Los Angeles, roughly.
But the math department does get to have an annual research retreat at the research station located in the Santa Rosa Island. So that's kind of neat.
KK: Oh, how terrible for you.
EL: Yeah. That must be so beautiful.
KK: I was in Laguna Beach about a week and a half ago, which is, of course, further south from there, but still just spectacularly beautiful. Really nice.
CF: Yeah, I feel really fortunate to have the opportunity to stay in the Southern California area. I did my PhD at UC Santa Barbara, where I studied the intersections of mathematical physics, partial differential equations, and harmonic analysis and has motivated what I'm going to talk about today.
KK: Good, good, good.
EL: Yeah. Well, and Cynthia was on another podcast I host, the Lathisms podcast. And I really enjoyed talking with her then about the some of the research that she does. And she had some fun stories. So yeah, what is your favorite theorem? What do you want to share with us today?
CF: I'm glad you asked. I have several favorite theorems, and it was really hard to pick, and my students have heard me say repeatedly that my favorite theorem is the fundamental theorem of calculus.
EL: Great theorem.
CF: It's also a very, I find, intimidating theorem to talk about on this series, especially with so many creative individuals pairing their favorite theorems with awesome foods and activities. And so I just thought that one was maybe something to live up to. And I wanted to start with something that's a little closer to to my research area. So I found myself thinking of other favorites, and there was one in particular that does happen to lie at the intersection of my research area, which is mathematical physics, PDEs and harmonic analysis. And it's known as Heisenberg's Uncertainty Principle. That's how it's really known by the physics community. And in mathematics, it's most often referred to as Heisenberg's Uncertainty Inequality.
CF: So, is it familiar? I don't know.
EL: I feel like I've heard of it, but I don't--I feel like I've only heard of it from kind of the pop science point of view, not from the more technical point of view. So I'm very excited to learn more about it.
KK: So I actually have a story here. I taught a course in mathematics and literature a couple years back with a friend of mine in the in the foreign languages department. And we watched A Serious Man, this Coen Brothers movie, which, if you haven't seen is really interesting. But anyway, one of the things I made sure to talk about was Heisenberg's Uncertainty Principle, because that's sort of one of the themes, and of course now I forgotten what the inequality is. But I mean, I remember it involves Planck's constant, and there's some probability distribution, so let's hear it.
CF: Mm hmm. Yeah, yeah. So I was, I was like, this is what I'm going to pair it with. Like, I'm going to pair the conversation, like the mathematics description, physical description, with, basically I was thinking of pairing it with something Netflix and chill-like. I'm really glad that you brought that up, and I'll tell you more in a little bit about what I'm pairing it with. But first, I'll start mathematically. Mathematically, the theorem could be stated as follows. Given a function with sufficient regularity and decay assumptions, the L2 norm of the function is less than or equal to 2 over the dimension the function's defined on, multiplied by the product of the L2 norm of its first moment and the L2 norm of its gradient. And so mathematically, that's the inequality.
This wasn't stated this way by Heisenberg in the 1920s, which I believe he was recognized with a Nobel Prize for later on. Physically, Heisenberg described this in different ways it could be understood. Uncertainty might be understood as the lack of knowledge of a quantity taken by an observer, for example, or to the uncertainty due to experimental inaccuracy, or ambiguity in some definition, or statistical spread, as, as Kevin mentioned.
And actually, I'm going to recommend to the listeners to go to YouTube. There's a YouTuber named Veritasium, I'm not sure if I'm pronouncing that correctly.
EL: Oh, yeah, yeah.
CF: Yeah, he has a four minute demonstration of the original thought by Heisenberg and an experiment having to do with lasers that basically tells us it's impossible to simultaneously measure the position and momentum of a particle with infinite precision. The infinite precision part would be referring to something that we might call certainty. So in the experiment, that the YouTuber is recreating, a laser is shown through two plates that form a slit, and the split is becoming narrower and narrower. The laser is shone through the slit and then projected onto a screen. And as the slit is made narrower, the spot on the screen, as expected, is also becoming narrower and narrower. And at some point--you know, Veritasium does a really good job of creating this sort of like little "what's going to happen" excitement-- just when the slit seems to completely disappear and become infinitesimally small, the expectation might be that the that the laser projecting onto the screen would disappear too, but actually at a certain point, when the slit is so narrow, it's about to close, the spot on the screen becomes wider. We see spread. And this is because the photons of light have become so localized that the slit and their horizontal momentum has to become less well defined. And this is a demonstration of Heisenberg's Uncertainty Principle. And so according to Heisenberg--and this is from one of his manuscripts, and I wish I would have written down which one, I'm just going to read it-- "at the instant of time when the position is determined, that is, at the instant when the photon is scattered by the electron, the electron undergoes a discontinuous change in momentum, and this change is the greater the smaller the wavelength of the light employed, in other words, the more exact determination of the position. At the instant at which the position of the electron is known, its momentum, therefore, can be known only up to magnitudes which correspond to that discontinuous change; thus, the more precisely the position is determined, the less precisely the momentum is known, and conversely." And there's also, so momentum and position were sort of the original context in which Heisenberg's Uncertainty Principle was stated, mainly for quantum mechanics. The inequality theorem that I presented was really from an introductory book to nonlinear PDEs, which is really what I study, nonlinear dispersive PDEs specifically. So you could use a lot of Fourier transforms and stuff like that.
But it has multiple variations, one of which is Heisenberg's Uncertainty Principle of energy and time, which more or less is going to tell you the same thing, which is going to bring me to my pairing. Can I can I share my pairing?
CF: My pairing--I want to pair this with a Netflix and chill evening with friends who enjoy the Adult Swim animated show Rick and Morty.
CF: And an evening where you also have the opportunity to discuss sort of deep philosophical questions about uncertainty and chaos. And so really the show's on Hulu, so you could watch the show. I don't know if either one of you are familiar with the show.
KK: Oh, yeah. Yeah. I have a 19 year old son, how could I not be?
CF: And actually my students were the ones that brought this show and this specific episode to my attention, and I watched it, and so I'll see a little bit about the show. It's a bit, it's somewhat inspired by The Back to the Future movies. It's a comedy about total reckless and epic space adventures with lots of dark humor and a lot of almost real science. I mean, Rick is a mad scientist and Morty is in many ways opposite to Rick. Morty is Rick's grandson and sidekick. Rick uses a portal gun that he created. It allows him and Morty to travel to different realities where they go on some fun adventures. And there are several references to formulas and theorems describing our everyday life. And so particularly the season premiere of season two, Rick and Morty pays homage to Heisenberg's Uncertainty Principle as well as Schrodinger's cat paradox and includes a mathematical proof of Rick's impression of his grandkids.
I won't spoil what he proves about them. I'll let the reader, the listeners, go check that out. But basically season one ended with Rick freezing time for everyone except for him and his two grandkids Morty and Summer. In this premiere, Rick unfreezes time and causes a disturbance in their reference timeline, and any uncertainty introduced by the three individuals gets them removed entirely from time and causes a split in reality into multiple simultaneous realities. The entire episode is following Heisenberg's Uncertainty Principle for energy and time and alluding to the concept from the quantum world that chaos is found in the distribution of energy levels of certain atomic systems. So I'm going to back it up a little bit. We talked about Heisenberg's Uncertainty Principle in terms of momentum and position. Heisenberg's Uncertainty Principle for energy and time is for simultaneous measurements of energy and time, and specifically that the distribution of energy levels and uncertainty and its measurement is a metaphor to chaos within the system. So within a time interval, it's not possible to measure chaos precisely. There has to be uncertainty in the measure, so that the product of the uncertainty and energy and the uncertainty and the time remains larger than h over 4π, the Planck's constant. In other words, you cannot have both, you cannot simultaneously have both small uncertainty in both measurements. In other words, lots of certainty, right? You have small uncertainty, you have lots of certainty.
So you can't have that both happening. So less chaos leads to more uncertainty and vice versa. Less than certainty (or more certainty) leads to more chaos. And so this, you know, this episode, if you watch it, seems to present the common misconception that more uncertainty leads to more chaos. And this is where I've thought about this really hard and even tried to find someone who put it nicely, maybe on a video, I couldn't. But I think--this is just my opinion--I think the writers really got it right on this episode, because the moment that the timeline merges in the episode is the moment when the main character Rick has given up on his chances of fixing a broken tool he was counting on for fixing the timeline. So in fact, in this episode, he's shown doing something which is unlike him. He's shown praying and asking God, or his maker, for forgiveness, you know, as the timeline is, as all of these realities are collapsing. And in my opinion of Rick, this is the largest amount of uncertainty he's ever displayed throughout this series. And this happens right at the moment that he restores the timeline and therefore reduces the chaos. So I really think the writers got it right on that.
EL: That's really neat.
CF: Yeah, yeah, I loved it. And so for me, I also find this as a perfect time to, you know, hang out with friends, Netflix and chill it up, and then afterwards talk, you know. I would really like to challenge the listeners to observe this phenomenon in their real life. And for some people, it might be a stretch. But to some extent, I think we observe Heisenberg's Uncertainty Principle in our daily lives, like in the sense that the more sure that we are about something, or the more plan that we've made for something, the more likely were observe chaos, right? The more things we've gotten planned out, the more things that are actually likely to go wrong. I get to see this at the university, right, with so many young minds planning out their future. And I really see that the more certain a student feels about their plan, the more likely they're going to feel chaos in their life, if things don't go according to plan. So I really enjoy Heisenberg's Uncertainty Principle on so many levels, mathematically, physically, maybe even philosophically, and observing it in our real lives.
EL: Yeah, I really like the the metaphorical aspect you brought here. And if I can reveal how naive I am about Heisenberg's Uncertainty Principle, I didn't know that it was applicable in these different things other than just position and momentum. Maybe I'm the only one. But that's really interesting that there, so are there a lot of other places where this is also the case?
CF: Well, mathematically, it's just a function defined, for example, on Rn that has regularity and decay properties, and so that function's L2 norm--so the statement of the theorem is under those decay and regularity assumptions, that function's L2 norm has to remain less than or equal to 2 over n multiplied by the product of the L2 norm of the first moment of the function times the L2 norm of the gradient of the function. And so in some sense, we can view that as talking about momentum and position, and so that has applications to various physical systems, both momentum and position. But in some sense, whenever you have a gradient of a function, it can also relate to some system's energy. And so mathematically, I think we have the position to view this in a the more abstract way, whereas physically, you tend to only read about the the momentum and position version and less about the energy and time version. So that's why it took me a long time to think about did the writers of Rick and Morty get it right, because they're basically relying on, it seems they're giving the impression that uncertainty is leading to chaos. Because every time someone feels uncertainty, the timeline gets split and multiple, simultaneous versions of reality are going on at the same time, introducing more chaos into the system. And I kept thinking about it: "But mathematically, that's not what I learned, like what's happening?" And so I really think it's at the end where where Rick merges all the timelines together and basically reduces the chaos in the system. I really think that's the moment where we're seeing Heisenberg's Uncertainty Principle at play. We're seeing that in the moment where Rick was the least certain about himself and his abilities to fix this is the moment where the timeline were fixed. I really think someone had to be knowing about the energy and time version of Heisenberg's Uncertainty Principle.
KK: I need to go back and watch this. I've seen all the first two seasons, but I don't remember this one in particular. My son should be here. He could tell you all about it. We could be having this, oh yeah this yeah. It's a bit raw of a show, though, so listener warning, if you don't like obscenities and--
EL: Delicate ears beware.
KK: Really not politically correct humor very often. It's, you know, it's
CF: I agree.
KK: it's a little raw.
KK: It's entertaining. But it's
CF: Yeah, it's definitely dark humor and lots of references to sci-fi horror. And, you know, some references are done well, some are just a little, I don't know.
CF: But I definitely learned about this show from undergraduate students during a conference where we were, you know, stuck in a long commute. And students found themselves talking to me about all sorts of things. And they mentioned this episode where something was proven mathematically, and I'm a huge fan of Back to the Future, I hadn't watched--I only recently, you know, watched the show, even though it's been around for some time, apparently. And I'm a huge fan of Back to the Future. And they're telling me that there's mathematical proofs. And so I'm course I'm like, "Well, I'm gonna have to check out the mathematical proofs." Any mathematician that watches the show could see that the mathematical proof, I'm not sure that it's much of a mathematical proof.
So it got it got me to watch the episode. And once I was watching the episode, what really drew my attention was that I realized they're talking about chaos and uncertainty.
EL: So going back to the theorem itself, where did you encounter that the first time?
CF: First year grad school at UC Santa Barbara. And it's actually--I never told the professor who was teaching a course turned out to eventually go on and become my PhD advisor. And that first year that I was a graduate student at UC Santa Barbara, I was much more interested in differential geometry and topology than I was in analysis. And this was in one of our homework assignments sort of buried in there. And I don't remember exactly who, if it was the professor himself, or maybe one of his current graduate students, or a TA for the course, that explained that inequality and its physical relationship to chaos and uncertainty. And I'm pretty sure that the conversation with whoever it was, it was about chaos and uncertainty. And it wasn't about momentum and position, which I think would have turned me off at the time. But we were talking about this, relating it to uncertainty and measurements and chaos present in the system. And for me, since that moment, I think I've lived by this sort of mantra that if I plan things out, more things are going to go wrong the more planning that I do. But, you know, I kind of have to keep that in mind that I can only plan so much without introducing some chaos into the system. And so it made a huge impression on me. And I asked this professor who assigned this homework assignment, I'm sure it was the first semester, I mean, the first quarter, of graduate real analysis, if he had more reading for me to do. And he became my advisor, and I went into this area, mathematical physics, PDEs, and harmonic analysis. So it made a huge influence on me. And that's why I wanted to include it as my favorite theorem.
EL: Yeah, that's such a great story. It's like your superhero origin story, is this theorem.
KK: Yeah. So surely this L2 norm business, though, came after the fact. Like Heisenberg just sort of figured it out in the physics sense, and then some mathematician must have come up with the L2 norm business.
CF: Right. I actually think Heisenberg came up with it in the physical sense. There was someone who wrote down something mathematically, and I actually haven't gone--I should--I haven't gone through the literature to find out which mathematician wrote down the L2 norm statement of the inequality.
But in the book Introduction to Nonlinear Dispersive Equations by Felipe Linares and Gustavo Ponce-- Gustavo's my advisor--on page 60, it's Exercise 3.14. It proves Heisenberg's inequality and states it the way I've stated it here in this podcast, and it's a really neat analysis exercise. You know, you have to use the density of the Schwartz class functions, you use integration by parts. It's a really neat exercise and really helps you use those tools that PDE people use. And yeah, my advisor doesn't know how much that exercise influenced my decision to study the mathematical physics, PDEs, and harmonic analysis.
KK: Good, then now our listeners have an exercise, too. So,
CF: That's right. Yeah. So my recommendations are watch Rick and Morty, try exercise 3.14 from Introduction to Nonlinear Dispersive Equations, and have a deep philosophical conversation about uncertainty and chaos with your good friends as you Netflix and chill it out.
EL: Nice. Yeah, wise words, definitely. Thanks a lot for joining us. I really need to brush up on some of my physics, I think, and think about this stuff.
CF: I'm happy to talk about it anytime you like. Thank you so much for the invitation. I've really enjoyed talking to you all.
KK: Thanks, Cynthia.