Kevin Knudson: Welcome to My Favorite Theorem, a podcast that starts with math and goes all kinds of weird places. I'm Kevin Knudson, professor of mathematics at the University of Florida and here is your other host.
Evelyn Lamb: Hi. I'm Evelyn Lamb. I’m a freelance math and science writer in Salt Lake City. Happy New Year, Kevin.
KK: And to you too, Evelyn. So this will happen, this will get out there in the public later. But today is January 1, 2019.
EL: Yes, it is.
KK: And Evelyn tells me that it was it's cold in Utah, and I have my air conditioning on.
KK: That seems about right.
EL: Yeah. Our high is supposed to be 20 today. And the low is 6 or 7. So we really, really don't have the air conditioner on.
KK: Yeah, it's going to be 82 here today in Gainesville. [For our listeners outside of the USA: temperatures are in Fahrenheit. Kevin does not live in a thermal vent.] I have flip flops on.
EL: Yeah. Yeah, I’m a little jealous.
KK: This is when it's not so bad to live in Florida, I’ve got to say.
KK: Anyway, Well, today, we are pleased to welcome Robert Ghrist. Rob, you want to introduce yourself?
RG: Hello, this is Robert Ghrist.
KK: And say something about yourself a little, like who you are.
RG: Okay. All right, so I am a professor of mathematics and electrical and systems engineering at the University of Pennsylvania. This is in Philadelphia, Pennsylvania. I've been in this position at this wonderful school for a decade now. Previous to that I had tenured positions at University of Illinois in Urbana Champaign and Georgia Institute of Technology.
KK: So you've been around.
RG: A little bit.
EL: Oh, and so I was just wondering, so it that a joint appointment between two different departments, or is it all in the math department?
RG: This is a split appointment, not only between two different departments, but between two different schools.
EL: Oh, wow.
RG: The math appointment is in the School of Arts and Sciences, and the engineering appointment is in the School of Engineering. This is kind of a tricky sort of position to work out. This is one of the things that I love about the University of Pennsylvania is there are very low walls between the disciplines, and a sort of creative position like this is is very workable. And I love that.
KK: Yeah, and your undergraduate degree was actually in engineering, right?
RG: That’s correct I got turned on to math by my calculus professor, a swell guy by the name of Henry Wente, a geometer.
KK: Excellent. Well, I see you’re continuing the tradition. Well, we’ll talk about that later. And actually, you actually have an endowed chair name for someone rather famous, right?
RG: That’s true. The full title is The Andrea Mitchell PIK professor of mathematics, electrical and systems engineering. This is Andrea Mitchell from NBC News. She and her husband Alan Greenspan funded this position. Did not intend to hire me specifically, or a mathematician. I think she was rather surprised when the chair that she endowed wound up going to a mathematician, but there it is, we get along swell. She's great.
EL: Nice. That's really interesting.
KK: Yup. So, Rob, what is your favorite theorem?
RG: My favorite theorem is, I don't know. I don't know the name. I don't know that this theorem has a name, but I love this theorem. I use this theorem all the time in all the classes I teach, it seems. It's a, it's a funny thing about basically Taylor expansion, or Taylor series, but in an operator theoretic language. And the theorem, roughly speaking, goes like this: if you take the differentiation operator on functions, let's say just single input, single output functions, the kind of things you do in basic calculus class. Call the differentiation operator D. Consider the relationship between that operator and the shift operator. I’m going to call the shift operator E. This is the operator that takes a function f, and then just shifts the input by 1. So E(f(x)) is really— pardon me, E(x) is really f evaluated at x+1. We use shift—
EL: I need a pencil and paper.
RG: Yeah, I know, right? We use the shift operator all the time in signal processing, in all kinds of things in both mathematics and engineering. And here's the theorem, here's the theorem. There's this wonderful relationship between these two operators. And it's the following. If you exponentiate the differentiation operator, if you take e to the D, you get the shift operator.
KK: This is remarkable.
RG: What does this mean? What does this mean?
KK: Yeah, what does it mean? I actually did work this out once. So what our listeners don't know is that you and I actually had this conversation once in a bar in Philadelphia, and the audio quality was so bad, we're having to redo this.
KK: So I went home, and I worked this out. And it's true, it does work out. But what does this mean, Rob? Sort of, you know, in a manifestation physically?
RG: Yeah, so let me back up. The first question that I ask students, when they show up in calculus class at my university is, is, what is e to the x? What does that even mean? What does that mean when x is an irrational number, or an imaginary number, or something like a square matrix, or an operator? And, of course, that takes us back to the the interpretation of exponentiation in terms of the Taylor series at zero, that I take that infinite series, and I use that to define what exponentiation means. And because things like operators, things like differentiations, or shifts, you can take powers of those by composition, by iteration, and you can rescale them. Then you can exponentiate them.
So I can talk about what it means to exponentiate the differentiation operator by taking first D to the 0, which of course, is the identity, the do-nothing operator, and then adding to it D, and then adding to that D squared divided by 2 factorial [n factorial is the product of the integers 1x2,…n], that's the second derivative, then D cubed divided by 3 factorial, that's the third derivative. If I can keep going, I’ve exponentiated the differentiation operator. And the theorem is that this is the shift operator in disguise. And the proof is one line. It's Taylor expansion. And there you go.
Now, this isn't your typical sort of my favorite theorem, in that I haven't listed all the hypotheses. I haven't been careful with anything at all. But one of the reasons that this is my favorite theorem is because it's so useful when I'm teaching calculus to students, when I'm teaching basic dynamical systems to students where, you know, in a more advanced class, yeah, we'd have a lot of hypotheses. And, oh, let's be careful. But when you're first starting out, first trying to figure out what is differentiation, what is exponentiation, this is a great little theorem.
EL: Yeah, this conceptual trip going between the Taylor series, or going between the idea of e to the x, or 2 to the x or something where we really have a, you know, a fairly good grasp of what exponentiation means in that case, if we, if we're talking about squares or something like that, and going then to the Taylor series, this very formal thing, I think that's a really hard conceptual shift. I know that was really hard for me.
KK: Yeah. So I, I wonder, though, I mean, so what's a good application of this theorem, like, in a dynamics class, for example? Where does this pop up sort of naturally? And I can see that it works. And I also agree that this idea of—I start calculus there, too, by the way, when I say, you know, what does e to the .1 mean, what does that even mean?
RG: What does that even mean?
KK; Yeah, and that’s a good question that students have never really thought about. They’re just used to punching .1 into a calculator and hitting the e to the x key and calling it a day. So, but where would this actually show up in practice? Do you have a good example?
RG: Right. So when I teach dynamical systems, it's almost exclusively to engineering students. And they're really interested in getting to the practical applications, which is a great way to sneak in a bunch of interesting mathematics and really give them some good mathematics education. When doing dynamical systems from an applied point of view, stability is one of the most important things that you care about. And one of the big ideas that one has to ingest is that of stability criteria for, let's say, equilibria in a dynamical system. Now, there are two types of dynamical systems that people care about, depending on what notion of time you're using— continuous time or discrete time. Most books on the subject are written for one or the other type of system. I like to teach them both at once, but one of the challenges of doing that is that the stability criteria are are different, very different-looking. In continuous time, what characterizes a stable equilibrium is when you look at all of the eigenvalues of the linearization, the real parts are less than zero. When you move to a discrete time dynamical system, that is a mapping, then again, you're looking at eigenvalues of the linearization, but now you want the modulus to be less than 1. And I find that students always struggle with “Why is it different?” “Why is it this way here, and that way there?” And of course, of course, the reason is my favorite little theorem, because if I look at the evolution operator in continuous-time dynamics—that’s the derivative—versus the evolution operator in discrete-time dynamics—that is the shift, move forward one step in time—then if I want to know the relationship between the stability and, pardon me, the stable and unstable regions, it is exponentiation. If I exponentiate the left hand side of the complex plane, what do I get? I get the region in the plane with modulus less than 1.
RG: I find that students have a real “aha” moment when they see that relationship, and when they can connect it to the relationship between the evolution operators.
EL: I’m having an “aha” moment about this right now, too. This isn't something I had really thought about before. So yeah, this is a really neat observation or theorem.
RG: Yeah, I never really see this written down in books.
KK: That’s—clearly now you should write a book.
RG: Another one?
KK: Well, we'll talk about how you spend your time in a little while here. But, no, Rob, I mean, so Rob has this—I don't know if it's famous, but well known—massive open online course through Coursera where he does calculus, and it's spectacular. If our listeners haven't seen it, is it on YouTube, Rob? Can you actually get it at YouTube now?
RG: Yes, yes. The the University of Pennsylvania has all the lectures posted on a YouTube channel.
KK: Well, I actually downloaded to my machine. I took the MOOC A few years ago, just for fun. And I passed! Remarkably.
RG: With flying colors, with flying colors.
KK: Yeah, I'm sure you graded my exam personally, Rob.
KK: And anyway, this is evidence for how lucky are students are, I think. Because, you know, you put so much time into this, and these these little “aha” moments. And the MOOC is full of these things. Just really remarkable stuff, especially that last chapter, which is so next level, the digital calculus stuff, which sort of reminds me of what we're talking about. Is there some connection there?
RG: Oh yes, it was creating that portion of the MOOC that really, really got me to do a deep dive into discrete analogs of continuous calculus, looking at the falling powers notation that is popular in computer science in Knuth’s work and others, thinking in terms of operators. Yeah, that portion of the MOOC really got me thinking a lot about these sorts of things.
KK: Yeah, I really can't recommend this highly enough. It’s really great.
EL: Yeah, so I have not had the benefit of this MOOC yet. So digital calculus, is that meaning, like, calculus for computers? Or what exactly is that? What does that mean?
RG: One of the things that I found students really got confused about in a basic single variable calculus class is, as soon as you hit sequences and series, their heads just explode because they get sequences and series confused with one another, and it all seems unmotivated. And why are we bothering with all these convergence tests? And where’d they come from? All this sort of thing.
EL: And why is it in a calculus class?
RG: Why is it even in a calculus class after all these derivatives and integrals? So the way that I teach it is when we get to sequences and series, you know, in the last quarter of the semester, I say, Okay, we've done calculus for functions with an analog input and an analog output. Now, we want to redo calculus for functions with a digital input and an analog output. And such functions we're going to call sequences. But I'm really just going to think of it as a function. How would you differentiate such a thing? How would you integrate such a thing? That leads one to think about finite differences, which leads to some some nice approaches to numerical methods. That leads one to looking at sums and numerical integration. And when you get to improper integrals over an unbounded domain? Well, that's series, and convergence tests matter.
KK: Yeah, it's super interesting. We will provide links to this. We’ll find the YouTube links and provide them.
KK: So another fun part of this podcast, Rob, is that we ask our guests to pair their theorem with something, and I assume you're going to go with the same pairing from our conversation back in Philadelphia.
RG: Oh yes, that’s right.
KK: What is it?
RG: My work is fueled by a certain liquid beverage.
RG: It’s not wine. It's not beer. It's not whiskey. It's not even coffee, although I drink a whole lot of coffee. What really gets me through to that next-level math is Monster. That's right. Monster Energy Drink, low carb if you please, because sugar is not so good for you. Monster, on the other hand, is pretty great for me, at any rate. I do not recommend it for people who are pregnant or have health problems, problems with hearts, anything like this, people under the age of 18, etc, etc. But for me, yeah, Monster.
KK: Yeah. There's lots of empties in your office, too like, up on the shelf there, which I'm sure have some significance.
RG: The wall of shame, that’s right. All those empty monster cans.
KK: See, I can't get into the energy drinks. I don't know. I mean, I know you're also fond of scotch. But does that does that help bring you down from the Monster, or is it…
RG: That’s a rare treat. That's a rare treat.
KK: Yeah, it should be. So when did your obsession with Monster start? Does this go back to grad school, or did it even exist when we were in grad school? Rob and I are roughly the same age. Were energy drinks a thing when we were in grad school? I don't remember.
RG: No, no. I didn't have them until, gosh, what is it, sometime within the past decade? I think it was when I was first working on that old calculus MOOC, like, what was that, six years ago? Six, seven years ago, is when I was doing that.
RG: That was difficult. That was difficult work. I had to make a lot of videos in a short amount of time. And, yup, the Monster was great. I would love to get some corporate sponsorship from them. You know, maybe, maybe try to pitch extreme math? I don't know. I don't think that's going to work.
KK: I don't know. I think it's a good angle, right? I mean, you know, they have this monster truck business, right? So there is this sort of whole extreme sports kind of thing. So why not? You know?
EL: Yeah, I'm sure they're just looking for a math podcast to sponsor. That's definitely next on their branding strategy.
KK: That’s right. Yeah. But not us. They should sponsor you, Rob. Because you're the true consumer.
RG: You know, fortune favors the bold. I'd be willing to hold up a can and say, “If you're not drinking Monster, you're only proving lemmas,” or something like that.
EL: You’ve thought this through. You've got their pitch already, or their slogan already made.
RG: That’s right. Yup.
KK: All right. Excellent. So we always like to give our guests a chance to to pitch their projects. Would you like to tell us about Calculus Blue?
RG: Oh, absolutely! This is—the thing that I am currently working on is a set of videos for multivariable calculus. I'm viewing this as something like a video text, a v-text instead of an e-text, where I have a bunch of videos explaining topics in multivariable calculus that are arranged in chapters. They’re broken up into small chunks, you know, roughly five minutes per video. These are up on my YouTube channel.There's another, I don't know, five or six hours worth of videos that are going to drop some time in the next week covering multivariate integration. This is a lot of fun. I'm having a ton of fun doing some 3d drawing, 3d animation. Multivariable calculus is just great for that kind of visualization. This semester, I'm going to use the videos to teach multivariable calculus at Penn in a flipped manner and experiment with how well that works. And then it'll be available for anyone to use.
KK; Yeah, I'm looking forward to these. I see the previews on Twitter, and they really are spectacular. How long does any one of those videos take you? It seems like, I mean, I know you've gotten really good at at the graphics packages that you need to create those things. But, you know, like a 10 minute video. How long does one of those things take to produce?
RG: I don't even want to say,
RG: I do not even want to say, no. I've been up since four o'clock this morning rendering video and compositing. Yeah, this is my day, pretty much. It's not easy. But it is worthwhile. Yeah.
KK: Well, I agree. I mean, I think, you know, so many of our colleagues, I think, kind of view calculus as this drudgery. But I still love teaching it. And I know you do, too.
KK: And I think it's important, because this is really a lot of what our job is, as academics, as professional mathematicians. Yes, we're proving theorems, all that stuff, that's great. But, you know, day in, day out, we're teaching undergraduates introductory mathematics. That's a lot of what we do. And I think it's really important to do it well.
EL: Well, and it can help, you know, bring people into math like it did for Rob.
KK: That’s right.
RG: Exactly. That's exactly right. Controversial opinion, but, you know, you get these people out there who say, oh, calculus, this is outdated, we don't need that anymore, just teach people data analysis or statistics. I think that's a colossal error. And that it's possible to take all of these classical ideas in calculus and just make them current, make them relevant, connect them to modern applications, and really reinvigorate the subject that you need to have a strong foundation in in order to proceed.
KK: Absolutely. And I, you know, I try to mix the two, I try to bring data into calculus and say, you know, look, engineering students, you’re mostly going to have data, but this stuff still applies. You know, calculus for me is a lot about approximation, right? That's what the whole Taylor Series business, that's what it's for.
KK: And really trying to get students to understand that is one of my main goals. Well, this has been great fun. Thanks for taking time out from rendering video.
EL: Yeah, video rendering.
RG: Yes. I'm going to turn around and go right back to rendering as soon as we're done.
KK: That’s right, you basically have a professional quality studio in your in your basement, right?Is this how this works?
RG: This is how it works. Been renovating, oh, I don't know. It's about a year ago I started renovations and got a nice little studio up and running.
KK: Excellent. Do you have, like, foam on the walls and stuff like that?
RG: Yes, I'm touching the foam right now.
KK: All right. Yeah. So Evelyn I aren't that high-tech. We've just now gotten to the sort of like, multi-channel recording kind of thing.
KK: Well, yeah, well, we're doing this now, right, where we’re each recording our own audio. I'm pleased with the results so far. Well, Rob, thanks again, and we appreciate your joining.
EL: Thanks for joining us.
RG: Thank you. It's been a pleasure chatting.