Hurwitz zeta function
Episode 5 – An Improbable Combination
A sophomore solves an unsolved math problem and bonds with a legendary professor.
Guest: Ross Lippert, MIT Class of ’93, PhD ’98, Course 6, 8, and 18, Senior House.
How does the mind of a genius approach learning? Can we incorporate this in our own lives?
Ross’ Solution (click on View PDF at top of page): A Probabilistic Interpretation of the Hurwitz Zeta Function
Video: The Riemann Hypothesis, Explained (16 min)
Video: The Simplest Math Problem No One Can Solve – Collatz Conjecture (22 min)
Host: While at MIT, a good friend of mine insisted that I attend 18.313, an undergraduate math course on probability. This course was taught by a beloved professor named Gian Carlo Rota. Rota was legendary for his enthusiasm, charisma, and clarity.
18.313 was probability for hard core mathematicians and physicists. As a mechanical engineering student, I had no business being there. But I went anyway to see what all the fuss was about.
Rota loved Coca Cola and students had a can ready and waiting every time he entered the room.
While sipping a can at the end of class one day, Rota made a casual remark about an unsolved problem. To his utter surprise, a sophomore with just the right combination of courage and genius solved the problem.
I’m not talking about the fictional janitor in Good Will Hunting. I am talking about my real classmate, Ross Lippert, MIT Class of 93, PhD 98, Course 6, 8, and 18, Senior House.
Today, for the first time, we will hear the story directly from Ross.
I’m Ravi Patil. And this is Institrve, true stories about MIT, a trove of wonder discovery and madness. This podcast explores the diversity of the human experience. The question of what it means to be human is a timeless one. By hearing the stories of others, we just may find a piece of ourselves and be inspired to transcend our own limitations.
What attracted you to math and science?
Ross: My physics class. I just like then already fast-tracked in math, but I didn’t think anything of it. I was taking algebra in seventh grade, calculus in 11th, but then, when I took my physics class and I realized that the calculus could do something then that opened up the possibility of, wow, I can understand the world what else can I do?
Host: Hmmm, Ross’s mother must have subconsciously wondered the same thing as she unwittingly pushed him into the deep end of the pool.
Ross: I actually transferred from honors physics to AP physics but they didn’t have any more AP books. But my mom, not to be deterred, went to the University of Miami and got me a physics book.
But that physics book used calculus instead of dumb quadratic formulas and trigonometry that a high school book does so it really fit completely well. She has no idea what she did when she did. So I learned Maxwell’s equations in integral form in 11th grade.
Host: Maxwell’s equations are a set of partial differential equations that describe the behavior of electric and magnetic fields. To learn them in integral form in 11th grade on your own is quite an achievement. It requires thinking in calculus and understanding how vector fields behave.
Ross: I had just a book that was one level up but it was at the math level that I had so I could do it.
Host: That’s interesting serendipity. In high school, calculus is where I fell in love with math, because it was so weird and powerful. But later in the day, I’d go to physics and they didn’t use calculus, but in my mind, the gears are turning. It seems like it should apply here, but it never did.
Ross: There’s a rate! A rate of some sort going on.
Host: And so it was only at MIT when it all came together for me.
Ross: Oh, yeah, I think that really is a shame. I knew that I had this super textbook. And it basically had all the same chapters the high school textbook had. The high school textbook had these hacky formulas. Where does 1/2 at2 come from? In my book, when I did the same thing, it would just be like, oh, and here’s how to get it from calculus.
Host: So let’s fast forward to the spring of ’91. You’re in 18.313 Probability. Had you met Rota before that class?
Ross: No. Now Rota sometimes taught in 18.03 differential equations, a section, but no, I didn’t know who this guy was and actually, for years after, I still didn’t know who that guy was.
I was there with a friend and we would just watch how he came in, the way he wore three glasses.
Host: One pair each to see the chalkboard, his notes, and his students.
Ross: [sfx: soda can opening] He insisted on having a diet Coke, ready for him at all times, cold, fresh from the machine or whatever.
Host: He also gave Hershey bars to students who asked sharp questions.
Ross: He would take a painstaking amount of time to just clean the board thoroughly before he began his lecture.
Host: I sat in that class for a couple of weeks drawn by the allure of Rota. It was like walking into a performance art session.
Ross: His lectures were always very well-prepared. [sfx: writing on chalkboard] He would write something and then he’d slide the chalkboard up and write the next thing, raise that up, move on to the middle chalk boards, fill them in, raising things up all the time, until he had filled up nine sliding chalkboards worth of stuff. And then the hour was over.
Host: Rota laid the foundation for combinatorics, establishing it as a respected field of study and inspiring generations of mathematicians. I believe he is also the only professor at MIT to have had a joint appointment in Mathematics and Philosophy.
Let’s start with a probability question that you have an intuition for.
Ross: So it went like this. He was talking about different ways that people think about probability on the integers.
Host: Suppose I ask you, “What are the chances of picking an even number from the set of integers?”
Ross: Well, we want to say 50%.
Host: That’s correct as everyone intuitively knows, but as Ross explains, it isn’t so obvious how to calculate it.
Ross: It’s very difficult to make that 50% stick when you’re talking about an infinite number of integers. There are an infinite number of even numbers. There are an infinite number of integers. Infinity over infinity, that’s equal to one half. How do you make that work? How do you make sense of a statement like that?
He talked about ways that people made sense of it. The easiest way is that you say, “I have the set of numbers and I want to figure out what’s the probability that a random number lies in that set. Let’s just cut everything off at 10,000. Forget everything in your set that goes past 10,000. I count how many and divide by 10,000.
Okay, all right. Let’s go ahead to a million. You know how many, divide by a million. These are all nice numbers, go to a billion, a trillion, whatever. If that converges on something, then we say, “Okay that’s the probability.”
But if you try and use the axioms of probability, like and-ing an or-ing, you multiply two independent events, all the things we’re supposed to do with algebra in probability, none of that’s going to work if your definition of probability is this clunky sort of thing, just taking a limit all the time.
Host: For those that aren’t really into math, just hang on for a few more minutes and we’ll return to the human element. Understanding this next section is not needed for the remainder of the episode.
So, how do you approach this problem?
Ross: So he introduced something else. There’s a completely different way representing this sort of parameterized family of probabilities on the integers and it obeys all the laws because there’s a certain parameter in it that if this parameter is greater one, the distribution converges, like the total weight of the integers sums to something that’s finite so you have a finite denominator, right? So you’re going to have a finite numerator, everything’s finite, fine. It’s just a regular, old probability distribution as if you were dealing with finite sets.
But it’s a different way of getting there and it works with everything that you want. And then the expression you get at the end of the day, you have to take a limit in the parameter. As the limit exists, fine, you have an answer. If the limit doesn’t, then it doesn’t.
The way that you weight the integers is such that the sum weight of all the integers is the Riemann zeta function.
Okay. And he went through this. He went through why it works, how it works. And then at some point he wrote on the board, he said, “Okay, this is the Hurwitz zeta function.”
Host: Ross just mentioned 2 zeta functions, the Riemann and the Hurwitz.
The Riemann zeta function has this intriguing property that it reveals the distribution of the prime numbers.
It presents one of the most important unsolved problems in math and there’s a $1M bounty from the Clay Mathematics Institute for anyone who can prove or disprove the Riemann Hypothesis, which states that all non-trivial zeros of the Riemann zeta function lie along a critical line.
Researchers have checked the first 10 trillion non-trivial zeros by brute force using computers and so far, all lie on the critical line. Not a single one strays outside. And these zeroes are intimately connected with the distribution of prime numbers.
Seems promising, right? Yes, but 10 trillion is not even a drop in the ocean of infinity.
The reason why you should care is that factoring enormously large numbers as the product of primes is the basis of modern encryption, and therefore your digital security and privacy.
The Riemann zeta function is a special case of the Hurwitz zeta function, the function that Ross worked with.
Okay, we’re now finally ready to unveil that passing comment Rota made about the unsolved problem.
Ross: Rota said, “I don’t know of any probabilistic interpretation of the Hurwitz zeta function.”
Host: This simple comment from Rota inspired Ross to take up the challenge. Note that Ross is not hunting for prime numbers here. Rather, he is attempting to view the Hurwitz zeta function through the lens of probability.
Ross: All right, there has to be something. We could probably work something out. That evening, I went over the notes. He sketched out a little proof about that result of how you get one half for the probability of an even number or an odd number, that sort of thing. How it converges on the density of your number set.
Host: So when you went home, was it like the movies? You were struck by lightning with inspiration? Where are you cranking hard? Like, how long was that process of working it out before you thought you had something?
Ross: I think it just took an evening. I was sitting on my waterbed. I had a waterbed in Senior House. I think I was the only person who did. I inherited it from a previous resident and I was just working there. I think the window was open. It was a little cold, yeah, just grinding away there.
It was a lot of just teasing apart, looking for analogies in the sketch that he had given about how the zeta function worked. And just seeing where I could make a similar turn. There’s a lot of unpeeling of the definitions and then there’s one trick in the middle.
I wound up with something. So the next class, that Friday, I said, “Here, this is the probabilistic interpretation of the Hurwitz zeta function. Here’s an expression.”
And he said, “Okay, nobody’s ever done this before. You should talk to me later.” And I was like, “Okay, let’s talk later.”
And I met with him in his office. We talked about it some more. Now you have to understand, this is my sophomore year and I don’t know how to write math.
Host: Ross offered an interesting take on the psychology of mathematical writing, which made me smile.
Ross: I certainly had a very motivational, reasoning for how this works, but he wanted to work with me to get this into a much better form that sounds like it was written by a mathematician.
You know, that doesn’t have the sort of, “Now we do this and now we do this,” which is usually like how we think about math when we’re grinding things out whereas the mathematician always has to express himself as if he’s just stumbling across things.
“And this suddenly appears, and we can see this, right?” And you’re busy seeing, observing, and noting rather than actually pushing the equation around and twisting it about. I learned how to write and arrive at the result in a proper way.
Host: Ross published his solution in the Advances in Mathematics journal as a sole author. You’ll find the link in the show notes.
You worked on this in your free time. It wasn’t assigned and you had problem sets and everything else going on. Why were you drawn into this?
Ross: I certainly thought it would be really interesting to see if I could do something with it. And I don’t remember feeling really behind in other work. But it was certainly a more fun thing to try and explore than anything else that was around. It was more like very fun crossword puzzle or toy to play with.
Host: Ross indicates that he didn’t feel like he was behind, but it turns out he had a lot going on. He was actually pursuing a quadruple major. More on this later.
At the end of the day, Ross had a humble take on his achievement.
Ross: I had solved an unsolved problem. I more think that nobody really tried before and that’s why I was able to do it. I would say it was an unsolved problem of somewhat lesser importance.
Host: Did he make an announcement to the class that you had solved that problem?
Ross: No, it wasn’t a big deal. It wasn’t that much of a contest.
Host: Did you sense you had arrived on the scene with the course 18 department after publishing that paper?
Ross: I don’t think anybody in course 18 noticed me besides Rota. I’m not sure it made much difference.
Host: No glory to be found here.
Ross: Now you see after I had that result, then Rota told me the conjecture he really wanted to solve, which is something called the basis conjecture.
He said, “Can you do this one?” I can’t do the basis conjecture. Nobody’s done the basis conjecture. I’m glad I stopped trying to work on the basis conjecture because no one has gotten it.
But that’s not really the interesting thing. The interesting thing is that once I started working with Rota, I was on the map of the Rota people.
Ross: There was a little group of Rota people who would take all of his classes and I took all of his classes and I would visit him socially. I’ve been over to his apartment with some other, like, Rota fans. He would cook us spaghetti dinner. He would sometimes take the gang out to Henrietta’s Table in Harvard Square, which was amazingly fancy for me. Have you been to Henrietta’s Table?
Host: Yes, but not as a student.
Ross: Okay, you can be quite awestruck if you’re starving undergraduate and someone takes you there for Sunday brunch.
Host: Did your association with him change the way you viewed the world or yourself?
Ross: Other people around me were like, “Oh, you’re friends with Rota.” I was young. He was just this guy. I didn’t know that he was famous or anything. I actually wish I’d taken more advantage of knowing this guy because in my thirties, I read his book Indiscrete Thoughts in which he just describes the history of the MIT math department in the fifties and sixties. And I thought that was really interesting. And I had the guy sitting in front of me. He would mention certain things occasionally about some mathematician somewhere.
Here’s a strange story though. So ‘96, they had the Rotafest. This is the one I went to.
Host: Rotafest was a 4 day a four-day mathematical conference held at MIT in 1996 to celebrate Rota’s 64th birthday.
Ross: It’s a ghoulish story but these posters are all up. Like Rotafest, Rotafest.
Rota traveled a lot, so sometimes he just wasn’t around or took sabbaticals or whatever. I wasn’t seeing him around and I had a good friend, the guy who was taking the probability class, so I had known him since then and he was like, “Yeah. I haven’t seen him around.”
Why are they having a Rotafest? Oh, wait a second. We haven’t seen him around and there celebrating his career. Oh my God! He’s dead!
Some of my friends, I’m like, “Let’s see. What evidence do we have?”
Let’s go to his office. His office is locked. The lights are off. Okay, I went to his office again today. His office is locked. The lights are off.
We didn’t know what was going on so finally I actually sent Rota an email that said, “There’s this Rotafest going on. You’re still alive…aren’t you?”
And he wrote back and said, “That’s not very funny.”
Maybe that was a rude thing to say. I didn’t know what else to do, you know?
Host: When you interacted with him, you didn’t have that feeling of distance. This is a God-like being and I’m just an undergrad. You were pretty free and open.
Ross: Yeah, yes here’s a man in his sixties. I’m a kid in his twenties. So young people don’t know how to interact with old people. I know this now that I’m an old person. It’s very strange so I was probably just strange.
But yeah, here’s this guy. He would invite us over. We went out for a movie once after Henrietta’s Table, all right.
Host: What movie did you see?
Ross: We saw Unforgiven with Clint Eastwood.
Host: Okay. How random.
Ross: He enjoyed it.
Host: Unfortunately, Ross’ association with Professor Rota came to a premature end. That seemingly flippant email he sent earlier was prescient.
Ross: He died shortly after I graduated. I got my PhD in ’98. He died in the spring of ’99.
One thing that struck me about Rota was that he lived alone. And all of us Rota groupies wondered, “Is he married? Was he married? Is he widowed?” Of course, you don’t ask these questions, but we would spread rumors between each other.
And, yeah, I found out Googling this week that, oh, he had been married and he was divorced. His wife was at least alive when he died. He didn’t have any children. Okay, but none of us knew this. He never said anything about his actual personal life.
I was sad when I heard that the only way that people knew he’d had a heart attack and died was that he didn’t show up at a conference that he was supposed to go to. And a few days later somebody went into his apartment and found him dead. And that sounded lonely.
Host: So, Ross, let’s switch back to you. You have a whopping five MIT degrees, a quadruple major as an undergrad, and then a PhD in math. And it looks like you got your first four undergrad degrees in four years. What were they?
Ross: I got CS, EE, physics, and math.
Host: Oh, so you are Course 6-1, 6-3, 8, and 18. That is truly phenomenal.
Ross: I’m a good combiner. You know how people combine foods in order to [get] whatever dietary results they want. Well, if I take this class at the same time as they take that class, they cover pretty much similar things and I might do this with this and yeah, you can stack things up. And all you have to do is just keep track of all these.
Host: In order to do that, you must have tested out of a gazillion things coming in?
Ross: I had about 90 units of credit from AP. Yeah, yeah, I think so. Let’s see, I had AP chemistry. I had AP physics. I had AP calculus, but oh, no, but I didn’t use it. I had two different AP Englishes, the composition and the lit.
Host: Okay, so that’s basically freshman year in a way. Standard workload is 48 units per semester so that’s 96. You came in with 90.
Ross: I think I came in with 90. That’s the rough number.
Host: Were you in classes with upperclassmen right away?
Ross: Some of it, yeah, because I tested, that’s right, I had tested out of both Physicses, AP Physics 1 and AP Physics 2. Like I said, I had Maxwell’s equations in 11th grade. I didn’t quite understand what that little triangle was, because I didn’t know them in differential form, but I could still just figure it out from context.
Host: Let me clarify a bit as there is something quite remarkable in all of this. Already ahead of his time, Ross had learned Maxwell’s equations in integral form in high school. In order to receive credit for electromagnetism at MIT, he had to take a placement exam when he arrived on campus as a freshman. The only catch was that the equations in the exam were written in differential form. That’s what Ross meant earlier when he said he didn’t quite understand what that little triangle was.
Ross: Even though I got the AP credit, you still had to take a test that MIT would give you to get out of E&M, but I passed that test.
Host: Ross took an exam where he couldn’t decipher the equations being shown to him but he understood the subject matter so well that he reasoned his way through the problem statements and solved them using what he did know, the integral form of the equations. At the end of the day, both sets of equations are identical so you get the same answer, but the notation is different.
You’re definitely an outlier with the amount that you absorbed. Do you find that has generalized to other things like learning music or things that are non-math related? Curious.
Ross: Yeah, I pick up things very quickly.
Host: Okay, can you give me an example of something?
Ross: So recently, I’ve re-picked up piano. For the first time in 30 years, I’m living in enough space to actually sit in a room by myself with a keyboard and a piano as well. So I had two years, fifth and sixth grade, but I just wasn’t that interested in what I was learning and how I was learning it.
And now I said, “Okay, now I can start figuring out how to play things again.” And I looked things up on YouTube and then I get some sheet music, and like recently I’ve been able to Schubert’s Ave Maria. That’s been my latest thing.
Okay, and I challenged myself. I’m going to play Havana but with a really complicated baseline. And then I can spend weeks getting it wrong until I get it right, but all through the process, I keep learning about chords, chord structure, things like that.
And then again, if I have a question, I go to YouTube and I find out how all that fits together. And this is the sort of thing that I would have liked to have learned when I was in fifth and sixth grade taking piano class.
But instead in piano class, at least the way that I was taught, they treat you like a computer. [sfx: simple piano sequence] They hand you a sheet of notes and they say, “Type this and type it well,” and that’s all there was. I finally figured out what a chord is.
What I’ve learned about piano is all it takes is practice, but you must practice. You only get better with practice and if you practice, you will get better.
That’s a beautiful symmetry: whatever you put into it, you will get out of it. And then I began figuring out how guitar works, but I’m not going to play guitar. That looks too painful.
Host: What inspires you? What puts you in that space where you’re like, “Yes, I really want to do this!”
Ross: That’s a good question. Some of it is just you think, “Okay, anybody can do this.” In 2008, I wanted to really understand what had happened in the Great Depression because the stock market was crashing and we were all freaked out.
And then I just became a history buff. I read dozens of books and with history, there’s always something new to find out or a different opinion or somebody like glossed over this area. Sometimes, I just plunge into something and then just suck up everything I can.
Host: If you could solve another unsolved math problem, which one would it be?
Ross: I would love to solve the Collatz conjecture. Such a simple little dynamical system. No one’s been able to do it.
Host: There’s a wonderful video, I think Veritasium, that YouTube channel, on the Collatz conjecture and, basically, the refrain throughout this entire video is an older mathematician telling a younger one, “Do not waste your time on this problem.”
Ross: Yeah. And I would sometimes waste my time on that problem. Just, okay, how far out can I go? What can I say? But yeah, it’s a very easy-to-state problem and there’s just nothing you could do.
Host: I won’t describe the Collatz conjecture here. Check out the show notes for an absolutely brilliant, entertaining video about the failed attempts. This one is worth watching with your kids. Let’s say 9th grade or older. For some reason, kids often think that math is a settled field with nothing new to be done but this video will refute that misconception.
Given the way you at an early age absorbed a lot of advanced subjects, any thoughts on how you would change teaching of these subjects to high schoolers?
Ross: You have to align your physics book with calculus. That was a must for me, as you had the experience too, okay. You just have to align that. If most students are taking calculus their senior year, then they better be taking physics their senior year and the book should respect that.
My daughter is in fifth grade and she’s gonna be going to middle school. So I went to an open house and I got to see what the math curriculum pathways look like in the state of Florida.
And they’ve done something there that I like, which is that you don’t necessarily go into trigonometry and calculus. After you finish your algebra two, you’ve got probability and statistics. You’ve got linear algebra. You’ve got all these other things. There’s a lot more diversity and options post algebra two. So I would’ve recommended that people get these things.
Host: What message would you like to convey to those interested in math or science?
Ross: Remember that textbook story? I ended up patterning my undergraduate life on that. I realized that with the right textbook in your hand, with someone who speaks to you in your language or presents the problem your way, you can go a lot further than, perhaps, the crappy book that you were assigned to buy.
I would go to Wordsworth and I would look around there and I would find a different book. I didn’t know the subject, but I’d look at the table of contents, see that it’s the same stuff, read a little bit of it. See if you like the style better or worse and I’d just go ahead and buy it. And then when you have a question, because you didn’t understand the assigned book, you’d go to the other one. You can have multiple opinions, all right.
Don’t get your information just from where they want you to get your information.
Host: Such great advice. I suppose everyone uses YouTube now for that but back then, we tended to stick to the required text.
So here’s a silly question because you majored in essentially everything but if you had to do MIT all over again, what would you study?
Ross: Well, I liked what I studied. I think it is the same answer. Yeah, I’m pretty happy with it.
This was advice that I gave students when I was an upperclassman. I said, “Every term, take a class that you really wanna take. Give yourself a reason to get out of bed. Don’t have four required classes. Make sure that one class you’re taking is a class that you really have something that you’re really interested in and you’re not just grinding through your requirements.”
Host: That’s wonderful and something I didn’t do. I grinded through requirements.
After finishing his quadruple major, it’s no surprise that Ross couldn’t limit himself to just one topic during his math PhD. His doctoral thesis concerned nonlinear eigenvalue problems and spanned numerical approaches to linear algebra, optimization under quadratic constraints, as well as wavelets.
Throughout this research, he focused on making computation faster and more efficient, which is the common thread that runs throughout his non-academic career. Whether his prior work in computational biology or his current work with the financial markets, Ross has brought numerous innovations to market that lie at the intersection of computer science and math.
As I began work on this episode, human brilliance was the obvious theme but as I progressed, I discovered something mysterious and surprising underneath the surface: There exists a state of being that is simultaneously intense and focused yet relaxed and joyful. Ross exemplifies this. Recall that he referred to working on the Hurwitz zeta function as “a new toy to play with.”
We can do this too! So take time to explore your curiosity with the innocence you had as a child.
Choose happiness. See you next time.
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