Po-Shen Loh: Mathematics, Math Olympiad, Combinatorics & Contact Tracing | Lex Fridman Podcast #183

The following is a conversation with Po Shen Lo, a professor of mathematics at Carnegie Mellon University, national coach of the USA International Math Olympia Team, and founder of EXPY that does online education of basic math and science. He's also the founder of Novid, an app that takes a really interesting approach to contact tracing, making sure you stay completely anonymous and it gives you statistical information about COVID cases in your physical network of interactions so you can maintain privacy very important and make informed decisions. In my opinion, we desperately need solutions like this in early 2020 and unfortunately I think we will again need it for the next pandemic. To me, solutions that require large scale distributed coordination of human beings need ideas that emphasize freedom and knowledge. Quick mention of our sponsors Jordan Harbinger show on it BetterHelp8 Sleep and element. Check them out in the description to support this podcast, As a side note, let me say that Po and I filmed a few short videos about simple, beautiful math concepts that I will release soon. It was really fun. I really enjoyed Po sharing his passion for math with me in those videos. I'm hoping to do a few more short videos in the coming months that are educational in nature on AI, robotics, math, science, philosophy, or if all else fails, just fun snippets into my life on music, books, martial arts and other random things. If that's of interest to anyone at all. This is the Lex Friedman podcast and here's my conversation with Po Shen Lo. You know, you mentioned you really enjoy flying and experience the different people in different places. There's something about flying for me, I don't know if you have the same experience that every time I get on an airplane. It's incredible to me that human beings have actually been able to achieve this. And when I look at like what's happening now with humans traveling out into space, I see it as all the same thing. It's incredible that humans are able to get into a box and fly in the air and safely and land and the same it seems like. And everybody's taking it for granted. So when I observe them, it's quite fascinating because I see that cleanly mapping to the world where we're now in rockets and traveling to the moon, traveling to Mars. And at the same kind of way I can already see the future where we will all take it for granted. So I don't know. I don't know if you have you you personally, when you fly, have the same kind of magical experience of like how the heck did humans actually accomplish this?

So I do. Especially when there's turbulence which is, you know, like on the way here there was turbulence and the plane jiggled. Even the flight attendant had to hold on to the side. And I was just thinking to myself, it's amazing that this happens all the time and the wings don't fall off, you know, like given how many planes are flying. But then I often think about it and I'm like, you know, a long time ago I think people didn't trust elevators in a 40 story building in New York City. And now we just take it completely for granted that you can step into this shaft which is 40 floors up and down, and it will just not fail.

Yeah, again, I'm the same way with elevators, but also buildings. When I'll stand on the 40th floor and wonder how the heck are we not falling right now? Like how, how amazing it is with the high winds, like structurally just the earthquakes and the vibrations, I mean natural vibrations in the ground. Like, how is this, how are all of these, you go to like New York City, all of these buildings standing. I mean to me one of the most beautiful things actually mathematically too is bridges. I used to build bridges in high school from like toothpicks, just like out of the pure joy of like physics making some structure, really strong understanding, like from a civil engineering perspective, what kind of structure will be stronger than another kind of structure, like suspension bridges. And then you see that at scale, humans being able to span a body of water with a giant bridge and it's. I don't know, it, it's so humbling. It makes you realize how, how dependent we are on each other, sort of. I talk about love a lot, but there's a certain element in which we little ants have just a small amount of knowledge about our particular thing. And then we're depending on a network of knowledge that other experts hold. And then most of our lives, most of the quality of life we have has to do with the richness of that network of knowledge, of that collaboration, and then sort of the ability to build on top of it, levels of abstractions. You start from like bits in a computer, then you can have assembly, then you can have C, or you have an operating system, then you can have C and Python. Finally some machine learning on top. All of these are abstractions. And eventually we have AI that runs all of us humans. But anyway. But speaking of abstractions and programming, in high school you wrote some impressive games for Ms. Dos. I got a chance to in browser somehow it's magic. Got a chance to play them. Alien attack. 1, 2, 3, and 4, what's the hardest part about programming those games? And maybe can you tell the story about building those games? Sure.

I actually tried to do those in high school because I was just curious if I could.

Yeah, and that's a good starting point for anything, right?

Yeah, yeah, yeah. It's like, could you. But the appealing thing was also, it was a soup to nuts kind of thing. So something that has always attracted me is I like beautiful ideas, I like seeing beautiful ideas, but I actually also like seeing execution of an idea all the way from beginning to end in something that works. So, for example, in high school, I was lucky enough to grow up in the late 90s, when even a high school student could hope to make something sort of comparable to the shareware games that were out there. I say the word sort of like still quite far away, but at least I didn't need to hire a 3D CG artist. There weren't enough pixels to draw anyway. Even I can draw. Right. Bad art, of course. But the point is, I wanted to know, is it possible for me to try to do those things where back in those days, you didn't even have an easy way to draw letters on the screen in a particular font. You couldn't just say, import a font. It wasn't like Python. So, for example, back then, if you play those games in the web browser, which is emulating the old school computer, even the letters you see, those are made by individual calls to draw pixels on the screen.

So you built that from scratch, almost building a computer graphics library from scratch, yes.

The primitive that I got to use was some code I copied off of a book in assembly of how to put a pixel on a screen in a particular color.

And the programming language was Pascal.

Ah, yeah. The first one was in Pascal, but then the other ones were in C after that.

How's the emulation in the browser work, by the way? Is that trivial?

Because it's pretty cool.

You get to play these games that have a very much 90s feeling to them. Ah.

So it's literally making an Ms. DOS environment which is literally running the old exe file.

Wow. I didn't browser.

This is.

That could be more amazing than the airplane. So it wasn't so much about the video games, it was more about can you build something really cool from scratch?

Yes.

And you did a bunch of programming competitions. What was your interest, your love for programming? What did you learn through that experience, especially now that much of your work has taken a long journey through mathematics?

I think I always was amazed by how computers could do things fast. If I wanted to make it an abstract analysis of why it is that I saw some power in the computer. Because if the computer can do things so many times faster than humans, where the hard part is telling the computer what to do and how to do it, if you can master that, asking the computer what to do, then you could conceivably achieve more things. And those contests I was in, those were the opposite in some sense of making a complete product like a game is a product. Those contests were effectively write a function to do something extremely efficiently. And if you are able to do that, then you can unlock more of.

The power of the computer, but also doing it quickly. There's a time element from the human perspective to be able to program quickly. There's something nice. So there's almost like an athletics component to where you're almost like an athlete seeking optimal performance as a human being trying to write these programs. And at the same time it's kind of art because you're. The best way to write a program quickly is to write a simple program, is to have a damn good solution. So it's not necessarily you have to type fast. You have to think through a really clean, beautiful solution. I mean, what do you think is the use of those programming competitions? Do you think they're ultimately something you would recommend for students, for people interested in programming or people interested in building stuff?

Yes, I think so. Because especially with the work that I've been doing nowadays, even trying to control Covid. Something that was very helpful from day one was understanding that the kinds of computations we would want to do, we could conceivably do on like a four core cloud machine on Amazon Web Services, out to a population which might have hundreds of thousands or millions of people. The reason why that was important to have that back of the envelope calculation with efficient algorithms is because if we couldn't do that, then we would bankrupt ourselves before we could get to a big enough scale. If you think about how you grow anything from small to big, if in order to grow it from small to big, you also already need 10,000 cloud servers, you'll never get to big.

And also the nice thing about programming competitions is that you actually build a thing that works so you, you finish it, there's a completion thing, and you realize, I think there's a magic to it where you realize that it's not so hard to build something that works, to have a system that successfully takes in inputs and produces outputs and solves a difficult problem, and that directly Transfers to building a startup, essentially, that can help some aspect of this world, as long as it's mostly based on software engineering. Things get really tricky when you have to manufacture stuff. That's why people like Elon Musk are so impressive that they. It's not just software. Tesla Autopilot is just not just software. It's like you have to actually, like, have factories that build cars and there's like a million components involved in the machinery required to assemble those cars and so on. But in software, one person can change the world, which is incredible. But on the mathematics side, what if you look back or maybe today, what made you fall in love with mathematics?

For me, I think I've always been very attracted to challenge, as I already indicated with writing the program. I guess if I see something that's hard or supposed to be impossible, sometimes I say maybe I want to see if I can pull that off. And with the mathematics, the math competitions presented problems that were hard that I didn't know how to start, but for which I could conceivably try to learn how to solve them. So, I mean, there are other things that are hard called like, get something to Mars, get people to Mars. And I didn't. And I still don't think that I'm able to solve that problem. On the other hand, the math problems struck me as things which are hard and with significant amount of extra work I could could figure it out and maybe they would actually even be useful like that. Mathematical skill is the core of lots of other things.

That's really interesting. Maybe you could speak to that, because a lot of people say that math is hard as a kind of negative statement. It always seemed to me a little bit like that's kind of a positive statement that all things that are worth having in this world are hard. I mean, everything that people think about that they would love to do, whether it's sports, whether it's art, music, and all the sciences, they're going to be hard if you want to do something special. So is there something you could say to that idea that math is hard? Should it be made easy or should it be hard?

So I think maybe I want to dig in a little bit onto this hard part and say, I think the interesting thing about the math is that you can see a question that you didn't know how to start doing it before, and over a course of thinking about it, you can come up with a way to solve it. And so you can move from a state of not being able to do something to a state of being able to do something where you help to take yourself through that instead of somebody else spoon feeding you that technique. So actually here I'm already digging into maybe part of my teaching philosophy also, which is that I actually don't want to ever just tell somebody, here's how you do something. I actually prefer to say, here's an interesting question. I know you don't quite know how to do it. Do you have any ideas? I'm actually coming up with. I'm actually explaining another way that you could try to do teaching. And I'm contrasting this to a method of watch me do this now, practice it 20 times. I'm trying to say a lot of people consider math to be hard because maybe they can't remember all of the methods that were taught. But for me, I look at the hardness and I don't think of it as a memory hardness. I think of it as a can you invent something hardness. And I think that if we can teach more people how to do that art of invention in a pure cognitive way, not as hard as the actual hardware stuff, right? But in terms of the concepts and the thoughts and the mathematics, teaching people how to invent, then suddenly actually they might not even find math to be that tiresomeness hard anymore. But that rewardingness hard of I have the capability of looking at something which I don't know what to do and coming up with how to do it. I actually think we should be doing that, giving people that capability so hard.

In the same way that invention is hard, that is ultimately rewarding. So maybe you can dig in the that a little bit longer. Which is. Do you see? Basically the way to teach math is to present a problem and to give a person a chance to try to invent a solution without, with minimal amount of information first. Is that, is. Is that basically how do you build that muscle of invention in a student?

Yes. So the way that. I guess I have two different sort of ways that I try to teach. Actually, one of them is in fact this semester, because all my classes were remotely delivered. I even threw them all onto my YouTube channel. So you can see how I teach at Carnegie Mellon. But I would often say, hey everyone, let's try to do this. Any ideas? And that actually changes my role as a professor from a person who shows up for class with a script of what I want to talk through. I don't have a script. The way I show up for class is there's something that we want to learn how to do and we're going to do it by improv. I'm talking about the same method as improv comedy, which is where you tell me some ideas and I'll try to. Yes, and them. You know what I mean? And then together we're going to come up with a proof of this concept where you were deeply involved in creating the proof. Actually, every time I teach the class, we do every proof slightly differently because it's based on how the students came up with it. And that's how I do it when I'm in person. I also have another line of courses that we make that is delivered online. Those things are where I can't do it live. But the teaching method became also similar. It was just. Here's an interesting question. I know it's out of reach. Why don't you think about it? And then automatic hints, we feed automatic hints through the Internet to go and let the person try to invent.

So that's like a more rigorous prodding of invention. But you did mention disease and Covid and you've been doing some very interesting stuff from a mathematical but also software engineering angle of coming up with ideas. It's back to the I can. I see a problem. I think I can help. So you stepped into this world. Can you tell me about your work there under the flag of Novid and both the software and the technical details of how the thing works?

Sure, sure. So first I want to make sure that I say this is actually team effort. I happen to be the one speaking, but there's no way this would exist without an incredible team of people who inspire me every day to work on this. But I'll speak on behalf of them. So the idea was indeed that we stepped forward in March of last year when the world started to become our part of the world started to become our part, meaning the United States started to become paralyzed by Covid. The shutdown started to happen. And at that time, it started as a figment of an idea, which was network theory, which is the area of math that I work in, could potentially be combined with smartphones and some kind of health information anonymized. Exactly how? We didn't know yet. We tried to crystallize it and many months into this work, we ended up accidentally discovering a new way to control diseases, which is now what is the main impetus of all of this work is to take this idea and polish it and hopefully have it be useful not only now, but for future pandemics. The idea is really simple to describe. Actually, my main thing in the world is I come up with Obvious observations. So I'll explain it.

Now, Einstein did the same thing, and he wrote a few short papers.

But. So the idea is like this. If we describe how usually people control disease, for a lot of history, it was that you'd find out who was sick, you'd find out who they've been around, and you try to remove all of those people from society against their will. Yes, now that's the problem. The against their will part gives you the wrong kind of a feedback loop, which makes it hard to control the disease, because then the people you're trying to control keep getting other people sick. You can see already how I'm thinking and talking about this. Feedback loops. This is actually related to something you said earlier about even how skyscrapers stay in the air. The whole point is control theory. You actually want to. Or even how an airplane stays. You need to have control loops which are feedbacking in the right way. And what we observed was that the feedback control loop for controlling disease by asking people to be removed from society against their will was not working. It was running against human incentives. And you suddenly are trying to control 7 billion, 8 billion people in ways that they don't individually want to necessarily do. So here's the idea, and this is inspired by the fact that at the core of our team were user experience designers. That's actually, in fact, the first thing I knew we needed when we started was to bring user experience at the core. Okay, but. So the idea was, suppose there was a. Suppose, hypothetically, there was a pandemic. What would you want? You would want a way to be able to live your life as much as possible and avoid getting sick. Can we make an app to help you avoid getting sick? Notice how I've just articulated the problem. It is not, can we make an app so that after you are around somebody who's sick, you can be removed from society? It's, can we make an app so that you can avoid getting sick? That would run a positive feed. I don't know if I want to call it positive or negative, but it would run a good feedback loop. Okay, so then how would you do this? The only problem is that you don't know who's sick, because especially with this disease, if I see somebody who looks perfectly healthy, the disease spreads two days before you have any symptoms. And so it's actually not possible. That's where the network theory comes in. You caught it from someone. What if we change the paradigm and we said, whenever there's a sickness, tell everybody how many physical relationships separate them from the sickness. That is the trivial idea we added. The trivial idea was the distance between you and a disease is not measured in feet or seconds. It's measured in terms of how many close physical relationships separate you. Like these six degrees of separation, like LinkedIn. Simple idea. What if we told everyone that it turns out that actually unlocks some interesting behavioral feedback loops, which, for example, let me now jump to a non Covid example to show why this maybe could be useful. Actually, we think it could be quite useful. Imagine there was Ebola or some hemorrhagic fever. Imagine it spread through contact, through the air. In fact, pretend, pretend. That's a disastrous disease. It has high fatality rate, and as you die, you're bleeding out of every orifice. Okay, yeah.

Not pleasant.

Not pleasant. So the question is, suppose that such a disease broke. Who would want to install an app that would tell them how many relationships away from them this disease had struck.

A lot of people.

A lot of people. In fact, almost. I don't want to say almost everyone. That's a very strong statement. But a very large number of people.

That's fascinating framing. The more deadly and transmissible the disease, the stronger the incentive to install it in a positive sense, in the good feedback loop sense. That's a really good example. It's a really good way to frame it. Because with COVID it was not as deadly as potential pandemics could have been, viruses could have been. So it's sometimes muddled with how we think about it. But yeah, this is a really good framing. If the virus was a lot more deadly, you want to create a system that has a set of incentives that it quickly spreads to the population where everybody is using it and it's contributing in a positive way to the system.

Exactly. And actually, that point you just made, I don't take credit for that observation. There was another person I talked to who pointed out that it's very interesting that this feedb loop is even more effective when the disease is worse. And that's actually not a bad characteristic to have in your feedback loop if you're trying to help civilization keep running.

Yeah, it's a really. It's in this dynamic, like people figure out, they dynamically figure out how bad the disease is the more it spreads and the deadlier it is as the people observe it. As long as the spread of information, like semantic information, natural language information, is closely aligned with the reality of the disease, which is a whole nother conversation. Right. Maybe we'll chat about that. How we sort of make sure there's not misinformation where there's accurate information. But that aside, okay, so this is a really nice property.

Right. And just going on, on that, actually just talking more about what that could do and why we're so excited about it, it's that not only would people want to install it, what would they do if you start to see that this disease is getting closer and closer. We surveyed informally people, but they said, as we saw it getting closer, we would hide, we would try to not have contacts. But now notice what this has just achieved. The whole goal on this whole exercise was you got the people who might be sick and you got everyone else set A and set B. Set A is the people who might be sick. Set B is everyone else. And for the entirety of the past, contact tracing approaches, you try to get set A to do things that might not be to their liking or their will, because that's removing them from society.

Yes.

We found out that there's two ways to separate set A from set B. You can also let the people at set B, at the fringe of set A attempt to remove themselves from this interface. It's just the symmetry of A and B separation. Everyone was looking at A, we look at B, and suddenly B is in their incentive to do so. Beautiful.

So there's a virus that jumps from human to human. So there's a network, sometimes called graph of the spread of a virus. It hops from person to person to person to person. And each one of us, individuals are sitting or plopped, plopped into that network. We have close friends and relations and so on. It's kind of fascinating to actually think about this network and we can maybe talk about the shapes of this kind of network. Because I was trying to think exactly this, like how many people do. I'm kind of an introvert. Not kind of. I'm very much an introvert, but so can I be explicit about the kind of people I meet in regular life, say, when it was completely opened up? There's no pandemic. There is a kind of network, and there's maybe in the graph theoretic sense, there's some weights or something about how close that relationship is in terms of the frequency of visits, the duration of visits, and all of those kinds of things. So you're saying we might want to be. To create on top of that network a spread of information to let you know, as the virus travels through this network, how close is it getting to you? And the number of hops away it is on that network is really powerful. Information that creates a positive feedback loop where you can act essentially anonymously and on your own, like nobody's telling you what to do, which is really important. It's decentralized and not whatever the opposite of authoritarian is. But you get to sort of the American way, you get to choose to do it yourself. You have the freedom to do it yourself and you're incentivized to do it. And you're most likely going to do it to protect yourself against.

You getting.

The disease the closer it gets to you based on the information that you have. But, but can you maybe elaborate? First of all, brilliant. Whenever I saw the thing you're working on. So forget for Covid. This is of course really relevant for Covid, but it's also probably relevant for future diseases as well. So that was the thing I'm nervous about is like, if this whole, if our society shut down because of COVID like, what the heck is gonna happen when there's a much deadlier disease like this? This is disappointing. The whole time, 2020, the whole time I'm just sitting like this, like, is the incompetence of everybody except the people developing vaccines. The biologists are the only ones that.

Got their stuff together.

But in terms of institutions and all that kind of stuff, it's just been. It's just been terrible. But this is exactly the power of information and the power of information that doesn't limit personal freedom. So your idea is brilliant. O mathematically, can you maybe elaborate? What are we talking about? How do you actually make that work? What's involved?

Sure. First I'm going to reply to something you said about the freedom inside this, because actually that was the idea. The idea is this is game theory. And effectively what we did is analogous to free market economy as opposed to central planning, if you just line up the set of incentives correctly so that people have, in their purely selfish behavior, are contributing to the optimization of the global function. Yes, that's it. And the point of what we do, I guess, in mathematics is we try to explore the search space to go and find out as many possibilities as there are. And in this case it's an applied search space. That's why the inputs from design, user experience, design and actual people are important. But you asked about, I guess, the mathematical or the technical things underpinning it. So I think the first thing I'll say is we wanted to make this thing not require your personal information. And so in order to do that, what gave me the confidence to, I guess, lead our team to run at the Beginning is, we saw that this could be done without using GPS information. So technically what's going on is if two smartphones, it's a smartphone app. If two smartphones have this thing installed, they just communicate with each other by Bluetooth to go and find out how far they can detect nearby things by Bluetooth. And then they can find out that these two phones were approximately such and such distance apart. And that kind of relative proximity information is enough to construct this big network.

Okay, so the physical network is constructed based on proximity, that's through Bluetooth. And you don't have to specify your exact location. It's the proximity.

I'm not using the Pythagorean theorem, basically. I mean, if I just knew the GPS coordinates, we could use the Pythagorean theorem too. Sorry, that's just how I call it. Distance formula, whatever you want to call it.

Yeah, so we're not doing the old Pythagorean based violation of privacy.

Okay.

But so is that enough to form, to give you enough information about physical connection to another human being? Is there a time element there? Is there? So, okay, that sounds like a really strong, like low hanging fruit. Like if you have that, you could probably go really, really far. My natural question is, is there extra information you can add on top of that, like the duration of the physical proximity?

So first of all, we actually do estimate the duration. But the way we estimate the duration is like how a movie is filmed in the sense that every so often, every few minutes, we check what's nearby. It's like how a movie is filmed, you take lots of snapshots. So there's no way in a battery efficient way to really keep track of that proximity. However, fortunately, we're using probability. The fact is, the paradigm that we're using is it's not super important. If you run into that person only for 10 minutes at the grocery store store. If that's a stranger that you run into 10 minutes in this grocery store, that's not going to be relevant for our paradigm. Because our paradigm is not telling you who were you around before and might therefore have gotten infected by already. Ours is about predicting the future. We change from. I mean the standard paradigm was what already happened, quick damage control. Ours is predict the future. If you run into that person once in the grocery store today and never see them again, it's irrelevant. For predicting the future and therefore for ours, what really matters is the many hours around the other person at which point if you're scanning every five to.

Eight minutes, that's going to come out in the probability. Like, statistically speaking, it's going to come out as a strong relationship, and a person in the grocery store is going to wash out is not an important physical relationship. Amir, this is brilliant. How difficult is it to make work? So you said one, there's the mathematical component that we just kind of talked about, and then there's the user experience component. So how difficult is it to go just like you built the video game Alien Attack from zero to completion? What's involved? How difficult is it?

So I'm going to answer that question in terms of building the product, but then I'm also going to acknowledge that just having an app doesn't make it useful, because that's actually maybe the easy part, if you know what I mean. There's all of this stuff about rollout, adoption and awareness, but let's focus on the app part first. So that's again, why I said the team is incredible. So we have a bunch of people who. Let's just say that the technology that we use to make it is not the standard way you make an app. If you think about a standard iOS app or Android app, those are a user interface that contacts a web server and sends some information back and forth. We're doing some stuff that has to hook into the operating system of saying, let's go use Bluetooth for something it wasn't really meant for. Right. So there's that part.

And by the way, what is the app called?

Oh, it's called Novid Covid with an N. Very nice.

So you have to hook into Bluetooth. You're saying you have to do that beyond the permissions that are, like, at the very surface level provided on the phone?

Well, I don't want to call them permissions. I just want to say that's not what you usually do with Bluetooth.

Gotcha.

Usually with Bluetooth, you say, do I have headphones nearby?

Yes.

Okay, I'm done. You don't go and say, do I have headphones nearby? Or do I have another phone nearby which is doing something? And then keep asking that same question. Keep asking the question. Right. So it's actually not easy. And I mean, there were some parts of it which actually a lot of people had tried unsuccessfully. Actually, it's known that, for example, the UK was trying to do something similar, and the problem they ran into was when you program things on iOS, iOS is very good at making it hard to do things in the background. And so there was quite a lot of effort required to go and make this thing work.

So the whole point, this thing would run in the background and iOS, I mean most Android probably as well. Right. But yeah, iOS certainly makes it difficult for something to run in the background, especially when it's eating up your battery. Right.

Well, we wanted to make sure we didn't eat up the battery. So that one, we actually are very proud of the fact that ours uses very little battery actually, even if compared to Apple's own system.

Beautiful. So what else is required to make this thing work?

Right. So the key was that you had to do significant amount of work on the actual mobile app development, which fortunately the team that we brought was this kind of general thinkers where we would dig in deep into the operating system documentation and the API libraries. So we got that working. But there's another angle which is you also need the servers to be able to compute fast enough, which is tying back to this old school computer programming competitions and Math Olympiads. In fact, our team that was working on the algorithm and backend side included several people who had been in these competitions from before, which I happen to know because I, I do coach the team for the math.

Yes.

And so we were able to bring people in to build servers, a server infrastructure in C actually, so that we could support significant numbers of people without needing tons of servers.

Is there some distributed algorithms working here or. You basically have to keep in the same place the entire graph as it builds because especially the more and more people use it, the bigger, the bigger the graph gets. I mean this is very difficult scaling problem. Right.

Ah, so that's actually why this computer algorithm competition stuff was handy. It's because there are only about 7 to 8 giga people in the world. Yeah, that's not that many. So if you can make your algorithms linear time or almost linear time, a computer operates in gigahertz. I only need to do one run, one recalculation every hour in terms of telling people how far away these dangers are. Yes. So I suddenly have 3600 seconds and my CPU cores are running in gigahertz and at most they're eight giga people.

Well, you're skipping over the fact that there's N squared potential connections between people. So how do you get around the fact that. The potential set of relationship any one of us could have is 8 billion. So it's 8 billion times squared. That's potential amount of data you have to be storing and computing over and constantly updating.

So the way we dealt with that is we actually expect that the typical network is very sparse. The Technical term sparse would mean that the average degree or the average number of connections that a person has is going to be at most like 100 strong connections that you care about. If you think of it almost in terms of the heavy hitters actually in most people's lives, if we just kept track of their top hundred interactions, that's probably most of the signal.

Yeah, yeah. I'm saddened to think that I might not be even in a double digits.

But, oh, I was intentionally giving a crazy number to account for college students you call.

Those are the who you call in the heavy hitters, the people who are like the social butterflies.

Yeah, yeah, yeah.

I need to. I'd love to know that information about myself. By the way, do you expose the graph like how many like about yourself, how many connections you have?

We do expose to each person how many direct connections they have.

That's great.

But for privacy purposes, we don't tell anybody who their connections. Like how their connections are interconnected.

Yes, gotcha.

But at the same time, we do expose also to everyone an interesting chart that says here's how many people you have that you're connected to directly, here's how Many at distance 2, meaning via people. And then here's how Many at distance 3. And the reason we do that is that actually ends up being a dynamic that also boosts adoption. It drives another feedback loop. The reason is because we saw actually when we deployed this in some universities that when people see on their app that they are indirectly connected to hundreds or thousands of other people, they get excited and they tell other people, hey, let's download this app. Yeah, but you know, we also saw in those examples, especially looking at the screenshots people gave that is hit as soon as the typical person has two or three other direct connections on the system, because that means that our app has reached a virality r naught of 2 to 3. The key is we were making a viral app to fight a virus spreading on the same network that the virus spreads on.

So you're trying to out virus the virus.

That's right. That's exactly right.

Okay, great. What have you learned from this whole experience in terms of, let's say for Covid, but for future pandemics as well? Is it possible to use the power information here of networked information as the virus spreads and travels in order to basically keep the society open? Is it possible for. For people to protect themselves with this information or do you still have to have most like in this overarching policy of everybody should stay at home all that kind of thing.

We are trying to answer that question right now. So the answer is we don't know yet. But that's actually why we're very happy that now the idea has started to become. Become more widely known and we're already starting to collaborate with epidemiologists again. I'm a math. I'm just a mathematician. Right. And a mathematician should not be the person who is telling everybody this will definitely work. But because of the potential power of this approach, especially the potential power of this being an end game for Covid, we have gotten the interest of real researchers and we're now working together to try to actually understand the answer to that question. Because, you see, there's a theory. So what I can share is the mathematics of here's why there's some hope that this would work. And that's because I'm talking about endgame now. Endgame means you have very few cases. But everywhere we're always thinking, once there's few cases, then does that mean we now open up? Once you open up in the past, then the cases go up again until you have to lock down again. And now when we talk about the dynamic process that makes. It's guaranteeing you always have cases until you have the great vaccines, which is, you know, we both got vaccinated. This is good. But at the same time, why I'm thinking this is still important is because we know that many vaccine makers have said they're preparing for the next dose next year. And if we have a perpetual thing where you just always need a new vaccine every year, it could actually be beneficial to make sure we have as many other techniques as possible for parts of the world that can't afford, for example, that kind of distribution. Yeah.

So actually, no matter how deadly the virus is, no matter how many things, whether you have a vaccine or not, it's still useful to be having this information.

Yes.

Is to stay home or not, depending on how risk. Like, I'm a big fan, just like you said, of having the freedom for you to decide how risk averse you want to be. Right. Depending on your own conditions, but also on the state of like, what you just how dangerously you like to live.

So I think that actually makes a lot of sense. And I also think that since we're. When we. When you think of disease spreading, it spreads in aggregate in the sense that if there are some people who maybe are more risk tolerant because of other things in their life, well, there might also be other people who are less risk tolerant. And then those People decide to isolate. But what matters is in the aggregate that this R0 of the infection spreading drops below 1. And so the key is if you can empower people with that power to make that decision, you might actually still be able to drive that R0 down below 1.

Yeah, and also this is me talking. People get a little bit nervous, I think, with information somehow mapping to privacy violation. But first of all, in the approach you're describing, that's respecting anonymity. But I would love to have information from the very beginning, from March and April of last year, almost like a map of like where it's risky and where it's not to go. And not map based on sort of the exact location of people, but where people usually hang out kind of thing. Just it may be not necessarily about actual location, but just maybe activities like just to have information about what is. What is good to do and not, you know, in terms of like safety. Is it okay to run outside and not is it okay to go to a restaurant or not. I just feel like we're operating the blind. And then what you had is a very imperfect signal, which is like basically politicians desperately trying to make statements about what is safe and not. They don't know what the heck they're doing. They have a bunch of smart scientists telling them stuff. And the scientists themselves, also very important, don't always know what they're doing. Epidemiology is not. Is as much an art as a science. You're desperately trying to predict the future, which nobody can do, and then you're trying to speak with some level of authority. I mean, if I were to criticize scientists, they spoke with too much authority. It's okay to say I'm not sure, but then they think like if I say I'm not sure, then there's going to be a distrust. What they realize is when you're wrong and you say I'm sure it's going to lead to more distrust. So this is imperfect. Like just chaotic, messy system of people trying to figure out with very little information. And what you're proposing is just a huge amount of information. And information is power. Is there challenges with adoption that you see in the future here? So there's maybe we could speak to. There's approaches, I guess from Google. There's different people that have tried similar kind of ideas. Not you have quite a novel idea actually. But speaking the umbrella idea of contact tracing. Is there something you can comment about why their approaches haven't been fully adopted? Is there challenges there? Is there reasons why Novid might be a better idea moving forward in general, just about adoption.

Yeah. So first of all, I want to say I always have respect for the methods that other people use. And so it's good to see that other people have been trying. But what we have notice is that the difference between our value proposition to the user and the value proposition to the user delivered by everything that was made before is that unfortunately the action of installing a standard contact tracing app will then tell you after you have already been exposed to the disease so that you can protect other people from you and what that does to your own direct probability of getting sick. If you think about it, suppose you were making the decision, should I or should I not install one of those apps, what does that do to your own probability of getting sick? It's close to zero.

This is the sad thing you're speaking to. Not sad, I suppose it's the way the world is. The only incentive there is to just help other people, I suppose. But a much stronger incentive is, is anything that allows you to help yourself.

Yes. So what I'm saying is that let's just say free market capitalism was not based on altruism. I think it's based on if you make a system of incentives so that everybody trying to maximize their own situation somehow contributes to the whole, that's a game theoretic solution to a very hard problem. And so this is actually basically mechanism design. We've basically come up with a different mechanism, different set of incentives which incentivizes the adoption. Because actually whenever we've been rolling it out, usually the first question we ask people like say in a university is, do you know what Novid does? And most of them have read about the other apps and they say, oh, Novid will tell you after you've been around someone so you can quarantine. And we have to explain to them. Actually, Novid never wants to ask you to quarantine. That's not the principle. Our principle isn't based on that at all. We just want to let you know if something is coming close so that you can protect yourself.

If you want.

Yeah, if you want. If you want. If you want. And the quarantine is like, yes, in that case, if you're quarantining, it's because you're shutting the door from the inside, if that makes sense.

Yes, exactly.

Exactly.

I mean, this is brilliant. So what do you think the future looks like for future pandemics? What's your plan with Novid? What's your plan with these set of ideas?

I am actually still an academic and a researcher. So the biggest work I'm working on right now is to try to build as many collaborations with other, other public health researchers at other universities to actually work on pilot deployments together in various places. That's the goal. That's actually ongoing work right now. And so, for example, if anyone's watching this and you happen to be a public health researcher and you want to be involved in something like this, I'm just going to say I'm still incentive thinking there's something in it for the researchers too. This could open up an entire new way of controlling disease. That's my hope. I mean, it might actually be true. And people who are involved in figuring out how to make this work, well, it could actually be good for their careers too. I always have to think like, if a researcher was getting involved, what are they getting out of it?

Oh, so you mean like from a research perspective you can like publications and sets of ideas about how to, from a sort of network theory perspective, understand how we control the spread of a pandemic?

Yes, and what I'm doing right now is this is basically interdisciplinary research where maybe our side is bringing the technology and the network theory. And the missing parts are epidemiology and public health expertise. And if the two things start to join also, because everywhere that you deploy, let's just say that the world is different in the Philippines as it is in the United States. And just the natures of the locality would mean that someone like me should not be trying to figure out how to do that. But if we can work with the researchers who are based there now, suddenly we might come up with a solution that will help scale in parts of the where they aren't all getting the Moderna and Pfizer vaccines which cost like $20 a pop in the US so.

If they want to participate, who do they reach out to?

Oh, that would just be us. I mean the novid.org website has novid.org it has a feedback reach out form and actually we are. I mean again, this is the DNA of being a researcher. I am actually very excited by the idea that this could contribute knowledge that will outlast all of our generations, like all of our lifetimes.

There you go. Reach out to nova.org what about individual people? Should they install the app and try it out? Or is this really geographically restricted?

Oh yeah, I didn't come on here to tell everyone to install the app. I did not come to tell everyone to install the app because it works best if Your local health authority is working with us.

Gotcha.

There's a reason. It's because this is back to the game theory. If anyone could just say, I'm positive, the high school senior prank would be to say that we have a massive outbreak on finals week. Let's not have final exams. So the way that our system works, it actually borrows some ideas. Not borrows. We came up with them independently. But this idea is similar to what Google and Apple do, which is that if the local health authority is working with this, they can, for everyone who's positive, give them a passcode that expires in a short time. So for ours, if you're on the app and saying, I'm positive, you can either just say that and that's called unverified, or you can enter in one of these codes that you got from the local health authority. So basically, for anyone who's watching this, it's not that you should just go and download it, unless you want to go and look at it, that's cool. But on the other hand, if you happen to know anyone at the local health authority, which is trying to figure out how to handle Covid, well, then, I mean, we'd be very happy to also work with you.

Gotcha. So the verified there is really important because you're maintaining anonymity. And because of that, you have to have some source of verification in order to make sure that it's not possible to manipulate, because it's ultimately about trust and information. And so it could be. Verification is really important there. So basically, individual people should ask their local health authorities to sign up to contact you. I hope this spreads. I hope this spreads for future pandemics, because I'm really. It's the amount, the millions of people who are hurt by this. I think our response to the virus, economically speaking, the number of people who lost their dream, lost their jobs, but also lost their dream entrepreneurs, you know, jobs often give meaning. There's people who financially and psychologically are suffering because of our, I'll say it, incompetent response to the virus across the world, but certainly United States, that should be the beacon of entrepreneurial hope for the world.

So.

I hope that we'll be able to respond to these kinds of events much better in the future. And this is exactly the right kind of idea. And now is the time to do the investment. Let's step back to the beauty of mathematics. Maybe ask the big, silly question first, which is, what do you find beautiful about mathematics?

I think that being able to look at a Complicated problem which looks unsolvable, and then to be able to change the perspective, to come from a different angle and suddenly see that there's a nice solution. I don't mean that every problem in math is supposed to be this way, but I think that these reframings and changing of perspectives that cause difficult things to get simplified and crystallized and factored in certain ways is beautiful. Actually, that's related to what we were just talking about with even this fighting pandemics. The crystal idea was just quantify proximity by the number of relationships in the physical network instead of just by the feet and meters. Right. If you change that perspective now, all of these things follow. And so mathematics to me is beautiful in the pure sense just for that.

Yeah. It's quite interesting to see a human civilization as a network, as a graph, and our relationships as kind of edges in that graph, and to then do outside of just pandemic, do interesting inferences based on that. This is true for like Twitter, social networks and so on. How we expand the kind of things we talk about, think about sort of politically. If you have this little bubble quote unquote of ideas that you play with, it's nice. From a recommender system perspective, how do you jump out of those bubbles? It's really fascinating. YouTube was working on that, Twitter is working on that, but not always so successfully. But there's a lot of interesting work from a mathematical and a psychological, sociological perspective there within those graphs. But if we look at the cleanest formulation of that, of looking at a problem from a different perspective, you're also involved with the International Mathematics Olympiad, which takes small, clean problems that are really hard, but once you look at them differently, can become easy. But that little jump of innovation is the entire trick. So maybe at the high level, can you say, what is the International Mathematical Olympiad?

Sure. So this is the competition for people who aren't yet in college math competition, which is the most prestigious one in the entire world. It's the Olympics of mathematics, but only for people who aren't yet in college. Now, the kinds of questions that they ask you to do are not computational. Usually you're not supposed to find that the answer is 42. Instead, you're supposed to explain why something is true. And the problem is that at the beginning, when you look at each of the questions, first of all, you have four and a half hours to solve three questions. And this is one day. And then you have a second day, which is four and a half hours, three questions. But when you look at the questions, they're all asking you explain why the following thing is true, which you've never seen before. And by the way, even though there are six questions, if you solve any one of them, you're a genius and you get an honorable mention. So this is hard to solve.

What about, is it one person? Is it a team?

So each country can send six people and the score of the country is actually unofficial. There's not an official country versus country system, although everyone just adds up the point scores of the six people and they say, well, now which country stacked up where?

Yeah, so maybe as a side comment, I should say that there's a bunch of countries, including the former Soviet Union and Russia, where I grew up, where this is one of the most important competitions that the country participates in. Like it was a source of pride for a lot of the country. You look at the Olympic sports, like wrestling, weightlifting, there's certain sports and hockey that Russia and the Soviet Union truly took pride in. And actually the Mathematical Olympiad, it was one of them for many years, is still one of them. And that's kind of fascinating. We don't think about it this way in the United States. Maybe you can correct me if I'm wrong, but it's not nearly as popular in the United States in terms of its integration into the culture, into just basic conversation, into the pride. Like, you know, if you win an Olympic gold medal or if you win the super bowl, you can walk around proud. I think that was the case with the Mathematical Olympiad in Russia. Not, not as much the case in the United States, I think. So. I just want to give that a little aside because beating anybody from Russia, from the Eastern republic or from China is very, very difficult. Like, if I remember correctly, you know, there's people, this was a multi year training process. They train hard and this is, this is everything that they're focused on. My, my dad was, was, was a participant in this. And it's, I mean, it's, it's as serious as Olympic sports. You think about like gymnastics, like young athletes participating in gymnastics. This is as serious as that, if not more serious. So I just want to give that a little bit of context because we're talking about serious high level methods athletics almost here.

Yeah. And actually I also think that it made sense from the Soviet Union's perspective because if you look at what these people do eventually, even though, let's look at the USSR's Olympic International Math Olympiad record. Even though they, I say even though they won a lot of awards at the high school thing, many of them Went on to do incredible things in research mathematics or research other things. And, and that's showing the generalizability of what they were working on because ultimately we're just playing with ideas of how to prove things. And if you get pretty good at inventing creative ways to turn problems apart, split them apart, observe neat ways to turn messy things into simple crystals. Well, if you're going to try to solve any real problem in the real world, that could be a really handy tool too. So I don't think it was a bad investment. I, I think it clearly worked well for Soviet Union.

Yeah. So this, this is interesting. People sometimes ask me, you know, you go up and under communism, you know, was there anything good about communism? And it's difficult for me to talk about it because it's not. Communism is one of those things that's looked down on like without in absolutist terms currently. But you could still, in my perspective, talk about the actual. Forget communism or whatever the actual term is. But, you know, certain ways that the society function that we can learn lessons from. And one of the things in the Soviet Union that was highly prized is knowledge. Not even knowledge, it's wisdom and the skill of invention, of innovation at a young age. So we're not talking about a selection process where you pick the best students in the school to do the mathematics or to read literature. It's like everybody did it. Everybody. It was almost treated as if anyone could be the next Einstein, anybody could be the next, I don't know, Hemingway, James Joyce. And so you're forcing an education on the populace and a rigorous, rigorous, deep education. Like, as opposed to kind of like, oh, we want to make sure we, we teach to the weakest student in the class, which American systems can sometimes do, because we don't want to leave anyone behind. The Russian system was anyone can be the strongest student, and we're going to teach you the strongest student and we're going to just to pretend or force everybody, even the weakest student to be strong. And what that results in, it's, obviously, this is what people talk about is a huge amount of pressure. Like it's psychologically very difficult. This is why people struggle when they go to mit, this very competitive environment. It can be very psychologically difficult, but at the same time it's bringing out the best out of people. And that mathematics was certainly one of those things. And exactly what you're saying, which kind of clicked with me just now, as opposed to kind of a spelling bee in the United States, which I guess you Spell. I'm horrible at this, but it's a competition about spelling, which I'm not sure, but you could argue doesn't generalize well to the future skills. Mathematics, especially this kind of mathematics, is essentially formalized competition of invention, of creating new ideas, and, and that generalizes really, really well. So that's quite brilliantly put. I didn't really think about that. So this is not just about the competition. This is about developing minds that will come to do some incredible stuff in the future.

Yeah, actually I want to respond to a couple of things there. The first one is one which is this notion of whether or not that is possible in a non authoritarian regime. I think it is. And that's actually why I spent some of my efforts before the COVID thing actually trying to work towards there. The reason is because if you think about it, let's say in America, lots of people are pretty serious about training very hard for football or baseball or basketball. Basketball is very, very accessible. But lots of people are doing that. Why? Well, actually, I think that what was going on with the authoritarian thing was at least the message that was universally sent was being a good thinker and a creator of ideas is a good thing. Yes, exactly. There's no reason why that message can't be sent everywhere. And I think it actually should be. So that's the first thing. The second thing is what you commented about this thing about the generalizable skill and what could people do with Olympiads afterwards? So that's actually my interest in the whole thing. I, I mean, I don't just coach students how to do problems. In fact, I'm not even the best person for that. I'm not the best at solving these problems. There are other people who are much better at making problems and teaching people how to solve problems. In fact, when the Mathematical association of America, which is the group which is in charge of the US participation in these Olympiads, when they were deciding whether or not to put me in back in 2013 as the head coach, I had a conversation with their executive director where I commented that we might do worse because my position was, I don't, I mean, I actually didn't want to focus on winning. I said, if you're going to let me work with 60 very strong minds as picked through this system because the coach works with these, gets to run a camp for these students. I said, I'm actually not going to define my success in terms of winning this contest. I said I wanted to maximize the number of the students that I read about in the New York Times in 20 years. And the executive director of the Mathematical association of America was fully in support of this because that's also how their philosophy is. So in America, the way we run this is we're actually not just training to win, even though the students are very good and they can win anyway. One reason, for example, I went and even did the COVID thing involving quite a few of them is so that hopefully some of them get ideas, because in 20, 30 years I won't have the energy or the insight to solve problems. We'll have another catastrophe and hopefully some of these people will step up and.

Do it and ultimately have that long term impact. I wonder if this is scalable to, because that's such a great metric for education. Not how to get an A on the test, but how to have. How to be on the COVID of New York Times for inventing something new. Do you think that's generalizable to education beyond just this particular Olympia? Like it's even you saying this feels like a rare statement, almost like a radical statement as a goal for education.

So actually the way I teach my classes at Carnegie Mellon, which I will admit right away is not equivalent to the average in the world, but it's already not. It's already not just the top 60 in the country as picked by something. Let me just explain. I have exams in my class which are 90% of the grade. So the exams are the whole thing, or most of the whole thing. And the way that I let students prepare for the exams is I show them all the problems I've ever given on the previous exams. And the exam that they will take is open notes. They can take all the notes they want on the previous problems. And the guarantee is that the exam problems this time will have no overlap with anything you have seen me give in the past, as well as no overlap with anything I taught in the class. So the entire exam is invention. Wow. But that's how I go, right? My point is, I have explained to people, when I teach you, I don't want you to have remembered a method I showed you. I want you to have learned enough about this area that if you face a new question, which I came up with the night before by thinking about what could I ask that I have never asked before? Oh, that's cute. I wonder what the answer is. Aha. That's an exam problem. That's exactly what I do before the exam. And then that's what I want them to learn. And the first exam. Usually people have a rough time because it's like, what kind of crazy class is this? The professor doesn't teach you anything for the exam, but then by the second or third, and by the time they finish the class, they have learned how to solve anything in the area.

How to invent.

How to invent in that area. Yeah.

Can we walk back to the Mathematical Olympiad? What's the scoring and format like? And also, what does it take to win?

So the way it works is that each of the six students do the problems, and there are six problems. All the problems are equally weighted, so each one's worth seven points. That means that your maximum score is six problems times seven points, which is the nice number of 42. And now the way that they're scored, by the way, is there's partial credit. So the question is asking you explain why this weird fact is true. Okay, if you explain why, you get seven points. If you make minor mistake, maybe you get six points. But if you don't succeed in explaining why, but you explain some other true fact which is along the way of proving it, then you get partial credit. And actually, now this is tricky because how do you score such a thing? It's not like it was. The answer was 72 and you wrote 71. And it's close, right? The answer is 72 and you wrote 36. Oh, but that's pretty close, because maybe you're just off by it. By the way, they're not numerical anyway. But I'm just giving some numerical analog to the way the scoring might work. They are all essays. And that's where I guess I have some role, as well as some other people who helped me in the US delegation for coaches, we actually debate with the country which is organizing it. The country which is organizing the Olympiad brings about 50 people to help judge the written solutions. And you schedule these half hour appointments where the delegation from one country sits down at a table like this. The opposite side is two or three people from the host country, and they're just looking over these exam papers saying, well, how many points is this worth? Based on some rubric that has been designed. And this is a negotiation process where we're not trying to bargain and get the best score we can. In fact, sometimes we go to this table and we will say, we think we want less than what you gave us. These are our principles. If you give us too much, we say, no, you gave us too much. We do that. However, the reason why this is an interesting process is because, if you can imagine, every country which is participating has its own language and so if you're trying to grade the Mongolian scripts and they're written in Mongolian, if you don't read Mongolian, which most people don't, then the coaches are explaining to you, this is what the student has written. It's actually quite interesting process.

It's almost like a jury. In the American legal system, you have a jury where they're deliberating, but unlike a jury, there's the members of the jurist speaking different languages sometimes. That's fascinating, but, I mean, it's hard to know what to do because it's probably really, really competitive. But your sense is that ultimately people like, how do you prevent manipulation here? Right?

Well, we just hope that it's not happening. So we write in English. Therefore everything that the US does, everyone can look at. So it's very hard for you, it's very hard for you to manipulate. We don't manipulate. We only hope that other people aren't. But at the same time, as you see, our philosophy was we want to use this as a way to develop general talent. And although we do this for the six people who go to the International Math Olympiad, we really want that everyone touched at any stage of this process get some skills that can help to contribute more later.

So I don't know if you can say something insightful to this question, but what do you think makes a really hard math problem on this Olympiad? Maybe in the courses you teach or in general, what makes for a hard problem? You've seen, I'm sure, a lot of really difficult problems, what makes a hard problem.

So I could quantify it by the number of leaps of insight, of changes of perspective that are along the way. And here's why. This is like a very theoretical computer science way of looking at it, okay? It's that each reframing of the problem and using of some tool, I actually call that a leap of insight. When you say, oh, wow, now I see I should kind of put these plugs into those sockets, like so, and suddenly I get to use that machine. Oh, but I'm not done yet. Now I need to do it again. Each such step is a large possible large fan out in the search space. The number of these tells you the exponent. The base of the exponent is like how many different possibilities you could try. And that's actually why, if you have a three insight problem, that is not three times as hard as a one insight problem. Because after you've made the one insight, it's not clear that that was the right track necessarily, unless you're very still.

A branching of possibilities. You're saying there's problems like on the Math Olympia that requires more than one insight.

Yes, those are the hard ones. And also I can tell you how you can tell. So this is how I also taught myself math when I was in college. So if you are taking a. Not taught myself. I was taking classes, of course, but I was trying to read the textbook and I found out I was very bad at reading math textbooks. A math textbook has a long page of stuff that is all true which after you read the page, you have no idea what you just read.

Yeah, this is just a good summary.

Of a math textbook. Okay. Yeah. Because it's not clear why anything was done that way. And yes, everything is true, but how the heck did anyone think of that? So the way that I taught myself math eventually was the way I read a math textbook is I would look at the theorem statement, I would look at the length of the proof, and then I would close the book and attempt to reprove it myself. Now, the length of the proof is telling you the number of insights because the length of the proof is linear in the number of insights. Each insight takes space. And if I know that it's a short proof, I know that there's only one insight. So when I'm doing my own way of solving the problem, like finding the proof, I quit. If I have to do too many pluggings, it's equivalent to a math contest. In a math contest, I look, is it problem one, two or three that tells me how many insights there are. This is exactly what I did.

That's brilliant. Linear in the number.

I don't know.

I think it's possible that that's true.

Approximately. Approximately.

Approximately, yeah.

Just, I don't know.

Somebody out there is going to try to formally prove this.

Oh no, I mean, you're right. There are cases where maybe it's not quite linear, but in general.

Well, some of it's notation too and some of it is style and all those kinds of things.

But within a textbook, within the same book. Within the same book? Yes, within the same book on the same subject. This is what I was using.

That's hilarious.

Because you know, if it's a two page proof, you just know this is going to be insane. Right.

But that's the, that's the scary thing about insights. You look like Andrew Wiles working on the Fermo Oslo's theorem is you don't know. Something seems like a good idea and you have that idea and it feels like this is a leap, like A totally new way to see it. But you have no idea if it's at all useful. Even if you think it's correct. You have no idea if this is like, going to be. Go down a path that's completely counterproductive or not productive at all.

That's.

This, that's a crappy thing about invention, is like, I have. I'm sure you do have. I have a lot of really good ideas every single day. But like. And I'll go inside my head along them, along that little trajectory, but it could be just a total waste. And it's that you know what that feels like. It just feels like patience is required not to get excited at any one thing.

So I think this is interesting because you raised Andrew Biles. He spent seven years attacking the same thing. Right. And so I think that what attracts professional researchers to this is because even though it's very painful that you keep fighting with something, when you finally find the right insights and string them together, it feels really good.

Well, there's also, like, short term, it feels good to whether it's real or not to pretend like you've solved something in the sense, like you have an insight and there's a sense like this might be the insight that solves it. So at least for me, I just enjoy that rush of positivity, even though I know statistically speaking, it's probably going to be a dead end.

I'm the same way. I'm the same way. In fact, that's how I know whether I might want to keep thinking about this general problem. It's like, if I still see that I'm getting some insight, I'm not at a dead end yet. But that's also where I learned something from my PhD advisor. Actually, he was a real big inspiration in my life. His name is Benny Sudakov. In fact, he grew up in the former Soviet Union. Excellent. He was from Georgia, but he's an incredible person. But one thing I learned was choose the problems to work on that might matter if you succeed. Because that's why, for example, we dug into Covid. It was just, well, suppose we succeed in finding some interesting insight here. Well, it actually matters. Then it's worthwhile.

Yeah. And I think Covid, the way you're approaching Covid, has two interesting possibilities. One, it might help with COVID or another pandemic. But two, I mean, just this whole network theory space, you might unlock some deep understanding about the interaction of human beings that might have nothing to do with the pandemic. There's a Space of possible impacts that may be direct or indirect. And the same thing is with Andrew Wiles proof I don't understand, but apparently the pieces of it are really impactful for mathematics, even if the main theorem is not so along the way, the insights you have might be really powerful for unexpected reasons.

So I like what you said. This is something that I learned from another friend of mine who's also, he's a very famous researcher. All these people are more famous than I am. His name is Jacob Fox. He's Jacob Fox at Stanford. Also a very big inspiration for me. We were both grad students together at the same time.

Well, most importantly, you're good at selecting good friends.

Ah, yeah, well, that's, that's, that's, that's the key. You got to find good people to learn things from. But his thing was he often said, you know, if you solve a math problem and have this math proof, math problem for him is like a proof. Right. So suppose you came up with this proof. He always asks, what have we learned from this that we could potentially use for something else? It's not just, did you solve the problem that was supposed to be famous? And is there something new in the course of solving this that you had to invent that we could now use as a tool elsewhere?

Yeah, there's this funny effect where just looking at different fields where people discover parallels, they'll prove something. It'll be a totally new result. And then somebody later realizes this was already done 30 years ago in another discipline in another way. And it's really interesting. We did this offline in another illustration you showed to me. It's interesting to see the different perspectives on a problem. It kind of points like, there's just like very few novel ideas that everything else that most of us are just looking at different perspective on the same idea. And it makes you wonder. This old silly question that I have to ask you is, do you think mathematics is discovered or invented? Do you think we're creating new idea, we're building a set of knowledge that's distinct from reality? Or are we actually like, is math almost like a shovel where we're digging to like this core set of truths that were always there all along?

So I personally feel like it's discovered. But that's also because I guess the way that I like to choose what questions to work on are questions that, that maybe we'll get to learn something about. Why is this hard? I mean, I'm often attracted to questions that look simple but are hard. Right. And what could you possibly learn from that. Sort of like probably the attraction of Fermat's last theorem, as you mentioned. Simple statement, why is it so hard? So I'm more on the discovered side. And I also feel like if we ever ran into an intelligent other species in the universe, probably if we compared notes, there might be some similarities between both of us realizing that PI is important because you might say, why humans? Do humans like circles more than others? I think stars also like circles. I think planets like circles. They're not perfect circles, but nevertheless, the concept of a circle is just point and constant distance. Doesn't get any simpler than that.

It's possible that, like an alien species will have, depending on different cognitive capabilities and different perception systems, we'll be able to see things that are much different than circles. And so if it's discovered, it will still be pointing at a lot of same geometrical concepts, mathematical concepts. But it's interesting to think of how many things we would have to still align, not just based on notation, but based on understanding. Like, just like some basic mathematical concepts, like how much work is they're going to be in trying to find a common language. This is, I think, Stephen Wolfram and his son helped with the movie Arrival, like the. Developing an alien language. Like, how would aliens communicate with humans? It's fascinating because math seems to be the most promising thing. But even like math, like, how do you visualize mathematical ideas? It feels like there has to be an interactive component. Just like we have a conversation, there has to be. This is something we don't, I think, think about often, which is, like, with somebody who doesn't know anything about math, doesn't know anything about English or any other natural language. How would we describe. We talked offline about visual proofs. How would we, through visual proofs, have a conversation where we say something, here's the concept, the way we see it. Does that make sense to you? And, like, can you mess with that concept to make it sense for you and then go back and forth in this kind of way? So purely through mathematics, I'm sure it's possible to have those kind of experiments with, like, tribes on Earth that don't. There's no common language through math. Like, draw a circle and see what they do with it. Right? Do some of these visual proofs, like the summation of the odds and adds up to the squares. I wonder how difficult that is before one or the other species murders. I hope that the curiosity for knowledge will overpower the greedy. This is back to our game theory thing that the curiosity of, like discovering math together will overpower the desire for resources and ultimately, like, you know, willing to commit violence in order to gain those resources. I think as we progress, become more and more intelligent as a species, I'm hoping we would value more and more the knowledge because we'll come up with clever ways to gain more resources so we won't be so resource starved. I don't know. That's a hopeful message from when we finally meet aliens.

Yeah, yeah.

The cool thing about the Math Olympiad, I don't know if you know, work from Francois Chollet, from Google. He come up, he came up with this kind of IQ test, slash. It kind of has similar aspects to it that also Math Olympia does for AI. So he came up with these tests where they're very simple for humans but very difficult for AI to illustrate exactly why we're just not good at seeing a totally new problem. We, sorry, AI systems are not good at looking at a new problem that requires you to detect that there's a symmetry of some kind or there's a pattern that's. That hasn't. It hasn't seen before. The pattern is like obvious to us humans, but it's not so obvious to find that kind of. You're inventing a pattern that's there in order to then find a solution. I don't know if you can comment on, but from an AI perspective and from a math problem perspective, what do you think is intelligence? What do you think is the thing that allows us to solve that problem and how hard is it to build a machine to do that? Asking for a friend.

Yeah, well, so I guess you see, because if I just think of the raw search space, it's huge. That's why you can't do it. And if I think about what makes somebody good at doing these things, they have this heuristic sense. It's almost like a good chess player of saying, let's not keep analyzing down this way because there's some heuristic reason why that's a bad way to go. Where do they get that heuristic from? Now, that's a good question. I don't know. Because that if you asked them to explain to you, they could probably say something in words that sounds like it makes sense. But I'm guessing that's only a part of what's really going on in their brain, of evaluating that position. You know what I mean? If you ask Garry Kasparov what is good or why is this position good, he will say something. But it's probably not approximating everything that's going on inside. So there's basically a function being computed. Yeah, but it's hard to articulate what that function is. Now the question is, could a computer get as good at computing these kinds of heuristic functions? Maybe I'm not enough of an expert to understand, but one bit of me has always been a little bit curious of whether or not the human brain has a particular tendency due to its wiring to come up with certain kinds of things which is just natural due to the way that the topology of the neurons and whatever is there for which. If you tried to just build from scratch a computer to do it, would it naturally have different tendencies? I don't know. This is just me being completely ignorant and just saying a few ideas.

Well, this is a good thing that mathematics shows is we don't have to be. So math and physics or mathematical physics operates in a world that's different than our descendants of a brains operate in. So it allows us to, to have multiple, many, many dimensions. It allows us to work on weird surfaces with like topology as a discipline is just weird to me. It's really complicated, but it allows us to work in that space. Differential geometry and all those kinds of things where it's totally outside of our natural day to day, four dimensional experience, 3D dimensional with time experience. So the math gives us, gives me hope that we can, that we can see we can discover the processes of intelligence outside the limited nature of our own, like human experiences. But you said that you're not an expert. It's kind of funny, I find that we know so little about intelligence that I think, I honestly think like almost children are more expert at creating artificial intelligence systems than adults. I feel like we know so little, we really need to think outside the box. And those little. I found people should check out Francois Chollet's little exams. But even just solving math problems, I don't know if you've ever done this for yourself, but when you solve a math problem, you kind of then trace back and try to figure out where did that idea come from? Like what, how do, like what was I visualizing in my head? How did I start visualizing it that way? Why did I start rotating that cube in my head in that way? Like what is that? If I were to try to build a program that does that, where did that come from?

So this is interesting. So I try to do this to teach middle school students how to learn how to create and think and invent. And the way I do it is there are these Math competition problems. And I'm working in collaboration with the people who run those. And I will turn on my YouTube live and for the first time look at those questions and live solve them. The reason I do this is to let the middle school students and the high school students and the adults and whoever wants to watch just see what exactly goes on through someone's head as they go and attempt to invent what they need to do to solve the question. So I've actually thought about that. I think that first of all, as a teacher, I think about that because whenever I want to explain to a student how to do something, I want to explain how it made sense, why it's intuitive to do the following things, and why the wrong things are wrong, not just by this one short fast way. Well, why this is the right way, if that makes sense. So my point is I'm actually always thinking about that, like, how would you think about these things? And then I eventually decided the easiest way to expose this would just be to go live on YouTube and just say, I've never seen any of these questions before. Here we go.

Don't you get, man, that's anxiety inducing for me. Don't you get trapped in a kind of little dead ends of confusion even on middle school problems?

Yes, that's what the comments are for. The live comments come in and students say, try this. Oh, wow, it's actually pretty good. I'll never get stuck. I mean, I'm willing to go on camera and say, guess what? Po Shundo can't do this, that's fine. But then what ends up happening is you will then see how maybe somebody's saying something. And I look at the chat and I say, aha, that actually looks useful. Now that also shows how not all ideas, not all suggestions are the same power, if that makes sense. Because if I actually do get stuck, I'll go fishing through the chat if I've got any ideas.

I don't know if you can speak to this, but is there a moment for the middle school students, maybe high school as well, where there's like a turning point for them where they maybe fall in love with mathematics or they get it? Is there something to be said about like discovering that moment and trying to grab them, to get them to understand that mathematics is no matter what they want to do in life, could be part of their life?

Yes, I actually do think that the middle school is exactly the right time because that's the place where your mathematical understanding gets just sophisticated enough that you can start doing interesting Things because if you're early on in counting, I'm honestly not very good at teaching you new insights. My wife is pretty good at that. But somehow, once you get to this part where you know what a fraction is and when you know how to add and how to multiply and what the area of a triangle is, at that point, to me, the whole world opens up and you can start observing. There are really nifty coincidences. The things that made the Greek mathematicians and the ancient mathematicians excited. Actually, back then it was exciting to discover the Pythagorean theory. It wasn't just homework.

So is there what. Which discipline do you think has the most exciting coincidences? So is it geometry? Is it algebra? Is it calculus?

Well, you see, you're asking me and I'm the guy who gets the most excited when the combinatorics shows up in the geometry.

Is it. Okay, so it's the combinatorics in the geometry. So first of all, the nice thing about geometry, this is the same nice thing about computer vision, is it's visual. So geometry can draw circles and triangles and stuff, so it naturally presents itself to the visual proof. Right. But also the nice thing about geometry, I think for me is the earliest class, the earliest discipline, where there's that's most amenable to the exploration, the invention through proofs. The idea of proofs, I think is most easily shown in geometry because it's so visual, I guess. So that to me is like, if I were to think about when I first fell in love with math, it would be geometry. And sadly enough, that's not used. Geometry only has a little. Appears briefly in the journey.

Of a.

Student and it kind of disappears and not until much later, which, you know, there may be, like differential geometry. I don't know where else it shows up for me in computer science. Like, you could start to think about, like computational geometry or even graph theory is a kind of geometry. You could start to think about it visually, although it's pretty tricky. But yeah, it was always that. That was the most beautiful one. Everything else, I guess calculus can be kind of visual too. That could be pretty beautiful. But is there something you try to look for in the student to see, like, how can I inspire them at this moment? Or is this like individual student to student? Is there something you could say there?

So first of all, I really think that every student can pick up all of this skill. I really do think so. I don't think it's something only for a few. And so if I'm looking for a student. And actually, oftentimes if I'm looking at a particular student, the question is, how can we help you feel like you have the power to invent? Also? Because I think a lot of people are used to thinking about math as something where the teacher will show you what to do and then you will do it. Yes. So I think that the key is to show that they have some. Let them see that they have some power to invent. And at that point, it's often starting by trying to give a question that they don't know how to do. You want to find these questions that they don't know how to do that they can think about and then they can solve. And then suddenly they say, my gosh, I've had a situation. I've had an experience where I didn't know what to do. And after a while I did.

Is there advice you can give on how to learn math for people, whether it's middle school, whether it's. Somebody as an adult kind of gave up on math maybe early on.

I actually think that these math competition problems, middle school and high school, are really good. They're actually very hard. So if you haven't had this kind of experience before and you grab a middle school math competition problem from the state level, which is used to decide who represents the state in the country, in the United States, for example, those are pretty tricky. And even if you are a professional, maybe not doing mathematical things and you're not a middle school student, you'll struggle. So I find that these things really do teach you things. By trying to work on these questions.

Is there a googleable term that you could use for the organization for the state competitions?

Ah, yeah. So there are a number of different ones that are quite popular. One of them is called Math counts M A T H C O U N T S and that's a big tournament which actually has a state level. There's also a mathleague.org Mathleague L E A G U E dot org also has this kind of tiered tournament structure. There's also the American math competitions, AMC 8AMC also has AMC 10, that's for 10th grade and below, and AMC 12. These are all run by the Mathematical association of America. And these are all ways to find old questions.

What about the daily challenges that you run? What are those about?

We do that too. But I mean, the difference was ours isn't that one's not free. So I should actually probably be careful. The things that I've just mentioned are also not free. Not all of Those things I mentioned just now are free either, but people can figure out what's not.

But this is really nice to know what's out there. But can you speak a little bit to the daily challenges?

Sure, sure. So that's actually what we did when I guess I was thinking about how would I try to develop that skill in people if we had the power to architect the entire system ourselves? So that's called the daily challenge with Postren Low. It's not free because that's actually how I pay for everything else I do. That was the idea, but the concept was, aha. Now let's invent from scratch. So if we're going to go from scratch and we're going to use technology, what if we made every single lesson something where first I say, hey, here's an interesting question. Recorded, of course, it's not live, but it's like I say, hey, here's an interesting question. Why don't we think about this? But I know you don't know how to do it. So now you think, and a minute later a hint pops on the screen, but you still think. And a minute later a big hint pops on the screen and you still think. And then finally, after the three minutes, hopefully you got some ideas. You try to answer, and then suddenly there's this pretty extended explanation of, oh, yeah, so here's multiple different ways that you can do the question. And by accident, you also just learned this other concept. That's what we did.

Is this targeted towards middle school students, high school students?

It's targeted towards middle school students with competitions. But there's a lot of high school students who didn't do competitions in middle school where they would also learn how to think. If you can see, the whole concept was, can we teach people how to think? How would you do that? You need to give people the chance to, on their own, invent without that kid in the front row answering every question in two seconds.

And people can find it, I think, what? Daily.it's daily.

Potionlo.com but if you go to find my website, you'll be able to find it. Beautiful.

Can we zoom out a little bit? And so day to day, week to week, month to month, year to year, what does the lifelong educational process look like? Do you think for yourself? But for me, what would you recommend in the world of mathematics or sort of as opposed to studying for a test, but just like lifelong expanding of knowledge and that skill for invention?

I think I often articulate this as, can you always try to do more than you could do in the past. Yeah, but that comes in many ways. And I will say it's great if one wants to build that with mathematics, but it's also great to use that philosophy with all other things. In fact, if I just think of myself, I just think, what do I know now that I didn't know a year ago or a month ago or a week ago? And not just know, but like, what do I have the capability of doing? Yes, and if you just have that attitude, it brings more.

See, the thing is, there's also a habit, like it is a skill. Like I've been using Anki, it's an app for. Helps you memorize things. And I've actually this few months ago, started doing this daily of setting aside time to think about an idea that's outside of my work. Like, let's say, let's pick. It's all over the place, by the way. But let's say politics, like gun control. Is it good to have a lot of guns or not in society? And just. I've set aside time every day I do at least 10 minutes, but I try to do 30 where I think about a problem and I kind of outline it for myself from scratch, from not looking anything up, just thinking about it, using common sense. And I think the practice of that is really important. It's the daily routine of it. It's the discipline of it. It's not just that I figured something out from that thinking about gun control. It's more that that muscle is built too. It's that thinking muscle. So I'm kind of interested in, you know, math has. Because especially because I've gotten specialized into machine learning and because I love programming so much, I've lost touch with math a little bit to where I feel quite sad about it. And I want to fix that. Even just not math like pure knowledge math, but math like these middle school problems. The challenge is, right. Is that something you see a person be able to do every single day, kind of just practice every single day for years?

So I can give an answer to that. That gives a practical way you could do it, assuming you have kids. No, no, I mean, you can do it yourself.

Step one, get kids.

No, no. I'm just saying this because I'm just thinking out loud right now. What could I do? What could I do to suggest. Because what I have noticed is that, for example, if you do have kids who are in elementary school or middle school, if you yourself go and look at those middle school math problems to think about interesting ways that you can teach your elementary school or middle school kid. It works. That's what my wife did. She never did any of those contests before, but now she knows quite a lot about them. I didn't teach her anything. I don't do that. She just was messing around with them and taught herself all of that stuff. And that had the automatic daily. I'm always thinking, how do you make it practical, Right? Yes. And the way to make it practical is if the timer on the automatically daily is that you are going to automatically daily do something with your own kids.

Yes.

Now it feeds back. Okay.

And that includes the whole lesson that if you want to learn something, you should teach it.

Oh, I strongly believe that. Yes, I strongly believe that.

So I currently don't have kids, so that's. Maybe I should just get kids to help me with the math thing. But outside of that, I do want to integrate math into daily practice. So I'll definitely take out. I'll definitely check out the daily challenges and see, because what is it? Grant Sanderson, we Talked about offline 3 blue 1 brown. He speaks to this as well, that his videos aren't necessarily. They don't speak to the thing that I'm referring to, which is the daily practice. They're more almost tools of inspiration. They kind of show you the beauty of a particular of problem in mathematics, but they're not a daily ritual. And I'm in search of that daily ritual. Mathematics. It's not trivial to find, but I hope to find that because I think math gives you a perspective on the world that enriches everything else.

So I like what you said about the daily also, because that's also one reason why I put my Carnegie Mellon class online. It's not every day, it's every other day. Semester is almost over. But the idea was, I guess my philosophy was, if I'm already doing the class, let's just like put it there. Right. But I do know that there are people who have been following it who are not in my class at all, who have just been following it because, yes, it's combinatorics. And the value of that is you don't really need to know calculus to follow it, if that makes sense. So it's actually something that people could follow. So again, and that one's free. So that one's just there on YouTube.

Well, speaking of combinatorics, what is it? What do you find interesting? What do you find beautiful about combinatorics?

So combinatorics to me is the study of things where they might be more finite and more discrete. What I mean is like if I look at a network, actually a lot of times the combinatorics will boil down to something. And the combinatorics I think about might be something related to graphs or networks. And they're very discrete because if you have a node, it's not that you have 0.7 of a node and 0.3 of a node over there, it's that you've got one node and then you jump one step to go to the next node. So that notion is different from, say, calculus, which is very continuous, where you go and say, I have this speed which is changing over time, and now what's the distance I've traveled? That's the notion of an integral where you have to think of subdividing time into very, very small pieces. So the kinds of things that you do when you reason about these finite discrete structures often might be iterative, algorithmic, inductive. These are ideas where I go from one step to the next step and so on, and make progress, I guess. I actually personally like all kinds of math. My area of research just ended up in here because I met a really interesting PhD advisor potential. That's honestly the reason I went into that direction. I met a really interesting guy. He seemed like he did good stuff, interesting stuff, and he looked like he cared about students. And I said, let me just go and learn whatever you do. Even though my prior practice and preparation before my PhD was not combinatorics, but analysis, the continuous stuff.

So the annoying thing about combinatorics and discrete stuff is it's often really difficult to solve from a sort of running time complexity perspective. Could you speak to the idea of complexity analysis of problems? Do you find it useful to find it interesting? Do you find that lens of studying the difficulty of how difficult the computer science problem is a useful lens onto the world?

Oh, very much so. Because if you want to make something practical which has large numbers of people using it, the computational complexity to me is almost question one. And that's again, that's at the origin of when we started doing this stuff with disease control. From the very beginning, the deep questions that were running through my mind were, would we be able to support a large population with only one server? And if the answer is no, we can't start because I don't have enough money. Yeah.

And there the question is very much linear time versus anything slower than linear time. As a very specific thing, you have a bunch of really interesting papers. If I could ask, maybe we could pull out some cool insights at the High level. Can you describe the data structure of a voting tree and what are some interesting results on it? You have a paper that I noticed on it.

Yeah. So this is an example of I guess how in math we might say, here's an interesting kind of a question that we just can't seem to understand enough about. And maybe there's something else going on here. And the way to describe this is you could imagine trying to hold elections where if you have only two candidates, that's kind of easy, you just run them against each other and see who gets more votes. But as you know, once you have more candidates, it's very difficult to decide who wins the election. And there's an entire voting theory around this. So a theoretical question became what if you made a system of runoffs, like a system of head to head contests which you structure like a tree almost looking like a circuit? I'm using that way of thinking because it's sort of like in electrical engineering or computer science. You might imagine having a bunch of leads that carry signal which are going through and gates and, or gates and whatnot. And you've managed to compute beautiful things. This is just from a purely abstract point of view. What if the inputs are candidates and for every two candidates it is known which of the candidates is more popular than the other. Now can you build some kind of a circuit board which says first candidate number four will play against five and see who wins and so on. Okay, so now what would be a nice outcome? Right. This is a general question of could I make a big circuit board to feed an election into? Like maybe one nice outcome would be whoever wins at least is preferred over a lot of people.

Yes.

So for example, if you ran in 1024 candidates, ideally we would like a guarantee that says that the winner beats a lot of people. Actually, in any system where there are 1024 candidates, there's always a candidate who beats at least 512 of the others. This is a mathematical fact that there's actually always a person who beats at least half of the other people.

I'm trying to make sense of that mathematical fact. Is this supposed to be obvious?

No, but I can explain it. No, I can't. The way it works is that. But think of it this way, every time imagine I have all these candidates and everyone is competed. Everyone is compared with everyone else at some point. Well, think of it this way. Whenever there's a comparison, somebody gets a point that's the one who is better than the other one. My claim is there's somebody whose score is at least half of how many other people there are.

Yeah, I'm just trying to. My intuition is very close to that being true. But it's beautiful. I didn't at first. That's not an obvious fact.

No, it's not.

And it feels like a beautiful fact.

Well, let me explain it this way. Imagine that for every match you didn't give one point, but you gave two points. You gave one point to each person. Now, that's not what we're really doing. We really want to give one point to the winner of the match, but instead we'll just give two. If you gave two points to everyone on every matchup, actually everyone has the same number of points and the number of points they get is how many other people there are. Does that sort of make sense? I'm just like saying.

No, no, everything you're saying makes perfect sense.

Okay. So the point is if for every comparison between two people, which I'm doing, for every two people, I gave one point to each person. Your score, everyone's score is the same. It's how many other people there are. Now, we only make one change for each matchup. You give one point only to the winner, so we're awarding half the points. So now the deal is if in the original situation everyone's score was equal, which is how many other people there are, now there's only half the number of points to go around. So what ends up happening is that there's always going to be like, the average number of points per person is going to be half of how many other people there are. And somebody is going to be above average.

Somebody's going to be above average.

At least. Average. Yeah. This is this notion of expected value that if I have a random variable which has an expected value, there's going to be some possibility in the probability space where you're at least as big as the expected value. Yeah.

When you describe it like that, it's obvious. But when you are first saying in this little circuit that there's going to be one candidate better than half, that's not obvious.

Yeah, that's funny. Math.

This is nice. Okay, so do you have this. But ultimately you want. You're trying to. With a voting tree. I don't know if you're trying this, but to have a circuit that's. That's like compressed, that's small.

Well, you'd like it.

That achieves the. Achieves the. The same kind of. I mean, the smaller it is. The. If we look at, practically speaking, the, the lower the cost of running, the election, of running through, of computing the circuit.

That is true. But actually at this point, the reason the question was interesting is because there was no good guarantee that the winner of that circuit would have beaten a lot of people. Let me give an example. The best known circuit when we started thinking about this was the circuit called Candidate one plays against candidate two, Candidate three plays against four, and then the winners play against each other. And then, by the way, five plays against six, seven against eight. The winners play against each other. You understand? It's like a giant binary tree where you.

Yeah, it's a binary. Like a balanced binary tree.

Okay. It's a balanced binary tree. 1, 2, 3, 4, up 20, 24, everyone going up to find the winner. Beautiful. Well, you know what? There's a system in the world where it could just be that there's a candidate called number one that just beats like 10 other people. Just the 10 that they need to be on their way up and they lose to everyone else, but somehow they would get all the way up.

Yes.

My point is it is possible to outsmart that circuit in one weird way of the world, which makes that circuit a bad one. Because you want to say, I will use this circuit for all elections and you might have a system of inputs that go in there where the winner only beat 10 other people, which is the people they had to beat on the way up.

So you want to have a circuit where there's as many. Like the final result is as strong as possible.

Yes.

And so what ideas do you have for that?

So we actually only managed to improve it to square root of N. So if n is number of vertices, N over 2 would be the ideal. We got it to square root of N versus log based 2. Yeah, exactly. Yeah.

Which is.

Well, that is.

It could be a lot, could be a big improvement. So that's okay. Cool. Is there something you can say with words about what kind of circuit, what that looks like?

I can give an idea of one of the tools inside, but the actual execution ends up being more complicated. But one of the widgets inside this is building a system where you have a candidate who plays one part of the whole huge tree. Is that that same candidate, let's call him seven. Seven plays against somebody. Let's make up some numbers. Let's call the others like letters. So seven plays against A. Seven's also going to play against B separately. And the winners of each of those will play each other. By the way, 7's also going to play C. 7's going to play D. And the winners are going to play each other. And the winners are going to play each other. We call this seven against all. Well, seven against everyone from a bunch of.

Got it. So there's some nice overlap between the matchups that somehow has a nice feature to it.

Yes. And I can tell you the nice feature because if at the base of this giant tree, at the base of this giant circuit, this is a widget, we build the things out of widgets. So I'm just describing one widget, but in the base of this widget, you have lots of things which are seven against someone, seven against someone, seven against someone. In fact, every matchup at the bottom is seven against someone. What that means is if seven actually beat everyone they were matched up against, well, seven would rise to the top. So one possibility is, if you see a seven emerge from the top, you know that seven actually beat everyone they were against. On the other hand, if anyone else is on top, let's call it F. If F is on top, how did F get there? Well, F beat seven on the way at the beginning. So the point is, the outcome of this circuit has a certain property. If you see a seven, you know that the seven actually beat a bazillion people. If you see anyone else, at least you know they beat seven.

Yeah. Then you can prove that it has a nice property. That's really interesting. Is there something you can say, perhaps, going completely outside of what we're talking about, is how we may have mathematical ideas of improving the electoral process?

That one. Note. No, I can't give you that one.

I mean, is there, like. Do you ever see it as. As. As there be. Do you see as there being a lot of opportunities for improving how we vote? Like from your. I don't know if you saw parallels, but, you know, it seems like if this actually kind of maps to your sort of COVID work, which is there's a network effect.

Right.

It seems like we should be able to apply similar kind of effects of how we decide other things in our lives.

And.

And one of the big decisions we make is who represents us in government. Do you ever think about, like, mathematically about those kinds of systems?

I think a little bit about those. Because where I went to college, the way we voted for student government was based on this. Is it called ranked choice, where you eliminate. You eliminate the bottom and there are runoff elections. So. So that was the first time I ever saw that, and I thought that made sense.

Yeah.

The only problem is it doesn't seem so easy to get something that makes sense adopted as the new voting system.

That's a whole nother.

There's.

That's not a math solution. That's. Well, it's math in the sense, it's game theory. So you have to come up with incentive. It's mechanism design. You have to figure out how to trick us, despite our basic human nature, to adopt solutions that are better. Yes, that's a whole nother conversation, I think. Can you, just because it sounded really cool, talk a little bit about stochastic coalescence and you have a paper on showing that we could describe what it is, but I guess it's super linear, super logarithmic time and you came up with some kind of trick that make it faster. Can you just talk about it a little bit?

Yeah. So this was something which came up when I was at Microsoft Research for a summer. And I'm putting that context because that shows that it has some practical motivation at some point. Actually, I think it's still. It doesn't need to. It doesn't need to.

It can be beautiful in its own right.

Yeah. So the easiest way to describe this is suppose you got like a big crowd of people and everybody knows how many hours of sleep they got last night. And you want to know how many total hours of sleep were gotten by this big crowd of people at the beginning. You might say that sounds like a linear time algorithm of saying, hey, how many hours you got? How many you got? How many you got? Add. Add. Add.

Yes.

But there's a way to do this. If you remember that there are people and they presumably know how to add, you could make a distributed algorithm to make this happen. For example, while we're thinking of these trees, imagine you had 10, 24 people. If you could just say, hey, person number one and person number two, you will add your hours of sleep, person number two will go away, and person number one is going to remember the sum. Person three and four add up and person three takes charge of remembering it. Person four goes away. Now this, like person one knows the sum of these two, person three knows the sum of those two. They talk. You see what I mean? It's like you're going up this tree, same tree that we talked about earlier.

Built up a tree from the bottom up.

Yeah, build up a tree from the bottom up. And the beautiful thing is, since everyone's doing stuff in parallel, the amount of time it takes to get the total sum is actually just the number of layers in the tree, which is 10. So now that's logarithmic time to add up the number of hours that people slept today. Sounds fantastic. There's only one problem. How do you decide who's person number one and person number two?

Yes.

So if, for example, you just went out into the downtown and said, hey, get these thousand people go. Well, if you're going to go and say, and by the way, you're one and you're two and you're three, that's linear time. Yes. That's cheating. So now the question is how to do this in a distributed way. And there were some people who proposed a very elegant algorithm and they wanted to analyze it. So I came in onto the analyze side. But the elegant algorithm was like this. It was like, well, we don't actually know what this big tree is. There isn't any big tree. So what's going to happen is first everyone is going to decide right now, oh, one important thing everyone is going to at the very beginning of the whole game, they will have delegated responsibility to themselves as the one who knows the sum so far. So the point is there's going to be people are all going to have like a pointer which says, you are the one who knows my, you've taken care of my ticket, my number. Yeah.

They select a representative for this particular piece of knowledge and at the very.

Beginning you're your own representative. The thing has to start simple, right? So at the beginning you're 100 representative.

You'Re pointing to yourself.

Got it? Yep. And now the way this works is that at every time step someone blares a ding dong on the, on the, on the town clock or whatever. And each person flips a coin themselves to decide, am I going to hunt for somebody to give my number to and let them represent me or am I going to sit here and wait for someone to come? Okay, okay. Well, they flip their coin. Some of the people start asking other people saying, hey, I would like you to be my representative. Here is my number. But the problem is that there's limited bandwidth of the people who are getting asked. It's like you can't go out to prom with five people. This is not what we're doing. We're adding numbers. Okay, but you can only add one number. So the person who has suddenly gotten asked by all these people, well, they'll have to decide who they're going to take it from. And they randomly just choose one. When they randomly choose one, all the others are rejected and they don't get to delegate anything in that round. But now if this person has absorbed this one who said, okay, here you take charge of my number. This person now updates their pointer. You're in charge. And this person adds the two numbers. That was the first round. In the next round, when they do the coin flipping, this person doesn't flip anymore because they're just delegating. It's that anyone who has the pointers themselves, that's like a person who is in charge of some number of informations, they flip the coin to decide, should I find other people who are agents or should I wait for people to ask me? Yes, brilliant. This is somebody else's idea. And now the idea is, okay, if you just keep doing this process, what ends up happening? Oh, yeah. And also, by the way, if you decide that you want to go reach out to other people, here's the catch. When you're one of these agents saying, okay, I'm going to go look for someone, you have no idea who in this crowd is an agent or somebody who delegated it to someone else. You just pick a random person. When you pick the random person, if it lands on someone and the person says, oh, I actually delegated it to someone, then you follow the delegation chain, walk up, delegation chain. And you can do, like, path compression in the algorithm to make it so you don't consistently do lots of walking up. But the bottom line is that what ends up happening is that you end up reaching out. Whenever you're one of the ones reaching out, you can think of it as each agent is responsible for some number of people. It's almost like they're the leader of a bunch. As the process is evolving, you have these lumps. Each lump has an agent, and when the agent reaches out, they reach out to another lump lump where the probability of them hitting that lump is proportional to the size of the lump. That is the one funny thing about this process. This is not that they can reach out to a uniformly random lump where every lump has the same chance of being reached out to. The bigger the lump is, the more likely it is that you end up.

Reaching that lump, which is a problem.

Let me explain why that's a problem. Because, you see, you're hoping that this has a small number of steps, but here's a bad situation.