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 XP 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 needed 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, Onit, BetterHelp, Aidsleep, 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 mentioned you really enjoy flying and experiencing 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. When I look at 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. Everybody's taking it for granted.

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. At the same

kind of way, I can already see the future where we will all take it for granted. I don't know

if you personally, when you fly, have the same kind of magical experience of how the

heck that humans actually accomplish this. I do, especially when there's turbulence,

which is on the way here. There was turbulence, and the plane jiggled, even the flight attendant

had to hold on to the side. I was just thinking to myself, it's amazing that this happens

all the time and the wings don't fall off. Given how many planes are flying, but then

I often think about it, and I'm like, a long time ago, I think people didn't trust elevators

in a 40 story building in New York City. 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.

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? How amazing it is with

the high winds, structurally, just the earthquakes and the natural vibrations in the ground.

How are all of these, you go to New York City, all of these buildings standing? To me, one

of the most beautiful things, actually mathematically too, is bridges. I used to build bridges in

high school from toothpicks, just out of the pure joy of physics making some structure

really strong. Understanding from a civil engineering perspective, what kind of structure

will be stronger than another kind of structure, like suspension bridges. Then you see that

at scale, humans being able to span a body of water with a giant bridge. I don't know.

It's so humbling. It makes you realize how dependent we are on each other. 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. Then we're depending on a network of knowledge

that other experts hold. 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

the ability to build on top of it levels of abstractions. You start from bits in a computer,

then you can have assembly, then you can have C++. You have an operating system, then you

can have C++ and Python, finally some machine learning on top, all of these are abstractions.

Eventually, we have AI that runs all of us humans.

But speaking of abstractions and programming, in high school, you wrote some impressive

games for Emma Stoss. I got a chest to, in browsers somehow, it's magic. I got a chest

to play them. Alien Attack 1, 2, 3, and 4. What's the hardest part about programming

those games? 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.

That's a good starting point for anything.

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 think 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,

those, 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 and the browser work, by the way? Is that, 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 have to do it. I didn't have to do it. 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 as 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 the best

way to write a program quickly is to write a simple program. You have to have a damn

good solution. So it's not necessary 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

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.

Also, the nice thing about programming competitions is that you actually build a thing that works,

so 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. 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 it's not

just software. Tesla autopilot is not just software. It's like you have factories that

build cars and there's 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, 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. I mean, there are other

things that are hard called get something to Mars, get people to Mars. 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 figure it out.

And maybe they would actually even be useful. 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 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

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 like 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 that a little bit longer, which is do you see basically the way to teach math

is to present the problem and to give a person a chance to try to invent a solution without

with minimal amount of information first is that 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'd 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 actually I don't have a script.

The way I show up for classes, 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. 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 automatically 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. This 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. 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 would find out who was sick, you'd

find out who they have 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 like 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 seven billion, eight 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 the, 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 pent, 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 changed 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 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, so 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 like 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 like the more deadly and transmissible the disease, the stronger

the incentive to install it in a positive sense, the 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 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 feedback

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? We, that's, we might, maybe we'll chat about that, how we sort of make sure there's

not misinformation, whether 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 you notice what this has just achieved. The

whole goal on, on, 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 tried

to get set A to do things that might not be to their liking or their will, because that's

removing them from society.

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, it's 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 that hops from person to person to person to person.

And each one of us individuals are sitting or plop, 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, I was trying to think exactly this, like how many people do I, so 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 of, 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 is decentralized

and, and not the, 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 incentivize to do it, and you're most likely going to do it to, to, to protect yourself

against you getting the disease as the, the closer it gets to you based on the information

that you have. 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 disease as well. So this, 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 going to happen when there's a much deadlier disease? Like this,

this was 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.

Okay. Mathematically, can you maybe elaborate? What are we talking about? Like 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, right? 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. 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. So is that enough to give you enough information

about physical connection to another human being? Is there a time element there? So okay,

that sounds like a really strong low hanging fruit. 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. Yes. 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. 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. 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 product, 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

as not an important physical relationship. I mean, this is brilliant. How difficult is

it to make work? So you said, one, there's a mathematical component that we just kind

of talked about, and then there's the user experience component. So how difficult does

it 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 like 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 it 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. 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 can, we

actually are very proud of the fact that ours uses very little battery. Actually, even if

compared to Apple's own system. So beautiful. So what else is required to make this thing

work? Right. So the key was that you had to do a 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 Olympians. 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 do coach the team for the math. 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? So that's actually why this computer algorithm competition stuff was handy. It's

because there are only about seven to eight 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 3,600 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 anyone of us

could have is eight billion. So it's eight billion times squared. That's the 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 100 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.

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

Yeah. Yeah. 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 two, meaning via people. And then here's how many at distance

three. 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. 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 R0 of two to three. 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 outvirus 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 a virus spreads and travels in order to basically keep the

society open? Is it possible 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 more widely known and we're already starting to collaborate

with epidemiologists. Again, 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 end game now. End game 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 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? And depending on your

own conditions, but also in 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 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 tolerance. And then those people decide to

isolate. But what matters is in the aggregate that there are not of the infection spreading

drops below one. And so the key is if you, if you can empower people with that power

to make that decision, you might actually still be able to drive that are not down below

one.

Yeah. And also this is me talking. I, is people get a little bit nervous, I think, with information

somehow mapping to privacy violation. But I, 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 and 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 artist of 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 there's this 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 have tried similar kind of ideas. Not in, you have

quite a novel idea, actually. But speaking, the umbrella idea of contact tracing. Is there,

is there something in comment about why their approaches haven't been fully adopted? Is there

challenges there? Is there, is there reasons why no vid 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 the other people

have been trying. But what we have noticed 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've 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, you're speaking to, not sad, I suppose it's the way

of the world is the only incentive there is to just help other people, I suppose. But

a much stronger incentive 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's theoretic solution to a very

hard problem. And so this is actually basically mechanism design, that 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 the 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. Yeah, 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. 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. 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

all 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 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? Also, 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 world 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 Novot.org website has Novot.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 Novot.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. 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 if you, 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.

So it could be, verification is a 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, are a 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. 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 edges in that graph. Outside of just pandemic, do interesting inferences

based on that. This is true for 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,

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's 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 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 still 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.

So 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?

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. It's 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 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. If I remember correctly, you know, there's people, this was a multi

year training process. They train hard. And this is everything that they're focused on.

My dad was a participant in this. And I mean, it's as serious as Olympic sports. You think

about like gymnastics, like young athletes participating in gymnastics. It's 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 math, 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 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 that's showing the generalization,

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 think

clearly it 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 can

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, 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 deep education.

Like as opposed to kind of like, oh, we want to make sure 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 as 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 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 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, you know, the generalizable skill and what could people do with Olympics afterwards?

So that's actually my interest in the whole thing. 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 Olympics, 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 cover of New York Times for inventing

something new. Do you think that's generalizable to education beyond just this particular

Olympia? 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 whole, 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 be

given 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 like, 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, 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, the, the mathematical Olympia?

Yes.

How to invent in 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, 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 you were, you know, that maybe you were just off by, but 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, you know, you schedule

these half hour appointments where the delegation from one country sits down at a table like

this 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, 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. This is how this is how our,

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, the coaches are explaining to you, this is what the student

has written. It's actually quite interesting process.

So it's almost like a, like a jury, you know, you have, you have a, in the American legal

system, you have a jury that where they're deliberating, but unlike a jury, there's the

members of the jury 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, so we write in English. Therefore, everything

that the U.S. does, everyone can look at. So it's very hard for, 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 at any, touch 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 bit of

looking at it. 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 big, how many different possibilities you could try. And that's, that's, that's

actually why like 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 still a branching of possible.

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, 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. Yeah. 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. Yeah. 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 plugings.

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. Yeah. I don't know if 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 is 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. Yeah. 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? That's the scary thing about insights. You

look like Andrew Wiles working on the Fermat's last 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 go down a path that's completely counter

productive or not productive at all. That's the crappy thing about invention is, like

I have, I'm sure you do. I have a lot of really good ideas every single day, but like, and

then I'll go inside my head along them, along that little trajectory, but it could be just

a total waste. You know what that feels like? It just feels like patience is required not

to get excited at any one thing. I think this is interesting because you raised Andrew

Wiles. He spent seven years attacking the same thing. Right? 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. 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 is probably

going to be a dead end. 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 insights, 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 on my life. His name is Benny

Sudakov. In fact, he grew up in the former Soviet Union. 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. That

is worth five.

Yeah. And I think COVID, the way you're approaching COVID has two interesting possibilities. One,

you might help with COVID or another pandemic. But two, just this whole network theory space,

you might unlock some deep understanding about the interaction with 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's 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.

Oh, yeah. That's the key. You've got to find good people to learn things from. But his

thing was he often said, 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? It was, 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 and another discipline in another way.

And it's really interesting. We did this offline in another illustration he 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?

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 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, will 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 there going to be in

trying to find a common language? I mean, this is, I think Stephen Wolfram and his son

helped with the movie Arrival, like developing an alien language, like how would aliens communicate

with humans? It's fascinating because like 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

the adds up to the squares. Yes, 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 Chalet from Google, he

came up with this kind of IQ test slash, it kind of has similar aspects to it that also

the math Olympiad 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 hasn't

seen before. The pattern is like obvious to us humans, but it's not so obvious to find

that kind of, it's 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. 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 did they get that heuristic from? Now that's a good question. I don't know.

Because 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 Gary 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.

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. And 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 descendant of a brain's operating. So it allows us to

have multiple, many, many dimensions. It allows us to work on weird surfaces with topology

as a discipline. It's just weird to me. It's really complicated. But it allows us to work

in that space or the differential geometry and all those kinds of things where it's totally

outside of our natural day to day four dimensional experience, three D dimensional with time

experience. So math gives me hope that we can discover the processes of intelligence outside

the limited nature of our own 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 I 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 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? 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 why this

one short, fast way, why this is the right way, if that makes sense. So my point is I'm

actually always thinking about that. 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, that's anxiety inducing for me. Don't you get trapped in a kind of like

little dead ends of confusion, even on middle school problems?

Yes, that's what the comments are for. The live comments come in as students, they 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? Potion

low 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.

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 something, 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 and counting, I'm honestly

not very good at teaching you new insights. My wife is pretty good at that, but if it's

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 pentagram here. It wasn't just homework.

So, which discipline do you think has the most exciting coincidences? 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.

Okay. So, it's the combinatorics in the geometry. So, first of all, the nice thing about geometry,

the same nice thing about computer vision is it's visual. So, geometry, you 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, the 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 there may be differential geometry. I don't know where else it shows up. For me,

in computer science, you could start to think about computational geometry or even graph

theories, the kind of geometry. You could start to think about it visually, although

it's pretty tricky. But yeah, 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, 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. 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 can 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.org also has this kind of tiered

tournament structure. There's also the American math competitions, A M C 8. A M C also has

A M C 10. That's for 10th grade and below and A M C 12. These are all run by the Mathematical

Association of America. And these are always defined old questions.

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

We do that too. But I mean, the difference was that one's not free. 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.

People can figure out what is free and 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 push and low. It's not free because that's

actually how I pay for everything else I do. So, 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 or something where first I say,

hey, here's an interesting question. Record it, 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 like this pretty extended explanation of, oh,

yeah, so here's like 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, with daily.

It's daily.potionlow.com. But if you go to find my website, you'll be able to find it.

Wonderful. 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 in 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 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. It is a skill. I've been using Anki. It's

an app for helps you memorize things. And I've actually, this a 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 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 outlined 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 challenges, 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. So you can do it yourself.

Step one, get kids.

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

were 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. 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, the three blue and one 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 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'd 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 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, 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 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. 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? 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. 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. 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, 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 an 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 manage 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

think of it this way. Every time, I think, imagine I have all these candidates and everyone is

competing, is everyone is like 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, like 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. 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 that's clear. No, everything you say 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 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 is going to be above that. 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, it's not. It's funny. Math, this is nice. Okay, so

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 like compared to small, that achieves the same kind

of, I mean, the smaller it is, if we look at practically speaking, 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 1 plays against candidate 2, candidate 3 plays against 4, and then the winners play against

each other. And then by the way, 5 plays against 6, 7 against 8, the winners play against each

other. You understand, it's like a giant binary tree. Yeah, it's a binary, like a balanced binary

tree. It's a balanced binary tree. One, two, three, four, up to 1024, 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 their 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 now what ideas do you have for that? So we actually only managed to improve it to square

root of n. So if n is the number of vertices, n over 2 would be the ideal. We got it to n,

we got it to square root of n. Versus log of n. Yeah, exactly. Yeah. Which is halfway. It could be a

lot. Yeah. It 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. Yeah. But the actual execution ends up being more complicated. But one of the widgets

inside this is building a system where you have like a candidate who plays like one part of the

whole huge, huge tree is that same candidate. Let's call them 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 who play each

other. By the way, seven's also going to play C. Seven'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 like 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,

like 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? No, I can't give you that one. Do you see there being a lot of opportunities

for improving how we vote? I don't know if you saw parallels, but you know,

it seems like this actually kind of maps to your sort of COVID work, which is there's a

network effect. It seems like we should be able to apply similar kind of effects of

how we decide other things in our lives. One of the big decisions we'll make is who represents

us in government. Do you ever think about 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 the bottom and there

runoff elections. That was the first time I ever saw that and I thought that made sense.

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

voting system. That's a whole nother. That's not a math solution. Well, it's math in the sense

that it's game theory. You have to come up with an 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.

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 you 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. This was something which came up when I was at Microsoft Research for a summer.

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 and

it's all right. Yeah, so the easiest way to describe this is, suppose you got 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. 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

1,024 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 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, built 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 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.

That's, 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, what 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 the representative for this particular piece of knowledge.

Right. And at the very beginning, you're your own representative. The thing has to start simple,

right? So the beginning of your own 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 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'd 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 get, you can't go

out to prom with five people. That 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.

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 walk up 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, 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

getting reached out to. The bigger the lump is, the less likely, 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

that could happen. Imagine if you had their end people that you're adding up. Imagine that you

have exactly square root of n lumps left, of which almost all of them are just one person

who's still their own boss, their own manager. Except one giant one. Now, what's going to happen

is going to be a huge bottleneck. Because every round, the giant one can only absorb one of the

others. And now you suddenly have time, which is about square root of n. The square root of n is

chosen because that is one where the lumps are such that you really are limited by this large

one slowly sucking up the rest of them. So the heart of the question became, well, but is that

just so unusual that it doesn't usually happen? Because remember, you start with everyone just

being independent. It's like a lot of lumps of size. How naturally do the big lumps emerge?

Yes. And so what that heart of the proof was, was showing that that was a joint work with

A. Lubecki, that one was showing that actually, in that thing, the lumps do kind of get out of

whack. And so it's not the purely logarithmic number of steps. But if you make one very slight

change, which is if you are one of the agents, and you have just been propositioned, possibly

relayed along by a couple of different people, if you just say, don't take a random one, but

accept the smallest lump. That actually does enough to even the whole thing up.

The distributes the lump size. I mean, yeah, it's fascinating how would the distributed algorithms

a little adjustment to make all the difference in the world? Yeah. Actually, by the way, this does

back to our voting conversation, this makes me think of like, these networking systems are so

fascinating to study, they immediately spring to mind ideas of how to have representation. Like, I

maybe as opposed to me voting for a president, I want to vote for, for like, for you to represent me,

maybe on a particular issue. And then you'll delegate that further. And then we naturally

construct those kinds of networks. Because that, that feels like I can have a good conversation

with you and figure out that you know what you're doing. And I could delegate it to you. And in

that way, construct a representative government or representative decision maker. That feels,

that feels really nice, as opposed to like us, like a tree of height one or something where it's

like everybody's just, it feels like there's a lot of room for layers of representation to form

organically from the bottom up. I wonder if there are systems like that. This is the cool thing about

the internet and the digital space where we're so well connected. Just like with the Novid app,

to distribute information about the spread of the disease, we can in the same way,

in a distributed sense, form anything like any kind of knowledge bases that are formed in a

decentralized way. And in a hierarchical way, as opposed to sort of old way where there's no

mechanism for large scale, fast, like distributed transactional information. This is really

interesting. This is where almost like network graph theory becomes practical. Most of that

exciting work was done in the 20th century, but most of the application will be in the 21st,

which is cool to think about. Let me ask the most ridiculous question. You think P equals NP?

Wow. I don't know. I mean, I would say, I know there are enough people who have very strong

interest in trying to show that it is. I'm talking about government agencies.

For security purposes. For security purposes. Yes. And most computer scientists would say,

believe that P equals NP. My question almost like, this is back to our aliens discussion. You want

to think outside the box, the low probability event, what is the world, what kind of discoveries would

lead us to prove that P does not equal to NP? There could be giant misunderstandings

or gaps in our knowledge about computer science, about theoretical computer science,

about computation, which allow us to think like flatten all problems.

Yeah. So I don't know the answer to this question. I think it's very interesting,

but actually I know, let's put it this way. By being at Carnegie Mellon and being around

the theoretical computer scientists, I know enough about what I don't know to say.

To be humble. I'm the wrong person to answer this question.

It's a great one. Well, Scott Aronson, who's now here at UT Austin, he used to be at MIT,

he puts the probability of P not equal to NP at 3%. I always love it when you ask,

it's very rare in science and academics because most folks are humble in the face of the mystery,

the uncertainty of everything around us. To have both the humor and the guts to say,

what are the chance that there's aliens in our galaxy, intelligent alien civilizations?

As opposed to saying, I don't know. It could be zero. It could be,

depending on the fact, you're saying it's 2.5%. There's something very pleasant about just having,

it's the number thing that's powered to the number. It's just like 42. It's like,

why 40? I don't know, but it's a powerful number. And then everything,

this is the power of the human psychology is once you have the number 42,

it's not that the number has meaning, but because it's placed in a book with humor around it,

it has the meme effect of actually creating reality. I mean, you could say that 42 has a

strong contribution of helping us colonize Mars because it created, it gave the whatever

existential crisis to many of us, including Elon Musk, when he was young, reading a book like that.

And then the now 42 is now part of his humor that he doesn't shut up about, it's constantly

joking about. And that humor is spreading through our minds. And somehow this like silly number

just had an effect. In that same way, after Scott told me the 3% chance, it's stuck in my head.

And I think it's been having a ripple effect on everybody else. The believing that P is not equal

to NP, Scott almost as a joke saying is 3%, is actually motivating a large number of researchers

to work on it. 3% is high. It's very high. Because for the potential impact that that would have.

But then 3% is not that high because it's only, you know, it's like, we're not very good. I feel

like humans are only able to really think about like 1%, 50%. And we kind of, I think a lot of

people around 3% up to 50%. Like in our minds, like 3%, this could happen. And it could happen.

And it's like, yeah, like half the time will probably happen. So we're not very good at that.

That's the other thing with the pandemic is we're not the exponential growth that we also talked

about offline is something that we can't quite into it. And that's something we probably should,

if we're to predict the future, to anticipate the future and to understand how to create technologies

that let us sort of control the future. Can I ask you for some recommendations,

maybe for books or movies in your life long ago, when you were Baby Poe or today,

that you found insightful or you learned a lot from it, you would recommend others?

Yeah. So I think, I don't necessarily have an exact name of these old things, but I was generally

inspired by stories true or fictional of campaigns. Like for example, like the Lord of the Rings,

that's a campaign, right? But the thing that always inspired me was it could be possible

for somebody who's crazy enough to go up against adversity, after adversity, after adversity,

and it succeeds. I mean, those are false. Those are fictitious. But I also spent a lot of time,

I guess, reading about, I don't know, I was interested somehow in like World War II history,

for whatever reason. That's a campaign which is much more brutal. But nevertheless,

the idea of difficulty, strategy, fighting, even when things, in that case, it was really

fighting, but just pushing on even when things are difficult. I guess these are the kinds of

general stories that made me, I guess, want to work on things that would be hard and where

it could be a campaign. It could be that you work on something for a year, multiple years,

because that was the point. Yeah, it starts with a single person. That's the interesting thing.

I've obviously been, I don't shadow about it recently about World War II, especially on the

Hitler side and the Stalin side. Some of that has really affected my own family, the roots of my

family very much. But it's interesting to think that it was just an idea and one person decided

to do stuff and it just builds and builds and builds. And you can truly have an impact on the

world, both horrendous and exceptionally positive and inspiring. It's agency of us individuals.

Sometimes we think we're just reacting to the world, but we have the full power to actually

change the world. Is there advice you can give to young folks? We give a bunch of advice on

middle school, high school mathematics. Is there more general advice you give about how to succeed

in life, how to learn for high school students, for college students, career or life in general?

So I think the first one would be to make sure that you're learning to invent and to make sure

you're not just learning how to mimic, because a lot of times you learn how to do X by watching

somebody do X and then repeating X many times with different inputs. I've just been very generic

in explaining this, but I guess this is just my own attitude towards the world. I didn't like

ever following anyone's directions exactly. Even if you told me this is the way to do

your homework is to write in pencil, I would say, but I think that is nice. Let's try.

So I've been that kind of a funny person. But I do encourage that if you can learn how to

invent as your core skill, then you can do a lot. But then the second piece that comes with that

is something I learned from my PhD advisor, PhD advisor, which was, well, make sure that what

you're working on is big enough. And so in that sense, I usually advise to people once they have

learned how to invent, ideally, don't just try to settle for something comfortable. Try to see if

you can aim for something which is hard, which might involve a campaign, which might be important,

which might make a difference. And it's more of, I guess, rather than worrying, what if you didn't

achieve that? There's also the regret of, what if I didn't try? That's how I operate. I don't

operate based on, did I succeed or fail? It was hard anyway. If I did this Novid thing and the

whole thing failed, would I feel terrible? No, it's a very hard problem. But would I have had the

regret of not jumping in? Yes. So it's that different mentality of don't worry about the

failing part as much of the make sure you give yourself the shot at those potentially unbounded

opportunities. You almost make it sound like there's a meaning to it all. Let me ask the

big ridiculous question. What do you think is the meaning of life? Or maybe the easier version of

that is what brings your life joy? So I'll just answer that one personally. For me, I'm a little

bit weird. I sort of see the pen and pencil discussion from earlier. Yeah. So I mean,

my thing is, I guess I personally just wanted to maximize a certain score, which was for how many

person years after I'm no longer here anymore, did what I do mattered. And it didn't matter if

it's necessarily attributed to me. It's just like, did it matter? And so that's what I wanted.

I guess that is very inspired by how scientists work. It's like, why do we keep talking about

Newton? It's because Newton discovered some interesting things. And so Newton's score is

pretty high. It's going to be infinity, right? Well, let's hope it's infinity, but pretty high.

Ah, yes. So the person years, you're going for triple digits. So Newton is like four digits

probably, like a thousand years. Or a person lifetimes. How do you like to think about what are we?

Sorry, I've met people times years. So then it's like, actually, his is huge. His is like going to

be billions or trillions, right? Trillions. But I guess, for me, I actually changed the metric

after a while. And the reason is because you may have seen, I found some simple way to solve

quadratic equations that is easier than every textbook. So my score might already be not bad,

which is why I decided then let's change it into the number of hours in the lifetimes as well.

So the way I was doing it before is that if a person was sort of remembering or using or appreciating

what I had done for like 10 years of their life, that would count as 10. So if there was one person

who for 10 years remembered or appreciated something I did, that counts as a score of 10,

and we add up over all people. And then that was with the hypothesis that the score would be very

finite in the sense that if I didn't come up with anything that might potentially help a lot of

generations in a forever way, then your score will be finite because at some point it's not,

people don't remember that you made like nice bottles or something, right? But then after

the quadratic equation thing, it was that there's some chance that that actually might make it into

textbooks. And if it makes it into textbooks, the chance that there will be an easier way

discovered is actually quite small. So in that case, then the score might get bigger. I was just

saying the score might actually already have been achieved in a non trivial way.

I see. Because it's fun to think about because it could be different. You can achieve a high score

by a small number of people using it for most of their lifetime and then generations and generations,

or you can have, if we do dissipate, if we do colonize, become multi planetary species,

you could have that little, a clever way to solve differential equations spread through

like trillions of people as they spread throughout the galaxy. And they would only use it each one

a few hours in their lifetime, but their kids we use it, the kids the kids we use it will spread

and you'll have that impact in that kind of way. Yes. So that's why I renormalized it. Because I

was like, well, that's kind of dumb because what's the importance of that? That'll save people 15

minutes. But so what I meant is I didn't want to count that as the main score.

Well, I'm going to have to try to come up with some kind of device that everyone would want to use,

maybe to make coffee, because coffee seems to be the prevalent performance enhancing chemical

that everyone uses. So I'll have to think about those kinds of metrics. Yeah. But you see,

that's just giving an idea of I guess what I found meaningful in general, like whether or not it's

like whether or not that quadratic thing is important or not. The general idea was I wanted

to do things that would outlast me. And that was what inspired me. And that's just how I choose

what problems to work on. And that's a kind of immortality is ideas that you've invented

living on long after you in the minds of others. And humans are ultimately not

are like meat vehicles that carry ideas for a brief for just a few years may not be the

important thing. It might be the ideas that we carry with us and invent new ones. Like we get a

bunch of baby ideas in our head. We borrow them from others. And then maybe we invent a new one.

And then you one might have a life of its own. And it's fun. It's fun to think about that idea

of living for many centuries to come, unless we destroy ourselves. But maybe AI will borrow it

and we'll remember Poe as like that one human that that helped us out before we of course killed

him and the rest of human civilization. On that note, Poe, this is a huge honor. You're one of

the great educators I've ever gotten a chance to interact with. So it's truly an honor that you

would talk with me today. It means especially a lot that you would travel out to Austin to talk

to me. It really means a lot. So thank you so much. Keep on inspiring. And I'm one of your

many, many students. Thank you so much for talking today. Oh, thank you. Thank you. It's

actually a real honor for me to talk to you and to get this chance to have this

really intellectual conversation through all of these topics. Thanks, Poe.

Thanks for listening to this conversation with Poe Shen Loh. And thank you to

Jordan Harmer to show on it, better help, eight sleep and element. Check them out in the description

to support this podcast. And now let me leave you with the words from Isaac Newton. I can calculate

the motion of heavenly bodies, but not the madness of people. Thank you for listening and hope to see

you next time.