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

the following is a conversation with poe 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 covet 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 on it betterhelp eight sleep and element check them out in the description to support this podcast as a side note let me say that poe and i filmed a few short videos about simple beautiful math concepts that i will release soon it was really fun i really enjoyed poe 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 friggman podcast and here's my conversation with po shenlo you know 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 and when i look at like what's happening now with humans traveling out into space i see it as all the same thing it's incredible that humans are able to get into a box and fly in the air and and safely and land in the same it seems like and everybody's taking it for granted so when i observe them it's quite fascinating because i see that cleanly mapping to the world where we're now on uh in rockets and traveling to the moon traveling to mars and at the same kind of way i can already see the future where we will all take it for granted so i don't know i don't know if you have uh you personally when you fly have the same kind of magical experience of like how the heck did humans actually accomplish this so i do especially when there's turbulence which is you know like on the way here yeah there was turbulence and the the plane jiggled even the flight attendant had to hold on to the side and i was just thinking to myself it's amazing that this happens all the time and the wings don't fall off you know like given how many planes are flying but then i often think about it and i'm like you know a long time ago i think people didn't trust elevators yeah in a 40-story building in new york city and now we just take it completely for granted that you can step into this shaft which is 40 floors up and down and it will just not fail yeah again i'm the same way with elevators but also buildings when i'll stand on the 40th floor and wonder how the heck are we not falling right now like how how amazing it is with the high winds like structurally just the earthquakes and the vibrations i mean natural vibrations in the ground like how is this how are all of these you go like new york city all of these buildings standing i mean to me one of the most beautiful things actually mathematically too is uh bridges i used to build bridges in high school from like toothpicks just like out of the pure joy of like physics making some structure really strong understanding like from a civil engineering perspective what kind of structure will be stronger than another kind of structure like suspension bridges and then you see that at scale humans being able to span a body of water with a giant bridge and it's i don't know it it's so humbling it makes you realize how how dependent we are on each other sort of i talk about love a lot but there is there's a certain element in which we little ants have just a small amount of knowledge about our particular thing and then we're depending on a network of knowledge that other experts hold and then most of our lives most of the quality of life we have has to do with the the richness of that network of knowledge of that collaboration and then sort of the ability to build on top of it levels of abstractions you start from like bits in a computer then you can have assembly and you can have c plus so you have an operating system that you can see plus plus and python finally some machine learning on top all of these are abstractions and eventually we have ai that runs all of us humans but anyway uh but speaking of abstractions and programming in high school you wrote some impressive games for amaz i got a chance to in browser somehow it's magic got a chance to play them alien attack one two three and four what's the hardest part about programming those games and maybe can you tell the story about about building those games sure i actually tried to do those in high school because i was just curious if i could and yeah and that's a good starting point for anything right yeah yeah it's like could you but the appealing thing was also it was a soup to nuts kind of thing so something that has always attracted me is i like beautiful ideas i like seeing beautiful ideas but i actually also like seeing execution of an idea all the way from beginning to end in something that works so for example in high school i was lucky enough to grow up in the late 90s when even a high school student could hope to make something sort of comparable to the shareware games that were out there not i say the word sort of like still quite far away but at least i didn't need to hire a 3d cg artist there weren't enough pixels to draw anybody 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 in the web browser which is emulating um the 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 the screen in a particular color and the programming programming language was pascal ah yeah the first one was in pascal but then the other ones were in c plus plus after that how's the emulation in the browser work by the way is that is that true real 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 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 to 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 as 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 like almost like an athletics component to where you're almost like uh an athlete seeking optimal performance as a human being trying to write these programs and at the same time it's kind of art because you're the best way to write a program quickly is to write a simple program you used 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 kovit something that was very helpful from day one was understanding that the kinds of computations we would want to do we could conceivably do on like a four core cloud machine on amazon web services out to a population which might have hundreds of thousands or millions of people the reason why that was important to have that back of the envelope calculation with efficient algorithms is because if we couldn't do that then we would bankrupt ourselves before we could get to a big enough skill 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 too big and also the nice thing about programming competitions is that you actually build a thing that uh works so you you finish it there's a completion thing and you realize i think there's a magic to it where you realize that it's not so hard to build something that works to have a system that uh successfully takes in inputs and produces outputs and solves a difficult problem and that directly transfers to building a startup essentially that can help some aspect of this world as long as it's mostly be based on software engineering things get really tricky when you have to manufacture stuff that's why people like elon musk are so impressive that they it's not just software tesla autopilot is just not just software it's it's like you have to actually like have factories that build uh cars and there's like a million components involved in in the machinery required to assemble those cards and so on but in software one person can change the world which is uh uh incredible but on the mathematics side what uh 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 the writing the program i guess if i see something that's hard or supposed to be impossible it certain sometimes i say maybe maybe i want to see if i can pull that off and with the mathematics the math competitions presented problems that were hard that i didn't know how to start but for which i could conceivably try to learn how to solve them so i mean there are other things that are hard called like get something to mars get people to mars and i didn't 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 like that mathematical skill is the core of lots of other things that's really interesting maybe you could speak to that because a lot of people say that math is hard as a kind of negative statement it always seemed to me a little bit like that's kind of a positive statement that all things that are worth having in this world are hard i mean everything that people think about that they would love to do whether it's sports whether it's art music and all the sciences they're going to be hard if you want to do something special so is there something you could say to that idea that math is hard should it be made easy or should it be hard ah so i think maybe i want to dig in a little bit onto this hard part and say uh 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 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 yes 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 this is i'm actually coming up with i'm actually explaining another way that you could try to do teaching and i'm contrasting this to a method of watch me do this now practice it 20 times i'm trying to say a lot of people consider math to be hard because maybe they can't remember all of the methods that were taught but for me i look at the hardness and i don't think of it as a memory hardness i think of it as a can you invent something hardness and i think that if we can teach more people how to do that art of invention in a pure cognitive way not as hard as the actual hardware stuff right but 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 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 um do you see basically the way to teach math is to present a problem and to give a person a chance to try to invent a solution without with minimal amount of information first is that is that basically how do you build that muscle of invention in a student yes so the way that i 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 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 class is there's something that we want to learn how to do and we're going to do it by improv i'm talking about the same method as improv comedy which is where you tell me some ideas and i'll try to yes and them you know what i mean and then together we're gonna 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 uh through you know 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 coven 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 uh so you stepped into this world can you tell me about your work there under the flag of novid and uh both the 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 coven the shutdown started to happen and at that time it started as a figment of an idea which was network theory which is the area of math that i work in could potentially be combined with smartphones and some kind of health information anonymized exactly how we didn't know yet we tried to crystallize it and many months into this work we ended up accidentally discovering a new way to control diseases which is now what is the main impetus of all of this work is to take this idea and polish it and hopefully have it be useful not only now but for future pandemics the idea is really simple to describe um actually my main thing in the world is i come up with obvious observations that's that's i'll explain it now einstein did the same thing and he wrote a few short papers but but so the idea is like this if we describe how usually people control disease for a lot of history it was that you'd find out who was sick you'd find out who they've been around and you try to remove all of those people from society against their will yes now that's the problem the against the 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 uh 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 um the idea was suppose there was a penta 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 however i don't know if i want to call it positive or negative but they would run a good feedback loop yes 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-covered example to show why this maybe it 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 that's a disastrous disease it has high fatality rate and uh as you die you're bleeding out of every orifice okay so yeah no not pleasant not pleasant so the question is suppose that such a disease broke who would want to install an app that would tell them how many relationships away from them this disease had struck like a lot of people a lot of people in fact almost i don't want to say almost everyone that's a very strong statement but a very large number of people that's fascinating framing like the the more deadly and transmissible the disease the the stronger the incentive to install it in a positive sense the in in in the good feedback loop sense that's a really good example it's a really good way to frame it because with covet it was not as deadly as uh as potential pandemics could have been viruses could have been so it's sometimes muddled with how we think about it but yeah this is a really good framing if the virus was a lot more deadly you want to create a system that has a set of incentives that it quickly expresses a population where everybody is using it and is 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 uh 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 while there's accurate information but that aside okay so this is a really nice property right and 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 we surveyed informally people but they said as we saw 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 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 six that b is everyone else and for the entirety of the past uh contact tracing approaches you try to get set a to do things that might not be to their liking or their will because that's removing them from society yes we found out that there's two ways to separate set a from set b you can also let the people at set b at the fringe of set a attempt to remove themselves from this interface it's just 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 it 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 was 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 cl and there's maybe um in the graph theoretic sense there's some weights or something about how close that relationship is in terms of the frequency visits the duration of business and all 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 uh feedback loop where you can act essentially anonymously uh and on your own like nobody's telling you what to do which is really important is decentralized and uh and not yeah whatever the opposite of authoritarian is but you get to sort of the american way you get to choose to do it yourself you have the freedom to do it yourself and you're incentivized to do it and you're most likely going to do it to to uh to protect yourself against um against you getting the disease as the the closer it gets to you based on the information that you have but uh can you maybe elaborate uh first of all brilliant uh whenever i saw the thing you're working on so forget for covid this is of course really relevant for covid but it's also probably relevant for future diseases as well so this uh that was the thing i'm nervous about like if this whole if our society shut down because of covid like what the heck is gonna happen when there's a much deadlier disease like this this is disappointing the whole time 2020 the whole time i'm just sitting like this like is the incompetence of everybody except the people developing vaccines uh the biologists are the only ones that got their stuff together but in terms of institutions and all that kind of stuff oh 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 yeah if you just line up the set of incentives correctly so that people have in their purely selfish behavior are contributing to the optimization of the global function yes that's it and the 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 apply 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 um i guess that the tech 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 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 [Laughter] uh yeah so we're not doing the old pythagorean based violation of privacy okay [Laughter] but so is that is that enough to form to give you enough information about physical connection to another human being is there a time element there is there so okay that sounds like a really strong like low hanging fruit like if you have that you could probably go really really far my natural question is is there extra information you can add on top of that like the duration of the physical proximity uh 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 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 we changed from i mean the standard paradigm was what already happened quick damage control ours 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 problem 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 that's not an important physical relationship i mean this is brilliant what uh 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 is it to go just like you built the video game alien attack from zero to 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 the the 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 novid covered with an ad very nice so you have to hook into bluetooth you're saying you have to do that um beyond the permissions that are like at the very surface level provided on the phone well i don't want to call them permissions i just want to say that's not what you usually do with bluetooth gotcha usually with bluetooth you say do i have headphones nearby yes okay i'm done you don't go and say do i have headphones nearby or do i have another phone nearby which is doing something and then keep asking that system keep asking the question right so so this is 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 iowa certainly makes it difficult for something to run in the background especially when it's to eating up your battery right ah 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 uh actually even if compared to apple's own system so beautiful so what else is required to make this thing work right so the the key was that you had to do significant amount of work on the actual mobile app development which fortunately the team that we brought was this kind of general thinkers where we would dig in deep into the operating system documentation and the api libraries so we got that working but there's another angle which is you also need the servers to be able to compute fast enough which is tying back to this old school computer programming competitions and math olympiads in fact our team that was working on the algorithm and back-end side included several people who had been in these competitions from before which i happen to know because i i do coach the team for the math yes and so we were able to bring people in to build servers a server infrastructure in c plus 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 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 uh scaling problem right ah so that's actually why uh 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 yeah i only need to do one run one one recalculation every hour in terms of telling people how far away these dangers are yes so i suddenly have 3600 seconds and my cpu cores are running in gigahertz and at most they're eight giga people well you're skipping over the fact that uh there's n squared potential connections between people so how do you get around the fact that uh you know that we you know the potential set of relationship any one of us could have 8 billion so it's 8 billion times uh squared that's you that's potential amount of data you have to be storing and computing over and constantly updating so the way we dealt with that is we actually expect that the typical network is very sparse the 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 if you think of it almost in terms of the heavy hitters actually in most people's lives 100 if we just kept track of their top 100 interactions that's probably most of the signal yeah yeah i i i'm saddened to think that i might not be even in a double digits but oh i i was intentionally giving a crazy number to account for college students you call oh those are the who you're calling the heavy hitters the people who are like the social butterflies yeah yeah yeah i need to uh um i'd love to know that information about myself by the way the that i do do you uh 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 yeah but you know we also saw in those examples especially looking at the screenshots people gave that is hit as soon as the typical person has two or three other direct connections on the system because that means that our app has reached a virality are not 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 out so you're trying to out virus the virus that's right that's exactly right okay great what have you learned from this whole experience in terms of um let's say for covid but for future pandemics as well is it possible to use the power information here of networked information as the virus spreads and travels in order to basically keep the society open is it possible for for people to protect themselves with this information or do you still have to have most like in this overarching policy of everybody should stay at home all that kind of thing we are trying to answer that question right now so the answer is we don't know yet but that's actually why we're very happy that now the idea has started to become become more widely known and we're already starting to collaborate with epidemiologists again i'm 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 kovid 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 yeah 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 yeah 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 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 no matter how deadly the virus is no matter how many things whether you have a vaccine or not it's still useful to be having this information yes because to stay home or not depending on how risky like i'm a big fan just like you said of uh having the freedom for you to decide how risk-averse you want to be right and depending on your own conditions but also on the state of like what you just how dangerously you like to live so i think that actually makes a lot of sense and i also think that um since we're when we when you think of disease spreading it spreads in aggregate in the sense that uh 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 this are naught of the infection uh 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 yes people get a little bit nervous i think with uh information somehow mapping to privacy violation but i 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 maybe not necessarily about actual location but just maybe activities like just to have information about what is what is good to do or not you know uh in terms of like safety is it okay to run outside and not is it okay to go to a restaurant or not i just feel like we're operating the blind and then what you had is a very imperfect signal which is like basically politicians desperately trying to make statements about what is safe and not they don't know what the heck they're doing they have a bunch of smart scientists telling them stuff and the scientists themselves also very important don't always know what they're doing epidemiology is not is as much an art as a science you're desperately trying to predict the future which nobody can do and then you're trying to speak with some level of authority i mean if i were to criticize scientists they spoke with too much authority it's okay to say i'm not sure but then they think like if i say i'm not sure then there's going to be a distrust what they realize is when you're wrong and you say i'm sure it's going to lead to more distrust so 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 um challenges with adoption that you see in the future here so there's uh maybe we could speak to there's approaches i guess from google there's different people that have tried similar kind of ideas not in you have a quite a novel idea actually but speaking the umbrella idea of contact tracing is is there is there something you can comment about why their approaches haven't been fully adopted is there challenges there is there is there reasons why novid might be a better idea moving forward in general just about adoption yeah so first of all i want to say i always have respect for the methods that other people use and so it's good to see that other people have been trying but what we have 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 have already been exposed to the disease so that you can protect other people from you and what that does to your own direct probability of getting sick if you think about it suppose you were making the decision should i or should i not install one of those apps yeah what does that do to your own probability of getting sick it's close to zero this is uh the sad thing you're um you're speaking to not sad i suppose it's the way the world is the only incentive there is to just help other people i suppose but a much stronger incentive is is anything that allows you to help yourself yes so what i'm saying is that uh 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 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 nova does and most of them have read about the other apps and they say oh no of it will tell you after you've been around someone so you can quarantine and we have to explain to them actually novad 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 yeah if you want if you want and then the 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's exciting yes exactly exactly i mean this is brilliant but so what um do you think the future looks like for future pandemics what's your plan with novid what's your plan with these set of ideas i am actually still an academic and a researcher so the biggest work i'm working on right now is to try to build as many collaborations with other 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 i always have to think like if a researcher was getting involved what are they getting out of it oh so you mean like uh from a research perspective you can um like publications and sets of ideas about how to from a sort of uh uh 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 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 modern and fizer vaccines which cost like 20 a pop in the u.s so if they want to participate who do they reach out to oh that would just be us i mean the novid.org website has nova.org it has 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 novanova.org uh 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 no borrowers 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 gives 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 kovid well then i mean we'd be very happy to also work with you guys so the the verified there is really important because you're 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 uh possible to manipulate because uh it's it's ultimately about trust and information and so it could be um verification is really important there so basically individual people should um ask their local health authorities to to just to sign up to contact you i hope this spreads i hope this spreads uh for future pandemics because i'm really i'm it's the amount the millions of people who are hurt by this i think our response to the virus economically speaking the number of people who lost their dream lost their jobs but also lost their dream entrepreneurs you know jobs often give meaning there's people who financially and psychologically are suffering because of our i'll say incompetent response to the virus across the world but certainly united states that should be the beacon of uh entrepreneurial hope for the world so i i hope that uh 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 of uh mathematics maybe ask the the big silly question first which is uh 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 crystallize and factored in certain ways is beautiful actually that's related to what we were just talking about with even this fighting pandemics the the crystal idea was just quantify proximity by the number of relationships in the phys in the physical network instead of just by the feet and meters right it's it's just if you change that perspective now all of these things follow and so mathematics to me is beautiful in the pure sense just for that yeah it's quite interesting to see a human civilization as a network as a graph and our relationships as kind of edges in that graph and to then do outside of just pandemic do interesting inferences based on that this this is true for like twitter social networks and so on how we expand the kind of things we talk about think about sort of politically if you have this little bubble quote-unquote of ideas that you play with it's uh nice from a recommender system perspective how do you jump out of those bubbles it's really fascinating uh 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 in 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 a small clean uh problems that are really hard but once you look at them differently can become easy but that little jump of innovation is is uh is the entire trick so maybe at the high level can you say what is the international mathematical olympiad sure so this is uh the competition for people who aren't yet in college uh math competition which is the most prestigious one in the entire world it's the olympics of mathematics but only for people who aren't yet in college now the kinds of questions that they ask you to do are not computational usually you're not supposed to find that the answer is 42 right instead you're supposed to explain why something is true yes and the 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 this is hard to explain a hard problem what about is it one person is it a team ah so it's 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 uh stacked up where yeah so maybe as a side comment i should say that um there's a bunch of countries including the former soviet union and russia uh 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 uh 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 uh just basic conversation into the pride like you know if you win an olympic gold medal or if you win the super bowl you can walk around proud i think that was the case with the mathematical olympiad in russia not not as much the case in the united states i think so i just want to give that a little aside because beating anybody from russia from the eastern republic or from china is very very difficult that like if i remember correctly you know there's people this was a multi-year training process they train hard and this is this is everything that they're focused on my uh my dad was was uh was a participant in this and it's i mean it's uh it's as serious as olympic sports you think about like gymnastics like young athletes participating gymnastics is this 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 ussrs or the 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 i think it clearly worked well for soviet union yeah so this this is interesting people sometimes ask me you know you go up and 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 uh communism is one of those things that's looked down on like without in absolutist terms currently but you could still in my perspective talk about the actual forget communism or whatever the actual term is but you know certain ways that the society function that we can learn lessons from and one of the things in the soviet union that was highly prized is knowledge not even now it's wisdom and the skill of invention of innovation at a young age so we're not talking about a selection process where you pick the best students in the school to do the mathematics or to read literature it's like everybody did it everybody it was almost treated as if anyone could be the next einstein anybody could be the next i don't know hemingway james joyce and so you're forcing an education on the populace and a rigorous deep education like as opposed to kind of like oh we want to make sure we um we teach to the weaker student in the class which american systems can sometimes do because we don't want to leave anyone behind the the russian system was anyone can be the strongest student and we're going to teach you the strongest student and we're going to just to pretend or force everybody even the weakest student to be strong and what that results in it's obviously this is what people talk about is a huge amount of pressure like it's psychologically very difficult this is why people struggle when they go to mit this very competitive environment it can be very psychologically difficult but at the same time it's bringing out the best out of people and that mathematics was certainly one of those things and exactly what you're saying which kind of clicked with me just now as opposed to kind of a spelling bee in the united states which i guess you spell i'm horrible at this but it's a competition about spelling which i'm not sure but you could argue doesn't generalize well to future skills mathematics especially this kind of mathematics is essentially formalized competition of invention of uh of creating new ideas and that generalizes really really well so that's 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 uh will come come to do some incredible stuff in the future yeah actually i want to respond to a couple of things there the first one this one um 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 covet 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 it's very very accessible but lots of people are doing that why well actually i think that what this what what was going on with the authoritarian thing was at least the message that was universally sent was being a good thinker and a creator of ideas is a good thing yes exactly there's no reason why that message can't be sent that's right 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 um you know the generalizable skill and what what could people do with olympiads afterwards so that's actually my interest in the whole thing uh i don't i mean i don't just coach students how to do problems in fact i'm not even the best person for that i'm not the best at solving these problems there are other people who are much better at making problems and teaching people how to solve problems in fact when i was when the mathematical association of america which is the group which is in charge of the u.s participation in these olympiads when they were deciding whether or not to put me in back in 2013 as the head coach i had a conversation with their executive director where i commented that we might do worse because my position was i don't i mean i actually didn't want to focus on winning i said if you're going to let me work with 60 very strong minds as picked through this system because the coach works with these gets to run a camp for these students i said i'm actually not going to define my success in terms of winning this contest i said i wanted to maximize the number of the students that i read about in the new york times in 20 years oh yeah 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 uh but how to have how to be on the cover of new york times for inventing something new and i do you think that's generalizable to education beyond just this particular olympia like it's even you're 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 um 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've seen me give in the past as well as no overlap with anything i taught in the class so the entire exam is invention wow but that's how i go right my point is i have explained to people when i teach you i don't want you to have remembered a method i showed you i want you to have learned enough about this area that if you face a new question which i came up with the night before by thinking about like what could i ask that i have never asked before oh that's cute i wonder what the answer is ah that's an exam problem that's exactly what i do before the exam and then that's what i want them to learn yeah 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 they intend in that area yeah can we uh walk back to the the mathematical olympia yes what's the scoring and format like and also what does it take to win so the way it works is that each of the six students do the problems and there are six problems all the problems are equally weighted so each one's worth seven points that means that your maximum score is six problems times seven points which is the nice number of 42. and now the way that they're scored by the way is there's partial credit so the question is asking you explain why this weird fact is true okay if you explain by 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 you that maybe you're just off by it by the way they're not numerical anyway but i'm just giving some numerical analog to the way the scoring might work they're all essays and that's where i guess i have some role as well as some other people who helped me in the u.s 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 schedule these half hour appointments where the delegation from one country sits down at the 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 mongolia 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 it's almost like um like a jury yeah you have you have uh 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 speak in different languages sometimes that's fascinating but uh 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 a it's very hard for me 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 uh say something insightful to this question but what do you think makes a really hard math problem on 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 heart 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 it's this is like a very theoretical computer science way of looking at it okay it's the is 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 yes 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 actually why like if you have a three inside problem that is not three times as hard as a one inside problem because after you've made the one inside it's not clear that that was the right track necessarily well unless there's still a branching uh of possible yeah [Laughter] 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 in if you are taking uh 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 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 that's brilliant 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 that 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 so i mean finding the proof i quit if i'm if i have to do too many pluggings it's equivalent to a math contest in a math context 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 i think um well it's possible that that's true approximately approximately approximately yeah just i don't know uh somebody out there is going to try to formally prove this oh no i mean you're right there are cases where maybe it's not quite linear but in general well some of it's notation too and some of it is uh 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 that's 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 uh that's the scary thing about insights uh you look like andrew wiles working on the firm oz 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 uh even if you think it's correct you have no idea if this is like going to go down a path that's completely counterproductive or not productive at all that's that's a crappy thing about invention is um 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 i'll go inside my head along them along that little trajectory but it could be just a total waste and it's that i that you know what that feels like it just feels like patience is required not to get excited any one thing so i think this is interesting because you raised andrew biles he spent seven years attacking the same thing yeah right and and so i think that what attracts professional researchers to this is because even though it's very painful that you keep fighting with something when you finally find the right insights and string them together it feels really good so well there's also like short term it feels good to to uh 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 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 sudokov in fact he grew up in the former soviet union he was from georgia but he he's an incredible person but one thing i learned was choose the problems to work on that might matter if you sub if you succeed because that's why for example we dug into cope it was just well suppose we succeed in finding some interesting insight here well it actually matters then it's worthwhile yeah and i think covid the the way you're approaching uh kovid has two interesting possibilities one it might help with the covet or another pandemic but two i mean 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 there's a space of possible impacts that may be direct or indirect and the same thing is like with andrew wiles's proof i don't understand but apparently the pieces of it are really impactful for for mathematics even if the main theorem is not so along the way the the insights you have might be really powerful for unexpected reasons so i like what you said this is something that i learned from another friend of mine who's also he's a very famous researcher all these people are more famous than i am his name is jacob fox he's jacob fox at stanford also a very big inspiration for me we were both grad students together at the same time well most importantly you're good at selecting good friends ah yeah well that's that's that's that's the key you've got to find good people to learn things from yeah but his thing was he often said you know if you solve a math problem and have this math proof math problem for him is like a proof right so suppose you came up with this proof he always asks what have we learned from this that we could potentially use for something else it's not just did you solve the problem that was supposed to be famous 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 you know there's this funny effect where uh 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 uh we did this offline in another illustration you showed to me it's interesting to see the different perspectives on a problem it kind of points like there's just like very few novel ideas that everything else that most of us are just looking at different perspective on the same idea and it makes you wonder this this uh this old silly question that i have to ask you is uh do you think mathematics is uh discovered or invented do you think we're creating a new idea we're building a set of knowledge that's that's distinct from reality or are we actually like it is math almost like a shovel where we're digging to like this core set of truths that are oh that that were always there all along so i personally feel like it's discovered but that's also because i guess the way that i'd 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 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 it doesn't get any simpler than that it's possible that that like an alien species will have depending on different cognitive capabilities and different perception systems we'll be able to see things that are much different than circles and so if it's discovered it would 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 the like um some basic mathematical concepts like how much work is there going to be in trying to find a common language i mean this is um i think stephen wolfram and his son help with the movie arrival like the 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 uh like the summation of the odds and the the adds up to the squares yes i wonder how difficult that is uh before one one or the other species murders i 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 uh ultimately like you know willing to commit violence in order to gain those resources i think as we progress become more and more intelligence 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 see the cool thing about the math olympiad um i don't know if you know work from francois chole from google he come up he came up with this kind of iq test slash it kind of has similar aspects to it that also math olympia does for for ai so he came up with these tests where they're very simple for humans but very difficult for ai to illustrate exactly why we're just not good at seeing a totally new problem we sorry ai systems are not good at looking at a new problem that requires you to detect that there's a symmetry of some kind or there's a pattern that's that hasn't it hasn't seen before the pattern is like obvious to us humans but it's not so obvious to find that kind of it's you're inventing a pattern that's there in order to then find a solution some 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 yeah 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 yes where did they get that heuristic from now that's a good question i don't know because that if you asked them to explain to you they could probably say something in words that sounds like it makes sense yeah 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 yeah but it's probably not approximating everything that's going on inside so there's basically a function being computed yeah but it's hard to articulate what that function is now the question is could a computer get as good at computing these kinds of heuristic functions maybe i 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 with 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 brains operate in so it allows us to to have multiple many many dimensions it allows us to work on on weird surfaces uh with like topology as a discipline it's just weird to me it's really complicated but it allows us to work in that space the differential geometry and all those kinds of things where it's totally outside of our natural day-to-day four-dimensional experience uh 3d dimensional with time experience so math gives us gives me hope that we can that we can see we can discover the processes of intelligence uh outside the limited nature of our own like human experiences but you said that you're not an expert it's kind of funny i i i find that um we know so little about intelligence that uh i think i honestly think like almost children are more expert at creating artificial intelligence systems uh than than adults i feel like we know so little we really need to think outside the box and those little i found people should uh check out francois charley's little exams but even just solving math problems i don't know if you've ever done this for yourself but when you solve a math problem you kind of then traced back and tried to figure out where did that idea come from like what how did like what was i visualizing in my head how did i start visualizing it that way how 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 that come from so this is interesting um so i try to do this to teach middle school students how to learn how to create 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 whoever wants to watch just see what exactly goes on through someone's head as they go and attempt to invent what they need to do to solve the question so i've actually thought about that i think that first of all as a teacher i think about that because whenever i want to explain to a student how to do something i want to explain how it made sense why it's intuitive to do the following things and why the wrong things are wrong not just by this one short fast way well why this is the right way if that makes sense so my point is i'm actually always thinking about that like how would you think about these things and then i eventually decided the easiest way to expose this would just be to go live on youtube and just say i've never seen any of these questions before here we go don't you get uh man that's that's anxiety inducing for me uh 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 contents come in and students say try this oh wow it's actually pretty good i'll never get stuck i mean i'm i'm willing to go on camera and say guess what potion though 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 [Laughter] 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 or they get it is there um is there something to be said about like discovering that moment and and trying to grab them to get to get them to understand that mathematics is so no matter what they want to do in life could be part of their life yes i actually do think that the middle school is exactly the right time because that's the place where your mathematical understanding gets just sophisticated enough that you can start doing interesting things because if you're early on in counting i'm honestly not very good at teaching you new insights my wife is pretty good at that but somehow once you get to this this part where you know what a fraction is and when you know um how to add and how to multiply and what the area of a triangle is at that point to me the whole world opens up and you can start observing there are really nifty coincidences the things that made the greek mathematicians and the ancient mathematicians excited actually back then it was exciting to discover the pythagorean theorem it wasn't just homework so is there what which discipline do you think has the most exciting coincidences so is it geometry is it algebra um is it calculus well you see you're asking me and i'm the guy who gets the most excited when the combinatorics shows up in the geometry is it okay so it's the combinatorics in the geometry so first of all the nice thing about geometry this is the same nice thing about computer vision is it's visual so geometry you can draw circles and triangles and stuff so it naturally uh presents itself to uh 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 uh that's most amenable to the exploration of the invention through proofs the idea of proofs i think is most easily shown in in geometry because it's so visual i guess so that that to me is like uh 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 of an um of a student and it kind of disappears and not until much later which you know to maybe like differential geometry i don't know where else it shows up for me in computer science like you can start to think about like computational geometry or even graph theory is the kind of geometry you can start to think about it visually although it's pretty tricky but yeah it was always um that that was the most beautiful one everything else i guess calculus can be kind of visual too that can that could be pretty beautiful but is there um 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 when i'm if i'm looking at a particular student the question is how can we help you feel like you have the power to invent also because i think a lot of people are used to thinking about math as something where the teacher will show you what to do and then you will do it yes so i think that the key is to show that they have some let them see that they have some power to invent and at that point it's often starting by trying to give a question that they don't know how to do you want to find these questions that they don't know how to do that they can think about and then they can solve and then suddenly they say my gosh i've had a situation i haven't i've had an experience where i didn't know what to do and after a while i did is there um 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 those school and high school are really good they're actually very hard so if you 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're 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 but by trying to work on these questions is there a a googleable term that you could use for the organization for the state competitions ah yeah so there are a number of different ones that are quite popular one of them is called math counts m-a-t-h-c-o-u-n-t-s and that's a big tournament which actually has a state level there's also a math league.org mathleaguelegue.org also has this kind of tiered the tournament structure there's also the american math competitions amc8 amc also has amc 10 that's for 10th grade and below and amc 12. these are all run by the mathematical association of america and these are all ways to find old questions what about the daily challenges that you run what are those about we do that too but i mean the difference was um ours isn't that one is not free so so this i should actually probably be careful the things that i've just mentioned are also not free not all of those things i mentioned just now are free either but people can figure out what it is but yeah 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 um 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 potion low it's not free because that's actually how i pay for everything else i do so that's that was the idea but the the concept was aha now let's invent from scratch so if we're going to go from scratch and we're going to use technology what if we made every single lesson something where first i say hey here's an interesting question recorded of course it's not live but it's like i say hey here's an interesting question why don't we think about this but i know 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 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 learn this other concept that's what we did so 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 dot but if you go to find my website you you'll be able to find it beautiful can we zoom out a little bit and uh 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 for yourself but for for me what would you recommend in in the world of mathematics or sort of as opposed to studying for a test but just like lifelong expanding of uh knowledge and that skill for invention i think i often articulate this as can you always try to do more than you could do in the past yeah but that comes in many ways and i will say it's great if one wants to build that with mathematics but it's also great to use that philosophy with all other things in fact if i if i just think of myself i just think what do i know now that i didn't know a year ago or a month ago or a week ago and not just know but like what do i have the capability of doing yes and if you just have that attitude it brings more see the thing is there's also a habit like it is a skill like i've been using anki it's an app for helps you memorize things and i've actually just a few months ago started doing this daily of uh setting aside time to think about an idea that's outside of my work like let's say let's pick put it it's all over the place by the way but let's say politics like gun control uh 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 kind of outlining 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 um even just not math like pure knowledge math but math like these middle school problems the challenges right um 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 i i'm just saying this because i'm just thinking out loud right now like 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 that i didn't teach her anything i don't 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 i'm always thinking how do you make it practical right yes and the way to make it practical is if the the timer on the automatically daily is that you are going to automatically daily do something with your own kid 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 and that so i currently don't have kids so that's uh maybe i should just get kids to help me with the math thing but outside of that i i do want to do great math into daily practice so i'll definitely take out uh i'll definitely check out the daily challenges and see because um what is it uh grant sanderson we talked about offline three blue one brown he he speaks to this as well that his videos aren't necessarily they don't speak to the thing that i'm referring to which is the daily practice they're more almost tools of inspiration they kind of show you the beauty of a particular of problem in mathematics but they're not a daily ritual and i'm i'm in i'm in search of that daily ritual mathematics it's not it's not trivial to find um but uh i hope i hope to find that because i think math gives you a perspective on the world that enriches everything else so i like what you said about the daily also because that's also one reason why i put my carnegie mellon class online it's not every day it was every it's every other day semester is almost over but the idea was i guess my philosophy was if i'm already doing the class let's just like put it there right but i do know that there are people who have been following it who are not in my class at all who have just been following it because yes it's combinatorics and the value of that is you could you don't really need to know calculus to follow it if that makes sense so it's actually something that people could follow so again and that one's free so that was just there on youtube well speaking of combinatorics uh what is it what do you find interesting what do you find beautiful about combinatorics so combinatorics to me is is the study of things where they might be more finite and more discreet 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 discreet because if you have a node it's not that you have point seven of a node and point three of a node over there it's like you got one node and then you jump one step to go to the next node so that that notion is different from say calculus which is very uh continuous where you go and say i have this speed which is changing over time and now what's the distance i've traveled that's the notion of an integral where you have to think of subdividing time into very very small pieces so the kinds of things that you do when you reason about these finite discrete structures often might be iterative algorithmic inductive these are ideas where i go from one step to the next step and so on and make progress i guess i actually personally like all kinds of math my area of research just ended up in here because i met a really interesting phd advisor potential that's actually 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 the the annoying thing about combinatorics and discrete stuff is uh it's often uh really difficult to solve uh from a uh sort of uh uh running time complexity perspective is there could you speak to the idea of complexity analysis of problems do you find it useful to find it interesting do you 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 um if you want to make something practical which has large numbers of people using it the computational complexity to me is almost question one and that's again that's at the origin of when we started doing this stuff with disease control from the very beginning the deep questions that were running through my mind were would we be able to support a large population with only one server and if the answer is no we can't start because i don't have enough money yeah and there the question is very much you know linear time versus anything uh anything slower than linear time um it's 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 um 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 like voting theory around this so a a theoretical question became what if you made a like a system of runoffs like a system of heads uh 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 a in electrical engineering or computer science you might imagine having a bunch of leads that carry signal which are going through and gates and or gains and whatnot and you've managed to compute beautiful things this is just from a purely abstract point of view what if the inputs are candidates and for every two candidates it is known which of the candidates is more popular than the other now can you build some kind of a circuit board which says first candidate number four will play against five and see who wins and so on okay so now what would be a nice outcome right this is a general question of could i make a big circuit board to feed an election into like maybe one nice outcome would be whoever wins at least is preferred over a lot of people yes so for example if you ran in 1024 candidates ideally we would like a guarantee that says that the winner beats a lot of people actually in any uh 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 uh no but i can explain it no no i can't the way it works is that think of it this way every time i 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 and it's it feels like a beautiful fact well let me explain it this way imagine that for every uh match you didn't give one point but you gave two points you gave one point to each person now that's not what we're really doing we really want to give one point to the winner of the match but instead we'll just give two if you gave two points to everyone on every matchup actually everyone has the same number of points and the number of points they get is how many other people there are does that sort of make sense i'm just like saying no no everything everything is same makes perfect 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 yes now we only make one change for each match up you give one point only to the winner so we're awarding half the points so now the deal is if in the original situation everyone's score was equal which is how many other people there are now there's only half the number of points to go around so what ends up happening is that there's always going to be like the average number of points per person is going to be half of how many other people there are and somebody is going to be above somebody's going to be 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 yeah when you describe it like that it's obvious but when you're first saying in this little circuit that there's going to be one candidate better than uh than half that's not obvious yeah that's funny math this is nice uh okay so do you have this but ultimately you want you're trying to with the voting tree i don't know if you're trying this but uh to have a circuit that's that's like compressed that's small well that achieves the the achieves the the same kind of um i mean the smaller it is the if we look at practically speaking the the lower the cost of running the election of running through of computing the circuit that is true but actually at this point the the reason the question was interesting yes is because the there was no good guarantee that the winner of that circuit would have like have beaten a lot of people let me give an example the best known circuit when we started thinking about this was the circuit called candidate one plays against candidate two candidate three plays against four and then the winners play against each other and then by the way five plays against six seven against eight the winners play against each other you understand it's like a giant binary tree where yeah the binary like a balanced binary tree okay it's a balanced binary tree one two three four up two thousand twenty four everyone going up to find the winner beautiful well you know what there's a system in the world where it could just be that there's a candidate called number one that just beats like 10 other people just the 10 that they need to be on their way up and they lose to everyone else but somehow they would get all the way up yes my point is it is possible to outsmart that circuit in one weird way of the world which makes that circuit a bad one because you want to say i will use this circuit for all elections and you might have a system of inputs that go in there where the winner only beat 10 other people which is the people they had to beat on the way up so you want to have a circuit where there's as many like the final result is as strong as possible yes and so what what ideas do you have for that so um we actually only managed to improve it to square root of n so if n is the number of vertices n over two would be the ideal we got it to and we got the square root of n versus log base two yeah exactly yeah which is well that is happening it could be a lot yeah it could be a big improvement so that's uh 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 the one part of the whole huge huge tree is that that same candidate let's column seven seven plays against somebody let's let's pick up some numbers let's call the others like letters so seven plays against a seven is also going to play against b separately and the winners of each of those will 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 yeah 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 you rebuild 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 match up at the bottom is seven against someone what that means is if 7 actually beat everyone they were matched up against well 7 would rise to the top so one possibility is if you see a 7 emerge from the top you know that 7 actually beat every one 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 beats 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 be the brazilian 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 uh how we may um have mathematical ideas of improving the electoral process that one no no i can't give you that one i mean is there like do you ever see it as um as as there be do you see is there being a lot of opportunities for improving how we vote uh like from your i i don't know if you saw parallels but you know it seems like if this actually kind of maps to your sort of covid work which is there's a network effect right it seems like we should be able to apply similar kind of effects of how we decide other things in our lives and and one of the big decisions we make is who represents us in government do you ever think about like mathematically about those kinds of systems i think a little bit about those because where i went to college the way we voted for student government was based on this is it called ranked choice where you're eliminated you eliminate the bottom and there's runoff elections so so i that was the first time i ever saw that and i thought that made sense yeah the only problem is it doesn't seem so easy to get something that makes sense adopted as the new voting system that's a whole nother there's that's not a math solution that's uh well it's math and sentences game theories you have to come up with incentive it's mechanism design you have to figure out how to trick us despite our basic human nature uh to adopt uh solutions that are better yeah that's a whole other conversation i think um can you just because it sounded really cool uh talk a little bit about stochastic coalescence and uh you have a paper on showing that something you describe what it is but i guess it's a super linear super logarithmic time and you came up with some kind of trick to make it faster just can you just talk about it a little bit yeah so this was something which came up when i was at microsoft research for a summer and that i'm putting that context because that shows that it has some practical motivation at some point actually i think it still it doesn't need to yeah it doesn't need to it could be beautiful and it's all right yeah so the easiest way to describe this is suppose you got like a big crowd of people and everybody knows how many hours of sleep they got last night and you want to know how many total hours of sleep were gotten by this big big crowd of people at the beginning you might say that sounds like a linear time algorithm of saying hey you how many hours you got how many you got how many you got add add add yes but there's a way to do this if you remember that there are people and they presumably know how to add you could make a distributed algorithm to make this happen for example while we're thinking of these trees imagine you had 1024 people if you could just say hey person number one and person number two you will add your hours of sleep person number two will go away and person number one is going to remember the sum person three and four add up and person three takes charge of remembering it person four goes away now this like person one knows the sum of these two person three knows some of those two they talk you see what i mean it's like you're going up this tree same tree that we've talked about earlier built up a tree from the bottom up yeah build a retreat 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 yeah there's only one problem how do you decide who's person number one and person number two yes so if for example you just went out into the downtown and said hey get these thousand people go well if you're gonna 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 nut 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 uh they will have delegated responsibility to themselves as the one who knows the sum so far so so the point is there's there's going to be people are all going to have like a pointer which says uh you are the one who knows my you you've taken care of my ticket my number yeah okay they select the representative for this particular piece of knowledge and at the very beginning you're your own representative the thing has to start simple right so at the beginning you're pointing to yourself you've got it yep and and how the way this works is that at every time step someone blares a ding dong on the on the on the tongue clock or whatever yes 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 they flip their coin some of the people start asking other people saying hey i would like you to be um 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 this is not what we're doing we're adding numbers okay but you can only add one number yeah 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 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 a 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 and now the idea is okay if you just keep doing this process what ends up happening oh yeah oh and also by the way if you decide that you want to go reach out to other people here's the here's the here's the 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 i actually delegated it to someone then you you you follow up you walk through the delegation chat 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 yeah 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 going to yeah the bigger the lump is the less like the more likely it is that you end up reaching that lamp 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 yeah but here's a bad situation that could happen imagine if you had like 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 their own manager except one giant 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 yes 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 the lumps are such that um you really are limited by this large one slowly sucking up the rest of them so the the heart of the question became well but is that just so unusual that we don't that it doesn't usually happen because remember you start with everyone just being independent it's like a lot of lumps of size well how naturally do the big lumps emerge yes and so what that part of the proof was was showing that that was a joint work with al lubetsky 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 distribute the lump size yeah that's i mean yeah it's fascinating how would the distributed algorithms a little yes make all the difference in the world yeah actually by the way this does this 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 uh for you paul 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 can 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 um 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 to distribute information about the the the spread of the disease we can the same way in a distributed sense form anything like any kind of knowledge bases that are formed in a decentralized way and uh in a hierarchical way as opposed to sort of old way where there's no mechanism for large scale fast uh like distributed transactional information this is really interesting this is where almost like network graph theory becomes practical yeah 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 uh let me ask the most ridiculous question you think p equals np wow i don't know i mean i would say i 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 and most computer scientists which would say believe that p equals np uh my question almost like this is back to our aliens discussion you want to think outside the box the the low probability event what is the world what kind of discoveries would lead us to to to prove that p does not equal to np like 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 i think it's very interesting but i 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 what to say i'm wrong to be humble i'm the only person to answer this question yeah yeah it's a great one well scott aaronson who's now here ut austin he used to be at mit puts the probability of uh p and p not equals to np at three percent he put it's i you know i always love it when you ask it's very rare in science in in academics because you're because most folks are humble in in the face of the mystery the uncertainty of everything around us to have the the both the humor and the guts to say um like what are the chance that there's um aliens in our galaxy intelligent alien civilizations as opposed to saying i don't know it could be you know it could be zero it could be depending on the fact you're saying it's 2.5 percent there's something very pleasant about just having uh it's the it's the number thing um this power 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 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 you know reading a book like that and then like now 42 is not part of his humor he doesn't shut up about 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 like the three percent chance it's stuck in my head and i think it's been having a ripple effect in everybody else the believing that p is not equal to np scott almost as a joke saying it's three percent is make it's actually motivating a large number of researchers to work on it three percent is high it's very high because for the potential impact that that would happen but then three percent 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 one percent 50 and we kind of i think a lot of people around three percent up to 50 percent like in our minds like three percent i think it could happen it could happen and it could happen is like yeah like half the time it'll probably happen so we're not very good at that that's that's the other thing with the pandemic is uh we're not the exponential growth that we also talked about offline uh is uh something that we can't quite intuit and that's something we probably should if we were to predict the future to anticipate the future and to understand how to create technologies that'll let us sort of control the future can i ask you for some recommendations uh maybe for books or movies in your life long ago when you were baby po or today that uh you found insightful or you learned you learned a lot from what you would recommend to 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 or you know like for example like the lord of the rings that's the content right but yes 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 uh fighting even when things that in that case was really fighting but just pushing on even when things are difficult i guess these are 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 that's that's the interesting thing i've obviously been don't shut up 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 the roots of my family very much but it's interesting to think um 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 uh horrendous and and uh like exceptionally positive and inspiring so yeah that's uh it's like it's uh 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 um advice you can give to young folks we talked we gave a bunch of advice on middle school high school mathematics is there more general advice you would 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'm just being 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 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 right 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 see 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 uh 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 i guess you can tell that but i see the pen and pencil discussion from earlier yes yeah 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 yeah and it didn't matter if it's necessarily attributed to me it's just like did it did it matter and so that's what i that's what i wanted um 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 yes so you're going for so person person years you're going for like triple digits you're going for it's like newton is like four digits probably like a thousand years yeah yeah or personal lifetimes like how do you like to think well what are we sorry i've met people times years people sometimes so so then it's like actually this is huge this is like going to be billions or trillions yeah right the trillions but um 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 yes 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 or or appreciating what i had done for for like 10 years if of their life oh that would count as 10 i see so if there was one person who for 10 years yes remembered or appreciated something i did that counts as a score of 10 and we add up over all people yes and then uh and 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 yeah right but then after the quadratic equation thing it was that there's some chance that that actually might make it into textbooks yes and if it makes it in textbooks the chance that there will be an easier way discovered is actually quite small yes so in that case then the score might get bigger i was just saying the score might actually already have been achieved in non-trivial way so 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 spread colonize become multi-planetary species you could have that little um a clever way to solve differential equations um uh spread through like trillions of people as they yes spread throughout the galaxy and they it would they would only use it each one a few hours in their lifetime but yes their kids would use it the kids the kids would use it it'll 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 will save people 15 minutes um but so so what i meant is i didn't want to count that as the main score well i'm gonna have to try to come up with uh some kind of device that everyone would want to use maybe to make coffee because coffee seems to be the prevalent uh performance performance-enhancing chemical that everyone uses so i'll have to think about in those kinds of metrics yeah but but you see that 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 yes and that was what inspired me and that's just how i choose what problems to work on and that's the kind of immortalities 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 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 that new one might have a life of its own and it's fun it's fun to think about that idea living from 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 that helped us out before we of course killed him and the rest of human civilization on that note well this is a huge honor you're one of the great educators i've uh ever gotten a chance to interact with so it's 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 thank you thank you it's actually a real honor for me to talk to you and and to get this chance to have this really intellectual conversation through all of these topics thanks bo thanks for listening to this conversation with paul shanlow and thank you to jordan harbor the 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 some 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

Channel: Lex Fridman

Views: 164,684

Keywords: math, math olympiad, lex math, lex podcast

Id: 6z1JwZbX4dQ

Length: 140min 15sec (8415 seconds)

Published: Fri May 14 2021

Reddit Comments

[u/meeshCosplay]

Starting around the 1:29:00 mark, Lex almost let the mask slip off:

Lex: "He [Francois Chollet] 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, umm 'we', sorry. AI systems are not good at..."

[u/OccamsNuke]

I didn't really have any expectations for this episode - one of my favorites, I really appreciated how well Po articulated his thoughts.

Would love to hear him again, perhaps exploring more of his work in more detail.

[u/theBirdu]

took discrete math last Semester, so now I know I have to listen to this podcast.