Alex Kontorovich: Improving math | 3b1b podcast #1

welcome to the 3b1b podcast i'm grant sanderson what you're about to listen to is my conversation with alex contour rich now alex is a research mathematician at rutgers university but more recently he's become active in a lot of different outreach related activities i first got to know him in part because of the work that he does with the mo math museum in new york he's on the advisory board for quantum magazine in recent months you might have seen him in a couple videos on youtube there was a collaboration he did with veritasium for pi day or a video he did with quanta about the riemann zeta function so in this conversation we talk about pretty much all things math whether that's education of young children research how the two of those are kind of related to each other how those relate to outreach um a long conversation about proof checking software like lean and the role that that has to play both for active research mathematicians and uh in education so if any of that interests you then stick around and if you're interested in more conversations like this we're going to start putting out a couple more at a hopefully decently regular cadence so subscribe to the feed [Music] when did you know that you wanted to be a mathematician sort of always so my mom is a mathematician she wrote a phd thesis she wasn't allowed to defend it that's a whole other story for maybe another time unless you really want to get into it we immigrated to this country and and i was eight years old when we arrived you know i go to school and i see all these strange i've never seen a word of english and so there's all these strange symbols and letters and so on they sat me down next to somebody and said you know just copy whatever he's writing so i copied everything that he wrote and i brought it to the teacher and she started laughing she's like no you put your name here not his name and then we got to math class and all of a sudden i was like oh my god these i know these symbols these are these are my old friends we studied these like two years ago but okay i'm happy to do it again you know the soviet system is uh a little bit ahead of the american system in many ways anyway so math was always something that i guess it did it did come easy it's hard to say a did it did it come easy or was it that uh everything else was infinitely more difficult in comparison um or was it that i had already seen all the things that i was seeing in school years before and that's sort of not only did that uh happen on arrival here but my mom always played all kinds of math games with us um you know driving to vermont to go skiing was a four-hour uh chance to you know play all kinds of brain games and give us sequences and see if we could figure out the patterns and you know that kind of stuff did you get the sense that your mom wanted you to become a mathematician that she was hoping you would um you know defend the thesis in the way that she never got the chance to she certainly wanted me to be able to solve math problems and whatever else came with that you know she she would be fine with do you want your kids to become mathematicians um you know there's the like my kids can do whatever they want they can do whatever they want when they grow up they can be topologists or geometers or number theorists like anything i'm a very i'm a free range dad you know i i want them to have the skill there's like the skill of being able to solve problems doesn't have to be math problems doesn't have to be there's when you write a wonderful work of fiction there's a lot of problems that have to be solved and there's a skill to sitting down and having the uh willpower to think for long periods of time and and upload lots of different things into your head and have them interweave with each other so all of the complex things that modern life involves one very very good way i think to train those things very early on is through the patterns that are afforded by mathematics uh not not what we teach in school but uh real mathematics well this cuts actually you made just a very brief side comment there which was how the soviet system was in a lot of ways ahead of the us system how does that look different or how does learning math and russia today look different than what most um like western european or american students would think of it today i have no idea back then you know i only finished first grade in russia so uh that's what i can speak to with as much authority as someone who was seven at the time and um you know and and i'm pulling back however many years uh to get there but the experience was that being good at math was something that was like really important you would not if you were grown up say oh my gosh math that's that's hard stuff i never really got the hang of that like that was like you know oh reading i'm who reads for fun these days right nobody you know it's like oh you're kind of an idiot you know uh whereas here it's culturally it's almost shameful to say that you're good at math it's like that's not something you should be proud of you should hide that if it's really true if there's a cultural answer which is like how valued it is um presumably that might manifest into a more tactical answer about what specifically you do with your students once you value it for example is it the case my understanding is that with russian math exposition problems actually play a much more central role in the way that books are written or you'll often have like books that are dedicated that are just written as a sequence of problems in a way that's not really as common among books written by american mathematicians one is that a true impression and then are there other ways that um this kind of cultural emphasis on the subject translates to a different way that it's learned not just a different emphasis on its importance there are lots of books in the u.s also that introduce things as problems maybe they're translations or something but you know there's this great art of problem solving uh beast academy series for for kids so these things exist of course there's the whole martin gardner for for the generation before uh ours you need both you need problems to excite you and then you need the skills that we teach in school which the whole purpose of those skills is to make thinking obsolete like what we teach in school is things like you know here's the algorithm for addition and subtraction multiplication and you have to get those well how should i put this politely um well so so here's several models model a you have to learn how to add subtract multiply and divide and you have to know these operations with you know arbitrary digits and precision and decimals and fractions and all this stuff so that when someone says two-thirds divided by four-ninths you're not sitting there going okay two-thirds if i think of like the unit and a third of the way and two-thirds of the way and then how many four ninths will go into that two-thirds if you actually understand what fraction division is i mean ideally you would also understand what fraction division is trying to accomplish and be able to answer it instantly but to answer it instantly you have to drill and so it allows you to free your brain to think about the hard things once things become easy in other words if two-thirds divided by four-ninths doesn't take any thought whatsoever you just spit out the answer because you know how to do you know what the process is that frees your mind to now it's uh 2x over 3y divided by 4 z over 9 x whatever and all of a sudden someone who had trouble with dividing fractions now there's algebra involved too like you got to be kidding me how am i ever going to going to get this so we need the fluency of being able to just do the operations we also need the understanding of why those are the operations and then we need free discovery and that's what problems are so problems are you don't have tools you got to make up your own tools you got to make sense of this thing so all of those things are important it sounds like you're implying like the american system or maybe a lot of other systems over emphasize the fluency component of it potentially at the expense of the deeper understanding and there's a risk that by in the service of trying to make things easier for people by having them just drill on their multiplication tables or the fraction division they lose sight of what it's actually supposed to mean i get the impression that then a lot of students kind of react against that or let's say you're enthusiastic about math by the time you go into an undergrad you want to like shake free of the shackles of rote stuff and really deeply understand why something's true and with each proof you come across you're like i really want to understand why do you think that there's a risk or have you seen in any of your students who kind of push too hard to try to deeply understand everything while they're calculating it and don't don't drill the way that they should don't drill the way that we do with elementary school students more specifically are there things that you as a research mathematician find make you better at your job because you just did something extremely rote to drill on them like i don't contour integrals or something like that where it is it is not reflective of deep understanding when you're doing the calculations but that's a that's an important part of actually moving forward yeah um great questions first of all i would say these days as far as i can tell from talking to teachers at mo math and and so on it's almost gone the the flip it's almost that we're we're putting uh understanding like what does it actually mean to divide by fractions over the fluency of actually doing it again you need both you can't you can't have just if before it was just drill and nobody understood what they were doing and then you switch to okay everyone should understand what they're doing but no one can solve these problems in a reasonable amount of time so you don't get any fluency with it neither of those things is actually going to help for developing you know as early on math really is like the trunk of the tree of course later it it becomes a tree but at first it's this trunk you have to build up uh you know understanding of numbers before you build up algebra before you build up geometry trigonometry and so on so we've almost had a backlash to the the drill and kill whatever that whatever that's called to uh understand everything and understand it from 10 different points of view even if you understand it from two different points of view no you have to understand the other eight also and you're going to be quiz done on it and you can't do your favorite uh algorithm for multiplying numbers you have to use the algorithm that i say like there's there's all this stuff that's happened as far as what it what it feels like at the at the research end i definitely have experiences where i think i understand something and then someone asks me a question and i have to sit there and calculate and i realize i'm not that good at calculating these things so then i go and i calculate 100 of them and i realize i have a much better theoretical understanding of what's going on because i can see all kinds of patterns that i didn't see before as a result of doing the problems and sort of getting that spidey sense that intuition for how it's going to come out so they really play into each other it's like the more intuition you have the easier it'll be to do a calculation and the more calculations you've done the more intuition you build on a given day when you're like you're only going to do research that day that is your one goal you're not teaching you're not engaged with outreach it's just i want to discover new things how many hours are going to go towards that kind of rote drilling on the things is it 1 of your time 50 of your time depends on the problem depends on the problem depends on the day a lot of my time is spent in collaboration so these days i'm on zoom you know 10 hours a day and i'm talking to a collaborator and we're discussing some problem that we're worried about and uh you know we'll just sit down and compute together and when there and we'll have different strengths and so if if one of us is stronger in xyz then we start like i was just having something uh a discussion with uh some friends of mine they're better at reading cocksetter diagrams and i'm better at turning those coccidor diagrams into spherical configurations and so like we would constantly race against each other not not race but like we're trying to understand what's really what's actually going on here is this this finite volume quotient or an infinite volume quotient what does the space actually look like and they start counting they have ways of doing this by counting points in a cocksetter diagram and they're sort of racing it that way meanwhile i'm over here drawing a bunch of circles on my online and then i say look it's going to be this because of this and they say wait a second we see it from this point of view and so they're faster at that calculation i'm faster at this calculation both of us come together and realize we're saying the same thing in a different language and then i learn their language and they learn mine and we continue building our intuition of what's going on so so that's the kind of example i'm not drilling for the sake of drilling i'm i'm sort of uh when i'm drilling it's because i'm looking for something like i want this behavior and i think it should happen here no it's not quite happening the way i thought it would maybe it would happen over there that has one good feature but another bad feature and so i keep looking and so on and that looking is is the practice it's not it's not a purposeful practice it's not i'm gonna sit down and do a hundred uh cocksetter diagram calculations today it's the um wait how does this go where is it you know i'm sort of it's like walking around you know the wiles like to say he was walking around the room blind and eventually he found the light switch but he already knew where a lot of things were by by feeling and so uh yeah you get faster and faster at feeling you get faster at walking and not falling over and not you know having some major issue in mathematics because you computed something and it turned out that your computation wasn't right so when you're thinking of mapping that uh version of all of this at the highest level of this dichotomy between the uh the rote just to get it in your bones and then the understanding down to say like teaching your kids because you want them to understand this stuff does that mean that instead of a multiplication worksheet that has something that is a little misdirected you try to say well you do have to drill all this stuff in some capacity but it should be in the service of other problems or is there a role for that worksheet that's just it's going to be what it is it's not entirely fun but this is going to be a much more efficient way to learn the multiplication table than trying to do it in context a bunch of times so far with them i really try to take advantage of those rides to vermont four hours in the car where they have nothing to do i have nothing to do so you know we'll play uh the times two game i start two times two four okay back to me eight your turn 16. and it's like you know we'll do however many of those and it's just it's just kind of fun it's just uh you know killing time and um so so that has both the the game aspect to it and the we're actually getting a whole lot of practice in um i'm getting to the age where he i just need to stick a worksheet under his nose and be like just do these these problems because because he's not seeing them in school in the way that i would want and so i have to that resonates so much actually so while i don't have kids um one of my fondest memories in the last month or two was uh being on a car ride with my niece who uh is just barely able to count and trying to teach her addition and it's like this heavily social thing where i'm like holding up some fingers and saying like if i have two on this hand and one on this hand how many are there and then asking if she can do it without me holding up my hands and what struck me the most was that by the time we got to where we were going and we're supposed to be like having fun we're out on a boat on a lake um she kind of turns she's like can we play the number game and i'm there like heck yeah we can i'm like the fact that it uh is the thing that she wants to like opt for as opposed to playing around in the water and like do you think that at that point part of the reason that the the games with your mom like stuck out or the games on long drive to vermont with your kids like what makes that different from a worksheet is not so much the content but the fact that they're just playing with another human and like making eye contact while they do it and that's a lot more engaging than a blind piece of paper yup and you're a person who's excited about it so they see your excitement and they get excited about it like kids just kids will do see this is the amazing thing about kids and this by kids i mean phd students they don't know what's hard so i give them something which to me looks really hard but everything is hard for them so they just go and then they do stuff and they come back and they show me like oh my god how did you do that like i thought that was gonna be really hard and like oh i don't know i just did it like it wasn't any harder than all the other things are and they said you know you said that those things were easy so i just did this and i thought you would say this was easy do you have any specific stories in mind of a particular uh grad student who you gave a particular problem to and they they shocked you by answering what you didn't think they could almost every one of my students has has uh you know i sort of i have in mind for how it's gonna go but of course i haven't done all the calculations and uh and i tell them to to look at it and i think it's gonna go like this and then almost every time it's it's not exactly that there's some sometimes it's more or less that but there's some some subtleties along the way sometimes there's improvements on what i was suggesting that they discover on their own and uh and sometimes they go much farther than what i suggested so uh i've had instances where i said you know yeah you did this case now the next thing would be to do you know the induction and do all the cases but i i think that's too hard i don't you know i'm not gonna i'm not gonna discourage you from that but i'm also not gonna tell you go and do this now in order to graduate and then two months later he had the induction it's like okay i guess it wasn't uh you know looking back at it it was doable i didn't think it was going to be but uh there were a couple of things that sort of fall out in the induction that make it doable and he saw them and he did it so it's like fantastic there's your thesis so back to kids they also don't know what's hard you know even addition like i'm doing this now with with my five-year-old there's like five different kinds of addition the first is what you just said where there's two numbers that are less than five because you can put one of the numbers on one hand and the other number on the other hand and then all of a sudden you get two numbers but their sum is less than 10. so like six plus three so you can do six plus three and you can still get there but then you know how do you do six plus six like you run out of fingers so when i do six plus three i'll show them okay now you remember you didn't have to actually count six you already have the six so just put just hold up three and now count six seven eight nine and then well that means you can do six plus uh whatever is the seven because you can do six seven eight nine ten eleven twelve thirteen right so there's like five different steps in learning to count on your fingers that go from two numbers less than five the sum is less than ten and then starting to to skip the first count and go straight to the seconds count which means you can get up to 20. you know two numbers less than ten you can count and then you can you can add anything uh less than 10 to anything and so you sort of build up over over months of of playing these games you can build up you know all kinds of uh accounting with really really little kids who are not like uh sometimes you know like if a teacher or or uh some somebody uh on the ski lift or whatever overhears uh something that we're talking about and like oh my god this kid's really smart it's like no he's just done a hundred hours of math in the last you know there's an awful lot more hours of math that he's done than a lot of other kids his age and that's he's not smarter it's just time it's like it's like saying uh somebody's a much better skier than somebody else who's only been on the slopes for six minutes and somebody else has been there for six years like yeah time you know these are skills it's a skill can i shift gears a little bit to talk about outreach sure so you're a lot more active than the typical mathematician when it comes to trying to communicate your fields you're uh the editor-in-chief of experimental mathematics at the mo math museum you're on the advisory board of quanta um do you think more mathematicians should be more engaged in those ways or is it something that um is fitting for you but it's better if others kind of just stay focused on the core research work that they're doing and teaching i think everybody should do whatever they think they're good at this is not something i ever thought i would be good at i i always liked lecturing i always enjoyed talking to students i always uh cared about making sure that um people understood what i was doing so i would see professors who you know they're lecturing at the blackboard uh they never turn around they never actually talk to you they never ask you why is this the definition they just they just tell you there's the definition and this this goes like that and that's it end of story um i very fortunately had a lot of other professors who said what should the definition be what's the right idea here let's try out a couple definitions and i'll show you why this one doesn't work and why that one doesn't work and then we scratch our heads and together we come up with what ends up being the right definition and then there's no there's nothing to memorize not that memorizing that uh uh you know an archimedean field that satisfies the triangle inequality and it there's nothing uh you just know what it is right you the object already manifests itself inside of you and so uh you know how to how to use it how to like where it came from well maybe i should back up and say when we say editor in uh chief of experimental mathematics at mo math um what does that actually mean it's before so i think about bringing in lecturing what do you think those are two different oh those are two different things one is being editor in chief of experimental mathematics it's a it's a pretty uh so this is a completely standard research uh thing to do for a researcher to be an editor or editor-in-chief of a you know a major journal and the nice thing about experimental math the thing that i really love about experiment about the journal is uh it's a broad journal it's there's no definition of what what that means and so we get results from geometry topology algebra number theory discrete math uh pdes you name it you know there's there's a so we get to see sort of my my favorite experimental math paper is something that discovers a new phenomenon numerically or or theoretically but they but we don't know how to prove that the phenomenon is what it is it's been identified it's been sort of studied uh by models or by heuristics or by special examples and so on uh sometimes with a computer but definitely there are lots of papers that have no computer component and um and if someone could actually prove the things that they're seeing that would be an animal's paper so i think of uh something appears in our in our journal and then 10 years later i want it to be an animal's paper if the problem actually finds the tools so these are problems that don't yet have tools like there are lots of fields and fields exist because people develop tools that solve general classes of problems and then someone discovers some new phenomenon or some new uh idea that doesn't yet have the right tools maybe maybe they just maybe the tools exist but the people working in that field aren't aware of the tools and so this is a this is like the fertile you know i'd like to think of it as new orleans in in the time of you know the the birth of jazz and something like this where there's there's lots of different types of music all happening at the same time and they're all sort of listening to each other and so so that part of it is i wouldn't consider that outreach at all that's just the standard fair for a research mathematician um what i'm doing at mo math is they have this uh uh visiting professorship that they have uh that they started three years ago so monjol bhargava was the first one and pete winkler was last year's and uh next year it'll be steve strogatz and i got to do it this year so so that's an that's an outreach thing that's a i'm taking a sabbatical and um doing lots of outreach that's a mildly embarrassing um conflation on my end in terms of packing not those two together so i mean it sounds like the experimental mathematics paper is almost like or journal is a systematization of the phenomenon you were describing where as an individual to do a bunch of calculations to figure out the broader tools like this is something where that's happening where a pile of people do a bunch of calculations and a different pile of people later might have to find the tools exactly that's the hope yeah yeah this should be the place i mean it's funny when her journal was first started 30 years ago it was it was almost a laughing stock it was like people said oh yeah the journal of unapproved conjectures or something or or un you know unsolved problems it was it was like that's not what you publish you don't you don't show someone your dirty underwear you know it was like yeah i don't want to see your calculations give me a theorem you know and uh it's it's funny because that's not how so gauss uh dirichlet riemann these were all experimental mathematicians by by today's standards uh some hid their um calculations more than others uh we didn't learn until 100 years after riemann died that he actually computed the first bunch of the zeros of the zeta function by hand oh wow you can put those in his paper um but but they're there they're do you think there's a little bit of a resurgence in uh experimental math these days is there more appreciation for the fact that that's what progress can look like and sometimes even a necessary step on the way or or even if there has been like how much how much room is there to go between where the current um reputation that a good experimental math results gets and like what it should be based on the significance it represents i think these days based on the journals based on the papers we're getting in the journal and uh the people that we're getting them from i mean the top people are all submitting to to the journal and uh so you know the results that are coming out like i'm really really proud of the things that are that are coming out these days because uh people see the journal as a home for really new ideas i mean what we value more than anything is ideas and so some ideas lead to the final proof or the resolution of some long-standing problem fantastic that goes into animals and venciones jams whatever if if the idea is so new that there's not even a there yet like there's definitely something there there's there's some interesting new phenomenon no one has understood no one is processed uh how do you get at it i don't know nobody knows that should go to experimental math yet again i i feel a poll that maybe i shouldn't resist to just ask you more about about the research side of things but for the sake of scratching the itch that i know i have i like in the actual outreach um let's just focus on let's say like the role that you play at quanta um as an advisor first of all is that the right summary you're kind of on the advisory board for the magazine yeah there's a scientific board and really we're just the feelers i mean they have a lot of their own feelers these days uh you know a lot of the staff and the staff writers and the and the uh contributing writers they're like embedding themselves they're going to overwolf and spending a week just hanging around with mathematicians they go to the institute for advanced study and hang out for a conference and talk to people so they almost don't need us but uh there will still be things that in that i hear uh just through the grapevine that uh i think oh that's kind of a big deal we should really cover that and then i i call up one of the editors and say hey we should really you know look this is something you guys should consider covering that's basically the extent of and sometimes i'll say a little bit more about why i think there's an interesting result some of the people suggest people that they should to talk to about it that kind of thing do you feel like any part of your role is to um be a little bit of a gatekeeper on the final quality of things and if there's a risk that uh the math would be misrepresented that it's important to have a set of eyes from an expert kind of flagging the um if not misleading things the uh the things that would otherwise make experts kind of roll their eyes at the way it's presented thankfully i think thankfully i this is not one of my roles so i suggest things to them they go and they run with it i never see anything again and then something comes out um so if it comes out and some mathematicians aren't so happy with it then i get a little bit of hate mail uh which uh i'm happy to you know sort of correspond with people uh when when they have issues with what came out when they have um you know good suggestions that i pass those along for sure being on the board means that they can't cover anything remotely close to the things that i do so some of my close collaborators like hey come on this is a great result why can't we put you know why can't we suggest this to khanna and you know there's a there's the obvious uh conflict of interest there so um well let's say you have collaborators that want you know they've got a result that presumably if they want it in quanta it's because they want it to be read by more than just the um the usual subscribers to journals and and peers but instead like a broader audience what should they do if if not go to you and get it into quantum how do they get their work out there what's the best means of popularization uh actually you you're to blame for this uh for getting me on twitter uh one way of of popularizing your work is to is to get on twitter and to just uh you know write i sort of treat twitter as a as a blog because uh sometimes i'll write you know a pretty long thread trying to describe the ideas uh in a paper in some lay way and and put some some extra pictures that i wouldn't put in a paper or movies or whatever you know all kinds of different ways of communicating with people um yeah well the people that collaborate with me unfortunately uh for the for the time that i'm on the board uh if it's a paper that they've written with me it's it's not gonna get covered um that's well they should go right people with other people i guess they should ditch me as a collaborator um or like write the same sort of threads or write the same sort of things yes it's funny a lot of my work is anyway sort of on one hand interdisciplinary on the other hand sort of away from the mainstream so it's using tools it's sort of bringing together tools from different fields but in a way that uh it it's often more at least what i've been doing more recently is not solving other people's problems it's finding my own problems and sort of developing fields around that so uh so when you're doing that kind of thing you have to explain to other people why you care about those things because otherwise you're not going to be able to get anything into a good journal because they have to find a referee and a referee has to say oh yeah this is interesting work do you find that trying to um motivate it to the public uh is something that actually helps with the core research itself and sharpen your questions or is it a little bit more of a public service to um just make sure that the stuff actually is um getting out there and it frankly wouldn't make you much better at it had you tweeted it or not getting it out to the public does not i think do anything for my research career on the other hand uh writing it in a way that gets it close to being understandable by the public there's a chance that that someone who's reading a paper that they're asked to referee will stumble you know just through google will stumble on a post there and they'll be able to very quickly understand what's going on in a way that they wouldn't from the way math professional math papers are written so i would never write the way i write on twitter in a professional paper and conversely and so uh maybe i don't know since i don't know who's referring my papers uh but me for all i know they'll go and see a twitter thread about it and say oh okay i see what he's doing here okay that's kind of interesting let me let me go read the paper some more if that happens great i i have very little confidence uh that that's actually what's happening it's more for me to um you know one of the things i've been trying to do in mo math is uh bridge the gap between research mathematics and and school-age mathematics because there's there's tons of things that we think about that uh could be explained to kids and could be fun of course they have other things to learn of course there's all kinds of other issues with math education but this is just one place where i thought there would be room for interaction so it sounds like there's kind of two things there one is you know getting it to students or whether that's like young kids or others making it understandable and the other is that the culture of math writing as it is in a formal setting is not conducive to actual understanding sometimes where even if it's um another mathematician if it's from another field like having this outlet that's ostensibly for the public but really like wink wink it's because this is a more understandable way to write it um would that be an accurate kind of summary of what uh what the like ulterior function of some of these posts are whenever we have a colloquium speaker right colloquium speaker is supposed to be speaking to the broad audience in all the faculty not the expert in their field who is the person who likely invited them there they should speak to it to everyone so i always tell them please speak like you're speaking to first-year graduate students that's your target audience think back to first-year graduate students if you can speak to them inevitably they don't uh the first year the fifth year grad students are lost but if they really really make an effort to speak to first-year graduate students there's a chance that the faculty will be able to understand something so it's very hard to remember what other people don't know and it's very hard to remember how hard it is to know to pick up things that you didn't know before you're describing this phenomenon where it's very hard to remember what it's like not to know something and what that means is that you almost have to coach someone to talk to an early student even if the target audience is an expert in an adjacent field um so my impression is this isn't as much of a problem in other technical fields outside of math that like physicists by and large know how to talk to physicists and computer scientists are typically really great at like um presenting their work in an accessible way to others like how did this happen to math what's going wrong the underlying objects in physics are still particles or the cosmos or whatever it is that they're that they're talking about like all physicists can understand the standard model and expansion and the big bang or whatever uh black holes all computer scientists understand runtime algorithms and so on algebraic geometers love to start a talk with you know let's let's begin with an affine scheme over a totally real whatever right they just start with saying these words and they'll even define the nice ones will define the words but just because you define the words doesn't mean that i have the 10 years of intuition to then like give me a calculation that uh is the the most basic instance of what you that's what i try to do when i when i when i speak i try to give the absolute most basic calculation i try to aim for high school if i can you know something that's that's like algebra uh geometry high school algebra high school geometry if i can boil down the thing that i'm saying to something like that or something where uh you know you can really do some numerical computation and show a graph of here's what came out uh something that's a picture something that you can really um get your get your head around even if you've never seen these subjects now for the people that have seen these subjects i maybe look like a fool right because they're like everybody knows this what are you doing why are you wasting my time on the other hand all the times that i've given lectures like that in front of you know esteemed audiences even the experts come up afterwards and they say you know what i knew everything you were saying for the first 20 minutes but it was still nice to see it from that point of view it was still nice to experience it that way so hopefully over time we will move away from the uh i'd like to blame board by key for this i don't know if that's uh an appropriate uh attribution but at some point there developed a culture of everything needs to be the most general possible and you want to look very smart in front of people i would much rather someone understand something that i do then look smart in front of them well you brought up earlier the fact that like using the definition as an ending point rather than a starting point and like that's a that's a very concrete thing that a good lecturer or a good presenter can think about are there any other things that pop into your mind where when you're thinking of just concrete tactics for good explanation tools that you have in your back pocket things that you find yourself thinking about that a colloquium speaker who's not well trained in it might not be thinking about that takes the um the care to communicate better and turns it into an actually higher quality talk for me it's a lot of um uh trial and error so there are some talks that i've given 20 times where the first 10 i tried to present it in this way and it really i could just you know you can tell if you're paying attention that's that's really the thing to uh that i try to do very much is pay attention to what the the crowd is telling me with their body language with their are they leaning in are they are they falling asleep are they writing what i'm saying are they following along um do they look confused do they have their eyebrows like this these are all the all the different things that i'm uh i don't think i'm consciously doing this i'm i i think i'm just talking to people and i want them to respond and i make sure this is the thing especially when i'm teaching i make sure very early on in this like in the first lecture in the first five minutes i ask a question and then i sit in silence [Music] and it's like no no this isn't one of those i ask a question i wait five seconds nobody answers and then i tell you the answer like i actually want to talk to you people i want to know what's in your head and based on that i can adjust the pace at which at which i go that is such on-point advice because it's so hard to do it's such an uncomfortable five seconds oh yeah oh yeah it feels like an hour's gone by right like imagine a room with 500 people and i ask a question and then i just sit there it's like nope i am not breaking the silence one of you has to do it and then we will establish that this is a room in which we have a conversation it's not a lecture yeah i mean it's a lecture but it's the lecture is a conversation it's not me pontificating we're going to get somewhere so i try to do that in all my talks even even at the professional level as i'm speaking as i'm making definitions i you know i regularly stop and say like it's hard to ask them questions because then people don't want to look stupid and and uh be embarrassed there are some places where you can ask a question that you know someone's going to know the answer to it but then other people will sit there and think for a minute while the person is saying should i give it away or not that kind of thing which is by the way why i'm so impressed with guys like you because you're putting these videos out without someone there so i tried this over zoom i tried i gave my course last fall and uh and i ended it and i and there was something that i wanted to do and i was like all right i'll just record an extra hour and put it out there for whoever wants to look at it and i tried like three times to start the thing and there was no one there i'm talking to a black hole and uh there's no response there's no feedback and i just gave up i after like three or four false starts of getting five minutes into it and being like ah this is stupid let me just start again and like this is stupid because there's no one there so i don't know how you do it by the way it is exceedingly hard um well i i will say when it comes to um a three blue and brown video almost always it will be better if i have taught the exact lesson or my exact outline for it to someone in person just the the other day i was you're kind of struggling on a script i want to write about like the unsolvability of the quintic i'm like okay let's get my head clear and just like took an hour with someone to just like sit down over zoom and like write it all out and honestly doing that like three or four more times before actually making the video will definitely make it better and it's uh sort of an easy mistake to just try to record the thing when it comes to but that's scripted so that's like a whole other game what you're describing where you're saying i'm sitting down i'm recording myself um so it's not like i'm writing out what will be said and like producing it later with animations you're just recording in the moment what you're describing resonates highly because when i worked at khan academy i was part of a batch of a lot of other people that had just been hired as the content creators that aren't sal which was a novel thing at that time and universally actually sitting down to record the videos was this very uncomfortable thing and everyone because they could sort of dictate what they were working on just ended up writing a lot of articles instead because that was like this easier thing to do and more comfortable but when you say like okay sit down be in the room like record a thing um it took us all a while to realize that everyone else was exactly as comfortable as uncomfortable as we were but yeah i'm honestly envious when you know you can just be there with a lecture hall in front of you and if it's someone as charismatic as yourself where you can be comfortable in that five second silence that um sets the tone for engagement that will follow thereafter it's a whole other ballgame that someone like me like looks across at that pasture and sees just how green that grass is and says boy wouldn't that be lovely it doesn't take five seconds thereafter once they know that that's what you do here then they just then we're just having a conversation they interrupt me they they you know we're just having a banter i kind of wonder if this is related to another thing that i want to talk about which is the relation between um the work that mathematicians do and adjacent fields where i guess it sometimes seems like what a mathematician ends up being most known for um not not always but in like these peculiar cases it's things that are like kind of outside of math or a little bit trivial like i i found it of note that uh the the letter of acceptance for gauss into the royal society briefly mentioned some of his math and then went on for a long time about how he helped to track down ceres just like problem that astronomers had they lost the asteroid and they needed to find it again um or like terence tao's most cited papers about compressed sensing which i mean it's mathematical but it's it's probably not his like deepest result in the sense of um uh like pure math of the other stuff that he's worked on two questions one do you think it's like still in the culture of math to collaborate with people outside the math department like is this something you yourself do and then i'll hold off on the second second half of this it's not something that i've done so uh right now i'm trying to do some collaboration with the math ed department now that doesn't i don't know if that counts or not i mean um i would be very interested to talk to political scientists economists physicists uh musicians obviously yeah so i did a little bit of this was back in high school i did some like economics and then in college i did a little bit with with nash which was sort of more mostly math with applications to econ or something um but no i've i've been pretty um pretty secluded in uh in the department in that sense and this correct me if i'm wrong this is the norm you ask most members of missions and that would be the answer and it's this bizarre thing because mathematicians are some of the world's best problem solvers it's this filtration for people who are really interested in pure puzzles and highly capable at coming up with the creative insights to solve them and it feels like we've reached the strange societal equilibrium where perhaps not because anyone decided this was the best allocation of talent the best problem solvers in the world are all working on math maybe that's the best allocation of these kinds of problem solvers like maybe that is actually where they should go because you're pairing up the hardest problems with the hardest people and and to even like frame them as the hardest problems or to frame them as the best problem solvers i get that there's like that's wrong that's sort of an elitist thing or it's like a very purist thing but there is a way that like if you just stick peter schultz on a problem he's probably gonna solve whatever it is even if it wasn't necessarily in math or same with terence tower probably yourself and many others what if like if it was the case that i don't know nsf grants all came with a clause that says like 10 of your time or 10 of your papers has to be as part of a field that's not math it has to be part of an act of collaboration elsewhere do you think the world would be better would we actually be like a little bit farther along decades from now as a society if this is what we started to do with this absurd pool of talent that's going towards a weirdly specific set of questions i was about to say the nsf forcing people to do something is will rarely come to something good i would say on the other hand the way i got into any of this outreach stuff is that the nsf forced me to um i got something called a career grant and uh this is one of these nsf you know some fancy junior professorship uh grants and um one of the stipulations is that you're supposed to do some kind of outreach component to it that's more than just uh you know your broader impact so i'm gonna advise some grad students and maybe organize a conference or something no it has to be something more substantial this was when i was at yale and so we uh we had this series that was running monthly or something where we would bring in speakers for um you know school age uh kids in the new haven area and yada yada it was a some nice thing and then um i moved from from yellow to rutgers and i still had this grant and i didn't know what i was what i was going to do with it and i was thinking oh god am i going to start another program like this this seems like a lot of work and uh but mo math was around the corner and so i got in touch with them and said listen i have to do something for my grant can i uh can i just show up there and uh give a talk or something and they said who the hell are you but but then eventually uh you know this blossomed into what what it is now so thanks to the nsf for forcing me to do a little bit of outreach because it turns out i actually like it independent of whether the nsf strong arm is the right mechanism or not and maybe there's another mechanism would this like 10 clause to what it means to be a mathematician you know should we do that would that make the world better with mathematicians you you can't force them to think about anything we're so uh at least some of us i don't know maybe some people are better at this than than i am wherever it is that my mind goes i try not to restrict it i try to let it go there and and if these are the questions that i become interested in if nobody else is interested in them too bad it's still something that that i care about but presumably where your mind goes is a function of pretty deterministic like who you're with i mean a claude shannon type character being in bell labs like thinking about very concrete signals problems probably has no coincidence within doing this like important thing and when he was given more freedom to just say we're not going to restrict you like do whatever you want like you have the perfect career now because of how famous you are um just knocking over some lights over here because i'm so passionate uh he like i mean i don't want to say no significant work came later but like he spent a lot of time on things that didn't necessarily matter and maybe a little bit of the same forcing functions that he faced earlier in his career would have taken that like strong sense of ingenuity and genius and like creativity in like a more positive direction so like you definitely can determine what mathematicians think about with the right uh dials i don't know if you can force it in the sense of saying like we want biology results by mandate we're telling you like please work on biology and then do nothing about it but if you said um you have to work on a biology paper or something you have to like be around these people and just hear the problems they're working on surely some interesting problems would come up that you're like in bioinformatics or thinking about like improvements to the blast algorithm and you'd be like oh actually that's a highly this is like it's exactly the sort of patterns that i'm into but wouldn't have been thinking about if you were sitting in algebraic geometry colloquiums instead yeah um we should definitely have more cross fertilization and more colloquia where uh you know not a department colloquium but a university colloquium where where people have so this these exist at a lot of places um if they exist at rutgers i haven't been attending uh which is my fault not not theirs um yeah um well i'll give you one anecdote uh in this direction which is that uh i was at the institute for advanced study at the beginning of the year they always have these like 10 minute short talks by the junior faculty so that people learn who they are you know these are people fresh out of their phd and there's one great talk by a guy who explained this problem with cell towers and he said you see the cell tower it's receiving several signals uh from from several different phones and each of the phones is traveling so there's uh dilation there's there's a there's time uh you know from when the signal started to when it arrived they're also moving at different uh speeds and and the signal uh there's all these there's all these issues and there's this amazing uh representation theoretic uh process that he came up with that will um disambiguate the the signals and uh and and peter sarniac in the back of the room said oh great so did you sell this to at t like they should you know change all of their code and run your your algorithm and he said well it's funny i i did mention it to them and they said oh yes we know this problem we put a second tower and then that's it we just put a second tower it's cheaper to put a second tower than to change the the hard wiring on on all of the towers and so so that's sort of a lesson in mathematicians dreaming up these amazing ways that that solve all of life's problems and then in reality they actually don't do very much at all i mean it's mathematicians love knowing what the game is but life uh is much more interesting than that you solve one game you you inevitably there's there's these you know i don't know if they're unintended consequences or life is more complicated than that this cuts actually to the second half of the question rant uh that i'm asking slash giving which is would it make math better would it steer the direction of the kind of definitions that mathematicians want to come up with or the kind of problems that they think are useful in a healthy direction because of some tether to reality where maybe this one you know would have realized that um this like clever representation theory approach to cell towers um isn't as directly applicable to that problem it seems unlikely that the second tower approach would have resulted in like a useful field of math but maybe that could be like one in a one brick in a large building of things that accumulate towards like instincts for what is useful what is not when it is useful when it's not where then if we had our big pile of some of the world's best problem solvers and we also like slowly tweaked their intuitions in the direction of what makes a problem matter or not is there any chance that that actually makes math itself a little bit better because presumably you've acknowledged that the definitions are an end and not not a beginning presumably the definitions we have right now are not the optimum ones for like what the field of math should look like and that 100 years from now there will be some progress on changing what those are would would this 10 clause or would this like enforcing be a healthy function for that or incidental oh i think it's absolutely i i don't think there's any question about uh whenever math interacts with other fields it finds all kinds of new things to do in math i mean it's happened time and time again right physics was developed calculus was developed for physics and then calculus took on its own life and then uh you know these things sort of dovetail right uh hyperbolic geometry was developed for pure thought and then einstein needed it in general relativity physics is a very common place where we get our our problems from but there's all kinds of other you know there's lots of math biology there's big data there's uh machine learning there's all kinds of things where mathematicians should be getting involved and and do i uh so certainly in our department uh some departments some math departments have a separation there's a pure math department and there's an applied math department uh at rutgers we're i don't know if we're one of the few we're one of the some departments where applied and pure in the same department and so actually we should be talking to each other uh especially to the people who we have people who are in math and cs we have people who are in math and electrical engineering uh math and biology and so on so there are people uh doing it i'm not one of them and it's not because i don't want to be like i know now how to do a little bit of outreach i know how to do a little bit of research and i'm so busy with those things but it would be interesting to learn about uh the other things and um if the opportunity came up then uh i would try to you know make time for it and see would see what i could what i could do with it let's say that three years from now i find myself talking to you and you're like hey you know what i actually did end up doing this totally unexpected collaboration in this non-mathematical field right now what's your like credence level on what's most like what that field would likely be so i don't know if this counts uh but i'm starting to play with this lean theorem prover interactive theorem prover so this is something that i never thought i would be involved in just like when i started learning math i never thought i would have to learn tech shop you know latex so this is just something that you don't know at all in high school at least i didn't and then in college you're like oh wow everybody's handing in their homework like nicely typed up and i'm still chicken scratching it i held out for as long as i could and eventually i had to learn tech because everybody knew tech and it was kind of embarrassing not to know tech that might be the case for lean in 20 years it might be the journal so we're doing an experiment now in experimental math which is an experiment in the journal process uh so we had a special issue that we're still in the process of uh finishing up the referee reports we have a special issue on interactive theorem proving whether it be lean or one of the other systems uh isabel one of these um uh interactive theorem provers to just to understand what's out there and and and how they can be used and maybe at some point they'll be hooked into each other so they'll they'll talk to each other but uh at the moment lean is having this absolutely uh exponential explosion in activity to the point that uh just just this week there was a nature article about so peter schulze and a collaborator um had this you know a standard kind of big massive work hundreds of page pages thousands of pages maybe and its notes and um and schultz has just said i'm not sure this is right yes i wrote all of it it all sits in my head but the argument is so complicated and and he said he's had this experience before where a number i think he said one of his imo solutions was marked correct six out of six or maybe it was a sixth problem ten out of ten whatever uh and then later he discovered that there was an error with the solution that of course he didn't catch but neither did the people uh refereeing the answers so so he said hey maybe this is a way to really learn if every single t has been crossed and i dotted and uh he put it out there and lean exploded and less than six months later the the main theorem that he was worried about is now completely formalized and this is like cutting edge research in the most abstract sense do you do you feel tempted to look at some of your old results or some of the new results you're doing and figure out what it takes to leanify them so uh yes and no i think at the moment any attempt to lean so one thing lean doesn't have right now is complex analysis and uh you know complex analysis is all about drawing these nice pictures and taking contour integrals and uh and it's i don't think it would be that hard to do of course that's my naive optimism people haven't done it but but people should i have ideas about how to do it you would need that you would need some fourier analysis you need you need a bunch more things to be there for me to be able to start even stating my theorems but i have definitely now that i've been been doing a little bit uh in there i've definitely had the experience where i get to some exponential sum these are things that arise in in my work all the time you have a bunch of things of absolute value one and you're adding them up and somehow by some miracle that they have to cancel out right there i need to find cancellation in an exponential cell and uh there's a million techniques for this but there's not really a million there's there's more like five or six that you can use in 10 different ways on all of these different variables and so if mathematics worked like a lab and i had grad students who i could assign these partial tasks then i would say okay you try you know this this and this and you over there you try xyz and you try abc and one of you i'm sure will find something that cancels that's not how we work that's uh the amount of time that it would take me to train them to do xyz or abc they'd be off doing their own thesis anyway and anyway that's not what we have them them do we have them do their own their own research i would like oh my god if there was some way for me to automate this to tell a computer here's the tactic just apply kashi schwartz come on one of these things has got to buy some standard arguments you know a there's a in process and a b process whatever those are uh one of these things has got to get me the cancellation that i want and then six weeks of calculating later yes it did now if i could code as it sounds like you you see it as a means of like actually making you think faster and a better researcher more so than a fact checker um yes yes just like mathematica that's what i use mathematica for now i'm like oh is this even true i don't know code it up five minutes later computer says no it's not true oh okay so what is true okay uh that means good thing i wasn't trying to prove that conjecture because it's false i've lost years of trying to prove conjectures that ended up being false by eventually finding a counter example and then correcting what the conjecture should be and then proving the corrected conjecture so um a little bit earlier you were talking about the benefits of going and calculating 100 things yourself and doing what might be a little bit rote but it actually builds your intuitions in a very important way that some modern math education is like not appreciating is there any risk that um a research mathematician leaning heavily on lean and on mathematica and like this collection as a suite of tools robs themselves of the opportunity to build the intuitions that would come from those six weeks of calculation that's a great question i've had some experiences with lean where because lean is so much like a video game when you start when you start coding you sort of you say this is the theorem i want to prove and then it tells you okay here's what you know and here's what you need to get to and it sort of starts getting automated like you're like okay i know how to help this i know what what things to do to get to the next step i'm no longer actually thinking about the mathematics i'm just playing the video game it's like you know this is the joystick and you have to jump up at the right time and then mario will will go underground and get some coins or whatever if it starts feeling like that it's like i'm i just did i just proved this theorem but i have no idea what i did i just did these these lean tactics that i know help uh one after the other break down the the goal to something simpler and then at the end i had something that i was trying to improve which was something that i already had established and then i just say there i'm done does that mean this is something that you would encourage any let's say there's an undergrad who in the same way you needed to learn tech they're like wow i got to get on top of this lean they start using lean in the same way they start using mathematica to like do their work better would you emphasize that they should uh supplement that with a little bit of old school approach or can we get the best of both worlds we should absolutely get the best of both worlds and the next time i teach real analysis which is the first time you know undergraduate real analysis with epsilons and deltas where the first time that they see that they're very confused first of all we don't teach it right we teach it as some abstract nonsense it's engineering right how much precision do you want the answer to be how much is it going to cost if you can always find a function if you have to go out far enough in your sequence so that you're within epsilon of whatever result you want that's that's all epsilon and deltas are but those are particularly amenable to leanifying to making a lean you make a lean program follow that lean program and have it verify that your proof is correct then go and write out that proof almost verbatim you can just read it off of the lien write it out as a proof there's your homework's done and you know for a fact that you're going to get 100 on it because lean said yes you solved it i remember first semester freshman year of college i took this epsilon delta class with uh eli stein the great uh stein and uh and i remember thinking that his ta who's this uh great guy rami shikarchi at out of time he's such a jerk he's constantly taking these little nitpicky points off uh of my homeworks which are basically right this you know some stupid little thing and he like took a couple points off and then i get to grad school and i'm moving into my my office in grad school and i have my all my old papers and i just started you know for fun leafing through them i don't know if you ever look back at the stuff you wrote freshman year and i look at him like what the hell was i writing oh my god this is garbage this is complete the epsilons and deltas are backwards you can't have the epsilons and deltas be backwards and he only took off three points that would have taken off you know nine or something like romney was so nice so if lean was around back then boy would it have straightened me out it's actually very inspiring to me because i feel one of the common pieces of advice that i'll give to someone if they like want to learn more math is learn programming because um one there's some crossover in terms of the math is well motivated by the programming tasks but more than that you have this um different kind of rigor to your thought that comes about when uh it's you against the compiler and it's not just like i put this out i'm pretty sure that it's right and one that like builds a kind of comfort with being wrong because you have to acknowledge that you are wrong in order to progress so you can't be stubborn and say ah the grader was a little harsh on me and for a while i just thought okay that has to be the supplemental thing like you take some math classes you also take some computer science classes and the crossover in terms of what it does to your um your personality and your maturity with respect to rigorous problem solving hopefully carries but if it can just be packaged in one like the the way the problem sets are even assigned or the way that they're graded is um with something like lean i i mean honestly that feels a little bit game-changing in terms of saving time you don't have to learn all that programming as well to get the same sort of crossover and making a little bit more accessible where you can be in this playful environment to learn about epsilons and deltas and you don't have to play this game with a like another human being that you know maybe they're judgmental maybe you're just uncomfortable with that maybe you don't have like the confidence at that point to so regularly see yourself have like points marked off from this human who you see every day in your class absolutely now now take this all the way to the research level and take it all the way to the journals half of what i do is try to chase down referees to tell me if they've checked that a paper is correct right someone has to sit there and read the paper and really understand what's going on and check the calculations and verify at the end of the day yes that you know to the best of my understanding what they've done here is is correct some very small part of it is uh i think this is doable i think this is interesting this has you know potential to lead to some kind of breakthrough like that's the kind of uh thing that you can look at the paper and in in an hour you should be able to decide whether or not it's really making that breakthrough and then it's to take you three to six months to really go through the details and and try to flesh everything out imagine that all you had to do was read if the definitions were entered correctly read if the theorems state what what they're supposed to state and if the if the theorem and the discussion and so on is interesting and then click a button does it compile yes it compiles okay the proof's good so this was the experiment we were running in experimental math like you know is this in 20 years how journals will work and is it well who knows we'll see 20 years from now what what these things look like well if you were to put a probability to it you're you know a betting man i offer you some stakes on this outcome uh what's the right odds here yeah 20 years there will definitely be journals that 20 years from now will not accept a handwritten proof will all the top journals require a formalized proof i don't know but uh i could definitely see journals which just say if this isn't formalized we don't consider it mathematics anymore so any discussion of lean uh i like i have to ask this question it's almost like a moral obligation to ask the question if you're talking to a mathematician about lean which is will your job get automated not just because we can proof check but also because like in parallel we have ai doing better and better at math like tasks like the game of go or things of that sort you know how are you feeling about your job security right now um i'm feeling really good and i'll tell you why um look at chess now okay chess is a game computers have been better than people for a long time more recently with machine learning they've gotten even better than the computers that we wrote ourselves to beat us the computer that knew nothing beats that computer program right uh stockfish was defeated by the alpha alpha zero and the amazing the most amazing thing there perhaps i think they did this with go first where they put in a million human games and then they had it learned from there and then the next iteration they didn't put any games in they just had it play itself at random and it did better than the one that it learned on human games so that like really blows your mind but anyway um what we want as mathematicians isn't just the answer we want we want a discussion we want to understand so let's say i hit a i hit a button and lean comes up with a proof of the riemann boxes it's like well that's great uh what is it explain it what's actually going on like i it's you know if someone oracle tells me the riemann hypothesis is true uh okay why like i want to know why so it's not enough for the computer to tell me yes it's true go on with your business it's the why so so that is part of what a good paper in the lean world would do it would explain like not just it would do the thing that we do when we give talks when you give a talk you say uh here's what here's the interesting part about the theorem here's how the proof goes except there's a million technical details that make the proof not go like this at all uh you have to break things up into little pieces and on each piece it's not quite right but there's a piece of it that does what i just said but then there's another piece they have to do something else too and you could just say that and then if you want to see the details the code is exactly what what does everything that i just said um so it would allow you to get that high level of explanation without needing to then write down all the details of the proof because the details will be in the code i mean you will have written down the details of the proof just not there just not in the paper where where the human being has to sit and slog through so in a world of entirely lenified math do the mathematicians start to become bloggers and youtubers in where they spend their time i don't think so i think even if lean even if you say solve this by some tactic and lean does if it's something that you don't know how to do you're not satisfied with that like right now what the what tactics do is they do all the stuff that you're like oh come on it's like a plus b equals b plus a yes there's a there's a thing that lets you commute addition there's a thing that lets you turn this inequality and you multiply both sides by two and that's also an inequality like i don't want to do that that's that is what it takes to machine code a proof and that is the thing that mathematicians get really turned off by because you have to do there you have to sit there and do this like god-awful crap and and see just how many steps it takes uh what was it uh um tarsky and uh tarsky and somebody wrote this book where they really went from the first axiom from from first principles and by page 500 they were able to prove one plus one equals two and it was like this you know it was it was really a joke it was really a joke because it's like if that's what it takes to do this no one's doing it we're not doing this like you'll never get anywhere if that's what you're if that's what math looks like but now the computer will do those 500 pages under the hood it'll really check everything and uh and you get to just say one plus one equals two buy some tactic so that's the kind of tactic that we want it's skipping all the steps like don't don't waste my time if it's something where there's something deep happening i want to understand it right i don't need to sorry to interrupt do you think the journals reward that level of like truly explaining why something's true um rather than just showing that it's true like this paper really successfully um you know i really like timothy chow had this paper that he was talking about forcing and described um described there being like an unsolved expository problem or something to that effect which is not really a concept the spirit of my question is is more that like if at the moment journals value what is true more than like the deeper understanding of why what's the outlet to measure the thing that mathematicians actually want which is understanding why like blogs don't actually feel like a bad way to try to measure that and like this is maybe why um you have like terry tau's blog on what's new that it's it's sort of solving that problem can we understand why more so than what a good paper will really tell you the why we'll have a nice long introduction or an expository paper that accompanies it that will explain the what the why the how the pieces come together and then you know there's like you know maybe section one is the introduction where it tells the story here's here's what we're going to prove here's why here's the context in which this this theorem belongs and so on section two if done right should be here's a high level overview of how the rest of the proof is going to go here are all the ideas that are going into it and then begin the technicalities and all the calculations and so on and then the formal proof so a good paper and referees absolutely react to this a referee will say i you know uh i read the introduction i didn't really understand why they're proving this and then there was no kind of good explanation of a good high level overview of how the proof goes and then i gave up no i don't think you should accept this paper it's not well written so i definitely get that those kinds of reviews i think there already is a system that rewards papers that are written well and explain what they're doing why they're doing it what are some of the ideas in the proof what can someone learn from from reading this paper i put out to the internet the fact that i would be talking to you and asked if people had any specific things that they wanted to know one of them asks what advice would you have for someone who's starting grad school looking to get into langlens what field should they focus on learning deeply how much algebraic geometry should they learn they should find an advisor who is very heavily in that program this is not something if you look at anyone who's made any progress it's because their advisor was someone who was a big name in the field uh this isn't something that you can casually or it's not impossible i suppose but uh it would be extraordinarily it's already extraordinarily difficult the amount of material that you have to learn to uh start to to do anything but uh if you want to do this without having someone who can say don't worry about this part it just goes like this just move on you know use the treat this as a black box like uh without those kinds of hints along the way without an advisor that can really guide you into it it's already such an uphill battle you're making your work a thousand times more more difficult what is what does a fledgling phd student do then if they're actively looking for a good advisor um how should they spend the you know the senior year of their undergrad to tee themselves up for success there oh senior year of undergrad i would still say just take the farthest apart math classes that you can if you've taken a lot of pde take a bunch of algebra if you've if you've been sitting in gower theory but you've never exposed yourself to geometry or differential equations or you know algebraic geometry or something undergrad is a time to get as broad as possible and same thing with you know first year maybe even second year of phd because uh those are the best times if you've never seen a course in in algebra and all of a sudden you need it and you're a fourth year phd student like by then it's too late because they're gonna go too slow you're gonna feel embarrassed to be sitting in a class with a bunch of undergrads but it's also hard to pick it up on your own if uh if someone's not teaching a course on it the thing that i do these days uh is i just teach classes about topics that i don't know and then by the end of the semester i know something about them i think it's very important to know that this is often what professors are doing they're teaching in order to learn it themselves oh yeah but presumably that undergrad they need to decide what programs they're applying to they need to decide specifically which ones they'll go to that highly limits who their advisors can be even if there's freedom to choose once they're in that department and they could transfer but there's higher friction there would i be wrong in inferring that a senior undergrad should be thinking very heavily about a potential future advisor even at that stage even if their main goal at that moment is to not specialize too much you know people arrive in grad school and they think they know exactly what they're going to do and a year later they're doing something completely different so i would say it's more important at least for me it was more important i was not choosing an advisor when i chose a grad school i saw a bunch of grad schools there were some where people are having a really good time but maybe they weren't learning that much there were others where people were learning a ton and it didn't look like they were very happy and columbia just happened to be a place where people were working super hard and learning a ton of math and they went out drinking every night and had a really collegial atmosphere and everybody was uh cheering everybody else on and so that's where i wanted to that's where i ended up going going to grad school so um you know find something that works for you if uh if princeton works for you you know uh at princeton you're basically i think it's changed since since i was there but uh you're sort of on your own and uh it's just assumed that you know how to how to make progress and in year one you you pass your qualifying exam there's no courses no nothing's uh nothing's assigned to you you're sort of dropped in the deep end and uh good luck and i'm sure you'll be fine since you made it to princeton um a lot of other programs so at rutgers we have uh you know sort of required first year courses unless you place out of them with an exam early in this in the semester so we try to make sure that people have a broad foundation on which to than in their second year so a typical grad school experience will be year one taking courses that are at a faster pace at a more uh more depth and more you know more as being asked of you and you're taking like three or four of them which in undergrad usually you're taking one maybe two math classes maybe three or whatever but uh they're still at the undergrad level uh now you're taking three or four graduate classes and they're moving much faster and a lot more is expected of you and this is your full-time job you're being paid to learn mathematics so in year two you start reading books and papers and you start talking to people more and there's no uh homework assignments anymore it's all about just getting ready for your qualifying exams your typically your oral exam and then after that you have an advisor and then you are really reading papers going to conferences and so on so so much can happen between being an undergrad and thinking i'm going to go into the language program and getting to first year and being like actually i really like this romanian geometry course that this professor taught so um i liked topics but already as an undergrad i realized much more than topics i like professors so if there's somebody who like anything john conway was teaching i was going to be signing up for conway's class uh i realized that sort of immediately i had him freshman fall for linear algebra and i never uh knew that math could look like that the fact that yeah having conway as a linear algebra professor that that just feels like that explains so much in terms of follow-on success um well he was my he was my first experience of of college math like i had you know bc calcon as a senior in high school and i showed up thinking i guess i'll learn some multi-variable or i don't know i don't know what you're supposed to do in college and i get to college and it's like okay this is where you're supposed to take this epsilon delta uh thing with stein and linear algebra with conway so it's you know i had no idea how i knew how good i had it but yeah i i right away i was like this these are very special people and and it's a great flavor privilege to learn with them this is very sound advice though to focus on the person more so than the topic because it's so easy you know you ask someone you know what do they study uh what's your favorite part of that right that's almost the framing it's like what's the in favor style questions you're going to land on a subtopic and it seems less common to have as a part of your identity like who your favorite professors are or who you work well with when that probably determines your future a lot more and it determines your relationship with the subject a lot more and find out what the yeah and find out what the graduate students what the other graduate students are that professor are like are they a happy bunch do they help each other out you know if there's someone who has a fourth year and a second year and a first year they're going to talk to each other the second year is going to ask the fourth year questions that they're too embarrassed to bring to the professor maybe there's a postdoc in there too like maybe there's a nice group you know we have like 10 people that are all organized uh without any of the professors being even allowed to attend they organize little uh reading groups and um talk to each other so yeah that's what you really want you want more than the professors of course are important but your peers will be that much more important to have people that you can talk to that you can come to with stupid questions with with things that you would be embarrassed to bring to a professor do you as an advisor try to like foster that sense of community among your grad students and postdocs or do you realize it has to happen organically um it doesn't have to happen organically you can force it you can force it you can sort of uh at the beginning of the semester say hey guys who's running the uh the you know graduate number theory uh seminar this semester and it's like oh we should do that yeah yeah you should do that yeah so who's who's gonna you know who's who's being appointed the uh the chair this semester um and then someone okay i'm chair so i have to take over so i have to get a room i have to you know organize people to give lectures and so on and then it happens so there's definitely uh you know and and uh and having a good seminar making sure that there's people coming every week making sure people go to dinner with the seminar speaker that they get to learn what kinds of things are out there uh if there's a good conference i'll make sure to blast an email to to the group and be like hey this guy there's this conference you guys should really apply yadda yadda so no i think there's a lot that that a professor can do to um grease some of some of those interactions all right last question uh also from the broader internet how well-versed do you keep yourself in branches of math that are outside your own field and how frequently do you find yourself applying or attempting to apply the methods and theorems from those those fields outside your own specialty i guess i probably have had these five year periods that's probably a good way to describe it uh i'm i'm 13 years past 14 years past phd and so you can almost like there's the five year period where i'm developing the things from my thesis then pretty drastically change directions i mean it's all organic but uh yeah i had to learn you know a bunch of other fields and tools and for there was another five year period and the last five year period i'm doing things completely unrelated to the previous five so it always comes for me from teaching either an undergraduate course uh on on the topic that i need to acquaint myself with and so maybe i'll start with an undergraduate course and then the following semester i'll do like a graduate reading course and then the following semester i'll do an actual graduate you know topics course and so by the time a year and a half has gone by i have been telling people these things as if i know them and then now i know them and now i can you know really use them in my own research so um yeah i definitely am constantly trying to every year there's something that i uh there's some new tools that i have now that i didn't have a year ago and things that i understand that i didn't understand a year ago and so just do that every year and see what happens right it's it's all it's all this uh little one percent incremental progress and and over long time scales uh you realize you're picking up tools and you start to be able to make connections that you wouldn't even known to look for in the past because wait a second i knew something about that story uh what if i used that tool to say something here would that have any impact that kind of um well alex i uh just want to thank you immensely for both the insight that you've offered and also the infectious enthusiasm um i'm actually gonna make myself a liar because i want to ask just one final thing but i i don't want you to explain your answer i just want you to offer a single number and then i'll point people to your twitter you know because they should be following you on twitter anyway what is your your best guess like your what probability would you place on the fact that the collapse conjecture is actually false if you were if you had to assign a number you're a betting man what's the probability that it's false i give it 50 50. really wow i give it 50 50. um and like i said there are days i don't want your days that yeah follow you on twitter or i'll point to the appropriate thread maybe you have more context than twitter and i shouldn't hold you off i'll just say some days i wake up and i think i'm going to find a counter example and other days i wake up and i think no that can't possibly be because here's an approach to a proof but of course that won't work very okay the fact that i'm intrigued i hope uh spreads itself to anybody listening to this um so with that i'll call it an end yeah thanks for uh chatting with me it's uh thank you for all the things that you do because the more people like you we have the better chance we have of mathematics uh surviving as something that that people enjoy and not what they see in school and then go home and say oh god i hated that stuff i can't believe i have to do this with my kids now so hats off to you that means a ton coming from you you

Channel: Grant Sanderson

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Id: C-i4q-Xlnis

Length: 84min 23sec (5063 seconds)

Published: Fri Jul 16 2021