 I'm delighted to bring up Colin Rust. He's a portfolio manager here in the city, but more than that, he's a longtime friend of the Museum of Mathematics. Thank you, Colin. Robbie McKeel is a professor of mathematics and a Packard University fellow at Stanford. I've had the privilege and joy of knowing Robbie since we were at college together at the University of Toronto a few years ago. Robbie's problem solving ability was legendary. Some of you may have heard of the Putnam Math Contest, perhaps the hardest and most prestigious math contest there is. Robbie was named a Putnam Fellow and almost unprecedented for consecutive times. Before that, he was twice a gold medalist, once with a perfect score at the International Math Olympiad. Of course, solving contest problems under time pressure is very different from doing research mathematics. There, Robbie has this help, too, making significant contributions to algebraic geometry from classical problems to string theory. Let me just try to give you a tiny flavor of Robbie's research. One of his most celebrated papers is called A Murphy's Law for Algebraic Geometry. It's about these fundamental objects called modularized spaces, which are very abstract, but it's in essence a space of spaces. It's a space where each point is itself a space. What Robbie showed is that there are some kinds of modularized spaces that are very well behaved, but that's only because there's a very good reason. Unless you can identify that good reason, they're going to be as badly behaved as possible. Murphy's Law, what can go wrong, does go wrong at least in modularized spaces. As always, with Robbie's work, it's beautifully written. I studied math, but it's not my area of math. I can't follow the arguments, but at least in any detail, but you still get a very good idea of what it's about and why it's important and it's exciting to read. Let me tell you that as much as I love math, that only represents a rather small fraction of research math papers. Robbie's generous and curious soul has led him to teach and contribute to the community in so many ways. For example, helping launch the proof school in San Francisco, a school for kids who love math, and math overflow, a software tool that empowers mathematicians to help each other with their research problems. Among as many awards, I'll mention only that he won the Chauviné Prize, the top prize for writing that explains mathematics. That was for an article titled, The Mathematics of Doodling. I'm thinking in May of some connection to today's talk. Please join me in welcoming Robbie Vakil. Thanks very much, Colin. I just want to check. Does that sound okay at the back? Excellent. Great. Thank you very much for the chance to be here and to be with you at MoMath. I should say this is my first time I've had a chance to be inside MoMath. I've been in New York many times when MoMath was closed and had been in the little nose marks on the window where I was peering inside, so I'm very happy to have a chance to come. Really, because for me, MoMath is, I think it clearly is not just for me, a center of the mathematical community in the entire world, something which over the decades will just get more and more true now that it's so well-established. I'm really pleased. I feel as though this is one of the homes for my tribe. I should say that much of what I'm going to say today has come from random conversations with people from my tribe from when I was smaller, through people like Colin I met at college, and up to interesting people I get a chance to meet with of all ages today. So what I'd like to do is to tell you about something, tell you about a certain kind of doodle that I did that's really kind of common, and it's going to connect to a great deal of mathematics, and I realize this gradually over time, and in particular there are a lot of things I've noticed, and there are many things I've noticed and have no idea why they're true, and there are many things I've not noticed yet. And so what I'm going to say is that this looks like play that we're just messing around with a doodle, but this really is what nature, what science is about, which is looking at nature, finding patterns in nature and explaining them, and when you do then you can extend them and discover new things. So in particular what I only discovered a little bit later after high school is that mathematics in school for a very good reason is about answering questions, but science and mathematics really is about, to break new ground, it's all about asking questions, and the real skill and the skill that every day I think hard about is how do you ask the right questions, often the question is said the right way, then the answer is really, is in some sense, simpler. The real genius is in looking out at the world and finding things that will tell you about more about the world. So I want to do that, and the things I want to talk about are going to be related to different parts of mathematics, some of it very deep, and much of it is not in my own area and is foreign to me, and I think I'm happy doing that until it's important as a mathematician or as a scientist not to feel tied to your area of expertise. You should be willing to look foolish by trying something new and having ideas which may be well known or wrong, and I think that's sort of thing which often is a hard lesson to learn even in graduate school, of being willing to go outside of your comfort zone and not look smart. Okay, so let me start with the doodle that I, actually maybe I'll do it here, that I would do, and so this doodle I did from when I was very, from really fairly small, I think even from when I was five or six, and so it was the sort of thing where you're sitting somewhere and you start doodling, and I should say that you should all have paper and pencils with you now. Does everyone have get paper and pencil there? So okay, so you're, I want to say you're encouraged to doodle while I'm talking, hopefully you'll be engaged as well, if not it's my fault not yours, but actually doing something with pen and paper, I want to emphasize that this is, it's the same with pacing when you're thinking can be useful, I think it's something good to do if you do it wisely, and in particular what I want throughout this discussion, what I want you to do is to be doodling, maybe some of the ideas I'm saying, having thinking them through, I'm going to ask your opinion on various things, and some people, this is again something which I feel like when I was in high school working with other high school students up through now working I guess up to graduate students, because once they get older they stop listening, it's an idea of you should use paper to write your ideas on and scribble, sometimes people, I guess mathematicians in particular tend to stare off into space and think, and also especially people coming into college tend to be drawing a little bit in the corner of a paper at most, and they try to save paper in some deluded idea that this is somehow going to save the environment, but paper is, I want to emphasize that paper is, it's like an external storage, memory storage device, it's an external part of your brain, if you treat it as part of your brain you want to make as much use of it as possible because it gives you a great deal of leverage, so let me go back to the doodle, and so hopefully you're now happily willing to doodle, and what I would do is I would find some shape on the page and I would start drawing around it as close as I could without touching, and then once you start doing this you notice things, I notice things, you notice different, you'll notice different things, but I have a quick question which is how many people here have done a doodle that's this or essentially like this, can you put your hand in the air, okay so it's a really strong majority if your hand was not in the air you're the weird one, so I did not make this up just to tell this story, this is something which really is sort of out there in nature that you notice it's not reverse engineered, okay good so let me start with the first things I did start thinking about when I was just doing this doodle, and one thing I noticed early on is that when I would do this a lot things would get rounder and rounder or more and more circular, and so I wondered why or what was going on and that's a vague question and it's a good and it's an excellent five-year-old question it's not precise but that's not where questions start, they start by being kind of vague and then you start to try to pin them down and things can and this pinning down can take a long time so I was in my mind for a long time and when I was much older I found out why, so let me go to something which happened long after I was five which is to figure out to say precisely what we're doing and I'm doing this partially to make a point about language which is that I can say this in math speak or even in equation speak but this is just a translation of the English this is a more a translation of the of this English and in some sense we're making things more precise but it's really the same thing and shouldn't be thought of as phasing you so let me say it but don't let it bother you which is I'm thinking that you have something on paper so a plain set x and you're doodling and you're doodling you're trying to go as close as you can to the shape and that's r which I'm thinking of a really small number and then I'm going to doodle around x and so what does that mean it means I've got a shape and I want to include all the points that are close to some point of x that are within distance and most are for some point inside the shape so that's just I want you to be I'm not going to even use this language a whole lot I just want to make clear that this language is not to be frightened of that we have a shape and it's really I'm thinking the inside and then we have a doodle as close as possible and I'm only doodling the outside of the shape and now I have a new question or I reworded it which is that I start with my shape and I doodle and I doodle and I doodle and I keep doing it a long time and it doesn't become more and more circular and so the question is more precise we've defined what this is we have dot dot dots which are completely okay maybe I should say limits are completely we we somehow get the idea that limits are something really scary and you only see in in high school or later but but certainly when you're very small you understand limits you understand in in the sense that when things get to some extreme there's some behavior it's not shocking making them precise is hard but talking about them is is the idea should not be thought of as hard so this is still vague I don't know if we're more circular means the dot dot dots are alarming but it's but it's but it's progress and okay but inside this question now we made it more precise there's this r and I want to ask a natural question even if we can't answer the question we actually thought we cared about or do care about which is how what does that r have to do with anything and here's my here's my my point which is if I let me point out something obvious which is here's my shape x which I really mean everything inside it and I'm going to doodle around it and when I doodle if my doodle is of size two in other words this distance is two and I'm doodling within two of x and your doodle you're more coordinated than I am so you can do a doodle of size one then obviously your doodle is inside my doodle because everything within two units of x but with sorry everything within one unit of x is certainly within two units of x so that's not okay that's fine I'm glad to have said that but what I really want to ask about is if I if you do a doodle twice so you have a another doodle of so if you do a doodle of size one followed by another doodle of size one how does that compare whoops to a doodle of size two and I'm going to ask your opinion on that in a moment so hopefully you can doodle on the page and get some idea and the question so I guess I'll state this in the form of a of something to vote on and so the one one question is so my question is is it is the guy is the double doodle of double small doodle at least as big as the single big doodle or is the single big doodle at least as big as and one of as the double small doodle and again don't let the language hopefully the picture makes it clear and hopefully you're doodling and one and thinking about stuff I guess I want to be clear that you should feel free to ask me questions if I'm being fuzzy or there's things you want to pin me down on or if you want me to repeat something and so I'm getting closer to what I'm going to ask you to take a take a pick between one of these two and so I'm stalling a little bit to give you a chance to collect your thoughts and now I think I am yeah great I think I'm okay so I think I'm ready to ask what your opinion is and I want to be clear that you shouldn't be afraid of being wrong there's nothing wrong with being wrong so long as you're willing to change your mind when faced when when faced with contrary evidence so so so hopefully you have no fear and now I'm ready to take a vote so how many people sit here so no questions before oh sure question you're the back yep are you talking about the surface area in that line or are you talking about the line itself fair question I mean the sure the oh yes actually please always remind me to do that because otherwise it won't be caught so the question was am I talking about the sort of the area the surface area of the shape or the the length of the line or what do I mean okay so what I mean I don't mean the length of the line I do mean the I do mean the area inside but I'm being a little bit you've you've got me on being vague about what I mean as big as so something like this yeah size is that good or area and so I don't mind that it's a fuzzy question but as long as it's yes right I want the okay let me let me you're forcing me to make a decision so let's so let's say you let me know if you prefer something different let's say the actual area inside so the area inside this red not that it looks red versus the area inside the green no I want even the inside everything on these so I want including the inside so everything inside so one of them I have no idea which is red and which is green but I want that's I want to include all that area inside the dude is that is that I don't mind being fuzzy but I don't want to be fuzzy I want to be fuzzy for good reason not for bad reasons and so I don't is that help is that yes or do I think vaguely or do I think wonder about this but you're not you don't throw it in the if you look at the page after it's not a scribble it's not as my summits a scribble scrabble there's something there there's structure and then you your brain does something say oh look and then you look at something and so you wonder stuff so you may wonder different things and this is a and but I should say we're led to this wondering by the roundness like once we know what we're doodling is then you could reasonably wonder about well how does the size matter and then you might be one led to this question as to whether if you are much more coordinated I'm less how will our doodles look different so so it is and then you might try to make it precise and then you might choose perimeter instead of area but so you'll you'll follow something you won't you so you'll have you'll have ideas which won't be these ideas necessarily but they'll be interesting ideas which is kind of the main thing the doodles they are struck right they are struck there's there's structure they don't follow formal rules in general but they're okay so ready oh yes a question good question okay so I'm that's a question for you that when you doodle around x you're taking all the things that are let's say the doodle of size two you're taking all the things that are two centimeters away from x and so something interesting is going to be happening around the corners I agree with you about that are we getting more sure okay great so I had a good idea from here that some of you you're expecting things to get you're getting or you're even certain that they're big circular pieces coming into play great so this thing he says is okay well he's criticizing my drawing but uh but aside from that he's pointing out correctly that this is that when you take things that are a certain distance when you go around the corner you're going to end up having the whole circular like a part of a circle so that's a new that's already incorrect in my picture and we have some more clarity of understanding of what's going on great so I think you're getting closer and closer to feeling qualified to have an opinion yes I think both of them mm-hmm that's a reasonable guess that's a okay let's take a vote and find out who thinks this is the least as big raise your hand who thinks this is at least as big raise your hand okay so and now who did not vote okay you're wrong you're guaranteed so if you haven't if you didn't vote you're guaranteed to be wrong if if you did vote you are actually uh right in fact if you voted twice I'm sure you voted twice I guess that you were right twice are you great you voted twice Chicago style then you could uh I then then you can then you uh you were right both times so uh so maybe I should emphasize that why I want you to have an opinion which and there's a reason why in empirical science when you want to study something and learn some not to say learn something you have an idea you have a hypothesis you don't just do an experiment say I'm gonna mix the red stuff and the green stuff and see what happens you that's that's not a way to zero in on all do you have an opinion and you don't mind being wrong so now even if you only voted for one you think the other wasn't true there's something interesting going on and this leads you forces you to ask more questions and the other thing I want to say is that this is something which you can do the lawn paper by hand but it you can also write a computer program uh or get get a friend who's a graphic designer who's more coordinated to do a bunch of examples and you could actually get data by doing examples so in particular mathematics is even pure mathematics is an empirical science you learn things about the world by going to look at the world not just not just imagining uh not just thinking uh vaguely and when you do that you'll discover an amazing fact that not only are they the same area but in fact I think you even said it what's that the same perimeter to okay and he's magically guessing they're the same perimeter and even the same shape they're exactly the same shape and when you do some examples you'll say if you don't see why and some of you do see why you don't see why you'd say wait something is happening and I want to know what's going on so and if you think something's going on you realize that in fact they're the same shape and what matters about one and one and two is that one plus one is two uh and so in general you should expect the right thing to show is that if you do a doodle of size b I don't know if you can see it probably not a doodle of size b fall by doodle of size a that's exactly the same as doing a doodle of size a plus b and so when you wonder that then you want to know why it's true so to explain why it's true I want to get to one of the most important facts in mathematics period something which in what I people which you know from a very young age although not in this language which is at the shortest distance between two points is a straight line so to say it more to say it more mathematically if we're walking from point a to point c you want to go in a straight line and if you go by way of point b it's definitely going to take you longer unless b is on the road from a to c so and that looks like a triangle so hence it's called the triangle inequality uh and so that means that uh so so why does that what does it have to do with what we were doing before actually I'm really vaguely okay before I do this maybe can I ask what was going through your head when you said they're the same shape um because I know because when you do one around it's setting a new shape and if you do one around again it's kind of like being two like you're doing one and then you're doing another one yeah like you said month what's wrong with you great I can buy what you're saying partially because I can see what you're thinking and if you're listening to what he said and you're you're you're and you don't follow yet then you should think that he used that to come to a correct conclusion so his and of course the correct conclusion comes from intuition first uh and so something is in his head that you would want to figure out how is he thinking what's leading him to this correct insight uh and so okay great so let me make this precise in this way where one plus one equals two will become relevant so we start with our shape there it is it's everything inside and how do I show that if I do two smaller doodles I'll make a a's and b's that's the same as doing one bigger doodle and the answer is we'll just essentially run our election run our vote and we'll show this guy's at least as big this guy contains that guy and that guy contains this guy and that means they have to be the same so what I mean is the following thing which is one of the two things one of the two parts of the vote the argument that why people would say that anything in here is inside here is because anything sorry anything that's inside here is inside here is because if any place I can get to in a step of at most a plus b the big step most a plus b I can certainly get there in a step of size b followed by a step of size a right I just sort of cutting my step into pieces so anything I can get to this way I can get to in that way uh a step of a plus b can be broken into two smaller steps how about the other direction well anything I can get to in a step of size b followed by a step of size a can I get there in a single step of size a plus b at most and the answer is yes because that's at most that plus that so in other words what we've figured out the reason these two are the same thing the reason this is contained inside there or this is contained inside there and that means they're the same shape is the triangle inequality it's logically equivalent to the triangle inequality asking the questions we've been asking has by following this particular path led us inevitably to discovering the triangle inequality and it's even equivalent this is actually logically equivalent to the triangle inequality as the argument kind of shows so if you were to do the same kind of question with doodling on a basketball doodling on a on a cylinder or anything like that any place the triangle inequality holds whatever that means here it's kind of clear what it means in terms of shortest distance uh then this is going to work so we've discovered that this is kind of a very robust kind of fact there's a question to call it ah good question am I assuming the original doodle has one piece interestingly enough I think I would have assumed it to begin with but nowhere did we use that so I would have also assumed that it was had no that it was con that was sort of a polygon made but nowhere are we using that in this is uh so we've accidentally discovered that that doesn't matter yes sure I strongly believe in dumb questions I I mean I I feel like yeah sorry am I not so I I do repeat the question no I believe I people should ask that people actually ask questions they think are done repeat the question yes the question was do I believe in dumb questions and I am a believer and uh and I think they're really important yeah so I know oh yes that's what yes oh yes I'm sorry yes uh I wasn't sure whether that was intended to be the question please do ask are you saying am I understanding you're saying that the line from A to C having said that the short assistance between two points yeah is the same length as A to B and B plus C no it's the shorter length guaranteed to be shorter unless maybe if you're going from A to C and this maybe in this picture it's you're guaranteed the faster ways to go from A to C this way if you're going to drive from A to C you want to take this road you don't want to take it right yeah that is exactly right yeah so it sounds like you're happy so it sounds like you're making not a dumb question but a smart statement great that's a good question so again you can ask the dumb question later on but that was a good question so the so so so great so what I mean to say is the following thing which is I claim that any if I do little anywhere I can get to by taking a step of size B followed by a step of size A in different directions even I could get to in a single step of size of most A plus B ah great that's exactly the sound I love hearing okay yeah exactly it's the ah-ha you're when you hear it in your head that's a really good well it's scary a bit but it's uh it's good but it's your own voice in your head are those things for the final example good question what do you mean by to which I will great the way to answer good questions is what do you mean by spaces so but but the answer is you could answer that question you I mean I said what do you mean by spaces I'm not asking what's the dictionary I'm not sure sure there are I get like on the earth right I mean the earth happens to be round fairly I mean and so that so when you fly from here to London you want to fly the fastest way so the answer is yes it does work there so anything reasonable that should be so this is a vague answer but any reasonable kind of space it should be true it could be there's a there's the distance in New York the taxi cab distance of you can only walk on the streets you're not allowed to break through people's property and so that's that there it still makes sense there's like a fastest way to get some places yep so the answer to your question is yes that's a good question and depending on your answer to what space is you'll have an answer so it's kind of a robust kind of question yes here comes a dumb question finally good sure if I get if if I manage to remember what it was which I may not let me try to draw it again because I can't remember what it was good okay good so there's my shape and I think this is the you're criticizing my drawing again okay this is so okay so I did two smaller doodles let me do one and one and two rather than a and b and there's my single bigger doodle and I'm hoping if you can't see colors you can see at least darkness and lightness of the so the statement is that anywhere anything inside the red anything inside the big green doodle is also inside the red doodle and let me translate that this is purely a restatement that anything inside the double green doodle means anything I can get to in two steps two small steps I can also get there in one big step in other words any way any place I can get there if I get if I can get there in two steps of size one I can certainly get there in one step of size two by by taking the straight road the other guy so sorry I shouldn't have interrupted please go on I'm happy so far with me because square is smaller the next circle is smaller yeah and then so I don't understand therefore the two why the two equations above help you out by the equations I'm never sure ah good so so what we know is that so it's hard to see but because there's two lines here one is the single big step and one is a result of two small steps and it's hard to draw because they end up being the same line but fortunately I'm I'm not good at drawing so they don't look like they're the same line so what I want to say is that the the so one of these two things is where I can get to that's the boundary of where I can reach in two steps two small steps the other one is the boundary of where I can reach in one single big step and the remarkable fact that we're in the process of discovering is that in fact and they're the same thing anywhere I can get to in two step two small steps I can get to in one big step and why is that true well I'm going to show you that anywhere I can get to in two small steps I can also get to in two big one big step I'm going to show you that anywhere I can get to in one big step I can also get to in two small steps in other words and there's no difference between what I can get to in one big step or what I can get to in two small steps. That's what I want to do. Can I include the two scrolls, just together, for a little longer? Exit. OK, great. You're talking about the general version, exactly. So OK, so so far, I'm still waiting for the dumb questions. So you're sound completely right so far. But so I think something of my brain is just not safe. OK, forget the equations then. All I mean is, it sounds like you're already happy and the equations are a translation of what you're. So you agree, I think, that anywhere you can get to in the two small steps is exactly the same thing as you can get to in one big step so long as. And you said the key thing which kind of proves your understanding of the situation is that provided the size of the two small steps are the big step. In that case, forget that. Because that's meant to be proxy for the real thought which you've already had. So you already understand. And if you don't understand, if the issue is a language issue, then that doesn't matter. That's just language. In terms of substance, you already understand this fact, which is maybe you would obvious fact. But I think it's not an obvious fact for most people. Are you happy or you still want to ask a dumb question? You can ask me a dumb question later if you want. So far, the questions are good. I think it's the beginning of a good discussion. So we should talk more after. But really, I feel like this is a great example of how this can obscure what this language can obscure the truth. And the truth is what you said. It might be. That is a reason why some people think they're not good at math. It's because of issues of that. So that is a reason why some, not everyone, they can think that they're not good at math. It depends on whether you think visually, or formally, or in different ways, too. But so far, it sounds good to me. OK, so you can ask this stupid question later on. So for now, we'll take our understanding and see what it takes us and where it takes us. And armed with what we know, we can save ourselves a lot of time, because instead of taking a million small steps, instead of doing a doodle a million times, we might as well just take one big step of size one million. Because we know it doesn't matter whether we take a bunch of small steps or one big step, so long as this number is, if there are a million ones there. So that means that's simpler. We got rid of the dot, dot, dots. And so now I have a reworded question, which is, and I'll say it in math speak. But this is not the way a five-year-old would say it. The five-year-old would say it correctly, which is, would you do a really big doodle? But r is really big. Does this thing, a single really big doodle, get more circular? And that's still vague. I don't know what more circular means, exactly. But that's OK. There's a principle here, which is, the question is shorter, which means we're closer to the truth. And that's a common, that really is amazingly true lots of times. So whenever you can get the question shorter, you're closer to the truth. You've removed, you've peeled the onion and gotten rid of some of the mystification. And now I'm going to tell you why it's true. And so here's the, this I realized long, long after I was five. But let me, so I'm not pretending this is something short. But it relies on this insight that I want you to get, which is not so, which is really not so hard unless you're psyched out by it. So let me explain it. So we have a shape inside a bigger shape. So a is the smaller shape, for example. And b is a bigger shape. So I claim that if I did a doodle around a, I get a shape. And if I do a doodle around b, I get another shape. My statement is that if a is inside b, then the doodle around a is inside the doodle around b. And let me say that another way, which I find more enlightening, which is, anything I can get to, this is anything I can get to from a step from somewhere in a, right? And this is anything I can get to in a step from somewhere in b. But everywhere in a is in b. So anywhere I can get to in a step from a is somewhere I can get to in a step from somewhere inside b. So let me say that again. But then I'm going to ask how you feel about what I'm saying, which is I have a shape. Let me make another example. There's a, and maybe b is just a tiny bit bigger. And now when I doodle around a, I get a shape. But when I doodle around b, I'm guaranteed to get a bigger shape. Because everything within a step of the red of this thing is certainly within a step of this, of b. So now I want, if you think you're kind of OK with that, can you nod your head? No, you don't have to be OK. It's not like it's required. Yeah, oh, question or comment? If it's over the edge of b, you mean? It's not allowed to be outside of b. Yeah, it's not allowed to be outside of b. So I'm saying I'm assuming a is inside b. So as long as it, if it's over the edge, I agree. The deal's off. R is the reason things are that far. And the people will be outside of b, you can realize it's inside. So I want the same R, so the same size doodle. So b is, let me just try to, sure, oh, yes. Always make me do that. I think I've lost some. People may think there were dumb questions asked earlier because they didn't hear them. But I should have repeated the smart questions. The question is, what happens at A near the edge of b and R's, and you take a bigger doodle around A? And I'm saying, I think that was the question. And my answer to what I thought was a question was that if I want the same size doodles around both A and B. Great, I think that was the question. But if there are follow-ups, please let me know. Well, I want her question. Oh, this is not. No, so this is going to get somewhere. I want to do a doodle around A. So I want to do a doodle around A and B. And so now I have four things in this picture. And I want A to be in two shapes, yeah. And you said I want to draw another one around A. Exactly, great. Another one around A and another one around B, yeah. Yeah, so I had started with two shapes and I had two more. I agree with you so far. And that is what I want to do. So if you draw a big doodle around A. Oh, I see. My drawing was bad. So not grammatically, but my drawing is OK. So if the doodle was big, what happened around B? Oh, yes, I'm happy with it. OK, good. That's a really good point to say. So John's point is this, which is if I take, and this may or may not address the issue, but let me try anyway. So I have a shape inside another shape. And I'm going to take a really big doodle around A, this big. And look, it happens to be around B. But forget that for a second. I'm going to do an equally big doodle around B. And my point is that the doodle around B is bigger than the doodle around A because B is bigger than A. So there are four things in the picture and the size of the other four shapes in the picture indeed. A, B, doodle, doodle. And my statement is that if A and B, if this shape's inside that shape, then this shape's inside that shape. But you could have drawn another one around A. Does that drawing get around B, doesn't it, by invading the territory? Yeah, if I did a small r, then I'd do it. Yeah, and then it sounds like you'd still be happy that the doodle around A, this is true for any size doodle. Yes? Well, she's talking about she's saying the potential, no matter how big A gets, there's always a potential for a bigger D. So A, you can keep getting as big as it wants, but B, since it's bigger, its potential is always going to be bigger than A. Whether or not he draws it. I have no idea that would happen. Well, I think that's what it sounded like. It's just saying, because he didn't draw the bigger D. No, I don't understand saying I'm going to draw a big one around A, so I may have grammatically said it incorrectly. So let me try one more thing, and then I'll draw, then I'll, let me say one more, let me make one more attempt. And then if it doesn't quite work, then we should talk. Because I feel like it's something where sitting and drawing makes things clearer, and it's harder to do in a semi-model line. Oh, is there a comment or question? What is your definition of more circular? Oh, good question. Oh, way back there. I don't know. Right, that's the right question to ask. Yes, I don't know. Yeah, I want a definition. Do you have one you want to give me, or? Well, I guess if your original shape has angles, it would be less angularity. OK, I like that, less angularity. Great. So OK, good. So we have a tentative definition or refinement of more circular. And I want to come back to that later, because if we're going to be done, if I'm going to convince you of this, I better tell you what more circular is at some point, right? And now we have a potential angle. Right, so that's a vagueness that is still to fill in. Yep? But if you had a triangle, you inscribed a triangle around it and another triangle, and another. Why would it get similar? So the question was, sorry, let me repeat the earlier question, because I forgot to do that too. The earlier question was, what does more circular mean? What do I mean by that? And I said, I haven't decided yet. And then she gave me a possible definition, which sounded like an improvement on the vague, more circular. And I was tentatively happy with that. And then the next question was, if I have a triangle inscribed in another triangle, in another, in another. So I'm saying just pick any two of those triangles. One is inside another. If you doodle around the smaller guy and you doodle around the bigger guy, the smaller doodle is inside the bigger doodle. No, that's not what I'm saying. You'll say it gets more circular as you keep it. Oh, that, yes. So the point is, if you could mimic the angles exactly out to forever, it'll never get similar. So we have a proposed more circular and a counter argument against that. So let's table that, because we have to, that's a good thing. At this point, we don't know what it means, and we want to figure out what it means. I should say that the argument of what we actually mean, this may sound like pedantics, because you want to know what the right definition is, but we don't get to make up the definition. We want to say something true, and if we say something false, then that was not so wise. So when you do science, new words are made up to represent concepts that need names. And sometimes the names come before the precise concepts. So let me make a, let me, before, let me now take on, so maybe nod your head if you're kind of happy with this, I'm not only hoping for over 50%, let me find out whether, oh sure, oh yeah. So if something's inside something else, then the doodle is inside the, then the doodle around the something is in the doodle around something else. So nod your head a bit, oh sure, question? It has to be absolutely equal to the increase inside the doodle. Absolutely, I want it to be exactly the same. So that sounds like you're happy, yes. The size of the doodle has to be exactly the same. So I'll take that as the head nod then, since you seem great. Okay good, so everyone who's nodded their head has agreed and can't take it back. And if you haven't nodded your head, then maybe take it on faith, but I realize one of the reasons you haven't nodded might be some confusion that you're not sure what you're taking on faith. So I definitely am aware of that. So let me now use this to do something. So we have our shape X. I want to pick any point inside, call it P. And I want to now talk about circles around P, centered at P. So C sub T I mean to be a circle of radius T. And I want to pick an R, capital R, such that X is inside the circle. So let me repeat, X is given to us and I'm just going to choose a P and I'm just going to choose a big circle centered at P, that's it. Okay, so now that means, let me just restate that in math speak. I picked the point is in X, X is in the circle, that's it. And now I'm going to doodle around the point, doodle around X and doodle around the circle. So when I doodle around a point, what do I get? A circle, great, so there it is. When I doodle around X, I get to doodle around X. When I doodle around a circle, what do I get? A bigger circle. So in other words, here's my picture. Here's what you've agreed. And so you said the doodle around a point is a circle, doodle around a bigger point is another circle. So what you've told me that you agree is that there are two circles and the doodle we care about is somehow squeezed between them. So let me draw a picture. And I think of my doodles as being really small, but now I want my doodle to be really big. So little R is a really big number and big R is a very small number, which is confusing and I apologize. So let me do an example. Let's say this is a million, little R is a million and big R is one. So this is a circle of radius a million and that's a circle of radius a million and one and I will attempt to draw that. So there's a circle of radius a million and there is a circle of radius a million and one and now my, great, it's so close, you can't believe it. And then the actual doodle, somehow the border is contained between those two. It contains a smaller circle and it's contained in a bigger circle. What's that? It's exactly, my drawing is not to scale. The drawing is terrible. Yeah, yes, exactly. Everything, right, when I doodle, I mean everything inside. Yes, exactly, that's right. So that's the doodle after I do a doodle with a size one million around X and that's the thing I'm hoping looks more and more circular and to me, that looks more and more circular. So now I will define more and more circular. I will say that, well, maybe it means that it fits between two circles, their ratios or radii should be really close to one. And then I say, and then I'm done and I've explained it, to answer the question. And you will, some of you will be very happy because you'll say, yes, look, it's more and more circular. Some of you will be unhappy. Some of you will be confused, but some of you will be actively unhappy, in particular if you have a different definition of what it means to be more and more circular. And there are other excellent definitions and in fact, that definition or at least a variant of it is a great definition. Then you have another thing to know why it's true. So I'm not saying that this is God's definition of more and more circular, but it's certainly a good one. And you may have another good one and that's another thing which you will discover is true or false. So we are led to maybe not this definition, but this or something close to it. And that tell, and so we, and it may again, seem like cheating that we said what the question was only after we'd answered it. But that's the way science works. You don't tell nature what's true. You guess, and then nature may tell you something you didn't know. And a great example I think of is mass and weight. Before Newton, there's no reason to have two separate words, but nature tells us that we need to separate these notions into mass and weight. And that's again, nature telling us that they need words. And similarly, if we have a vague idea of what it means to be more and more circular, there are different possible answers and we have to figure out what, it's our job as humans to figure out what it works. How it works. So let me, let's see, in the interest of times, say a few other things, one of which is, so I want to use the language of, oh sure. Oh, a question in the back? Oh, yes, please, yes. How do you locate the center? How do you locate the center? Oh, of x, you mean? I didn't say center. I just said pick, the question was, I have x and I've got a point in it. And when I draw a circle, okay, the question was, I think, how do you locate the center of the circle? Or, ah, there's, okay, if I understand the question, I may not, because I can't quite hear all the words you said, but you have a shape. And I say, just pick a point and then doodle around it. Never do I use the phrase center. As it becomes more circular. Oh, yes. Oh, great. Let me go back to my beautiful drawing. Here it is. So that's obviously my statement about using lots of paper, I should have used a big sheet of paper. So the inside one is the circle centered at p of radius a million. And the bigger circle on this side of it was a circle centered at p radius a million and one. And that's the picture of it. So I just picked them to be set in. Does that answer your question? Good, great. So we'll talk about that later to figure out. So, okay, so let me go back to a question I heard when I was 10. But first let me make a definition, which is a polygon is something with just edges as psi, finally many edges. And whoops, convex means no dense. And non-convex means it has dense. So definition by example, that's convex. That's not convex. So that's all you need to know about the definition. And then here's, now here's a cute, so here's a fun question, which is I will now draw you a pentagon. And this pentagon, because I'm not get a drawing as we've already agreed on, is not a regular pentagon, but it's a pentagon and it has no dense. And p, if you measure it's perimeter, it's perimeter is exactly p. So you can measure it to check that I'm right. And my question is, if I tell you the perimeter, tell me how long the perimeter of the doodle, in other words, tell me how long the doodle is. And it seems like there's not enough information for me to answer the question, because I haven't told you what these angles are, the lengths or anything at all. But that's other than the total perimeter. But that's enough to know. And the reason why, sure, far away. That's good. If you say it's the p, if you figure out the size. Yeah. Plus pi times r, we start what we're doing yet, right? Right. And the time of two pi times r times seven. Great. So he made a lucky guess. And indeed that happens with this particular pentagon to be true. But he said more as to why that was not just a lucky guess. What he said is, actually, again, he's criticizing my drawing. But that in fact, this is like a straight line. And so I buy part of what he's saying. And again, our goal is to understand what he's thinking and then we can be happy with it too. And then we have these little, as that gentleman in the second row said, bits of circles here. And so this length, whatever it is, is the same as this. This is the same as this. This is the same as this. This is the same as this. So immediately, these five things have length adding up to p. And then we've got these bits of a circle, which now, why did you quickly say a circle? Why did you say that? What was in your? That's OK. You don't have to. It's nothing wrong with being wrong. I'm trying to think of explaining my intuition. This is more just intuition. That's going to be, if you have a point that you're clearly on the circle, take a point and actually add it in some straight line. So great. So I should say it's important to, right, first you have an intuition, then you try to make, you try to, this is exactly what this thinking I want to sort of be clear. This is how ideas happen. This is how creativity happens. So if you have these guys look like, if you did it with a point, it would be a circle. And this is kind of like a blown out version of it. So let me say that in a different way, where these puzzle pieces can be brought together and blown back in, and you get a circle. These five slices of pie turn into this. Again, my drawing is bad. And feel free to ignore this if you're less happy about these geometric things. And the added bonus of what he did by adding these lines and saying, think of it in this way, is, for example, if I wanted to know the area of this shape. And at first I should say the area of the pentagon I measured and it's capital A. And then what's the area of the doodle? What's everything inside the doodle? Well, the area of the doodle is going to be equal to, well, the old area plus these rectangles and the doodles of size r. So this rectangle is area this times r plus this times r plus this times r plus this times r plus this times r, we get the perimeter times r. Again, just for the people who are very happy with this kind of thinking. And then finally, we have these pieces of a circle. I don't know what any of their areas is, but they fit together to form a really nice circle of radius r, which is pi r squared. So we magically get the error. So we get that 2. And I want to point out something, which is we just take, for those of you who know calculus and I have a side comment, which is, first I want to point out that these guys, there are a lot of letters here. But almost all of them are actual numbers. That's an actual shape I had. So that some number, p is some number, pi is a number. The only thing I'm in vague about is the size of my doodle. So if you squint your eyes, this is a polynomial. This is something some number plus some number times r plus some number times r squared. For those of you know calculus, the derivative of the area is the perimeter. It's got to be, that can't be a coincidence. So why is that true? But now I will cover that up so it doesn't bother anyone. But I also want to notice something else in general, which is if you look at this and forget the r's, these numbers that are the coefficients, everything is important in geometry. The area of a shape, the perimeter, and pi is certainly important in geometry. OK, let me now generalize and say something different, which is, let's say that x is convex, but not a polygon. So it's like some egg-shaped thing. And then I will do my doodle around it, which I think I'll do in pencil to make it more clearly contrasting. And so I want to tell you an amazing fact, which is that the same formulas hold. And why do I say this is amazing? Because, where'd it go? His entire argument, which I've now lost, another work of art, lost to the sands of time. Great. There we go. There's my beautiful description. And so his entire argument relied on these edges, and it does not work. So basically, his argument applies not at all. And so how does something so wrong feel so right? I mean, look at the look at the look. It's just the limit of the polygon. You make limits sound so easy. That's great. I think you cut up the sides of the circle into little pieces, and you raise those up. That was just absolutely great. Everything from the I get it and then the substance after that. That's exactly not just the facts are right, but the getting there is right. So you can actually see why this is true. I could say there's a really good way of seeing it, depending on who you are, you'll have different ideas. One is you could take undergraduate calculus and use Green's theorem in a clever way, and you would do it, and you would prove it, and you manipulate stuff. The other thing is to instead say, wait a second. And what they both just said, and then what you're doing is following the footsteps of Leibniz and Newton in making sense of these things. And that's hard to do, but frankly, going to your calculus book, unless you go back in time and read why that's true, you're cheating equally. So I feel like we were forced in our discussion to discover the roots of the calculus in this way. So one more example of this from a puzzle I heard when I was, now this I heard probably when I was around 10, give or take. So, okay, so the way this works is as follows. The ball, wow, thanks, it always gets away. Great, okay, good. So for the rest of this puzzle, the earth is a perfect sphere. So string is wrapped around the earth, around its equator. And then I cut the string, add exactly one meter more string, and then raise the earth, there is a string above the earth, like tie it up again, and raise it, so it's like levitating above the earth at a constant height. So hopefully I have a picture of how like moons, like how like the rings of Saturn or what that looks like. And my question then is how high off the ground is the string? And now, okay, and so I'm gonna give you some, is it closer to a millimeter, a micrometer, a nanometer? In case you're wondering how big they are, that's a millimeter, that's a micrometer, and that's a nanometer. Okay, so some comments. If there are pure mathematicians in the audience, the earth is not a perfect sphere, but we'll assume it is. Secondly, this is not the earth, it's a basketball. And so you may reasonably say, well, I don't really know how big is the earth so I can get an idea of what this is, but you live here, you have some idea how big the earth is, so that should be fine too. So now I'm ready for your, I want you to have a vote and give me your ideas, and again, there's nothing wrong with being wrong, but there is something wrong with off voting. So the question is, if I take, I'll say it again, and then you can vote. Earth, string, cut, add one meter, levitate how high? Now, okay, I'm ready for an opinion. So who thinks it's closer to a millimeter? Raise your hand. Who thinks it's closer to a micrometer? Raise your hand. Who thinks it's closer to a nanometer? Raise your hand. Okay, so we have lots of, that was well under 100% total voting, so a good chunk of you are just wrong. You can't, the people who vote again all will be right, but the vote seemed to me roughly maybe 10%, maybe 15, 15, 30% among the people who actually voted. And so in fact, amazingly, it's this. And it's actually about this high off the ground. But exactly, it's kind of amazing. And it's only amazing if you made a guess. If you guessed this, it's amazing and you are excited and happy. And if you didn't vote, you're just sad. So it's a, so it's a, so great. So why is this true? I want to say it, not in the way I would do it when you're 10, where you know about the circumference of a circle. I'll deliberately do it just to connect to things we're talking about. And so here's the earth. And here is the string above the earth. And again, pure mathematicians, this is not the earth. This is a drawing. And the, so, and what I know is that I've added one meter of string. And now it's levitated above the earth by distance of r. And now we already know that that looks like a doodle to me. And the perimeter of the doodle is the old perimeter plus two pi r. But we added exactly one meter. The new perimeter is the old perimeter plus one meter. So in fact, r is one over two pi meters, no matter what the earth is, although the earth is the earth, but you can replace it with Mars or a basketball. You always get the same answer. And something even better comes out of it, which is that in fact, you can do anything at all. In fact, any shape at all, this works. What's that? You could do a penny. You could do a Minecraft world. You could do like anything like this, because when you wrap string around the equator, it's wrap it tight and then levitate it. You always get a picture like this. And if we believe Leibniz and Newton and the suggestions from the audience, these things are always true. We could, but after we need enough string, but we could then do it with this triangle. So afterwards we'll do it if you can find some string. So let me say what else comes automatically from these ideas, which is, okay, maybe we start to give things up and try to generalize and see what breaks. And as we would always do in a good experiment, you add the weirdness, one bit of weirdness at a time to see how each individual ingredient changes what's going on. So now I'm gonna draw something with a dent, with a really big dent. And when I doodle around it, now it runs into itself. And when I do that, I will tell you an amazing fact that the total perimeter, I'm gonna include these guys in the perimeter, the same formula holds. And then now, as far as the area goes, in fact, the area, this works only if you double count this area. It wants to cover over itself, so it needs to be double counted. So this means that nature is telling us that we should be counting areas of multiplicity. It's forcing us to discover area with multiplicity. And so to, let me generalize this a bit, let me ask you a question to see if this is better yet. I'm gonna tell you an answer, and I want the question, which is, if we do this with a figure eight, let me redo my definition of doodle just to make my life easier, although you may reasonably ask why I'm changing my definition. I'm gonna walk around the outside of the thing, counterclockwise at the start, and stick out my arm. My arm has length exactly R, and I've got a marker. And so the marker goes down like this. And that's my doodle. And so I do my doodle. And now, because it's a figure eight, the bottom half is shrinking while the top half is growing. And I get asked, what happens to the perimeter? And again, I just say, this is an empirical thing. You could write a computer program to do examples. You could ask your, you could like do it on paper and then take string to measure it. And the amazing thing is, you get a nice formula for the perimeter, and it's different. You've lost the pie part. And I will now tell you the answer for the area. It also is different. And now, but you may say, what is the area of the figure eight? It's not what you think. It's not the top plus the bottom. That's not the area. And so what is the area? Well, you know you have the right answer when that's true. So now I'm gonna ask you to guess the right answer. And the good thing about it is, you might be right or you might be wrong, and I'll tell you. And that means you learn something and you could, you can imagine that I'm the computer oracle that could check things. And then once you know what's true, you could prove it or understand why it's true. So all I want is a guess. What's the area of this figure eight? And it's not the top plus the bottom. Yes, now you're catching me on changing my definition. So yes, now you are absolutely right. So I need to change my definition of doodle. So my old doodle would have done this. Yes, that you're absolutely right. And now I'm deliberately guess. And so in order to make this, to deal with things running over each other, I'm gonna change my definition and you may say why, and I would say just to make an interesting, it's a natural, make an interesting, that makes the question interesting. So changing, if you notice, I'm playing fast and loose with a lot of definitions, but I think that's the right way. That's good. It isn't like the definitions are, you can look them up in a book, but someone had to put them there. And someone had, and people argued about what the right definition was. And so we're gonna end up discovering something by making good different definitions. So in this case, I made a definition. And now I have a question and the question has an answer. Or better yet, I have an answer and the answer has a question. Yes? I'm guessing that the area of the figure eight is the top part plus the outside half of the bottom part. Great. So he's saying that the top part plus the outside half of the bottom part. And I'm saying is like if you cut it at the part where the line intersects itself, right there, the bottom part gets the stuff outside and the top part gets the stuff inside. Great. So you wanna count, okay, great. So what is the answer? Is that, that, so you said this? That would come as the area of the part. Great. So his suggestion is this. And then what do you want me to do with it? Not count it at all or? Count the part outside. Great. Good. So here's what he's saying, which is actually really interesting because at first it's gonna sound really crazy. Which is count this and he says no, this is the inside. And the sounds, okay, well that's infinite. I wonder if that makes no sense. But actually he's exactly right. If you, let me not say why you're right. But you can ask me later, but this is actually, the idea of renormalization in physics of subtracting infinities is wrong. But this is like a, so okay, he's right. But I'm gonna say, we want an answer that people in the audience will be less bothered by. Sure. Top minus bottom. Top minus bottom. So you're just subtracting, so that minus that. And if you take that definition, it's right. And his definition had infinities in it, but it was still right, I should say. But that's great. So in fact, the bottom figure eight has negative area. So what's forced on us is the notion of negative area just by asking these questions. And we also even lost the pi r squared. So let me now, let me now do a more complicated shape to show a link to physics, which is here's a more complicated shape. In fact, they should have made that darker. And now it crosses itself. And now if I do a little around it, sticking my arm out, doing the same, my same new definition, my arm I stick out, then when I do that, I get a new, I get now a four pi r and a two pi r squared. Something changed as well. That was the old answer here. And what changed? Well, let me just tell you the answer in the interest of time. How do you know what these numbers are? Well, it turns out that you could actually, the reason this is two and not a one, is because if I put this on the ground, you go around twice, exactly. So great, so it's on the ground. So it's really, from your point of view, watch how many times I turn. Going around the outside of one, there's a little loop in the bottom one. Here, that was two, right? What's that? Yeah, good, yes, good. So that was a two. There was the two, and that was twice that. But if I did a figure eight on the ground, or here's a circle on the ground, or a convex shape, that's one turn. Figure eight? Okay, let's see. I'll go around the top of the figure eight. And I go around the bottom, undoing my turn. And zero, net zero turn, so that's why there is a zero. So that's something called a winding number. It's something which comes up in topology in a graduate class. It comes up in electricity and magnetism. In a way that you run electricity through a wire, and we're forced by just drawing this picture and asking the question, how long is it? Which is something you have to ask about, something you just draw. That's a curve. We are led and forced to be led to this notion. So let me now ask in what happens in more dimensions. And I should say it's very natural to want to go to three dimensions first. And if you're really sophisticated, if you think you're sophisticated, you'll first ask what happens in n dimensions, any number of dimensions. But if you're super sophisticated, you'll ask what happens in one dimension. If you do it in one dimension rather than in two dimensions, because that is confusing. But I'll leave that open. And instead, let me go to the three-dimensional case and how do we discover what's going on? We have to do an experiment. We have to do examples. So here's a box. This box, MoMath provided standard issue MoMath boxes have height h, length l, and width w. And so I want to doodle around it. Now I need to figure out what I mean. So let's consider, go back to the first definition and have all the points at most one or are from it. So I picture it like a dust cloud or some clay on the surface of thickness exactly one. So it's flat here and then roundish here. And so my question is, how big is the volume of that doodle? And now the great thing about having done a two-dimensional case where we had a picture that looked like this that we broke up into pieces, it tells us the answer in 3D. So the volume of the doodle is going to be, well, imagine this layer of clay on top and including in the box. And what's the volume? Well, it's a box itself. So there's the volume of the box. There's also slabs of clay on top of each of these guys. So that's, so just like before, this slab of clay has volume, this area times R, this area times R, this area times R. So we are gonna get R times the surface area. And then just like in the two-dimensional cases, we got these little bits of spheres, eight octants of spheres that we can fit together to get a sphere which has, we know it's volume. So there we go. Ah, okay, so another complaint about my, great. Yes, so in fact, so what's missing, you're exactly right. They're also, so there's like a quarter cylinder on this and quarter and a quarter and a quarter which goes to get a cylinder which is exactly right. And so really that was wrong and I need to add the cylinders in and I get pi, that's the volume of one of the cylinders another and the third. And that's the correct answer. And I wanna point out, the reason I deliberately did it this way is that's the one people almost all, that's the one, if you're gonna forget one, that's the one you forget. And the clue that you missed it might be that you're missing this R squared. There's an R cubed and R and a constant. That's maybe a clue. And now the amazing thing is that this works for a convex body, a no-dent body in general. In other words, if you take any shape and you take its doodle, the clay around the outside, then you're gonna get something. I ask you the volume. It's going to be the volume of this clay is going to be the original volume plus the surface area times R plus four thirds pi R cubed plus something R squared. Something involving, it's not high plus one plus width. What does that mean in this case? So nature, it's something. Nature is telling us that this is something that happens to be high plus one plus width for a box. And if you believe that volume and area and spheres are important, you've got to believe that this is something equally on the same footing as well. Okay, so let me end for the sake of time on two things. One, I'll leave the Hilbert problem until later. I want to say one thing which looks fun, and then one thing which is serious and fancy and recent, but also fun. So here's the problem I heard about when I was 17 or 18. And this is a problem from St. Petersburg or Moscow depending on who you believe and I don't want to get into that fight. And the statement is you have a company with the rule of Russian train company that you're not allowed packages whose sum of dimensions, length plus width plus height is more than something, let's say one meter. So, and the question is you have a present you want to bring it's illegal. It's length plus width plus height is bigger than a meter and so you want the question is, can you smuggle it onto the train by putting it into a box that's bigger? And if you nest it in the corner, no problem, then you can't cheat because the smaller guy's length is less than this guy's length, the width is less than width and the height is less than the height. But if you fit it maybe, just maybe, you could fit it in some diagonal way where this illegal thing can fit in a box with smaller length plus width plus height. So if you set it up, squeeze it in, then it might work and the question, so two comments about it, which is, first you may think this is like a crazy made up thing or some crazy Russian thing, but I flew here on United and you can, this is where the boarding passes go and it says carry on the luggage may not exceed 45 linear inches length plus width plus height. So this is real applied math. Secondly, the way the problem is set up, you know what the answer has to be because just as you never argue with Sicilians when death is on the line, if you argue math with a Russian, the Russian is always right. So you cannot cheat the Russian train conductor. And now here's why, and I love this problem, it's one of my favorite problems because the answer is surprising and comes out of nowhere. So here's why. If you have a box in another box and I hope more math doesn't need this box, pretend this is a regular box, if it's in another box and so if it's fitting in the bigger box, that's what this means, then you and many of you agreed to this, the doodle around the smaller box is contained in the doodle around the bigger box. If you bought that, you told me what the volume of the dust cloud is around the box. There's the volume of the clay around the smaller box and that's a bigger box. Being a mathematician, I can't stop myself from canceling. And now if you squint, all these guys are numbers, even though I've wrote them as letters. These are actual physical boxes we're talking about and the only thing that's not a number is the R. So that really, that's the size of our doodle. And now I want to do what we did when we were five years old and let the doodle get really, really big. So if this guy, for any R, always beats this guy, that means this is maybe more for people who are algebra happy. This means the only way that this leading term has to be at least as big as this leading term, otherwise he'll eventually win the race. So this leading term is at least as big as this leading term. Again, I just can't stop myself from canceling the pies. And then out of nowhere, we've just won. And the problem is solved. So that's, so it's a completely, so this problem ends up coming up there. And let me, so I want to end with something modern, which is that this sort of thing works, but this is so far I'm going to be in the 19th century. It works in n dimensions. It's a great, similar, magical fact that if I have a high dimensional shape. So if I have a five dimensional shape, which I keep back here, and this five dimensional shape has a hyper volume. And if I hyper doodle around the hyper shape, then what's the hyper volume of the hyper doodle? Well, it turns out to be the original hyper volume plus the hyper series times R, that much we've talked about before. The last term is the volume of the unit hypersphere. But now, and it's always a polynomial, and now we have more and more mysterious sisters in between these two extremes, and that nature is trying to tell us something. And I will give you just one last example and then sum up, which is about a recent, last decade revolution of mathematics. And now I'm going to get, I'm going to speak even more metaphorically because I want to get across something that's in some sense kind of a highfalutin, but the idea is something that can be spoken about metaphorically. So this is, if you're interested in surfaces, here's an example of a kind of surface. I'm not going to say things really precisely. It's got three holes, and so that's called the genus. So if you're interested in things, they may have a certain number of holes. And if you're interested in these shapes, you might be interested in all of these shapes. A good way to think, you don't have to think about one, you think about them all together. And a magic fact is that these form some space. Colin mentioned the notion at the start of my talk about modularized spaces. There's a space of such guys, and it's got a dimension, and its dimension is six G minus six. So just, I feel like it's like Rubelike surfing. Like try to just sort of stay on the surfboard and let things wash over you if need be. And so Riemann, who is the greatest mathematician of his generation in the mid to late 1800s, knew this. He didn't know what he meant by space, but he knew its dimension. He knew the answer, but he didn't know exactly, couldn't figure out the question. David Mumford is an American mathematician toward the end of last century who then figured out what the space actually meant. And so he defined the space, and for this he got a Fields Medal. And the Fields Medal is being called some sort of analog of the Nobel Prize in mathematics, although it's not quite. It's hard to get, fewer than one per year are given out on average. And so this is a big deal. And then a few years later, Deline and Mumford, Deline is a Belgian mathematician. Now at the Institute for Advanced Study, who defined a new kind of space. It's a kind of space that is used now in many different places in mathematics and physics and certain special parts of other fields too. And it was a new kind of space and that this place was more properly. And then Deline also went a Fields Medal, if I'll just circle the name. And then now that you had the space, you get asked questions about its size and its shape. And there are a lot of ideas and open questions that meant a lot of important facts about these surfaces. And then in Ed Whitten, who's a physicist at the Institute for Advanced Study, who as a physicist is a better mathematician than almost any mathematician who's ever lived, who also went a Fields Medal for his work on the mathematics coming out of strength theory, one example, a flavor of which was, he could finally tell you about the shape of the space. And it was an idea, a conjecture, some of you didn't know why it was true coming from strength theory and it connected to a completely different part of mathematics. And the story I heard from Mumford is that apparently that is that God Crank called Whitten in his office at the Institute, told him the conjecture and then hung up before he could ask any questions. So there was this completely mysterious thing which empirically was true. You could test it out in all small cases. And then in the early 1990s, Maxim Konsevich, a Russian mathematician, won the Fields Medal for finding out in some physical sense why it's true. And then Andrei Kunkov, who's actually now at Columbia University along with his collaborator, Pandar Pandey, gave another explanation that was insightful in a completely different way. A Kunkov won a Fields Medal for his work related to this. And then there are even more. And now where this last story of my, of our discussion is going to end is at a graduate student a few years after. So if you are doing a PhD in mathematics, the worst thing to do is to go in the area that the smartest people in mathematics have been thinking about for many generations. So there's a graduate student, her name is, her name was Maria Mersikani, who decided that she had some ideas and she followed them and followed them. And what she did is, well, I should say her, like in pure math at least, the best theses, if it's really, really good, it'll appear in one of these top three journals in the subject, her thesis was so amazing, it was cut up into three parts. Each one went to a different one of the top journals. And I wanna give you, so I should say, I should sort of say the follow-up to the story, which is that she, what her ideas did is they completely explained everything in retrospect and made them much simpler. She did win a Fields Medal in the last round. It ends, I should say, when I first had these ideas and I spoke about them, these are things that you don't make up, they kind of come to you. And at this time, she was a graduate student. And I could say, here's someone who, it was a real pleasure in saying, here's someone with amazing talent and let's see what they can do. She then, later, due to our good fortune, she was a colleague at Stanford. And as you may know, she passed away of breast cancer just this past summer. So this ends on more of a unhappy note and I find it, one of the many reasons I'm unhappy is not just on a personal level, but on a level of the, what she did by age 35, thinking about what she would have done by age 75. At 35, she did more than a number of typical careers, but she didn't have a chance to really start a school of thought fully. Her, the story was only really beginning. So somehow there's like a really, really, really big loss. And I guess there's a lot of romance about mathematicians dying young, but I don't really see much romance in this. But she was, but let me go back. I wanna end on a higher note, which is she, when she was five, she thought like this. When she was 35, she thought like this. And I'm gonna give you the idea of her arguments and show you how they are exactly what we've been doing while doodling. They're exactly the same and she thought of them as completely analogous. So here's her, some of her ideas. She said, okay, if you're thinking of these shapes, let me complicate things by even adding holes to the shapes. A bunch of holes, so these have, and these holes have some lengths. The advantage of cutting holes in fabric is you can sew them together in interesting ways and relate different pieces of different shapes. So she said, okay, I'll make the question harder. And now I've added these things which I think of as little doodles. And if I don't want them, if I don't want holes, I'd make the doodles be zero or I can make them really big doodles if I want to too. So she said, first she would do that. Next she said, she worked out the size of the spaces. Now that she had these things, these doodles, she worked out the size of the spaces and there's a magic polynomial just as we saw when doodling around a pentagon. And then she said, okay, if I have a magic polynomial which she figured out in an analogous way what it was by cutting into pieces the same way that he did with the pentagon. Then she said, okay, she can actually work out what it is by cutting and pasting and by gluing so she could figure out how to compute them. Thereby answering lots of questions that people have been stuck on for more than half a century as soon as we knew what that space was. And then she said, okay, if I'm doodling and I have a magic polynomial that when you're five years old you do small doodles but eventually they get really big. So the question is what happens when the doodles get really big? When these RIs, the holes get super big and there's almost nothing left of the holes. She said, something interesting must happen. Those numbers must mean something. They turn out to be exactly the mystery numbers Whitton was talking about and all the things, the way they behaved were exactly what she already knew how they behaved. Then everything kind of fell right into place including these later things as well. All of a sudden the way they were thinking became, well, obviously that's true because if you look at it the right way and draw the right line, it's true. So despite the fact that if you look at the papers it will be not obvious on the surface but her ideas and the way she thinks is exactly the same sort of thing we've been doing where you start with intuition try to make precise, figure out what the right definitions are change your definition if it makes you feel better and then you really get a chance to see something new. So let me just, so I apologize for going longer than I intended. Let me just sum up by saying, reminding you what we've actually been doing which is we started with one doodle, a single doodle and we from this, from this we ended up seeing lots of other things. We discovered the triangle inequality. We saw things from electricity and magnetism and topology. We saw recent work of many fields metals. There's a Hilbert problem I can tell you about privately afterward if you're curious. There was, I'm forgetting much more. There's a beautiful Russian train problem. There are things at higher dimensions. So all this comes by just starting in one place and you're led inevitably somewhere, interesting. And I think all the things I described you are inevitably led to but as many of you have pointed out there are many other directions you can go and they're gonna take you interesting places too. And this for me is kind of what why science and mathematics for me is so much fun because when you really deeply, deeply look into something you can find things of amazing beauty and power. So thank you very much for your patience.