 So thank you very much for the introduction. Thank you for the opportunity to speak. I'm very excited to be here at Nerd Night. I'm going to tell you about Poincaré Conjecture or how I learned to stop worrying and lock my bike. Is this obvious? Yeah. So story begins around 1904. French mathematician Henri Poincaré asked a question that stumps the mathematics world for about 100 years. 2003, my bike was stolen. Coincidence? Well perhaps this is something that we're going to talk about. Poincaré's question in somewhat technical terms. Is every simply connected closed three-dimensional manifold homeomorphic to the three-dimensional sphere? So I'm going to come back to this and hopefully explain most of the words. In the statement, 2003, the same year that my bike was stolen, Perlman announces his proof of the result. He answers Poincaré's question and he built on work of mathematicians over the previous several decades. And there's like intrigue surrounding the proof and he's this like weird guy. So he gets his name in the New York Times, among other things, and mathematicians go nuts. And that's in like a positive way. My mathematicians are crazy. This is a really exciting moment. So they can be forgiven. The answer is yes. So Poincaré's conjecture holds. So what does all this mean? What does it have to say about the quote-unquote shape of the universe? And what does it have to do with locking your bike? So here is a different picture of Perlman. I actually chose not the one that Rick showed you. This one makes him look a little bit more normal. It is a little bit older. Here's a picture of Poincaré. And in the abstract, I know some people have read it. I did promise a little bit of a love story. This is a guy named Paul. So I don't know if I'm the only one who can see the resemblance. Paul was the guy that I bought the bike that replaced the stolen bike from in 2003. And we fell for each other and dated for several years in our 20s and we continue to be friends. So he's actually using the middle picture as his current Facebook profile photo. And I was like, dude, send me a picture of yourself for real. And he sent me the one on the right, which is very nice. Okay, so that's it for the love story. Back to math. So here's a picture of Leonard Oiler. I have no idea whether or not he's crazy. He's a mathematician in the 1700s, a Swiss mathematician. And he is commonly credited with being the father of topology, which is the field in math that I work in, and the rule in which the statement of the Poincare conjecture finds itself. On the right we have a cube. So you might know, or you might be able to count, a cube has eight vertices, 12 edges, and six faces. So the vertices are the... Vertices, these zero-dimensional pieces, edges, these one-dimensional pieces, and faces, these squares, two-dimensional. So for now I'm thinking about this as being sort of a hollow box that contains all of its sides and its top and bottom, but it's empty on the inside. So this is not a discovery of Oiler. It didn't take until the 1700s for people to discover this. But one thing that he did was he took this alternating sum, so that means plus, minus, plus. Eight vertices, eight minus 12, plus six, this is two. This is something that is now known as the Oiler characteristic of the cube. It's two. Okay, so where can we go from here? Well, this Oiler characteristic doesn't change if we change the shape of the cube. So I can deform it in this continuous way. I can stretch it. I can make it really, really big or really, really small. And it doesn't change the number of vertices, edges, or cubes, and so the Oiler characteristic doesn't change. So the Oiler characteristic of these cube-like things is also two. Likewise, I can make it super wiggly, and it doesn't change the Oiler characteristic. Something that is perhaps a bit less obvious is that it can decompose the faces. So here I am changing the number of faces, edges, and vertices, but I'm not changing the Oiler characteristic. This alternating sum is still two. Something's going on here. So for some of you who may have missed that, let me put this in terms a nerd can understand. So the Oiler characteristic of cubes are two, and a cube can mean anything that you can deform to be a cube or decompose to be a cube. So let me focus on the dodecahedron. The dodecahedron has 20 vertices, 30 edges, 12 faces. It's not a cube from that point of view. However, this alternating sum, the Oiler characteristic still gives us two. So Rick stole my joke. So here's something maybe a nerd can understand. Okay, so there's some property that the cube and the dodecahedron are sharing, but it's certainly not like the number of vertices, edges, and faces, because those are different for those two shapes. But this alternating sum, the Oiler characteristic is still two. So you could ask, maybe the Oiler characteristic is always two. And I remember seeing this in high school and like completely missing the point. In high school, the answer was yes. You had this formula like V minus E plus F is two. I don't know if anyone else saw that. I'm from Canada, so maybe we study it there. In any case, here's a donut. And what I'm going to do is find the Oiler characteristic of the surface of this donut that topologists call a torus. And we saw with the cube that it doesn't matter how you decompose the surface. You're always going to get the same Oiler characteristic. So I'm going to choose a decomposition that's easy for me. And it might be hard to count in this picture, but as long as I got everything right, I've got four vertices, eight edges, and four faces. So the Oiler characteristic is zero. Zero is not equal to two. So the high school answer was somehow wrong. But it was wrong for the reason that we were like excluding the Oiler characteristic for shapes like this. So now that we have this... Oh, wait. Oh, shit. Where are you going? Let me put this in terms a nerd can understand. There we go. I was like, don't screw up the jokes. So here we have our famous donut lover and a donut. The surface is called a torus. It has Oiler characteristic zero, not two. So our three examples, the cube, the dodecahedron, and the torus, and the Oiler characteristics, we have two and zero here. So what's going on here? There's something that the cube and the dodecahedron share with one another, but not with the torus. So there's some property that is invariant under deformation and decomposition that they share. And this is the Oiler characteristic. So I don't know if it's obvious to people, but because I qualify myself as a topologist, I can see that these are both topologically spheres. This is the thing that they share. It doesn't matter about the number of vertices, edges, and faces. What matters is this topological type. You can deform both of those in some continuous way to spheres. And now the Oiler characteristic is invariant under these continuous deformations. I should say continuous and vertical deformations. I mean something precise by that. So these guys, you can deform one to the other, but you can't deform a guy over here to the torus. And how do I know that? Because they have different Oiler characteristics. So they're topologically different. And the Oiler characteristic is telling us that that you can't deform in a continuous and vertical way one to the other. So this wouldn't be an intro to apology talk if I didn't say the following thing. Many people might know this. The saying goes that a topologist is someone who can't tell the difference between a donut and a coffee cup. So there's a reason for this. They're topologically equivalent. So imagine that both of these objects are made from something like infinitely stretchy, Plato or something. Then with your hands you can actually deform one to the other in some continuous invertible way. I've actually done this with Plato with students. It's like, I think it's fun. I guess they think it's probably pretty boring. So what's going on here? The coffee cup seems to have these two holes, one here and one here. But I'm a topologist. So I know that one of the holes is topologically insignificant. It's what keeps my coffee from spilling on me. This hole is just really a well. It doesn't go the whole way through. So it's topologically insignificant. That hole is coming from the handle. So in this continuous deformation, this handle here goes to the handle here. Is that something people have heard before? Okay. If you take one thing away from this talk, this should be this. All right. So we have shapes with Euler characteristic. Two shapes with Euler characteristic. Zero. My next example is going to be a shape of Euler characteristic. Negative two. Let's ask for what's about to go up there. It's the most delicious two hole donut you've ever seen. Do you see why I like topology? So we started with this going from a sphere to a donut. We decreased Euler characteristic by two. Going from a one hole donut to a two hole donut, we decreased Euler characteristic by two. Now you can maybe imagine a whole class of examples. I don't have pictures of those. I'm sorry. I don't have pictures of donuts with more holes and different Euler characteristics. In fact, there's a linear relationship between the Euler characteristic and the number of handles. So effectively, Euler characteristic means number of handles. So there's a classical theorem in topology, the classification of surfaces, and this is a very nice type of theorem in math. So before I state it, maybe I can tell you the type of theorem that I mean. So in math, a lot of the time, you make some definition. I haven't given you a definition of surface, but I've given you a bunch of examples. So you make some definition of the kind of thing that you want to study. And then you define what it means for two of those things to be equivalent. So you have a bunch of objects, and then you can sort them into the ones that are equivalent and the ones that are not. So imagine putting, now I'm talking about surfaces here and continuous deformations. So I can put all of the surfaces that are topologically equivalent, continuously deformable, into a box. And I can sort all of my surfaces into all of these boxes. And that tells me my topological type of surfaces. I've got all this list of boxes. And now what I'd like to be able to do is somehow know what boxes I have. So how many different kinds of surfaces can I have? So I can imagine, in this example, taking the Euler characteristic and labeling each of the boxes by the Euler characteristic of the surfaces inside. The surfaces inside of one box are all deformable to one another. So they all have the same Euler characteristic. So what the classification theorem says is I do this labeling of boxes and every box gets a unique label. So the Euler characteristic is a robust enough invariant to tell the difference among surfaces of different topological type. So this is like a good kind of thing that you might want in math. You've got all these different kinds of objects. How many are there? What's your list of objects? And you don't always get to have one of these classification theorems, but it's nice in surface topology that you actually have one. So what does this have to do with the Poincare conjecture? So on one of the first slides, I said that the Poincare conjecture was a statement about three-dimensional manifolds. And that's the classical version of the Poincare conjecture. There's actually a Poincare conjecture for all dimensions, at least two. And dimension two, the dimension two version of the Poincare conjecture is a statement about surfaces. So it's a lot easier to state and draw pictures of and convince people of than the other ones. So here I've got a torus and a sphere. So those are two examples of kinds of surfaces that we've talked about. And essentially the Poincare conjecture says you can lasso a donut, but you can't lasso a ball. That's really what the statement of the conjecture is in dimension two. Now we can make this a little bit more precise. What do I mean by lasso? I'm considering loops on the surface. So the way I picture these things are elastic bands just sitting on the surface. So sitting on the surface of the donut, sitting on the surface of the ball. And the ball has this property that if I look at any loop on it, I can stretch and deform. Remember this is topology, so this is allowed. I can stretch and deform the loop keeping it on the surface to be arbitrarily small. So using sophisticated technology, I have a movie of the deformation of this loop on the sphere to become arbitrarily small. And that's what I mean by shrunk to a point. But so notice that I've got this loop on the torus and I can't do that to that loop. I can drag it around on the surface of the donut as much as I want. I can stretch it and shrink it. But I can't make it arbitrarily small. It's somehow fixed. I can't make it arbitrarily small without breaking it or breaking the surface. And breaking is not allowed. That's not a topological move. So in this case, I can distinguish the sphere from the torus by this property. Every loop on the sphere can be shrunk to a point. And that there are loops on the torus that cannot be. So what the two-dimensional point create conjecture says is that this property distinguishes spheres. Topologically, a sphere is the only surface that has this property. We can shrink every loop to a point. So in some sense, this is like an unsatisfying theorem because you already know about the classification of surfaces using Euler characteristic. If all your surfaces sorted into boxes and you know that you can uniquely label those boxes by the Euler characteristic. Forget the Euler characteristic for a second. Think about instead this property of being every loop is shrinkable. So I can instead label the box by surfaces in this box satisfied by the property that every loop is shrinkable. And what the theorem says, what the point create conjecture says in dimension two, is there is only one box that gets the label that yes, every loop on the surface is shrinkable. So if all you care about is spheres, then that's a pretty good theorem that you can detect spheres by this loop shrinking property. So this is maybe the most technical slide. So I said the classical point create conjecture is about dimension three. And in dimension two, I'm just restating the theorem that every loop can be shrunk to a point. That means you've got a sphere, a two-dimensional sphere. A surface is something, again I haven't given you a definition, I've given you a bunch of examples. A surface is something that's locally two-dimensional. So a very nice example of a surface is the surface of the Earth. And I'm not thinking about anything like complicated like rivers and valleys and whatnot, but like, let's call it a globe. So locally this is two-dimensional. What does that mean? Well, so I just learned actually that the Greeks knew that the Earth was round. I thought that like nobody for Christopher Columbus like even had that idea, but I was wrong. In any event, like anyone could have been forgiven for thinking that. Like the Earth is really, really big and you're stuck to it, right? And as far as I can tell, as far as I've ever seen, the Earth looks flat, right? It looks two-dimensional. I can go north, south, east, west. And same for everyone in the audience. So it wasn't till, was it 1969 that we could actually get off the surface of the Earth and look back at it and be like, oh yeah, my sphere. So three-dimensional manifolds are generalizations of surfaces where I replace the word two-dimensional with the word three-dimensional. So here it's really hard to draw a picture of a three-dimensional manifold. Like I have this picture part of the sphere, but really I'm drawing a picture of a two-dimensional thing in three-dimensional space. And if I were to try to draw a picture of a three-manifold, I might try to draw it in four-dimensional space. And I'm just not that smart. So all I've got here is what it looks like locally. I can go up, down, left, right, backward, forward. Those are my three-dimensions of movement. So there is a version of a three-dimensional sphere. So this sphere, the one that we know the surface of the Earth, is a two-dimensional sphere sitting inside three-dimensional space. There's a version of a three-dimensional sphere sitting inside four-dimensional space. And that's the three-dimensional manifold that the Poincare conjecture is detecting. So this is a... I don't know any physics. But I put stars on this to make it look like this was a slide about physics. I can make this heavy statement that the universe is a three-dimensional manifold. And again, I'm not saying space-time, I'm saying space. I don't know anything about quantum effects, like whatever. All I know is about what I see. So from where I look, the universe that I live in looks pretty three-dimensional. I can go left, right, up, down. And the other one. And I imagine that it's the same for you, and I imagine it's the same for aliens living wherever they live. And so what that means is that the universe is locally three-dimensional. So it's some three-dimensional manifold, and we're just too stupid to know which one. It's really hard to get on a spaceship and leave the universe and look back at it like we did with the surface of the Earth. So what we have to do instead is perform these local calculations. I think there are people who actually work on this, and there's quite an extensive Wikipedia page on the shape of the universe. So because you're stupid and you're stuck inside of it, it's hard to perform the actual calculation. So let me propose a completely impractical experiment, because I'm a mathematician and I can. Let's use the Poincaré conjecture to figure out if the shape of the universe is a three-dimensional sphere. So what I propose is checking to see if you can send, if you've got a lasso, you're a Wonder Woman, you can send your lasso to like any point in the universe. You just always check to see if it comes back. If it comes back, if you can shrink it, that means there are no non-trivial loops. By the Poincaré conjecture, that means that the universe is a three-dimensional sphere. That's all you need to do. So I don't know what's taking the physicists so long. Okay, that's it for physics. I did want to mention that there is a really nice connection between the Poincaré conjecture over its history and a bunch of people who have lived in these parts of the world. So all of these pictures, except for this guy, this is a painting, but every other picture was taken in Berkeley. So there are lots of people who I've not listed, and I probably even miss some who have connections with Berkeley, but who've worked in the Poincaré conjecture over time. And so I said there exists a Poincaré conjecture in all dimensions. So we call dimension five and higher. That's a high-dimensional manifold. Five is a high number for topologists. So all of these guys, these guys were professors. This guy visited. This guy has a funny connection to Berkeley. So the legend goes that he enrolled as an undergraduate at Berkeley. And then this is like the late 60s. Maybe as a first-year student, wrote a letter to a topologist at Princeton who then invited him to come speak, or to come talk about math in Princeton, and he just stayed and got his PhD. So the legend is that he actually never got an undergrad degree. So I'd say the guys in the top row, I would call topologists. The guys in the bottom row are geometers. And I have a little bit more to say about Bill Thurston on the next slide, but I thought I would also mention all of these guys are Fields Medalists, which is, as Rick said, basically equivalent of a Nobel Prize. So I think that's pretty cool. There's all these hardcore people working in this field. So Bill Thurston. My thesis advisor was a postdoc at Berkeley when Thurston was a graduate student. And ever since then, he's referred to him as the master. And my thesis advisor is like a famous rock star mathematician. He calls this guy the master. And this is one of the reasons. Perlman didn't really prove the Poincare conjecture in dimension three. He proved Thurston's geometrization conjecture. So Thurston, the master, came up with a conjecture in geometry, which is like way harder than topology. That's why I studied topology. But it implied the Poincare conjecture. So if Thurston's geometrization conjecture were true, that would mean that Poincare's 100-year-old conjecture were true. And that's what Perlman proved. That's why one of the reasons this proof is so hard is because it's a provable way harder thing. It's essentially a classification theorem for three-manful folds, which is crazy. So I'm not going to give you a statement of the geometrization theorem. It's super cool. But this is one of my favorite pictures of Thurston right here. This is a fashion designer whose name I forget because I'm a mathematician and I don't know about that stuff. But one of them got in touch with the other one. He maybe heard wind of Thurston, and Thurston was totally on board for this collaborative project. There was a whole fashion line based on Thurston's geometrization conjecture. And it's totally worth googling. It's really neat. So I think this is a pretty cool thing about Thurston. Oh, right. Okay, so... So this is my bike in 2003 in Toronto, and what I believe I locked it to. This was after some drinking. The obstruction to stealing this bike, I would say, is geometric, not topological. What I mean here is it depends on the sizes of things. It depends on how big this lock is and how big the stop sign is. So remember, in topology, you can stretch something as far as you want. As long as it's this continuous deformation, it's topologically the same thing. And so you can stretch this lock to be as big as you want, and you can just lift it up over the top of the stop sign and steal the bike. So like I said, maybe this is what I actually did. And there is, of course, no obstruction stealing this bike or the lock. This is somehow a better situation here. If you can find one of these posts that actually connects to the ground in these two places, then you can have a topological obstruction to stealing the bike. And I'm not making any claims that it's harder to make a mistake. It's just harder to steal the bike in this way. It's locked correctly. So I claim that there is a connection between this and the point-gray conjecture. So I'm going to imagine, remember, a three-manifold is something that is locally three-dimensional, like the universe. So I'm going to imagine the three-dimensional manifold that consists of everything outside what you see in the picture, the earth, the post, and the bike. So it looks like that. So this is my picture of my three-dimensional manifold. It's not a three-dimensional sphere. It's just not. And so what the point-gray conjecture tells us is since it's not a three-dimensional sphere, there exists a non-trivial loop. There exists a loop in this manifold that cannot be shrunk to a point. And well, okay, there's an asterisk here because actually this is, like, total BS. But hopefully there's... There's some topologists and geometers in the audience, right? Call me out on it. In any case... the point-gray conjecture doesn't actually apply to this three-dimensional manifold. In any case, I'm pretending like it does. And so there's a loop that I can actually exploit to lock the bike correctly. So the upshot is topological obstructions in stealing a bike are, in general, in my perspective, better than geometric obstructions. I don't really like to stop signs that much anymore. The point-gray conjecture guarantees, kind of, that topological obstructions actually exist. And so my advice to you is to find a post with lots of topology, lots of holes, and use the topology to create a non-trivial loop to lock your bike. And then, of course, finally, actually lock your bike. So thanks very much. I do have two suggestions for further reading if you're interested. This is a really nice treatment of the history of the point-gray conjecture. And this is a little bit more technical, but a very nice introduction to the topology and geometry of three-dimensional manifolds. Jeff Weeks is known for being, like, totally awesome, and this book is totally awesome. And I would say it's quite accessible, like, if you have a patient and interested high schooler, he could read this book. Like, you don't need a lot of background. So that's it. Thank you. So Kate will answer some questions, but I want to remind people in theater, too. There are neighbors over there. Hey, guys. That Kate will be over there during break to answer your questions. If you have something that's just burning to get it on tape and you want it on the Internet, video will come over here and ask, but otherwise Kate will answer questions from theater one. Why can't you take the two-fold donut and just pull it back on itself so it's a one-fold donut? So that's continuous, but it's not invertible. So what I mean is a continuous deformation, an invertible continuous deformation should be one that when I, like, replay it backwards is also continuous. And so replaying that folding over backwards would involve breaking, and that's not allowed. So, yeah. So that's a good question. It's just not invertible. Right here. I understand your bicycle was stolen in Toronto. No, the current mayor of Toronto should be able to campaign to get rid of bicyclists from downtown. Yeah, I think he took it. But I have this question. He's kind of informal in between his co-aid story and everything. But, you know, if you have a shape that you can't visualize, how do you manipulate it rigorously? I guess you have to do some algebraic, please repeat the question. Ah, so the question is if you have a shape that you can't visualize, how do you manipulate it? Basically, how do you, like, treat it rigorously? And so, uh, so one thing that you can do is, you know, so one thing that you can do is call yourself a geometer and then you have, like, equations. Yeah, and so that's where the algebra comes through. And, uh, that's one of the things to me that's hard about geometry. I'm actually, like, really bad at equations. But, like, I gotta post back at Berkeley in math. So from my point of view, there are ways of defining things um, the sort of amount to using equations and in topology we would have some representatives like that and then say, like, oh yeah, anything that you can deform to that. So, uh, so yeah, so sometimes showing explicitly, actually I do this to my students, uh, when they're starting to learn topology, because they're so uncomfortable with, like, how not rigorous it seems. I'm like, draw a picture, it's an answer. Like, undergrads at birth, they don't like that. So I make them write a few easy examples down and they're so awful. Like, they're so hard to do that they don't ask for it anymore. But, uh, you should say, like, um, you know that, uh, you always have the potential to write something down. It might just be hard. And algebraic topology, which is what I study is sort of an exercise in, uh, making those things easier. So, like, Euler characteristic, I can write down the number, like, two. And that's completely rigorous. We do things like that, attach more algebraic convenience to these spaces. Can I ask the following question about the Euler characteristic? Yes. Oh, hi. So, the Euler characteristics you presented were integer value. Can you just could, if somebody wrote down a number and it's like 1, 23, 5, would you build a shape that has that Euler characteristic? So, that's not something I can do. Oh, sorry. Can you, uh, have a space with Euler characteristic that is not integer valued? Um, it's not something that I could do. And because of the definition of it, uh, where you take the number of, uh, vertices minus the number of edges plus the number of faces, you'll always get an integer because all of those are actual numbers. But there are generalizations of these things. Um, there are generalizations of the spaces that they refer to. Um, so, uh, so the easy answer is no and the hard answer is yes. Thank you. Hi. Hi. So, your shape with the Euler characteristic that you gave it to, with the two holes in it, uh, couldn't you put, uh, a loop around the center of it? Absolutely. And then shrink it around to form that loop and have it go to, you know, a small loop and then a point. So, this one that goes around? Sorry, the question is about... This one. Oh, that one. Ah, okay, yeah. Um, so this is the one that goes around? I mean, I could draw the loop with you. Did I just get it right? No. Let's try it. So, what do I wish? So, if there were a loop right here. So it doesn't go around the back? Uh, it would go around the back of the... Ah, okay. Okay, yes. So this one. Um, that one is actually... Uh, so the question is about the triviality or non-triviality of that loop. That one's actually non-trivial, but that's really hard to see. Um, and so, in this way of, like, writing things down rigorously, we have ways of representing these things, um, that are, like, effectively equations that's non-trivial. But yeah, these things get pretty complicated pretty fast. Um, so, uh... So, follow up question. What's the distinction between trivial and non-trivial? Deformable to a point or not. Okay. Yeah, so that one is not deformable to a point, meaning that it's non-trivial. That's right. So, there's, like, a lot of shapes and things that we experience that we kind of relate this to, that we see and touch or something. So, uh, like, is topology used to solve practical problems? Or is it mostly a... What? It's a campus liar. I mean... It's like getting into a soul. Ooh! It's cute. Is topology practical? Ooh. Well, I had a proposal about the shape of the universe. Sure. I called it impractical. Um... In other words, is it connected? Do people then try to apply this to things that are connected? Yeah, actually, I have a reasonable answer to that question. So, is topology practical? Uh, the answer is yes. I'm not an expert. But there are some, like, really interesting things that are going on, and we're really lucky to have fantastic topologists at that other university on the other side of the bay. Um... In the heart of Silicon Valley. Uh, one of the topologists there has got a start-up. Uh, he's developed a whole field of applied topology called persistent homology, which is effectively studying the topology of large data sets. And, uh, they just got, like, a huge run of funding, and, uh, I don't know a lot about it, but it's, like, super cool. So that's, like, vague, but hopefully that's the answer you were looking for.