 I'd like to bring up someone who will be introducing tonight's speaker, who himself has a rather interesting relationship with tonight's speaker, which he may let you know about. But Dennis Sullivan is an award-winning mathematician from Stony Brook University, and he's come tonight with some students from the university and from the Stony Brook area. We're delighted to welcome everyone here and let me turn it over to Dennis to do a full introduction of tonight's speaker. Thank you. There's some interesting things about Maury Chass I'd like to share with you. First of all, she's not from around here, New York, she's from Buenos Aires, which is like the last interesting city at the end of the earth, and it's a very interesting city. And she did her growing up there and her bachelor's degree, and then she studied graduate mathematics in Barcelona, Spain, which is another wonderful city, got her PhD there, and then now she has traded those two wonderful cities for an even more wonderful city, Stony Brook Long Island, and she's there as a mathematician, associate professor at Stony Brook. Now, at Stony Brook, besides her research, Maury does the math club and also instigated and energizes a summer workshop for students in geometry and topology, and in some sense many of the things you'll hear about in the talk are the kind of things that were motivated and developed in that workshop. We collaborated on a math paper called String Topology, and I wanted to tell a little bit about the story of that, and I have to put this down and I'll go into charade mode, okay? So these are two strings, circles, and they're moving around. As you've heard of string theory, and like particles or circles, and they come and they might collide and then they join and then become one string, like that. Or it feels weird, but anyway. And one string could be moving along by itself and it sort of gets too tangled and interacts with it, crosses itself, and then breaks apart into two strings. These are always circles moving through space, okay? So one used to hear about that when physicists would talk about string theory and they would talk that way and then there is a mathematical discussion called String Topology that makes an algebraic theory out of that. And actually this research that we did started in 98. There was a famous problem all since I've been thinking about math research for like 50 years, 54 years, and there's a famous problem that all through this period called the Poincare conjecture, it's about what's the nature of a space, a three-dimensional space which has the property that every loop in it can be deformed to a point. And that was reduced in the 60s to a problem about curves on surfaces about which this talk you will hear concerns. And the problem was a famous problem and it was equivalent to this problem about curves and surfaces, so we started talking about that. And in 2005 and to 10 this Poincare conjecture was solved and it took, you know, five years to vet the proof and it was finally vetted in like a 500 page proof in a book. And it shows that this problem about curves on surfaces is true, a problem that we worked on because the conjecture is now proven and that problem was equivalent to the conjecture. And this is still an open problem now. So these curves on surfaces, if you go to this talk, if you went to the first talk, you might think this is like just fun and playing around but it's serious business and it's complicated. And one person after the talk will, what are the applications of this? Well, one application would have been had we solved it at the time is get the $50 million, I don't know, one million. No, so nothing. It's one of these clay prizes. So anyway, that would be one application to give a new proof. Anyway, it's still a good problem that mathematicians aren't just happy to prove something. They want to give the right proof and good proofs and understand it. So that's still a good problem. Mora, our math pass diverged at that point. I went into more algebraic aspects of this string topology and she's continued with the harder aspects which are these curves on surfaces which today she's going to present like it's fun in games maybe, but it's very hard. And Mora, I wanted to say, is also a writer. And one play that she wrote, she wrote a play about letters between two sisters, Alicia Bull and her sister Lucy. These are two of the five daughters of George Bull. And I happened to be around so I got to proofread some of this and I'd proofread it like five or six times over a two-week period. I cried every time. I cried every time. And so I invite you to read this play. It's really, you know a lot about Mora Chass if you read it. She's also wrote a nice article about Miriam Mirtacani for the Quantum Magazine, which is nice to read. So as I said, our past diverged from this string topology stuff, but then we decided to get married and they converged in some other sense. Okay, so I give you Mora Chass. Okay, well it was intimidating to have these presentations, but now it's over so we can start talking about math, which is what we all like. So what I like to discuss occurs on surfaces and first I want to discuss surfaces. So what is a surface? I mean what do I mean when I say surface? First I'm looking at surfaces from the topological point of view. There's a joke that you might have heard that says a topologist can distinguish a coffee mug from a donut. And what we mean by that is, well, from a topologist two objects are the same if you can deform one into the other. And here we are thinking this is like a hollow donut and here we have, when you see the mug, it's very, very thin, you know, and here it has a hollow handle and they can be deformed one into the other. So these are our objects and this little cat, you know, in your imagination you can probably, hopefully, deform it also into a donut. You know, stretching, you are not allowed to break. You can stretch and contract and this material is not a material that exists in real life. You know, it's a mathematical material. You know, sometimes we have visualizations but it's the deformation but you can visualize it as a movie. So these are more surfaces. Again, my surfaces that I always present here, they are hollow. You have to think, you know, this is just the outer layer of a solid object. That's a surface and it's, you know, incredibly, incredibly thin. And all these surfaces, you know, they can be deformed one into the other and you see they have two handles. They are made of like two donuts fused together. Here I have more surfaces, now there are three donuts fused together and here is one more that you can think. It does three handles and four handles. Take a picture and think or look at it later in the video. It's a nice problem to think about. And here I have more surfaces that are slightly different from the surfaces. They have a different property than the surface I showed before. Here I have two types that I want you to compare. What's the difference? This is not a rhetorical question. What's the difference? Yeah, yeah. It's not like I'm going over itself so you can see every part of it. Okay, so this one, which one? This one is not going over itself and this one is. There's something there. Yes? Loud. Yeah. The orange-sided hollow and the arc on the top forms the turquoise hollow. So there's two holes, two within the... There's some holes. Oh, okay. There's two tunnels within the... There are tunnels, but yeah, okay. One more. Once decided, one's one-sided. Well, here we see the two-sided. That's a property because there are different colors. But this one is also two-sided. It's just you don't see the side. That doesn't mean it's not there. There's an inside and the outside. We couldn't pay the inside. Maybe there's a little dwarf there running around. There's two sides. Now the difference, and my question was a big debate, the difference I wanted to point out is that there's an inside. You know, they're almost the same from, you know, but I want to show you how we can look at them as the same. So this one, it's a little bit like the one on the left. And now I'm going to attach... This is from a topology. They're not like a disc. And I'm going to glue it here. Edge with edge. Edge with edge. And I imagine that I'm gluing. I'm not gluing because I want to reuse this for my next talk. And now we have, like, a donut that has, you know, a big bump here on this side. So, in fact, these two are almost the same. It's said when this has a hole, somebody took a big bite out of our surface, an enormous bite. You know, this took the bite. We open and we have a hole. You know, remember that I'm starting things up to the formation so size of holes are not important. So these are surfaces with boundary. There's this edge here in each of them. Here I have more surfaces with boundary and here I have a surface that you might have heard of, a Möbius band. A Möbius band is a very interesting surface it has a property that if you start walking from a point here and you keep walking, walking, walking, at some point you're coming back for the back. You start in the top and we keep, you know, without changing, just keep walking from your point of view you are going straight. So that's what we call a non-orientable surface if you have a situation like this. It's a Möbius band. Now I'm not going to discuss non-orientable surface because I have only 50 minutes. You know, not because they're not beautiful and fascinating and interesting but maybe next talk we can talk about them. So now here we are going to look at the surfaces that don't have a Möbius band, orientable surfaces. So orientable surfaces, you know, they can be deformed into a sphere. They can be deformed into a donut. They can be deformed into a donut. They can be deformed into a double donut, a triple donut and so on. That's one list of possibilities. They can also be deformed as a sphere with a hole or as a donut with a hole. You know, since I'm a topologist for me holes can have any size, you know, either form one into the other. Two holes, two handles, one hole, three handles, one hole and so on. And you know, you can imagine how this is going. It keeps going. And here I have a list that continues. Now, it continues, you know, in this direction and in this direction, forever and ever. Now here I have some surfaces and you know, here I have more kind of physical rendition of surfaces here and here and here. Now you can see how many and how wild they can be. You know, if you can picture them all, it's an enormous amount that they can be extremely complicated. But this is a theorem that says that no matter how complicated is your orientable surface, you can deform it into one of these lists. So this is one of these type of ideas that mathematicians want. You know, we create this infinite, incredibly large list of objects. And then you say, well, yeah, they're a little complicated, but in fact, each one of these objects is one of this. I really like it. Every time I look at the theorem, I say, ah, this is a good, beautiful theorem. Okay, now that we have our surfaces, we're going to draw maps on our surfaces. So this is a story that, you know, they said that Möbius posed this problem to his students. He said that there was a king in a faraway land that had four, he said sons, now I changed it to children. And he wanted to divide it, but he wanted the children to remain friends. So he wanted that every pair of children should be able to visit each other without touching the line of the third. So they have to share a piece of boundary so you can walk from one land to the other and not just a little point. Now they have to share a piece of boundary, every pair of children. So is it possible to divide a kingdom in such a way? He asked. And you can maybe make some pictures in your head and think if it's possible. We are going to talk about this soon. Now the king made his will, and then he had another child. So we're going to see what happens if he can also divide the kingdom among five children. So this is the problem. I'm going to change instead of a flat country to a sphere, it's more or less the same problem. So can we divide the sphere, this is our kingdom now into two regions, that share an edge? Yeah, of course we can. Here I have an example. Sizes again are not important. The only thing that is important is I need connected regions. So here I have my two regions and they share an edge. They share a lot of all these edges, in fact. What about if I say three regions? Well, we can do it with a previous picture. We just add one more region. We have our two regions, we add a third and we check that every pair of regions do share an edge. Can we do four? And the picture already suggests, you know, when I look at this, what I can add one just there. Now by looking at this picture, can we do five? I mean, if you look at it, there's not an obvious way to add a fifth region and have everybody touches to everybody. For instance, you know, I should go to a place like here, I have three regions touching in this point. If I put a little region here, well, it's not going to touch the red. If I go, you know, if I go a place where there's two regions only, I'm bad, I'm only touching two. So I try to go on the other moves, but there's no way I can add a fourth. So you can say, well, maybe there's a clever way that I cannot see because I'm nervous given a talk, but it's a clever way that I can put five. There is. Let's think about that for a second. So I'm going to translate the problem of these regions to another problem. So to translate the problem, at each region, I'm going to put a point, a little yellow dot. And if the regions share an edge, I'm going to put a segment between these two points. So here it is. And now I'm going to forget the colors of the regions and just keep the two points on the edge. And think, you know, like, now if I give you this picture, the one on the right, you can reconstruct, you know, approximately, in some sense, the one on the left because you just make this thing, each of the points go bigger and bigger and you will have your two regions share an edge. The same thing happens, you know, with three regions, do the same thing, I put three dots, there's an edge, so I put an edge between every pair and here we have the three regions. I do the four now, same thing, same thing. Now if you look at this picture, it's very clear here, there's no way I can add to this particular picture a fifth point because whatever I put a point, if I put it here, I mean the edges cannot intersect each other because if they intersect, the two regions are not sharing an edge properly. So there's no way to add, and if you stop and think, and we don't have time, but you should go to your home and think, basically, if you have four regions, it has to look more or less like this, the picture. You know, there's no other way, because you have four points and each point has a segment to the other points and these segments don't intersect, the picture should look like that, the other fifth. And this is kind of the proof, you cannot put five. So when the fifth children were born, the country collapsed, there's no way of dividing the land in that way. Now what if we change our planet? Now we are not more in a spherical planet, we are in a bagel-shaped planet. So here, you can put seven regions, so every pair of regions share an edge. So I have one here that I'm going to show you in a second, but first I'm going to tell you, I knew the number was seven. So here what I did to construct it is, here I have seven rectangles that I arrange it, you know, in a circular way. Now each rectangle has two edges here on top and two edges here on bottom. Each rectangle, say, we are looking at here the orange, the orange is touching the red and the yellow. Remember, I have seven colors. So I wanted to touch six, each color should touch the other six, so here it's touching two. And I want to glue this in such a way that there's going to be two colors here in the bottom and two colors here in each edge of the top. And the way we do it is I'm going to twist this. So I have, you know, the two sides coming like this. And now, once I have it twisted, I have a corner here and I have a few spots where I can put my corner. So for instance, I put my corner here, like this. But this one would not be good because this corner that I, it's going to be touching the blue and it's already touching the blue. So when again, when I twist, I have to touch other two colors, I'm twisting here, that I didn't touch before. You know, in fact, I have something to see it better, which is like this. You know, the twisting will happen like this. And I'm going to organize, I'm going to put it like this, like this, well, there's one way of gluing it, you know, one should think, where you touch colors that you, you know, that you're not already touching and everybody touches everybody. So I'm going to pass this around. This is not the only way. In this case, we get this very beautiful, I mean, I like it, it's beautiful, not because I made it, because all the regions are the same. They don't have to be the same. This is just other symmetries, a bonus for mathematicians. But you can look at it and see that every, pass it around. And this is a different way of doing it. You know, it's also with rectangles, but they are tiling different ways. So, so this is, you know, every region, every region share a good piece of boundary. And we're going to look at this problem drawing these dots. Again, at every region I put a dot, there are seven dots, and just for you to see, I put lines between one dot and all the others. So here, the two neighbors, you know, they have this segment. Here, for instance, I'm going to have, from this point, I'm going through the green, I'm going in the back, and you go through the top. That's my pencil. So you go in the back, you go in the back, go around, and go to the top, and each of this is there, and they are not touching each other. And again, if you try to think of this picture, now I have all of them, and finally I have this picture. If I think of this in a column, I start expanding it until, you know, until they cannot anymore, since they're an edge at some point, you know, say this color is expanding and there's another color here expanding, maybe I'll show you here. So by thinking that you expand this dot, you see the translation between this picture and this picture. This is, you know, I have this problem, the regions, and I translated my ability of painting the map in this way to drawing this graph here, where I have all these points and the lines between them. There are different problems, and, you know, they both give me the answer, so in math it's very useful sometimes to think these issues from different point of view. So, well, we can ask the same question for another surfaces, and this idea of thinking this from a different point of view, you know, translate the four regions to this edge with four and the seven regions with this, you know, graph with seven points, this is an important idea. You know, translate your problem to another problem. If you're going to take something out of your talk, this is a good thing, you know, really change your point of view and looking at different. And I have to do a little, I always do a little speech, but it's my, this is dear to my heart. So I was trying to find a picture of idea, and there's a Google idea, and you go to images, and of course you see light bulbs and white males with light bulbs. Apparently for, you know, the images they're only white males allowed to have ideas. But that's not true. If you didn't know it, I'm informing you. So it was hard to find a girl, a woman with an idea. You couldn't find, you know, and I was around pages and pages, I couldn't find, you know, black or brown girls with idea. No. So anybody can have ideas, you know. I mean it takes, you know, math is beautiful and interesting and it gives you what you give her. You know, you need to work hard to do something. And a person who works really hard and spends time gets something, okay. Preach it and then back to math. So here is the number of handles and the maximum number of regions you can have with the property. So if you have no handles like a sphere, you can put at most four regions, exactly four regions. That's the maximum number of regions you can put, so every pair of regions share an edge. If you have a bagel, you can put seven and no more. If you have two handles, you can put eight. If you have 309, 410, 511, 612. So what is seven? I'm tricking you. That's it. It's a trick, you know. This is totally true, but the next one is 12. And there is a formula. You know, we can compute this. By using this idea of the graphs, you can compute it. So what we know is, it's seven plus the square root of one plus 48 times the number of handles divided by two. If you remember the formula when you solve the quadratic equation, this is found by solving the quadratic equation. That's why it looks like that. And of course, this is not always a whole number. So the answer is the integer, the largest integer that is smaller than this number. So that's the answer for this problem. Any question, by the way? Yes, very loud. Also, so, like the results are, you know, won't you see it? It got awkward. How do you figure this thing out? Well, how do I figure, how does one figure this is out? A little bit of the idea is using, I mean, I don't know the whole proof, to be honest, but, you know, think of what we did in the very beginning when we look at the graph and we picture on the sphere the four dots with the edges between them. And you see you cannot add a fifth. So that idea of translating my problem to a problem of dots with edges between them, you know, you can, using that idea only, you can prove that you cannot put more than this. Just playing around. I mean, but is it graphical analysis or are you summing up going on? It's just counting, it's a little bit counting combinator. It's not graphical, it's abstract. You count and, you know, you see you cannot do this. We can talk a little at the end if you want, but it's a little counting and first you have to prove that well, you cannot put more than this and then you have to prove that you can put exactly one like this. So now we're going to see how to encode surface and I'm going to tell you what I mean by encoding. So here I have a surface with topologies called a pair of pans. Now this is a pair of pans of a person who has very, very weird dimensions, you know, the waist is more or less the same as the legs and from the topology point of view is the same as this, this with two holes. And if you don't see it, well, think here I have things that I grab one of these boundaries and I stretch it, you know. All my things are made of this material I cannot stretch it, I stretch it and I make it very big. So it was going to look like this. If I keep stretching and stretching I can make it flat. This is a rare surface. In most surfaces you cannot stretch and make it flat, but this one we can. Two things from the topology point of view are the same. And now to study them, what I'm going to do is cut them and the cuts are segments that go from a boundary component to a boundary component or to a boundary component to itself. And I put different colors and labels because what I want to do is I'm going to cut, open up and open up even more. Again I deform and I make it straight. So after I cut along these edges we get something like this, a polygon that has some color edges and some non-color edges. The non-color edges, which correspond to this here or even this are pieces of boundary and the color edges are pieces that are glued to each other when we look at the surface. And I also put some labels that I'll tell you why. The color for now is enough to reconstruct the surface. So what we're going to do is read off the label. So here I have A A bar, which is capital A B bar or capital B and B. And we make a ring of letters that I call a surface word. So given a surface I have this procedure I start producing this I can do with any surface that has a boundary. I start cutting I open up and I say can I keep cutting and if I can't keep cutting I do it, now the cuts should not intersect each other and I am not allowed to disconnect the surface. So here I cannot make another cut because if I produce another cut I will have two pieces. So you have to stop them. Once you get something flat you stop. It's flat and with no holes you stop. So now we started with a surface and we obtain a surface word. This surface word has knows a lot about the surface. And now we're going to tell the surface word to reveal the secrets of the surface to us. So before let's let me do another example. This is in mouse we call it a torus. It's a bagel with a hole or boundary component. So I produce a cut. So here there is a cut this way and there is a cut another way. I don't have a picture for that but maybe for it's easier to see it. If you produce first this cut you're going to get like a cylinder and then you cut it again and you get like the flat thing. So here it is when you open it up. And again I have the labels or the colors to remember because if I give you the surface word you know how to construct the surface now. You know you just go backwards I give you a surface word here I have four letters so I get an octagon you know glue the edges that I have the same letter and that's it that's my surface. Is that clear? Good. So this surface has surface word this is the pair of pans A, A, B, B This surface the bagel with a hole has surface word A, B, A, B. Those are two these are two surfaces that the theorem tells us that you cannot deform one into the other and they have different surface words, yes. Question for the first one you said it's A, A, C, B Yeah. Does it matter that the order of A, would it be a different surface if it was lower A, capital A lower D, capital of A? That's a very good question right now she was asking is what if I exchange if I put capital A here instead of the little A and small A instead of the big A right now I have to start with one and I have to fix myself it's a choice I can choose whatever way I want but then I have to stick with it once I choose one I have to continue with that right now I only need the colors you know we are going to use a capital lower case in a little bit and any surface you can start with any of these wild surfaces and by gluing pairs of edges sorry by cutting you get a polygon so here there's a beautiful video by Charles Lees who shows you start with an octagon and we're going to glue the pairs of edges that have the same color and what we're going to get this is a surface with no boundary but it works similarly to the surface with boundary and this is a video in YouTube that one should go and see I saw it I don't know 50 times so we're going to get a surface the question is which surface at the very end you know when every pair of edges is glued so now it's like this is like a cylinder with two extra holes this is good it makes me happy so this is a two handle I know because we all face this difficulty when you start to study the things you know if you have this pair of pairs certain surfaces are easy to visualize by cutting out what do I get but complicated things like this you can cut up, get a polygon and then if you go back you get the surface later you go to YouTube and Google Charles Lees he has wonderful videos now let's look at curves on surfaces so I'm going to study closed curves so a closed curve you can think of it as a super thin this is a little too thick for be a closed curve but it's a super thin rubber band that wraps around the surface you know they can cross itself they're not allowed to leave the surfaces don't touch the boundary component and in the same way that we study surfaces up to the formation we're going to study closed curves up to the formation so here I have my curve is the forming all these curves again are the same for the topology point of view they are all in one class because they can all be the form one into the other so all the stages of the red curve now I put this picture here because we have the two different types of the formation now the formation of the curve so here now from now on we're going to fix the surface the surface is not moving anymore it's fixed, we don't deform it and there we're going to draw curves and in that fixed surface we have the form curves so this deformation of surface belong to the past we are over it now this is our world fixed surfaces deforming the curve on the surface okay so now let's use our cutting up of the surface to study the curves up to the formation so we cut out the surface and remember we get the surface world which is AB and now we're going to use the different capitalization so here I have a curve and I picture in both incarnation of my surface here in the real surfaces and here when it's already cut up now for certain purposes this picture is much more clear that's why one of the reasons why we do the cut up because when you see something flat it's sometimes easy to understand I digest so if we look at this surface and I'm going to look at it here sorry if I look at the curve I'm going to read off the edges that my curve crosses so I choose a point to start arbitrary say here and then I'm going here and I said oops I found the edge level by lower case A so I'm going to record A this point you know when I cut the surface is the same as this because this is you know when I cut so this is the same as this so I do A small b go here small b again back here and then I go back where I started so what I did is A A B B now of course that we associate to this class of curves you know to all the curves that can be deformed into this one now you may say well you know I enter here in the A and when I touch here I cut the capital A why don't you record the capital A well the reason is if I enter through the little A I know what comes after you know because the little you know every time you cross the little A you are going to see the big A so you know we are you know very smart people you know if I have the little A I know the big A is going to come so I don't put it there because I know it's there so this is the curve word and sometimes I'm going to write it as a you know not as a ring of letters because imagine you know organizing this letter as a ring of letters you know type writers and not computer are not very friendly for that so here I have another curve more complicated and now this is much nicer to see it here and this is the corresponding curve word again you can you know start seeing here I have small A go here I have to I cannot choose you know if this is the point here in you know like the second to last I have to go here for the second to last to follow the letter to construct the word so this is A A B I was doing here A A B and then I should do A and so on so this is the curve word so now let's review so we have surface words that label surfaces you know to do have a surface word the surface word has to tell you which edge glues with who so they have to you have to have a letter and a letter in a different case because it organizes pairs of edges now when we look at the curves a curve word can be of you know any length and they can have the letters can occur as they want I have no other restriction that you know a letter and a letter case have occurred exactly once for instance we see our curve word here has three A's and then a B and that's fine no uppercase A and so in the same way that we can list all the classes of surfaces well we can also you know label all the classes of curves by this curve word so let me explain this so we have this is an enormous bag that contains all the deformation classes of curves on a fixed surface and I'm going to organize in smaller bags each bag contains curves that are deformable one into each other you know this maximal all the classes of curves so this is a theorem that says each bag of curves can be labeled by one of these words in other words well before we could list all the surfaces you know it was just the number of handles and the number of holes here we cannot list of the curves but you know we can each has a nice label a curve word so in some sense this allows us to works with the curve so this is one thing in my math life that I make extensive use of because you know now I have words well words are something that I can input a computer I can count so you can do you reduce a problem to a combinatorial one and there you can do things so important step for my math life okay so now we're going to study crossings of curves so we have a surface is fixed and you know we have a curve you have a class of curves and to each class we're going to associate a number and the number is the smallest number of crossings a curve in the class can have so for instance if I look at the class of the red curve what is the intersection number what is the smallest number of crossings what do you think okay I need an answer before the question zero why zero because if it can go through a hole it doesn't have to cross itself okay zero because gentlemen said zero because if it can go through the hole it cannot cross itself well you can only move this is the rules of the curve they can only move on the surface the hole is the nothing they cannot go there so here there's no way here I have one cross and here I have two more crossings I can deform my curve I make and disappear this one no matter how you deform your curve you will not be able to make this appear because you're not allowed to pass over the hole the curve is so this has intersection number one of course this is you have to trust me one should be able to prove one could be but when I remove this one no the answer is no so the crossing number of this class of curves is one okay so now we're going to try to count the number of crossings of our curves by looking at the words and so for that I go back to the picture my curve word and the picture in this planar model I'm going to study this crossing here and I'm going to highlight these two segments these two pieces of curves that contain my crossing and then I want to connect you know these segments with the word how are they related so I look at this segment and you know what happens why do I have this segment because at some point my curve enter through this edge right here and then went back to the same edge now enter through this edge means that my word has to have an a because every time you enter through this edge you have to record a little a you come here and you do a little a again so having a segment going to this edge from this edge to this is the same having a word that at some point has a little a and a little a one a next to the other same thing here to here I have a b and a b two b's together and every time I have an a a and a bb I'm going to have this crossing and well this is a special word it has only an a a and a bb but my word can have a million letters but if two a's together and two b's are together they will be a crossing so now we find a combinatorial condition for a curve to have crossings you know if you have a curve that has an a a a b b the class of that curve has crossings the word told us something about the curve and again let's rewind a little and think of all the possible curves and complicated moving through the surface the fact that the word can tell you something it's a little mathematical miracle that I do appreciate so here I have more a more complicated curve and again here I have another example another instance of this a a a a and in this case it's capital B capital B you know the two appear in some spots of the word and this imply a crossing now you can ask well are all the crossings can you count the crossings of a curve by just looking at these pairs of two letter words the answer is almost not exactly but you have to do a little technical thing you know here I'm not going to go into the let me just say maybe I'll say so here I highlighted this I'm going to study this point the red point now these two segments here have an a and then an a again this segment is there because I have an a and then a small b now if I tell you okay you have a word that has an a a and an a b you cannot say there's a crossing or there's no crossing you cannot be sure because well in this case there is but I could have drawn them like this you know if I go from here to here and from here to here or not so what do we do what we go to the past of the curve you know we look at backwards so these two pieces come from this shh like this so if the words tell you you start from these two different edges come to the same and then maybe you can go together for 100 letters who knows and then at the end and you have a crossing or you do this and you don't have a crossing and that's the condition that can be studied it's a little more complicated but it can be studied just by looking at the words in a twist in a little complication of the a a b b thing so this is the theorem the theorem says each crossing point can be assigned with the pair of two subwords of my word a certain pair and you know just there's a list of pairs that are good ones that implies crossings so if I want to study how many crossings my word or curve word has I have to count how many pairs how many special pairs of subwords my subword has so here I highlighted the pairs of subwords of this word they're not pairs, some pairs with special properties analogous to that of the a a b b so here I have the list of pairs they are in color and those are the ones who imply a crossing and again this is this is something that you can put in a computer, I mean my last moment this is a condition that can really be analyzed by a computer so I analyzed program this and I made a table so in this table the rows are going to contain the words of a certain amount of letters so the row one contains the words of one letter the row two contains words of two letters and so on and the columns are going to give me the number of crossings so here for instance in the column one I'm going to have and say here I'm going to have the number of curve words I'm going to have one crossing given by the column and two letters so you know the first table one can do it by hand so for instance there are four words of one letter and no crossings you know just a single letter a b capital a capital b these are the four one letter words they have no crossings and you know you can keep filling the table it gets more complicated to do without a computer this is now the work of the computer and you keep doing this is more and more there are many many many words so here for instance you know as this is 16 letters there are approximately 3 to the 16 words that's an enormous amount of words but you know that's fantastic about computers they don't complain you know they say count the intersections of these three to the 16 words they say okay okay and they do it so here it is and well this table here is the table of course we can't read anything but you start to see patterns you know we are in the eternal search for patterns in math you know when I saw this table already you see one pattern the green and the white is non-zero so you see this nice curve that in fact looks like a parabola and it looks like a parabola because it is a parabola that we proved it so another way another feature is that here it's painted in black it's no curves it's square and these are the rows so this is a row say number 20 and you see that when you have let me start here so this tells me this is white when it's very light means that there's very few curves so there's very few curves that have zero crossings and 20 letters and you get when you have when you look at more and more crossings there are more and more curves here there's a lot of curves more or less in the middle that have the number of crossings here I think I don't know something it's a lot and then they start decreasing the amount less, less, less and less so I wanted to study better this table so I isolated a row A so these are all the words with 8 letters and you know I did a a histogram so this is for instance there are 16 curves with zero crossings there's a rectangle here and then I keep filling this with rectangles whose height is proportional to the number of curves you have here there's nothing very obvious but if you do more you see how it getting closer and closer to a bell curve so in 20 if you look at from far this is just a perfectly beautiful normal curve so I was just playing with Excel and oh do the histogram and then I saw this and I jump on my chair and there was I knew there was a mathematician in the University of Chicago Steve Lally who had working analogous problems so I said look I think this distribution of third intersection of curves sampling by world length I think it's a Gaussian and he answered no I don't think so so I just sent him this picture and he answered let's prove it and we did now it's a theorem so we know that if you look at the distribution how the curves of a certain number of letters how they are organized by self-intersection they are organized in a bell curve form and after this I have a high school and Rachel Zung started the same problem but now started intersections so you have a curve and you want to intersect it with other curves you take all the curves and you want to count how many curves intersect two times your other curve and this turned out to be also also have a normal curve if she did a totally different proof of this and I think I think that's it so now we can do questions and you have to wait for a wonderful microphone so I was wondering when you make the cuts to like flatten your surface does it matter exactly where you make those cuts when it comes to analyzing curves on the surface no I mean straight in the sense that it doesn't intersect no restriction the anthropology is a lot of freedom I was just thinking that I don't normally think in a way of a typologist in terms of what edges intersect and connect and what lines or curves can cross so I was just wondering how does this math apply to like physics in everyday life I love the pure abstract aspect of math I mean for me this is and I know it can be applied and I can tell you some of the things I know is that in DNA I mean this appears everywhere these are tools I'm constructing but I just like the beauty of the tools alright if there are no more questions let's give our speaker one last round of applause thank you so much that was wonderful