we're very pleased to have such a large crowd for math for everyone if you stick around at the end of the talk um if everything goes according to plan there should be Pizza in the hallway um so now let's get to the talk um we're very pleased to have Steve trle uh from the University of San Francisco who studies geometry and topology and he also likes computer graphics and long bike rides Steve please take it away all right thanks everyone can you hear me well enough is my microphone picking up sweet sweet okay well thank you all so much for coming to an evening math talk this is always a good way to like self- select a serious crowd is to like host your talk at night so I know y'all are serious so I want to teach you some serious math I want to tell you about something I'm really excited about and what I'm excited about is what's happened in Geometry recently so like anything it's good to talk about history first if I want to talk about recent we should talk about where we came from where did us geometr come from from a long long time ago um when you usually think about geometry you probably first thing that comes to mind is a scene like this like Greece and people drawing lines and circles maybe Pythagoras maybe uid or my boy Archimedes up there um and that style of math the first real like rigorous mathematical subject changed the world and was studied for almost 2,000 years in that form the geometry of the plane perfected by the Greeks and used in architecture and in art and in science in the development of physics by Newton and Beyond um and then and then something changed and then sorry is it not not picking up oh oh jeez let's let's see um oh that sounded better um I don't know it sounds so low um okay yell at me if it's not working again and then we'll okay okay okay great okay so so then um a big change in Geometry came in the 1800s with gaus studying surfaces like this surface near where I live bumpy surfaces not regular surfaces like the flat plane and this period of math taking things other than the flat plane seriously spawned a bunch of beautiful mathematics um we're actually at about the 200y year anniversary of bully ey writing a letter to his dad saying dad I discovered a new universe by that Universe he meant the hyperbolic plane which you'll see a lot of in this talk so the story when people hear about geometry they usually hear about the previous 23 centuries not the most recent two and I want to tell you about the geometry from 1823 until 2023 the stuff that really makes me excited the stuff that made me a mathematician so this math really changed the world one of the first changes we notice is reman came along generalized this idea of new geometry and that was used not that much later by Einstein um in relativity and these days this observation has kind of infected Modern mathematics it's all over the place it's hard I mean I'm biased I'm a geometer so I see geometry everywhere I go but it's hard to look at math without seeing the effect of the last two centuries of geometry I want to tell you about two applications that are near and dear to my heart today I want to talk a little bit about geometry in topology and then geometry in data science and machine learning but before doing that we have to talk about what changed what was the new thing geometry was essentially the same for 23 centuries and in the last two it went from circles on the plane to black holes what changed was curvature thinking about curvature thinking about spaces other than the plane as geometric spaces in of themselves so our question a question gaus asked himself when being a surveyor for the German government is how do I understand these surfaces like the Rolling Hills how do I quantify the fact that these are different than the plane that's our starting point taking that seriously well how do we do that um you see my phone's not clicking click here H go well in ukian geometry what do we know we know that the ratio of what's the circumference of a circle in ukian Geometry it's 2 pi r right like that's one of those things that like I don't know I memorized as a kid I think I learned it before I knew what it meant I could just like spit out Cals 2i R from grade school what it means is as you make a circle larger it's circumference grows linearly with its radius so we're going to think about it as this plot here Cal 2 pi r tell me how quickly space grows as I move away from a point the circumference grows linearly as I move away from a Point area a equals pi r s means area grows quadratically the amount of area I own if I like wall off a region of size R grows quadratically with r but not if I lived on a hill if I was building my castle at the top of some some Hill and and I measured radius along the hill and then I measured circumference around the hill and I asked how does circumference grow with radius on my little medieval castle Hilltown I would find that it grows slower imagine a sphere if you stood at the North Pole and you walked down to the equator and now you measured a circle the equator is a circle all those points are constant distance from the north hole but circumference over our diameter what circumference over our diameter there or circumference over radius the radius on the surface of the Earth is a quarter of the way around the earth right starts at the North Pole goes down to the Equator the equator is one time around the earth so c ided r is four not 2 pi smaller in fact if we did this on a sphere circumference is 2 pi * the S of R this graph tells us how circumference grows slower quantitatively than ukian space and if instead of living on a sphere we looked on a saddle on a little Pringle chip shape like this we could ask how does circumference change with radius and it Gres faster in fact if your space had constant negative curvature it grows with 2 pi hyperbolic sign grows exponentially you can see in this picture these circles get bigger because they have to go up and go down as they move around so how can we use this information we can use this information to assign a number to every sort of surface we can say when circumference grows linearly with radius then we're going to say the curvature is zero when it grows slower the curvature is positive when it grows faster curvature is NE negative now as mathematicians we just don't want like this is like a Vibes definition like I'm just saying like theph looks like this we say it's like this that's not good for math right we want like a quantitative definition not just like a feeling so how do we make this quantitative we can take a limit as we want to measure the curvature at a point on our surface we look at circles and as they get smaller we measure circumference over radius we see what happens also shout out to my geometry students they actually have an exam on this formula today and I told them I was leaving but I was going to talk about our class so good luck everybody um this is how we quantitatively measure curvature measure difference from the ukian plane um we take 2 pi r the ukian circumference of a circle and we measure its difference from the circumference on our space and then divide by a normalizing factor um and this idea of taking curvature seriously taking other spaces seriously allows geometers to claim many things in the world as objects of our own study instead of being stuck studying the plane or tilings on your bathroom wall geometers can now talk about spheres shapes with positive curvature uh corals kale Leaf Saddles shapes with negative curvature um but instead of these real world examples mathematicians to get a first handle on things we often like to idealize our situation so we'll look at ideal a perfect sphere a perfect plane and a perfect piece of negative curvature oops oh my gosh my presentation moved ahead two slides sorry here we go here are the perfect versions drawn by eer um uniform positive curvature uniform zero curvature and uniform negative curvature are three types of geometric space and we can build these in three dimensions as well so how do we build a space of zero curvature in 3D well this one's easy we live there we live in a space that looks like stacks of Cubes everywhere ukian threedimensional space linear algebra's favorite space when you're first learning about this stuff how do we build a space of three-dimensional space a space that we could live in that has positive curvature well if the sphere has positive curvature and is 2D I'd like to build an analog of the sphere and here's one way to do it the sphere what happens when I slice a sphere into slices can see a bunch of circles right there's a DOT of the South Pole gets bigger As you move up till we get to the equator biggest circle biggest disc and then they shrink to the North Pole that's how we can understand the two-dimensional sphere what if I replace those circles with spheres and stack them in the fourth dimension just like a stack of circles in 3D builds a sphere that we can see a stack of spheres in 40 Builds an analogous space which I don't expect anybody be able picture that's hard right stack of spheres in 4D I don't know what that looks like we can write down the math we will write down the math and I'll show you a way to picture it later and then a 3D negative curvature space we could take a space in 2D that we know has negative curvature spin it around build a space in 3D so there's examples of positive negative 0 curvature not just in the plane but in higher Dimensions as well but my goal is to try and seriously engage with these spaces I want us to leave this talk understanding what it would feel like or what it would look like to be inside of one of these spaces and to answer that question we need to talk about what is Vision how do we see objects like when I'm looking at my water bottle I see it in a certain direction in the room this is going to sound really pedantic but I I swear this is important for later when I look at the floor I see the floor down why do I see the floor down you might say it's because the floor is down that's that's also what I thought but being pedantic is good for mathematics being precise is good for mathematics the reason I see the floor down isn't because the floor is down that's also true but it's because the straight line connecting my eye to the floor leaves my eye in the downwards Direction that's why I see the flooor down if the straight line connecting my eye to the floor left my eye in the upwards Direction that'd be weird but I'd see the floor up so what is Vision vision is a geometric sense vision is a way of probing the world with G6 in the real world light comes into our eyes but it's easiest for us to picture kind of the time reverse of that like imagine like imagine your eyes were like Shining light out and seeing things um so where do I see the ball I see the ball where the lines the straight lines connecting the ball to me hit the ball okay I swear it's going to get more interesting I promise um but first let's do some high school trigonometry from that observation once I know that the way Vision Works is by thinking seriously about ukian geometry of straight lines I can ask myself how big does a ball look if a ball is distance D away from me and his radius R how big does it look in my vision and I can find a formula for that I can do the trigonometry and I can find Theta the angle it takes up in my eye is approximately 2 R over D what does that mean that means we can prove mathematically an observation that we all have known since childhood if something's twice as far away from you it looks half as big like I know the people in the back row aren't actually a quarter in tall even though I can cover you with my thumb like I know you're the same size as the people in the front row it's just you're farther away and my brain is doing this formula for me because my brain knows ukian trigonometry because that's where I live the other thing we can prove from this is you can see less than half the sphere from any point if you have a ball in front of you look where our lines of sight are hitting the ball they're hitting it before the equator right tangents coming from your eye form a cone that cone hits the ball and you see less than half of the ball when it's really close to you you see really less like if someone's like right up in your face you don't see like much of their face and someone's far away you see almost half of a sphere and so a good examples here is Apollo coming back from the Moon we're far away from the Earth and you can see slightly less but almost half of the earth you can see the South Pole but you can't simultaneously see the nor North Pole okay but what about if we weren't looking at flat space if we were looking at curved space well in positive curvature circles grow slower than ukia in space space grows slower as you move away from a point and what that means is lines of light straight lines move apart slower than they do in ukan space in negative curvature where circles grow faster than ukian space straight lines move apart from each other faster so what I've animated in the top here I have the Earth and imagine you're standing at that point where all those lines are coming from those lines are like lines of sight those are light rays connecting to you and I'm like a mad scientist in my geometry machine and I have a dial and I'm changing the curvature of the universe as this animation plays out when those lines of site are closing back in I've dialed the curvature of the universe up to positive when I move my dial over to negative lines of sight gd6 move apart I want you to imagine what would you see if you were standing there and I was running my crazy Universe machine what would the Earth look like to you does anyone have a guess as I change the curvature to become negative so light rays spread apart what happens to the Earth in Your Vision shrinks yeah yeah I'm seeing a lot of the right hand motions here people being like absolutely less angle of your vision hits the Earth so the earth looks smaller in your field of view when I turn my crazy dial up to positive and the light rays Bend inwards they move apart slower the earth looks bigger so you'd be standing at that point and the Earth would look like it's oscillating in size it's not the Earth the same size you're not also getting closer or farther but the angle the amount of light that hits the Earth is changing this tells us something about geometry causing almost an illusion and this is our first hint of just how weird the world's going to be when we curve it so let's think a little more seriously about this um it's easier to think in two Dimensions where we can see a space of positive curvature like straight up so here is a positively curved sphere um and I want to imagine that you're standing at the North Pole and you are watching this green ball as the green ball moves away from you at first it gets smaller in Your Vision less of these light rays are hitting the green ball but once it get gets past the equator what happens when it gets past the equator lines of light start moving back together more of them hit the ball again and the ball looks bigger so this is what we'd expect as you move away it gets smaller at the equator it looks smallest and then it starts looking bigger again oh we'll see it yeah um it wraps more than half the sphere at some point not right past the equator actually if we if we if we look at this if the ball is distance X away from you and if the ball is if it's like X away from you or if it's X away from the South Pole it looks the same size and we can work that out we can work out spherical trigonometry so anyone who's in who likes trigonometry here's a homework assignment for you um work out the formula on a sphere for how big a disc looks at distance DOA don't worry I did your homework for you I made you come to a talk at night but I'm not actually gonna make you do homework here's the answer it's approximately 2 R not over D but 2 R over the S of D so we can calculate this quantitatively and so one more thing to look at one weird thing here when I'm doing this calculation how much of the sphere are we seen when we're really far away from the sphere and the lights bending back in those two rays of light those yellow lines are hitting the sphere more than halfway around right they're hitting the sphere on the back side because they've curved around and bent in well what does that mean well instead of looking at a green ball moving away from you let's imagine what happens if you were looking at your friend if you were to walk away from your friend in a positively curved space at first just like the green ball your friend's face would shrink but once you pass the equator your friend's face would grow even though they're getting farther away from you they'd look bigger but worse you see more of their face right if light bends around and hits a sphere on the back side light bends around and hits your friend's face on the backside and the farther you go the more of your friend's face you see what happens when your friend makes it to the South Pole when you're if you're at the North Pole and your friends at the South Pole any line of site you look on what do you see you see your friend's face when you're friends at the South Pole their face unwraps and engulfs you it fills every point of the sphere if you look up you see your friend's hair if you look back you see the back of your friend's head if you look to the side you see your friend's ear because all lines of sight all lines on the sphere that start at the North Pole hit the South Pole your friend becomes the whole world so that's pretty terrifying but it'd be way more terrifying if I was bored and I decided to animate it so in this computer program what I've done is I've taught the computer program the laws of geometry in a positively curved space I've taught the computer program do you know how what rate tracing is like when people are like making a video game or like making a movie they simulate it by like shooting rays of light out into a computer scene to draw the picture I wrote a ray Tracer in positively curved space and I put a friend in there and so what we're going to do is we're going to walk away from our friend at constant speed no matter what you see your friend is just getting farther away from you the whole time at a constant speed I promise at first their face looks a little smaller that's good but then their ears start to wrap which is kind of not the best and then their face kind of appears inside out that's a good homework problem think about why their face looks convex before their ears engulf your world and now your friends at the South Pole and you cannot get away from your friend your friend's actually as far away as possible this is the farthest they could ever get from you and yet they look the closest I watch this video anytime I want to be thankful that I don't live in positively curved space What would the Apollo Mission see if they were back from the Moon in positive curvature they would see more than half of the earth lines of lights that manage to come apart and reconverge due to the positive curvature allow you to see the North Pole and the South Pole at the same time so if you if I put you on a crazy spaceship and I dropped you off in space and you looked at a planet and you saw the North Pole and the South Pole at the same time that's our hint that we live in positively curved space if you see only one of the poles like a normal world that's like flat space so this is the sort of intuition we want to develop from these sort of pictures light rays Recon converging let you see more of a sphere than you expect and so here's a collection of a bunch of balls in positively curved space all these balls are the same size and they're just moving away from you and towards you at a constant speed but what you see is the balls in the background look the biggest because they're near the South Pole they're like your friends they're like your friend's face they're about to fill your world so judging distance would be kind of different ult in a positively curved space instead of me being able to cover up the back row with my thumb the back row would look like gigantic huge people the middle of the auditorium would look the smallest the people near the equator you would not be able to get away with sleeping in class in positively curved space by sitting in the back row you'd have to sit in the middle but what happens if I take my crazy curvature dial and I whip it over from positive space to negatively curved space what happens in negative curvature is lines of light move apart faster things look smaller all these balls you see not too far out like they're almost like pixelated size in the background there yeah like they shrink really small all these balls are actually closer than some of the balls in that last scene the farthest balls in this picture are like five meters away from you um they just look really small there's a lot of balls within 5 met of you that tells us something else that tells us there's a lot of space within 5 met of you why is that well circles remember how did circles grow in negative curvature they grow faster than ukan space space grows exponentially fast in negative curvature so there's all this room for activities in negatively curved space thousands and thousands and thousands of balls and what do you see not not only the balls look small and we can quantify it this time using hyperbolic trigonometry instead of ukian or spherical trigonometry um but how much of the ball do we see we see less than half right if lines of sight are moving away from each other when they hit the sphere tangent you'll never see half of the sphere you'll always see less than half of the sphere you can actually work this out turns out you can see pi times the radius of curvature of space squared for people who like doing Romanian geometry exercises there's your homework gr students um but here's the Apollo mission in negatively curved space instead of seeing both poles at once Indonesia fills an entire hemisphere because you can see very little of the earth as the light rays are racing apart and as space grows exponent entially As you move away from a point you can build lots and lots and lots of things in negative curvature here's showing you how many cubes fit nearby to a central Cube as you build out instead of a ukian space where it grows with the cube of space in three dimensions this grows exponentially here's what I look on the inside you see lots and lots of Cubes so this is your sense if you wake up in a space and things look really really really small in the distance and there's lot of them that tells you you're in negative cature these are kind of our this is like getting our bearings for positive versus negative we have some Clues right if if somebody's head wraps around you you know it's positive if something looks giant when it gets far away you know it's positive if there's like a crap ton of things you know it's negative if things are really tiny you know it's negative but these aren't the only options I'm I I don't actually remember all my literature classes but I remember I Shakespeare said something like there's more ways for space to curve heratio than you dreamt of in your geometry there's more than just three ways for space to curve it doesn't just have to be all positive zero or all negative what could we do I could take a space of negative curvature and I could stack them on top of each other I could build a stack of two-dimensional negatively curved spaces this space is going to have some other weird properties well it's going to have negative curvature in two directions because I stacked negatively curved spaces but in the stacking Direction it's actually going to look flat there's going to be directions here there's going to be planes up and down that are tiled by squares and cross sections that are not the space is called by the very uncreative name H2 crossr H2 meaning hyperbolic space negatively c space and R meaning the real line that's the stacking Direction so if a space has negative curvature in some directions and zero curvature in other directions should try and imagine what we're going to see if we look in the negative curve directions things should get small really fast right that's what happens in negative cature if I look in the flat directions things should get small at a normal speed at what I'm used to because I'm from Flat space so it's fun to try and work that out but it's better to just simulate it um in this direction you look the balls get really really small really really fast but they're also stretched out into ovals and if you look in this direction the balls don't shrink so fast you can pick out inside of this weird space you can tell what directions are positively or what directions are negatively curved and what directions are zero curved by looking you can just look at the size of the balls and that gives you a sense instead of stacking negatively curved spaces we can also stack positively curved spaces to get another example when I stack positively curved spaces what am I going to see I'm going to see One Direction Where stuff looks normal and then in the positively curved Direction I'm going to see things getting really big maybe wrapping around the world this is actually the space I would want to live in least out of all of these things um it's a lot of weird properties but in this direction we're looking in the direction where there's positive curvature balls are getting stretched out and wrapping around your face and now as we rotate we'll look in the direction of zero curvature we'll see the balls looking normal coming out of the top corner of the screen here there we go that's the direction with zero curvature where the balls look normal so that's how good detecting curvature with your eyes works you can look at a space once you've had a little training in this and you can just tell by what it looks like what the curvature is positive negative zero depending on the direction but you can do even weirder things than this instead of just stacking you can stack things in strange ways my personal favorite three-dimensional space is kind of like a twisted stack of planes it's called nil geometry and what I'm drawing here is the Rays of light from your eye imag you're that white ball in the middle these curves are the straight lines in nil geometry these are the lines that light would come to your eye we see sometimes all the straight lines are collected together in One Direction they're all kind of converge like right there they're like a laser beam in one direction and then sometimes they're spread out all over the place what does that mean that means in some directions there's positive curvature they're all kind of together and in other directions there's negative curvature they're all splayed apart this space is very confusing to get used to the first time you see it so I want to do just one thought experiment exercise here I want to try and imagine we're on the Apollo Mission we're coming back from the Moon and we're looking at the Earth and we imagine what we would see so you're the white ball again these rays of light are going from you to the Earth and back what what happens in the middle well right in the middle of your field of view that middle ray of light hits the Earth so you see the Earth right in front of you that makes sense but there's some other rays of light that hit the earth too actually in your peripheral vision those orange rays of light the ones that are farthest away from your field of view those are curving around and they hit the Earth right so what do I do I see the Earth in the middle of my vision and I also see the Earth in the peripheral of my vision but what about in between what's happening with those yellow lines it missed the Earth does anyone have a guess what it would look like if I were in this crazy world what do I see for the Earth yeah like a bullseye absolutely I see the Earth right in front of me and then I see a whole ring where light misses the Earth and then in my peripheral vision I see the Earth again that's almost too weird to picture so let's plug this into our Ray Tracer as well here just like your friend we're just going to move away from the earth at constant speed no matter what it looks like and eventually we're going to stop when we get the Earth to the position that we were just simulating so at first the earth looks smaller as we'd expect we're moving away from something of course it gets smaller now it looks like it's getting bigger but we're still moving away uh oh now it looks like a bull absolutely right what happens is some rays of light that we're leaving very very far away reconverge due to positive curvature and rehit the Earth you see the Earth in the far part of your vision you see the Earth in the middle of your vision and the light rays right in between skipped on past the world I would not want to fly a spaceship here here's what it would look like in the asteroid belt um too many balls so you can see what direction you can see where we were seeing those Rings right that direction has positive curvature and then you can see in other directions the balls look really really really small instead of being magnified those directions have negative curvature and in general there's lots of spaces like this there's lots of spaces that have some directions positively curved and some directions negatively curved so here's another one the space has the uncreative name SL2 R because well it's the group of matrices SL2 R this is the space of 2x two matrices with the terminant one as a geometric space it has a metric on it that has positive curvature and negative curvature and this is what it looks like the space where it looks all spiral worm holy that's the direction of the rotation matrices in this space this is what I mean there's geometry everywhere guys even inside matrices here's another space called Soul geometry um this also has positive curvature in some directions where balls are big and stretched out negative curvature in others so this is the big wide world that we didn't grow up in because we grew up in flat space but this is the big world of mathematical possibilities out there all these sorts of mixes so this in my opinion this is the big advance that we made in geometry in the last two centuries was going from thinking about the plane to treating all of these as equal citizens in the world of geometry all of these are spaces that we need to understand and that perspective has allowed us to do a bunch of really cool math which is what I want to do with the rest of the talk is tell you about two spots outside of geometry that this sort of thinking matters oh first sorry I forgot Shameless plug if you want to hear about how to make these pictures if you're like wow those pictures look cool but like I'm a details guy and I want to know exactly what's going on I got great news for you we have like 140 page paper that explains everything so um you can check it out it's called Ray marching thirst in geometries um it's a collaboration with Remy kulan saeta monoto and Henry seerman and myself but okay back to the fun stuff so application of geometry my personal favorite stuff in math is when we think really hard about something for one purpose and then it turns out that that stuff is important somewhere where you least expect it somewhere totally different like we made all of this progress thinking about what could space be and then of course that's useful for understanding what space could be that's what I was after but it's really it really gets me going when that stuff shows up somewhere else so the first application I want to show you is thinking about topology thinking about shapes in general but not thinking about their metrics not thinking about SP um so here's a question that you would study if you were a topologist you'd be interested in surfaces you'd be interested in curves and something you'd like to do is know when is a curve stuck on a Surface versus when it's not um a curve is stuck if you can't like shrink it to a point or pull it off the surface or equivalently if you cut along the curve some curves disconnect your surface and some don't we want to try and understand all the curves cures that don't like just cut off a pcer surface that are kind of stuck on there um back in the 1890s Henry ponre realized that this was a really good way to think about surfaces to think about shapes that if we could understand the curves on our space we could understand our space very well but of course the problem is how do we find all the curves on our space Here's a loop that's stuck on the Taurus here's a loop that's stuck on a donut so here's one Loop but like what else is there there's probably a crap ton of loops on the donut and I want to understand all of them my goal is to find them all and list them all every single Loop that's stuck on the doughnut any curve you could possibly draw on a doughnut shape that when you cut along it the donnut doesn't fall into two pieces that seems like a hard problem because it starts with any curve and there's lots and lots and lots of curves that you could try and write down but here's a cool idea cool idea is maybe he's not going to make sense right away but I promise it will take your take your donut find one Loop that's stuck and we're going to unroll our space along that loop we're going to cut it and we're going to like unroll so here I can cut the Taurus and unroll it into a cylinder the Taurus there's two stuck Loops that I can see in my picture here the yellow Loop and the green Loop that are stuck on the Taurus of course there's probably much more but I know about these two I cut along one of them the yellow Loop unrolled it now there's only one Loop that's stuck I have a cylinder now what am I going to do I'm going to cut along the green Loop and unroll that into the plane now those two stuck Loops are gone and my space changed it changed from the doughnut into the plane seems like a weird game to play but said promise there's something cool going on here what's going on is this plane keeps track of all the loops if I have a loop on my donut that doesn't go all the way around the green Loop or the yellow loop I can draw it in one square in the plane but what happens to the green loop after I've unrolled the green Loop it goes from one square to the next Square because I had to unroll it as I cut the surface if I go around the green Loop twice when I unroll that it goes two times in the green Loop Direction the yellow Loop also got unrolled when I unroll the yellow Loop it goes in the vertical Direction what about this loop I didn't even cut along this Loop this is some crazy weird spiral loop on my Surface but what ises it do it goes one time around the green Loop and one time around the yellow Loop and so unrolled it goes one time in the green Direction and one time in the yellow Direction actually every single Square in this picture captures some number of times around the green Direction and some number of times around the yellow Direction and so the way we can find all the loops on the Taurus is to unroll the Taurus and realize the collection of all the possible Loops looks like the ukian plane it looks like a grid and then we can ignore the loops themselves and what we do in topology is we say we can study this grid this grid has all the information of my Surface I wasn't originally thinking about geometry but the ukian plane showed up when I tried to think about loops when the Taurus unrolled into a square and for people in the no we call this the fundamental group in topology um this is a way of calculating the fundamental group of your space you unroll it and you look at the pattern that unrolled and that's all great it worked for the donut but how do I unroll something like this well I can try my same trick I can find some loops on the surface that are stuck and I can try and cut along it right that's what worked for the doughnut I found two Loops that were stuck and I cut and then the doughnut fell apart into a square tiled the plane held all the secrets of the doughnut here how many Loops are there there's well I have two doughnut holes each doughnut hole has two Loops there's four Loops what happens when I cut on all these Loops how many sides am I going to get on the Taurus I cut along two loops and each Loop gave me two sides I got a square here I have four Loops so I cut along it and I get an octagon my shape falls apart into an octagon which is great this says oh my gosh all I have to do is is unroll with octagons unroll with octagons in this Direction that's going to capture all the loops that went that way but I also need to unroll with octagons in this direction right I need to unroll with octagons in every direction because I'm trying to unroll my whole shape this seems bad how many octagons need to fit around each vertex on the Square on the Taurus we we had two curves meeting at a point so four squares met there here we have four curves medic at a point so we can look at that point and count how many regions there are eight octagons need to fit around each vertex if a single vertex has two Pi worth of angle what do we need we need 2 pi over 8 to be the angle of our octagon we would need our octagon to have angle 45° which is like not the greatest because like octagons don't have angle 45 degrees right like ukian geometry sad in fact in ukan space you can't even fit three octagons around a point and I want to fit eight so old me would have said old me being mathematicians of the old geometry before 200 years ago would have said impossible can't do it I can't even fit three octagons definitely can't fit eight this approach won't succe new mathematicians post the 1800s now that we take curvature seriously say what have I learned I've learned that in flat space there's not enough room to put eight octagons around every Point what should I do should I make space positive or negatively curved if I want to fit more octagons negatively curved I should make it negatively curved there's more room in negative curvature so great let's do that let's let's pick the negative one you can do some math and you can figure out sure there are octagons that fit there's enough room to put the octagons when you put negative curvature and here's what you get you get a tiling of the hyperbolic plane by octagons I know this looks weird all these octagons are actually the same size this is kind of like the Mercator Projection but for hyperbolic space you know how like Greenland is like actually 14 times smaller than Africa but they look the same size on our like childhood map all these octagons are the same size even though the ones on the outside look tiny so we took the Taurus and we unrolled it into the plane we can study a Taurus instead by studying the geometry of flat space we took the double donut and we unrolled that into octagons octagons we needed to fit lots of them so we needed negative curvature actually for every other surface we can do the same thing thing a theorem from the early 1900s called the uniformization theorem says this always works you can study all surfaces Surprise by using geometry you can the sphere is the sphere that has positive curvature the Taurus unrolls into the plane and everybody else everybody else who's really complicated unrolls into negative curvature because it needs a lot of space so what happens in 3D in 3D we could build weird Twisted three-dimensional spaces just like you could build weird surfaces one of those spaces is the three Taurus instead of a square the space un rolls into a cube and you can unroll that into a tiling of space we can study a space called the three-dimensional Taurus using geometry it unrolls into flat Space full of Cubes we can also find weird threedimensional spaces that unroll into something like aedra where you need to put lots of jedra around now you can't tile space with do decahedra they're too big they don't fit so you need to make space negatively curved and if you make space negatively curved then you can fill it with do decahedra but this isn't all in 3D weird things happen in threedimensional space you could also build a space that's a stack of surfaces what geometry would this have if I have a stack of surfaces each surface unrolls into the plane right each surface unrolls into the hyperbolic plane a stack of surfaces unrolls into a stack of hyperbolic planes we need other spaces a stack of torai with a Twist unrolls into a twisted stack of planes that crazy space with the bullseye shaped Earth we saw earlier here's an unrolled twisted donnut and all of this math led to an amazing conjecture by thirst in the 80s saying that all threedimensional spaces could be understood this way could be understood using geometry that conjecture had a million doll bounty on it one correl of that conjecture was worth a million dollars it was a problem that had been open for about a century and it was solved using geometry by proman so this is an example to me of something that like starting out doesn't sound like it has anything to do with geometry but we secretly found geometry inside of the problem by unrolling spaces and that's the sort of math that really excites me so at the end I want to show you one more quick application of of this one that one that application has been going on for a while want to show something I'm personally very excited about and invested in these days finding geometry in high-dimensional machine learning contexts so here's what I mean instead of Starting by thinking about geometry if I'm thinking about data if I'm thinking about relationships between data usually where I'm starting is by thinking about a graph where the vertices are maybe data points that I have of some sort or some sort of information and edges are relationships between them so these are the fundamental objects I'd be thinking about if I was encoding data in a graph one thing you want to know is maybe how similar two things are I want to know these two points in the graph I want to know how similar they are how close they are together in the graph and the way we measure that is by something called the graph distance here's how I measure distance in a graph I find all the different paths I could draw in theph so here's a path of length four maybe the distance is four I can definitely get from Green Point to Green Point with four units of distance but I can also do it with three the graph distance is going to be the shortest possible path that you could go on here in this graph the shortest path between the Green vertices is length two so the graph distance is two and so the math question here is that's hard to compute right like if you have a graph like imagine this thing had 10,000 vertices in it and I give you two of these vertices and I ask you how far apart are they how many paths are there between them which one's the shortest path um or other quantities you might want to learn about a graph um things could be hard to compute but we can bring geometry into the problem we're measuring a distance let's try and make this distance easier to measure I could take a graph like this and sometimes I could find a nice way to put the graph in the plane here I've embedded My Graph nicely into the plane all the edges of My Graph line up with the plane very nicely now what could I do if I want to measure the distance between two points on this well this is the real distance but that zigzag curve is actually pretty close in length to this pink curve right so if I want to know approximately the distance instead of finding the distance of all Curves in this big complicated graph I could do one Pythagorean theorem between the end points if the graph was in the plane problem not all graphs fit in the plane this graph here if I take a graph that looks like this and I try and put it in the plane and I try and measure the distance between these two points the real distance in the graph is they're far apart but what does the plane say the plane says they're close together that's not good there's some mismatch between my data My Graph and the geometry it's sitting in there's too many vertices on my graph to fit in the plane what do I need I need more room I need room for my graph to spread out and feel comfortable I need negative curvature in a negative curv space My Graph fits perfectly fine and I can actually measure the distance on the graph by measuring the distance in my curved space so this is good news that graph what did I do I put it in the plane and I measured the distance everything works great this graph didn't work in the plane sad but we know why too many points needs more space what do we do give it more space now it works but this has trouble too because my original graph doesn't want that much space my original graph was happy at home in the plane my new graph loved having lots of space to spread out but my old graph can't live there this ISS there's not a one-size fits-all solution to this problem if I want to use geometry to speed up my understanding of graphs I have to for each graph I need to like look at that graph and say like where are you like to live and then I need to put you there that's hard because I have to look at the graph and figure out where it lives that's what I'm trying to avoid but what have we learned we've learned that some graphs like zero curvature and some graphs like negative curvature and we know from geometry there's plenty of spaces with both so just to see that this is a real world problem like not just like a mathematical problem here's a real data set where the vertices are legal cases in England and Wales and the edges between them are citations it's like when one case CES another case um there's some important cases that are cited by everything and then there's like tree like pieces of less important cases and here's a little piece of that graph this little piece of the graph has some trouble it doesn't want to live in ukian space because part of it looks like a little tree part of it spreading out a lot the part that looks like a grid is totally happy in ukian space in flat space but not the tree but it also doesn't want to live in hyperbolic space because now the tree fits nice but the grid does not so this tells us we can't even just look at every graph and ask you where you want to live some graphs don't want to live in any of those options but studying geometry over the last two centuries has taught us about all these spaces that have different curvature in different directions so now we know where graphs like this should live they should live somewhere like this they should live somewhere that has some negative curvature for the tree like part to spread out and has some flat spaces for the gridlike pieces to feel at home and there's spaces like these there's higher dimensional analoges of a stack of hyperbolic planes where you can actually make this work so this is a project I've been involved with for the last couple years with a big group at heidleberg and now Max plank leish um so if you want to read more about this stuff check out some of our papers there as well or ask me questions after the talk anyway that's the story that's both the geometry I love and two big use cases we found for it thanks so much for coming on the evening all right thanks very much Steve for a fantastic talk we do have time for a couple questions can you can you have a guess an idea of the dimension of the space where you fits the oh that's a good question um was can you guess the dimension of the space where the graph would fit yeah like depending on the size of the graph how big of a dimension space you need for the examples we've been doing we've actually just been like relatively low Dimensions we because we've been working with fixed data size so I don't I don't have any ASM totic like well you in in hyperbolic space you need two right yep yeah most of our guyss are putting in like 30 they might have lots of different flat and tree like pieces that like need different uh symmetric spaces particularly symmetric spaces for the simplec group okay let me let me see if people other people who aren't coming to dinner have questions too so anyone have questions all right if you have uh any questions Steve will be here yeah yeah feel free to just come up come down all right um let's have another round of applause for a fantastic talk