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LAURENT SALOFF-COSTE: Hello, everyone. I'm Laurent Saloff-Coste, the Chair of the Department of Mathematics at Cornell. And it's my pleasure tonight to welcome you to this lecture by Jeffrey Weeks.
David Gabai will introduce the speaker in a minute, but I just wanted to start by telling you a little bit about the conference we are having this week and a man, [? VP ?] on this conference, which is William Thurston, whose picture is in the corner. He [? dealt ?] [? with ?] the faculty at Cornell for the last 10 years of his life. He passed away two years ago, approximately. And he was an extremely original mathematician who contributed to make great advance and to change the vision of many of us for mathematics.
He was very aware of and very interested by communicating with others. Many of his former students are here. Jeff is one of them. And many of them also share this desire to communicate with others, perhaps no one more than Jeff. I hope we all enjoy the talk tonight, and I would like to pass the microphone to David Gabai.
[APPLAUSE]
DAVID GABAI: Thank you very much, Laurent.
So it's really just an incredible pleasure for me to be able to introduce Jeff Weeks. We've known each other since we were graduate students.
And actually, I'd like to just add something about Bill Thurston's interest in communication. He's actually always had tremendous interest in teaching and education-- a readiness. As an undergraduate at New College, he and his classmates went out and worked with disadvantaged kids. As a graduate student, he led the teaching of other TAs to become better and more enthusiastic TAs.
As a parent of young kids, he would go in every year to the classroom and teach kids something mathematical. And at Princeton Davis and Cornell, he really had his innovative course on geometry and the imagination.
So Jeff Weeks received his PhD from Princeton University, and as mentioned, Bill Thurston was his advisor. And as a student, he discovered what's now known as the Weeks manifold. And after 25 years of just intense effort by many people, it was discovered that the Weeks manifold is the smallest volume hyperbolic 3-manifold. And Jeff's work played a crucial role.
And Jeff, during his thesis and some time after that, developed this computer program will SnapPea, which is this phenomenal program which enables one to visualize and get information about hyperbolic 3-manifolds. And SnapPea, essentially, created a new subfield of mathematics whereby a practitioner would stare at SnapPea's pictures and data for a long time. And then they'd come up with a geometric insight and then spend a couple years in trying to prove a theorem. I've experienced this personally.
[LAUGHTER]
DAVID GABAI: So as you'll see in a moment, that Jeff is not only a master teacher but a very innovative in his technology and methods of teaching and transmitting information. In particular, his website, geometrygames.org, has this cornucopia of computer programs which are fun, and interesting, and educational, and they're also for free.
And there are such things as computer chess, a flight simulator around various manifolds, drawing a four-dimensional drawing program. And there's a program for understanding wallpaper patterns, which has practical uses. I used that program to design the floor of my bathroom.
[LAUGHTER]
DAVID GABAI: So Jeff also wrote this amazing book called The Shape of Space, which without a huge effort one could nevertheless gets some intuition and understanding of some fairly sophisticated mathematics. And I'd once owned a copy of The Shape of Space, but then I lent it out. And then I discovered that the Princeton University Library has five copies, which might be unprecedented for a math book.
[LAUGHTER]
And I also discovered that these five copes circulate a lot, which might also be uncustomary for a math book.
[LAUGHTER]
So finally, let me mention three of Jeff's honors.
He won the Conant Prize of the American Mathematical Society for expository writing. He won a MacArthur Foundation Fellowship. And so he perhaps the singular honor-- I know of no other mathematician that was so honored-- is that he was introduced by Bill Thurston at a major mathematics conference when-- at Bill's 60th birthday conference that Bill gave this beautiful introduction to Jeff.
So to please give Jeff Weeks a warm welcome for his lecture of the Shape of Space.
[APPLAUSE]
JEFF WEEKS: Thanks, Dave.
OK. Thank you, all. Can people hear me OK? Good to the back? Super.
OK. So on the one hand, it's sad that Bill can't be here in person for the conference at Cornell this week because we really all miss him both as a mathematician and also as a human being. But on the other hand, it's a joy to see that even though the man is gone that his ideas live on so vibrantly. So this evening's talk is dedicated to the memory of Bill Thurston with thanks for all that he taught me personally and for the legacy that he's left for all of us.
OK. Bill taught us a lot about three-dimensional spaces, so this evening I'd like to show you the connection between his discoveries and the shape of the universe. OK.
So here's a picture of deep space taken by the Hubble Space Telescope. There's an interesting story behind this picture that normally for researchers it's very, very difficult to get observing time on the Space Telescope, that literally every minute is precious. But the director of the project, as one of the perks of his job, was given a full two weeks of observing time to look at whatever he wanted.
So people are naturally curious to know what he was planning to observe. There are lot of interesting possibilities. For example, star formation, or black holes, the evolution of galaxies.
But do you know what he decided to look at? Nothing. He picked a little tiny patch of black sky near the handle of the Big Dipper where nobody had ever seen anything.
Now, with an ordinary camera, you might take an exposure of 1/60 of a second. He took 10 days of exposure on that little tiny patch of black sky, and this is what he saw. OK? But without Bill.
[LAUGHTER]
AUDIENCE: So Bill wasn't there?
JEFF WEEKS: No.
[LAUGHTER]
Not at the time, anyhow.
So anyhow, as you can see, the universe contains galaxies of various shapes and colors. A typical galaxy contains about 100 billion stars-- not 100 million but 100 billion. It's less than the federal budget deficit, but it's still a lot.
In this picture, if you count them, you'll find several hundred galaxies here. But remember, this is a patch of sky about the width of this dime as viewed from 75 feet away. So if you were to look at the whole sky at this same resolution, you'd see, coincidentally, about 100 billion galaxies out there.
So we're looking out into space. We're seeing roughly 100 billion galaxies, each with roughly 100 billion stars. So for sure, the universe is big.
But is the universe truly infinite, or might there be some limit to the volume of space, some limit to the number of galaxies? In other words, might the universe be finite?
Well, there's an old argument about this due to the Greek philosopher Archytas, who said, well, if universe were finite, we could go to the boundary, stick our hand through. Maybe I shouldn't do this in case an alarm sounds-- stick our hand through, and we'd expect to find something on the other side.
Now, Archytas interpreted this to mean that the universe must be infinite. But if you think about it, what it really tells you is you don't expect the universe to have a boundary. And being infinite and not having a boundary are two different things. And it turns out that they're actually quite a few ways to have a finite universe with no boundary, so that's what we'll be looking at tonight.
The plan for this evening is to first look at some toy two-dimensional universes. Once everybody's happy with the main ideas there, then we'll come back and look at the real three-dimensional universe again.
OK. So our toy two-dimensional universe is this patch of sea floor. It's finite. And at first glance, it looks like it has a boundary, but really it doesn't. When the flounder swims off to the right, she comes back from the left. And when she swims off the top, she comes back from the bottom.
So this sort of universe is called-- oops. Yeah. Oh, OK.
And this reminds me here of something I meant to mention. When I ask some questions here, especially near the beginning, if you hold a PhD in mathematics, you are prohibited from answering.
[LAUGHTER]
OK.
So this space is indeed a torus. And so a torus-- side-to-side, the torus is like a cylinder. So the flounder can swim off to the east, swim all the way around, and come back from the west. But simultaneously, the torus is like a cylinder going top-to-bottom as well, so the flounder can swim off to the north, swim on around, come back from the south.
Now, if you take a real piece of paper and you connect the top edge to the bottom edge, glue it up, and then try to connect the left edge to the right edge, you just get a crumpled mess. So we're going to stick with the software instead.
So, OK. It's a toy universe, and with toys, you should play games. So we'll play some games here.
I'd like to start with tic-tac-toe. And to get warmed up, I propose that we play with all of us against the computer. So where would you like to go first? And again, PhDs, zip those lips.
[LAUGHTER]
[INTERPOSING VOICES]
JEFF WEEKS: Don't we have more ringers [? for ?] [? you? ?]
OK. Center? Corner?
[INTERPOSING VOICES] Center? Corner?
JEFF WEEKS: OK. Well, we got the corner there. But I think that this should be OK. Because remember, side-to-side, the torus is like a cylinder. And we can take a cylinder and rotate it around. So we can take our torus and rotate it around as well. Or if we want, we can stop one square short.
Now, our torus is also a cylinder going this way. So we can take it and rotate it around this way. So there we go. Rotate it around or stop one square short.
So people who wanted the center, this is good now, right?
[LAUGHTER]
OK.
Where would you like to go next? We could go in the center again if you want.
[LAUGHTER]
[INTERPOSING VOICES]
JEFF WEEKS: OK. Good.
OK. Crucial move here. So where do we need to go here?
[INTERPOSING VOICES]
JEFF WEEKS: Yeah. Top right because we're being threatened by the Os there, so we better go in top right and block the Os. And-- oh good. So now we have a win.
[MUSIC PLAYING]
OK.
[LAUGHTER]
Very good.
What I would like to do now is switch-- take the computer out of this game and erase the board. I'd now like to propose a game between the-- on one team, the students from Ithaca High School, the middle school, and any other citizens of Ithaca who don't work here. And against our augustine, learned scholars who've come for the conference.
So could I please have one volunteer from each group? Don't worry. If you're the volunteer, you don't have to decide the move. You just need to get your group's consensus and mark it down. So could I please have one other student or other non-PhD-holder, non-graduate student?
Yeah? OK. Coming down.
So to use the program, by the way, you can tap with one finger to make your move like so-- I'll have to undo this now-- or you can-- oops-- you can drag with two fingers to scroll the whole board.
[LAUGHTER]
Yeah, they're big steps on that side.
[LAUGHTER]
OK.
AUDIENCE: Does he get to go first?
[LAUGHTER]
JEFF WEEKS: Of course he gets to go first.
[INTERPOSING VOICES]
JEFF WEEKS: OK. So you can get your team's consensus.
[INTERPOSING VOICES]
JEFF WEEKS: OK. And now I need a volunteer from the professoriate here. [? Saul-- ?] OK.
[LAUGHTER]
[INTERPOSING VOICES]
SAUL: I plan to-- I plan to lose with good grace.
[LAUGHTER]
[INTERPOSING VOICES]
SAUL: Oh, yes.
[LAUGHTER]
Professoriate, where should I move?
AUDIENCE: The [? far ?] [INAUDIBLE].
[INTERPOSING VOICES]
JEFF WEEKS: OK. Students?
[INTERPOSING VOICES]
[LAUGHTER]
AUDIENCE: The top.
Not that way.
[LAUGHTER]
[INTERPOSING VOICES]
JEFF WEEKS: OK, guys.
[INTERPOSING VOICES]
JEFF WEEKS: I'm counting on you.
SAUL: I assume, yes?
AUDIENCE: Yes!
[LAUGHTER]
AUDIENCE: I could lose now.
[LAUGHTER]
[INTERPOSING VOICES]
JEFF WEEKS: OK.
[INTERPOSING VOICES]
AUDIENCE: Scroll.
JEFF WEEKS: Yeah.
[LAUGHTER]
[INTERPOSING VOICES]
AUDIENCE: Oh, my god.
JEFF WEEKS: OK.
[LAUGHTER AND GASPS]
Not looking good for the students here.
[INTERPOSING VOICES]
[CHEERING AND APPLAUSE]
JEFF WEEKS: OK. Thank you for [? grace. ?] Thank you.
OK. While we're here, would you like to try one of the other games?
[INTERPOSING VOICES]
JEFF WEEKS: Hmm? What? Chess? OK.
I think for chess, we should probably play against the computer because it speeds the game up by quite a bit.
OK. So in chess here, the way it works is that we're playing the white pieces. We go first. The computer is playing the black pieces, goes second. All the pieces move as in ordinary chess except for the pawns. Which in the interest of symmetry, if a pawn is not capturing, it can move one square up, down, left or right. If it is capturing, it can move one square on any diagonal.
OK. Where would you like to go first?
AUDIENCE: [INAUDIBLE]
JEFF WEEKS: Pawn down one. OK.
AUDIENCE: [INAUDIBLE]
JEFF WEEKS: Oh, it doesn't have that rule.
[GASPS]
[LAUGHTER]
JEFF WEEKS: Ah ha OK. So.
[LAUGHTER]
Take the queen. Oh, OK. We'll take the queen. Take the queen here.
[INTERPOSING VOICES]
JEFF WEEKS: OK. Now what?
AUDIENCE: Take the bishop.
JEFF WEEKS: Take the bishop. With the pawn or the rook? Or the king? The pawn?
AUDIENCE: The pawn.
JEFF WEEKS: Oh. Actually, this is pretty good. We've actually lasted a fair number of moves into this.
[LAUGHTER]
We could go on the offensive here and put the king in check. Shall we do that?
AUDIENCE: No.
JEFF WEEKS: Oh, nope. That was a dumb move.
[LAUGHTER]
OK
AUDIENCE: Move the other rook up and [? trap ?] that way.
JEFF WEEKS: Oh, good idea.
[INTERPOSING VOICES]
[LAUGHTER]
JEFF WEEKS: OK. Well-- so-- and anyhow, you get the idea.
[LAUGHTER]
Actually, if you want, at the end I'll put the address up. You can download the program and play this in the privacy of your home without anybody to watch how it goes.
[LAUGHTER]
For now, just real quickly, let's do a crossword puzzle here. I tried to find a justification for why to do this thing. It's just because I like the crossword puzzles. But this one's too easy-- a three letter word for father. I think in this group, we can go to a more challenging puzzle. This one, for example.
OK. Pursue-- five letters.
AUDIENCE: Chase.
JEFF WEEKS: Chase. OK. How about fruit used to make wine?
AUDIENCE: Grape.
JEFF WEEKS: Grape. OK. How about citrus orchard?
AUDIENCE: Grove.
JEFF WEEKS: Grove. OK. Genetically identical copy.
AUDIENCE: Clone.
JEFF WEEKS: Oops. I have to spell it right here.
[LAUGHTER]
Oh, there we go. We got it. OK. Large wading bird or construction equipment.
AUDIENCE: Crane.
JEFF WEEKS: Crane. OK. Fad.
AUDIENCE: Craze.
JEFF WEEKS: Craze. Nation that invented printing.
AUDIENCE: China.
JEFF WEEKS: China.
[LAUGHTER]
Old buddy.
[INTERPOSING VOICES]
AUDIENCE: Crony.
SAUL: Crony. There we go. Oops. We're too far down here. Not poetry.
AUDIENCE: Prose.
JEFF WEEKS: Prose. Public square.
[INTERPOSING VOICES]
JEFF WEEKS: Plaza. They're getting easier, yeah. Sauce for meat.
[INTERPOSING VOICES]
JEFF WEEKS: OK. There we have it.
OK. I'd like to ask you a question now. Let's go back to our flounder and zoom out.
Now, my question for you is in this torus, what does the founder see when she looks straight ahead?
[INTERPOSING VOICES]
JEFF WEEKS: Yeah, she sees her tail. So her line of sight wraps around. She sees her own tail like so. So in effect, this flounder has the illusion of looking straight forward and seeing another copy of herself sitting out there one unit in front of her.
How about if the flounder looks to the north? What does she see?
[INTERPOSING VOICES]
JEFF WEEKS: Yeah. Her tail again. Her line of sight wraps around. She sees her own tail, so she has the illusion of seeing another flounder sitting up here somewhere.
How about if she looks to the northeast along a 45 degree line?
Oh yeah, a crab. You're right. Let's see. Well, we can fix that. Line of sight wraps right to left, top to bottom. We get rid of the crab.
[LAUGHTER]
And she sees her own tail. So in effect, she looks straight ahead, she sees another copy of herself sitting up here.
OK. So in effect, this flounder has the illusion of living in an infinite universe. So even though there's really only one crab, the flounder sees it in infinitely many directions. There we go.
OK. So what have we got here?
Well, we have this nice universe. It's finite. It has no boundary. But unfortunately, it's only two-dimensional. So my next question for the non-PhD-holders here-- not even graduate students, yeah-- is, how can we apply this same idea to make a finite three-dimensional universe for 3D people like us?
Yes. OK.
We've done this little trick with the square to make a torus. And this gives us a universe that's finite but has no boundary. So I'd like to do something similar to make a three-dimensional universe that's finite but has no boundary.
Yeah. We can use a cube. OK. In fact, it doesn't even have to be a perfect cube. Any rectangular block of space will work fine. For example, the space inside this room.
OK. So here we are in the room. When I walk off this way, I'm not going to crash into the wall. But what's going to happen?
Yeah. I'm going to come in from the opposite side. And if I walk through the front wall, what happens?
Good. Come in from the back wall. And if I fly up through the ceiling?
Come up through the floor. Great.
This universe is called-- done that. Is called a 3-torus, just like the flounder's world is called a 2-torus
OK. So here we are in our 3-torus. When we look straight ahead, what do we see?
Yeah. Our line of sight goes out the front wall, comes in the back wall, hits us on the shoulders. So we see an image of ourselves, a whole room full of people sitting in front of us.
And if you look to the right, yeah. Line of sight goes out the right wall, comes in the left wall. So we see our left sides over there.
And if we look down beneath our feet?
[INTERPOSING VOICES]
JEFF WEEKS: Yeah.
AUDIENCE: [? --through ?] [? the ?] [? top ?] [? right ?] [? wall. ?]
JEFF WEEKS: Yes. Exactly.
[LAUGHTER]
OK. And mine too, I think. Maybe even a little better.
AUDIENCE: --not sure. [INAUDIBLE] more bald.
JEFF WEEKS: Oh, I think I got you beat. I didn't know.
[LAUGHTER]
OK.
AUDIENCE: [INAUDIBLE]
JEFF WEEKS: Pardon? Pardon?
AUDIENCE: But actually, we would see up [INAUDIBLE].
[INTERPOSING VOICES]
JEFF WEEKS: Almost like having a mirror there? Subjectively, it's almost the same effect, as if you're in a room with mirrors on the walls. The only difference is that if you're in the room with the mirrors and you walk towards your mirror image, your mirror image walks towards you, so you get closer. But in the 3-toruses, I walk this way, I see myself moving away. But otherwise, it's the same sort of effect.
OK. We can use the computer to see this effect. Let's-- call it up here. OK. 3-torus. OK.
So here we are in a cube containing a single earth. Let's go full screen here. Oops. We don't want to crash into it.
OK. When we fly out across this wall, we'll come back from the opposite wall. And we can cut a little window to see where we're going. So our line of sight's wrapping all the way around this finite universe, which gives us the illusion of infinite space.
[INTERPOSING VOICES]
JEFF WEEKS: Yeah. OK.
Good. So this is a 3-torus. Would you like to try a maze in a 3-torus?
[INTERPOSING VOICES]
JEFF WEEKS: Let's see here. 3D maze. Here we go.
OK. The goal is to slide the blue ball along these tubes to reach the white box. And the computer makes a different maze each time, so I don't know how to solve it. So please help me.
OK. We don't have much choice so far. OK. Here it looks like-- oh no, that one here doesn't go anywhere. We have no choice but to go down.
OK. Let's get a better view.
OK. Here it looks like we have a choice. Going down is pointless, but shall we go up, forward, or rightward?
AUDIENCE: Forward.
JEFF WEEKS: Forward? OK.
Here, let me-- oops. Let me rotate this around so we can see it a little better. OK. Now I guess upward, huh?
[INTERPOSING VOICES]
JEFF WEEKS: Oh. Oh, you're right. We've almost got it here.
AUDIENCE: [INAUDIBLE]
JEFF WEEKS: OK.
AUDIENCE: [INAUDIBLE]
JEFF WEEKS: There we go.
[APPLAUSE]
Hey, good job. Thanks for the help.
OK. Let me make sure I'm not forgetting anything here. OK.
AUDIENCE: [INAUDIBLE]
JEFF WEEKS: Pardon?
AUDIENCE: [INAUDIBLE]
JEFF WEEKS: No.
[LAUGHTER]
I've tried of it. So let's see. Do I have-- I'm not even sure whether I have this enabled. Do I have this enabled for-- Yeah, I do have it enabled for tiling view.
[LAUGHTER]
JEFF WEEKS: OK. So you can grab a coffee and--
[LAUGHTER]
JEFF WEEKS: So. OK. Let's look at some other possible shapes for space. Maybe we should revert this before I get into trouble. OK.
OK. Let's look at some other possible shapes for space.
In two dimensions, we have the Klein bottle. So let's call one up here. We'll go back to our flounder, and we will switch from the torus to the Klein bottle.
OK. So what's a Klein bottle? Well, side-to-side, the Klein bottle is just like the torus, no difference. But top to bottom, the Klein bottle is glued with a flip. OK? So anything that goes out on the upper right comes back in on the lower left and vise versa.
OK. Our flounder here, if she swims up the middle, she's going to come back in the middle. But what's different about her?
AUDIENCE: [INAUDIBLE]
JEFF WEEKS: Yeah. She's mirror reversed. She comes back as her own mirror image here. OK.
Would you like to try a game on a Klein bottle?
[INTERPOSING VOICES]
AUDIENCE: Yeah.
[LAUGHTER]
JEFF WEEKS: OK. What will it be?
[INTERPOSING VOICES]
JEFF WEEKS: Pool? OK. Let's do a pool. Good idea.
[LAUGHTER]
Do we have any pool sharks in the audience here who would like to come and play for the humans? Anybody want to give it a try? Nobody? OK. That's fine.
AUDIENCE: [INAUDIBLE]
JEFF WEEKS: Oh. Rick. OK.
[LAUGHTER]
OK. So a word for how to use the program, you grab the cue stick and obviously the direction your laser pointer is pointing is the direction the ball is going to go. And the length of the cue stick determines how hard a shot it is. So please.
AUDIENCE: [INAUDIBLE]
JEFF WEEKS: Oh, I should say that these things are pool balls down here, but that black thing is not. That black thing is the lone hole. So we're playing eight ball here. So--
[LAUGHTER]
So it's stripes and solids.
[INTERPOSING VOICES]
JEFF WEEKS: OK. So I think-- oh, did you sink something? Oh, that was the computer's shot. I see. OK. So we're back to humans again.
[INTERPOSING VOICES]
JEFF WEEKS: OK. So the eight ball is the four ball in this game.
AUDIENCE: --two fingers to drag.
JEFF WEEKS: Yeah, two fingers to drag. And then you can take your--
[INTERPOSING VOICES]
JEFF WEEKS: Yeah.
[INTERPOSING VOICES]
JEFF WEEKS: Oh, OK. So we grab it, move it around. And I think I might be obligated to take this shot because when I let go, I think it's going to-- we don't want to make it too hard here, though. Let's see maybe that.
[GAME BEEPS]
OK.
[APPLAUSE]
OK. So with stripes and solid, this means that we humans are now going for the striped balls, and the computer's going for the solid balls.
[LAUGHTER]
[INTERPOSING VOICES]
AUDIENCE: --scratches. That's a scratch.
[INTERPOSING VOICES]
[APPLAUSE]
JEFF WEEKS: But there is a Gaussian error on it.
[INTERPOSING VOICES]
JEFF WEEKS: Oh, I see. If you-- with the trackpad-- yeah. There you go.
AUDIENCE: [INAUDIBLE]
[INTERPOSING VOICES]
[LAUGHTER]
[INTERPOSING VOICES]
JEFF WEEKS: Yeah. You want tiling mode? We could do tiling mode.
[INTERPOSING VOICES]
JEFF WEEKS: No.
[INTERPOSING VOICES]
JEFF WEEKS: They haven't seen the tiling Klein bottle yet, so I'd rather introduce it separately.
[INTERPOSING VOICES]
JEFF WEEKS: OK.
AUDIENCE: [INAUDIBLE]
[INTERPOSING VOICES]
[LAUGHTER]
[INTERPOSING VOICES]
[GAME BEEPS]
AUDIENCE: Do you want to stop now?
JEFF WEEKS: Yeah, maybe we should stop now.
[LAUGHTER]
[GAME BEEPS]
[SIGHS]
OK. Hmm?
[INTERPOSING VOICES]
AUDIENCE: How about make it longer again?
JEFF WEEKS: Yeah.
AUDIENCE: [INAUDIBLE]
JEFF WEEKS: OK.
[INTERPOSING VOICES]
[GAME BEEPS]
[LAUGHTER]
JEFF WEEKS: Good job.
[APPLAUSE]
OK. Before we move on, I'd like to do-- that was an impressive last shot. I wouldn't have thought it was possible. I'd like to do a crossword puzzle made especially for this evening's talk.
OK. So the first word here, the clue is diplomat's skill.
AUDIENCE: Tact.
JEFF WEEKS: Tact. Very good. OK. And a question for you-- when we take that word tact and slide it off the top of the board, what's going to happen?
[INTERPOSING VOICES]
JEFF WEEKS: Yeah. So it comes back reversed. OK. Let's do the next word here.
Oh, by the way, note the little yellow arrow tells you which way to read the word. So like now, the arrow tells you you're going right to left.
OK. Next word-- Leander's girlfriend.
[INTERPOSING VOICES]
JEFF WEEKS: Hero-- good. OK. Yeah, I didn't-- my wife suggested this clue. I didn't know this either. But if you go to Wikipedia and search for Hero and Leander, then you can read the full romantic yet tragic story.
OK. And of course, if we take the word hero and pass it up by the top of the board, then it comes back reversed as well. OK.
So let's do a long word here. Going vertically, revolutionized 3D geometry and topology.
[LAUGHTER]
OK. It's got to be Thurston, but do we have a problem here that Thurston has eight letters, and we're playing on a five-by-five board?
[INTERPOSING VOICES]
JEFF WEEKS: Well, let's try. T-H-- oops. I've got to reverse the cursor. OK. We have T-H-U-R-S-- we already have the T and the O. And N. OK. So it fits.
Another vertical word here-- crystal surface.
AUDIENCE: [INAUDIBLE]
JEFF WEEKS: Yeah, facet. Oops. Get this right here. Facet.
How about going left to right here, rotation?
AUDIENCE: Turn.
JEFF WEEKS: Turn-- that's an easy one good.
Oh, here's another hard one. I had to look this one up too. Church area with semi-hemispherical roof.
AUDIENCE: Apse.
JEFF WEEKS: Apse. Good. Let's see. What's missing here? Ah, fish eggs.
AUDIENCE: Roe.
JEFF WEEKS: OK. Very good.
I have another question for you here. Let's go back to our flounder and again zoom out. So in the Klein bottle now, when the flounder looks south here, what does she see? The pink shell or the red shell?
AUDIENCE: [INAUDIBLE]
JEFF WEEKS: Yeah. She sees the red shell because of the flip. So as in the torus, the flounder has the illusion of living in an infinite universe, only this time, the pattern of the images is different.
OK. So the pattern of the images reveals the shape of the space. So an observant flounder could tell not only that our universe is finite but also whether it's a torus or a Klein bottle.
OK. Let's see. I think we want to go to three dimensions now. Let me make sure we're not forgetting anything. Yeah let's go to three dimensions. OK.
OK. So at first here, we're sitting in a torus, so we can take the Earth, pass it through the walls. And we go pass it through this way.
Now, if we switch from the torus to Klein space, now the sidewalls are glued just like in the torus. But the floor and the ceiling here are glued with a flip. So if we take this earth. And pass it up through the ceiling, what's going to happen?
Yeah. It comes back as its mirror image. So if we look carefully here, we can see here's the Pacific. Here's Australia, Asia, Africa, the Atlantic, South America, North America. Ithaca's on the left coast now.
[LAUGHTER]
How about if we take the Earth and pass it up through the ceiling again?
There it is. So it comes back normal.
OK. I'd like to show you one other space that has no analog in two dimensions, truly three-dimensional here. OK.
In this space now, we can take the Earth, pass it through the walls. Side-to-side, it's just like a torus. Top-to-bottom, just like a torus. No flip. But the interesting thing now is we're going to take the back wall and glue it to the front wall with a 90 degree rotation.
OK. So if we take this Earth and pass it through that front wall, it comes back in on its side. OK.
So let's go for a little flight here. We'll park the Earth here.
So what's going to happen when we fly through that front wall?
AUDIENCE: [INAUDIBLE]
JEFF WEEKS: Yeah. We're going to go out through this front wall and come in the back wall rotated 90 degrees. So let's do it here.
So we're coming back in rotated 90 degrees. So here's the original floor on our right, the original ceiling on our left. How about if we fly through again?
AUDIENCE: [INAUDIBLE]
JEFF WEEKS: Yeah, we'll be upside down. We rotated 180 degrees. So there's the original floor above us, the original ceiling below us.
How about if we go through again? Yeah, 270. So here we go. The original floor on our left, the original ceiling on our right.
How about once more?
AUDIENCE: [INAUDIBLE]
JEFF WEEKS: Yeah. We'll be back to normal. There we go. One moment. OK. Back to normal.
If we cut a little window in the wall, what will we see?
AUDIENCE: [INAUDIBLE]
[INTERPOSING VOICES]
JEFF WEEKS: So even though we're still flying around this finite universe, we have the illusion of infinite space. So the repeating images tell us that this universe really is finite. The pattern of the images reveals the shape of the space, whether it's a 3-torus, a Klein space, a space with a 90 degree rotation, or something else. OK?
OK. I think we want to go to the real universe now. Let's-- OK.
So to test whether the real universe is finite, we could, in principle, just look out into space and look for repeating images of our Milky Way galaxy. But this is harder than you might think.
What's the catch? Why can't we just look out and try to find other images of our Milky Way sitting out there?
AUDIENCE: [INAUDIBLE]
JEFF WEEKS: Something about light. What's that?
AUDIENCE: [INAUDIBLE]
JEFF WEEKS: Yeah. So why is the speed of light important here?
AUDIENCE: [INAUDIBLE]
JEFF WEEKS: Yeah. Because of time.
So are we seeing all these other galaxies out there as they are today?
AUDIENCE: No.
JEFF WEEKS: Yeah.
AUDIENCE: [INAUDIBLE]
JEFF WEEKS: Yeah. Whatever. OK. Let's see, your today or--
AUDIENCE: My today is different [INAUDIBLE].
JEFF WEEKS: Yeah.
AUDIENCE: [INAUDIBLE] --but actually, she's a skeleton?
JEFF WEEKS: Pardon?
[LAUGHTER]
AUDIENCE: Isn't that what you use where you are looking at the image of the young girl, but actually she's a skeleton, right?
JEFF WEEKS: Oh, right. OK. Yes.
OK. So if we're here-- so we look out into space, we see nearby galaxies as they were recently. We see more distant galaxies as they were further in the past because it takes the light longer to reach us. We see even more distant galaxies just forming, just condensing, just being formed from clouds of hydrogen gas. We could look even further. We see the primordial plasma from which the hydrogen condensed.
OK. Now, this plasma is opaque, so we can't see any further. But please don't be too disappointed because we're not missing very much. We're seeing back 13.8 billion years, and the beginning of time is only 380,000 years beyond that. So we're seeing 99.997% of the way back to the beginning of time.
So no matter which direction we look, we're seeing into the past. So our horizon is this huge sphere filling the whole sky with us at the center.
OK. Great. Does this mean that after the lecture tonight, we can go out, wait for a break in the clouds, look up into the sky, and see this plasma on our horizon? Well, yes and no.
The plasma really is there. OK? The catch is that the radiation we receive from it arrives not as visible light but as microwaves. So if the human eye were sensitive to microwaves as well as to visible light, then the night sky might look something like this.
So we'd look out. We'd still say nearby galaxies as they were recently, then more distant galaxies as they were further in the past, then really distant galaxies just being born from clouds of hydrogen gas. But then behind it all, we'd see this plasma filling our horizon.
Now, as you can see here-- and it's just a false color image, but you can see that parts of the sky are warmer, and parts are cooler. And this is because of density variations in the early universe. So the microwaves coming to us from denser regions do a little bit extra work against gravity and arrive a little bit cooler. Microwaves coming from less dense regions do a little bit less work against gravity, arrive a little bit warmer. So in effect, we're looking out, and we're seeing density variations in the early universe.
Now, you might be wondering, what does this have to do with--
AUDIENCE: Is this [INAUDIBLE]?
JEFF WEEKS: I'm trying-- I think I might have cut it from the actual picture. Whether I actually took it from this part of the sky or not, I don't know. I think it's cut from the actual picture. OK.
Yeah. So we're looking out. We're seeing density variations in the early universe. And the question is, what does this have to do with whether the universe is finite or infinite? So that's the next question we're going to address here.
OK. Here's a cartoon view of the primordial plasma superimposed on the modern galaxies that it evolved into. This red circle represents our horizon.
Now, the plasma was denser in some regions, less dense than others. In an infinite universe, these density fluctuations can be as broad as you like. But in a small, finite universe, the density pattern repeats like the flounder's repeating images that you saw earlier. So the broadest fluctuation you can have would be the distance from one image of a galaxy to the next. And in particular, the fluctuations can't be any broader than what you see on the horizon here.
Excuse [? you. ?]
Yeah. On the horizon, they can't be any broader than what you see here.
OK. So in summary here, in-- oops.
In an infinite universe, we expect to see broad density fluctuations on the horizon, while in a small, finite universe, we don't. OK. So to test whether the real universe has these broad fluctuations or not, we just need to look at the sky with a microwave telescope.
OK. In 2001, this delta rocket launched the Wilkinson Microwave Anisotropy Probe. That's kind of a mouthful, so people just call it the MAPS satellite.
MAP orbited the sun in synchronization with the Earth and observed the microwave sky for nine years. And this is what it saw. And this is the real data, by the way.
The cool thing about this picture is that it lacks the broad fluctuations you'd expect in an infinite universe, suggesting that our real universe may indeed be finite. More precisely, the broad fluctuations are weak at the 1/3,000 level, meaning that even in an infinite universe, you could get such weak fluctuations, weak broad scale fluctuations, just by chance, but only one time in 3,000.
Now, it turns out that upon closer study, the fluctuations you see here don't really look so much like what you'd expect in a 3-torus either. They fit much better to the Poincare dodecahedral space. OK.
What is this? Well, let's call one up here.
OK. So the Poincare dodecahedron space it's like a 3-torus, but it's made from a dodecahedron instead of a cube. So let's go inside.
OK. So when we fly out across one of these walls, what's going to happen?
Here we go. Come back from the opposite wall. We can cut little windows to see where we're going. We can make the windows wider for a better view.
Of course, in the real universe, mother nature doesn't show us the walls at all, so we'd see something more like this. And it would be up to us to figure out the shape on our own.
So in effect, it's kind of like a 3-torus. But instead of having a space built from a cube with images repeating-- oops. Why is it ignoring my-- oh. Out of control spaceship.
[LAUGHTER]
There we go. OK. We're back in control.
OK. So instead of having a universe built from a cube, with images repeating along a cubic lattice, we'd have a universe built from a dodecahedron with images repeating along a dodecahedral lattice.
The kicker here is that a dodecahedron lattice cannot exist in flat space. Instead, it exists on a hypersphere. So it's like we're flying on the three-dimensional surface of a four-dimensional ball.
So this means that if the real universe really does have the shape of a dodecahedral space, that means that our universe is not only finite but also slightly curved.
OK. Here we go. Slightly curved. OK.
OK. So why should a dodecahedral universe be curved? Well, our e-torus universe-- let's go back to regular mode here.
So our 2-torus universe is flat. And the reason the 2-torus is flat is that it's made from a square. And the square's four corners fit together perfectly. Each has 90 degrees.
Now, if you instead try to build something from pentagons like so, you'll find it doesn't work, that you get a little gap here because the pentagons don't fit snugly. But there's a way to solve this problem. And the way we can solve it is to take these pentagons and just connect them up anyhow.
OK. It's still not flat because it has this corner point here. But there's a way to solve this problem.
OK. What we'll do is we'll keep adding more pentagons until we get a dodecahedron. Let's call one up on the computer here.
OK. So here's a nice dodecahedron. It has 20 corners, and we don't like having these corners here. So what we're going to do is take the dodecahedron and pop it out onto the sphere. OK.
So once it's on the sphere, these spherical pentagons have perfect 120 degree angles. So three spherical pentagons fit together snugly. They fit perfectly, unlike the flat ones. OK. So no gap there.
What do we want to do now? OK.
So if we come-- if we go back to three dimensions-- OK. Cubes fit together snugly. And this is why when we make a 3-torus, the 3-torus is flat precisely because cubes fit together snugly.
But if we take dodecahedra-- let's see if I can do this here. If we try to take three dodecahedra and fit them together, you'll see they don't fit snugly. There's a little bit of wobble in there. There's a gap. Can everybody see the gap there?
So what we're going to do is do the same trick just like we took to the pentagons. They didn't fit snugly either, so we took them, and we put them onto a sphere so that would fatten up their angles, and then they'd fit. So we can do the same trick with the dodecahedra if we take them and put them on a hypersphere. Then on a hypersphere, the spherical dodecahedra have perfect 120 degree angles, so three of them fit together snugly around each edge.
So in effect, the reason that the dodecahedral space is spherical while the 3-torus was flat was precisely this little gap with the dodecahedra not fitting together.
OK. Going back to two dimensions now.
With three pentagons, we had-- oops. I need to find my other pentagon. OK.
With three pentagons, we had too little angle, and there was a gap. If we now add a fourth pentagon, now we've got the opposite problem. We have too much angle, so it's kind of like a negative gap there. But we can do the same sort of trick. We can take them and connect them together anyhow. And notice that when I connect them together, I get a surface that's kind of saddle shaped. OK.
Now, if we keep going with more and more of pentagons, and then we take this, and we smooth out the whole thing just like we smoothed out the dodecahedron, then we get this really cool surface here, which is called a hyperbolic plane. OK.
So you can see it's made from these pentagons. And every pentagon has perfect 90 degree angles. So at each corner, you can see that four pentagons fit together snugly. OK.
Yeah. So-- yes. That's all I want to say about that.
Oh, yeah. The other point here is that this hyperbolic plane-- you'll notice it's saddle-shaped everywhere. So you can't really lay it flat on a table.
OK. The same trick works in three dimensions. If you try to take four dodecahedra and fit them together around an edge, it just doesn't even come close. There's no way you're going to fit them in there. But we can go instead to three-dimensional hyperbolic space, the analog of this hyperbolic plane-- OK. So in three-dimensional and hyperbolic space now, the dodecahedra have exact 90 degree angles, so four of them fit together snugly around every edge.
OK. Yeah. That's all we need to say about that.
So getting around to where all this is heading is I'd like to take the last few minutes and talk a little about Bill Thurston's contributions to geometry.
So among Bill's many contributions to geometry, my two personal favorites are that every finite three-dimensional space can be smoothed out to be uniform. This means that if you take some sort of space that, rather than being nice and smooth, this kind of irregular, and all lumpy and bumpy, and you make it-- any way you want, any kind of space-- you can always take it, and take all those lumps and bumps, and just smooth the whole thing out. And you're going to get a space that's either spherical, or flat, or hyperbolic, or one of five other possibilities that we're not going to get into right now.
So that's-- this is Bill's famous geomerization conjecture, which Grigori Perelman proved in 2003, and which in turn implies the famous Poincare conjecture.
Now, Bill also discovered that a typical space smooths out to be hyperbolic. So in this point of view, it says that only the very simplest, most symmetrical spaces can be spherical, or flat, or something else. They're the exceptions. The typical spaces are hyperbolic.
So for example, if we start with a space-- OK. We start with a space that's a little bit complicated and not too symmetrical, then it will smooth out to be hyperbolic. You wouldn't think it just looking at something like this. Why should it be hyperbolic? And before Bill came along, that's what everybody thought. Oh, why should that be hyperbolic? But it turns out this is a typical case.
And by the way, in 2007, Bill's former students Dave Gabai, who-- where did Dave go? Right here. Yeah. All right. One of our conference organizers here, and another of Bill's students, Rob Meyerhoff, and Dave's student Peter Milley, proved that this particular space is indeed the smallest-- in effect, the simplest space that smooths out to be hyperbolic. So they just get more complicated from there.
OK. Observational evidence suggests that the real universe is indeed already smooth, not irregular and bumpy, so we don't need to smooth it out. So the question becomes, can we tell whether our universe is spherical, flat, hyperbolic, or something else?
Now, if we could see-- if we could see a large enough portion of the universe, then it would be really easy to detect the curvature. On the other hand, if our horizon is small relative to the size of the universe-- so if our horizon is only this big-- then the amount of the universe that we see looks almost flat.
OK. So in effect, if you look at-- let's make sure I'm not forgetting something.
So in effect, a small part of a hyperbolic universe looks almost flat. A small part of a spherical universe looks almost flat. And of course, a small part of a flat universe really is flat.
So indeed, the results released last year from the European Space Agency's Planck satellite show that the real universe looks almost flat.
[LAUGHTER]
So this suggests that whatever shape our universe might have, it suggests that we're seeing only a tiny portion of it.
So in conclusion, the universe might be flat, or it might be slightly curved. It might be finite, or it might be infinite.
[LAUGHTER]
The only thing we know absolutely for sure is that mother nature has more surprises in store for us. So thank you all for your attention.
[APPLAUSE]
OK. And there's the address if you want to play the games some more at home. And I put a little note on for the mathematicians here that the first draft of the 4D drawing program just went up. So if any if you want to try it and send suggestions for making the second draft better, please do. But for non-mathematicians, that new ones not really very developed yet. The Torus Games are a little further along.
[INTERPOSING VOICES]
JEFF WEEKS: OK.
LAURENT SALOFF-COSTE: Does anybody wants to try a question?
AUDIENCE: Yeah. Can you define what does universe mean?
JEFF WEEKS: What the universe means. My immediate gut reaction is just to say no, but let me think.
[LAUGHTER]
JEFF WEEKS: Yeah. Just-- I'd say physical space-- I'm debating whether to say in its entirety. I'll say in its entirety and then put the footnote in. The amount of physical space that's connected to the part we see. So it might be beyond our horizon, so we'd have no signals from it. But I would-- when I say universe, I'd probably exclude other universes in the sense that ones that just aren't even connected to ours.
[LAUGHTER]
[INTERPOSING VOICES]
LAURENT SALOFF-COSTE: Yes?
AUDIENCE: When [INAUDIBLE] the Poincare dodecahedral space, was that a random example of what the universe could be, or was there evidence in some of the background radiation that pointed to that [INAUDIBLE]?
JEFF WEEKS: Yeah. Where that comes from-- I'll answer twice. One's the generic answer and then a couple words for the mathematicians here. The generic answer is when you look at the microwave sky, that's the one that seems to fit best. So there's no smoking gun evidence saying the universe has to be this way. And in fact, with the curvature now, it seems pretty much excluded on that grounds. But before the tighter curvature estimates were available, that was a best fit to the data.
OK. A word for the mathematicians here. What do you mean by best fit? Look at the microwave sky, decompose it into spherical harmonics, you get some nice little graph showing the strength of the fluctuations at different frequencies, in effect, and that was the space that best fit it. OK.
AUDIENCE: Could you say something about how you've implemented your game? Not so much from a programming language, but the underlying math or--
JEFF WEEKS: Yeah.
AUDIENCE: --more from that angle.
JEFF WEEKS: Yeah. So I guess the design decision is-- the goal was to create something where you'd really feel like you have a 3-torus in your hand, to give it a palpable reality so you can experience it directly. That's the broadest answer.
And, well, how would you do that? Well, let me think on it.
So I guess the thought is by having the little bezel there, I think, was a part of it. Because you want to have the illusion that-- see, you're making me confess that this is just all an illusion here. But you want to have the illusion. To give this the appearance of the reality, you want to have a board, or you want to have a bezel. And so if stuff scrolls off one side, you want to see it come back from the other, so you feel that it's a real, concrete object that you're sliding around.
Was that the nature of the answer?
AUDIENCE: [INAUDIBLE]
JEFF WEEKS: Yeah. Yeah. And the technical detail is it just stores one copy of this thing. And then--
AUDIENCE: I mean, in some ways, is it really implementing the space that you mathematically [? know ?] exist?
JEFF WEEKS: Yeah. Yeah. Yeah.
[INTERPOSING VOICES]
AUDIENCE: So it's just an illusion. It's in fact, [INAUDIBLE].
[INTERPOSING VOICES]
[LAUGHTER]
JEFF WEEKS: Yeah. Yeah. I think-- because I think-- so I think yes.
AUDIENCE: So with the microwave background radiation, if the universe were small and finite, we would expect to see same pattern over and over just like the fish.
JEFF WEEKS: Yes.
AUDIENCE: But we don't see that. So do we think that the universe is finite, but it's bigger than our horizons here, and that's why you don't see the repetition?
JEFF WEEKS: Yes. Now, the different schools of thought on this is when the data was not as good, then really the only thing you could say for sure is that the distance for the repetition had to be at least 1/10 the horizon radius, or else we would have noticed by then. But now the data gets better, and it kind of pushes things outward.
And there are two schools of thought. One school of thought is, well, given that we're not observing it on this really small scale, it seems like almost too much to hope for that it's going to be just right there at the border. And I think I'd be an adherent to that philosophy. That seems kind of unlikely that the repetition distance is going to come in at 90% of our horizon.
OK. Not everybody has given up on this because there are still these weak fluctuations to account for, and the idea of a finite universe is-- well, maybe the best explanation for lack of a better one. And so the thought there is that it could explain it even if this repetition distance is slightly larger than the horizon. So there are people actively working on this.
You can say, well, could you tell if it's larger than the horizon? It's kind of like if you were to say, take a guitar string. So say I have a guitar string, and I'm only going to show you the middle 80%. I'm not going to tell you it's 80%, I'm just going to show the middle 80%, and then I pluck it. You could figure out the exact length because you'd see a fundamental. You'd see a second harmonic, a third harmonic, and you could work out where those endpoints were if you're seeing enough of it.
So there's kind of that school of thought that, well, even if we're-- the universe is 120% the size of our horizon, we could work it out doing some [? FRA ?] analysis on the microwave sky. It seems like a long shot, but weird things are going on, so it seems worth looking into.
OK.
AUDIENCE: One more?
AUDIENCE: [INAUDIBLE]
Is there more than one universe?
JEFF WEEKS: Ah, good question. In fact, I like this question a lot.
AUDIENCE: Could you repeat the question?
JEFF WEEKS: Is there more than one universe? And may I answer your question twice? Once from my gut feeling, and once for what we know scientifically?
So in terms of actual science, and looking in telescopes, and observing, no way to know. OK? In terms of my gut reaction, I would say almost surely. That if you could have one, why not have lots and lots? So my gut reaction is probably yes, but it's nothing that we have any evidence for.
Thanks for asking.
LAURENT SALOFF-COSTE: OK. Last question.
AUDIENCE: So my problem is similar than the one before. It's if universe is finite, what's outside the universe?
JEFF WEEKS: Ah, OK.
If the universe is-- the question is, if the universe is finite, what's outside the universe? And the answer to that-- I'll answer once very quickly, and then once with a little more detail.
The very brief answer is, well, nothing. Like-- not a vacuum. Just-- it isn't there. How do you make sense of that?
Let me have a sip of water. Well, it's not that long an explanation.
[LAUGHTER]
The way to understand this is-- I could never explain this myself before. But once I was working at the local elementary school with a classroom full of fourth graders, OK? And they were working towards these kind of shape of space ideas, but they started doing stuff with flatland, and with a one-dimensional world-- Line Land.
And in particular-- oh, we actually have a chalkboard here.
In Line Land, they were given the task-- so they had Line Land, and they did some stuff with-- there were four Line Landers here. I forget the names. So something like Alice, and Bill, and Carla, and Dave, say. And they had some little exercises about, well, Bill could visit Carla and Alice, but Bill couldn't visit Dave because in Line Land, there's no way to get by.
So they got comfortable with those ideas. And then they had the exercise to work on, well, how could you modify this universe so that Alice could visit Dave?
So they all set to work in the little groups thinking about this. And eventually, they have greater or lesser amounts of success. And one group comes to present a solution and said, well, what we did in our group is we took Line Land and we extended it around to make it into a circle. So now Alice can just cruise around on and visit Dave. OK?
N minus one of the kids in the room we're happy with this explanation. But this one kid pipes up and says, hold on. Something's fishy here. Line Land is supposed to be one-dimensional, but a circle is two0dimensional.
Now, I have a PhD in mathematics. I understood the mathematics of what's going on here, but I was just at a loss for words. I couldn't figure out how to explain this. So while I'm there trying to think how to explain it, one of the other kids in the class pipes up and says, look at the chalk and ignore the chalkboard.
[LAUGHTER]
OK? And that was exactly the right answer in this case. Look at that chalk. Ignore the chalkboard.
Now, when it comes to a universe, if our universe is a 3-torus, for example, and you--
AUDIENCE: [INAUDIBLE]
JEFF WEEKS: Oh, no, no. Fourth grade.
[LAUGHTER]
Yeah. Yeah. It was an accelerated group.
So one second. Let me just finish.
So if our universe is a 3-torus, we can go running out this side, come back from that side, up through the ceiling, back from the floor. OK. If you say, well, but what's outside this 3-torus? Well, look at the chalk. Ignore the chalkboard. Look at the universe. The universe is all there is, so there is no outside.
Hmm?
AUDIENCE: [INAUDIBLE]
JEFF WEEKS: OK. How can it be?
And I think the answer to your question there is that the reason we want there to be an outside is that we all grew up in this relatively limited part of the universe where we think of everything in terms of just flat space, Euclidean space. OK? And so we want to visualize everything in three-dimensional Euclidean space.
If there's something that doesn't fit in three-dimensional Euclidean space, well, we'll be a little flexible. Four-dimensional Euclidean space is good, or five-dimensional. We want to visualize everything in an infinite Euclidean space, but that's just-- it's just our own prejudice, the temporary limitations of our own imagination. Mother nature doesn't suffer this limitation.
So for Mother Nature, having the universe be a 3-torus, or a dodecahedral space, or something else-- no problem. It doesn't have to sit in a Euclidean space. It doesn't have to sit in a flat space.
LAURENT SALOFF-COSTE: So I would like to thank every one of you for coming tonight. And, of course, I would like to thank Jeff again.
[APPLAUSE]
When we look out on a clear night, the universe seems infinite. Yet this infinity might be an illusion. Jeff Weeks uses interactive 3D graphics and satellite data to illustrate possible shapes for space, and explains what Bill Thurston's pioneering discoveries say about the curvature of space, June 24, 2014 at the Bill Thurston Legacy Conference.
The conference, "What's Next? The mathematical legacy of Bill Thurston," held at Cornell June 23-27, 2014, brought together mathematicians from a broad spectrum of areas to describe recent advances and explore future directions motivated by Thurston's transformative ideas.
