I want to tell you about exotic objects in topology. Before I do that let me just tell
you more generally what topologists do. We study geometric shapes, but unlike in ordinary geometry
we don't care about lengths, angles, areas, volumes - we don't care about any of that. We just care about
what properties geometric shapes have that are unchanged when you deform them; and by deform I
mean you're allowed to bend it and twist it and stretch it as if it's made of an elastic
material, but you're not allowed to break it. -(Brady: So a square for example, if we started manipulating)
(that it would very quickly not be a square anymore.) - That's right we have a square right
here, and I could turn it into a circle or more or less a circle by just bending it. And I could
also turn it into a triangle. But what I cannot do is turn let's say a circle into two circles.
The only way to do this would be to break it and then glue it back but that's not allowed.
Similarly I could have a ball and I could turn it into a cube. There's a saying that topologists
are people who cannot tell apart donuts from coffee mugs so, for example, here is a donut
and here is a coffee mug. I mean they're the same, they're the same for us because you could deform
one into the other, they both have this hole through them. But what I cannot do is turn the
donut into a ball because the ball doesn't have a hole so I don't want to punch a hole in it, that's
not allowed. And while I could do that by taking a bite out of the donut, but we topologists are
healthy people, we're not supposed to be eating donuts. The objects that we mostly care about
are called manifolds. These are objects that look- from any given point they look pretty much like
flat space or n-dimensional hyperspace. What is n-dimensional hyperspace? In dimension 1 we
have a line, in dimension 2 we have the plane, in dimension 3 we have space; and then you
can think about dimension 4, just add another variable say time and that's four dimensional
hyperspace. And you can just add more variables: five dimensional and six dimensional and so on;
it's not something that we can visualize but we can think of it mathematically anyway and we
can write down equations. - (And manifolds sit in this space? Or are manifolds the space themselves?)
They can stay- they can sit in the space but they could also look like that space around any point. So let
me tell you an example: the surface of the Earth is a great example. It's a sphere; more or less, it's
called a geoid, it has mountains and valleys - but remember we can all deform them and make it into
a sphere. Topologically it's a sphere; and it sits in 3-dimensional space but around any point
it looks like 2-dimensional plane because I can move in basically two directions, I can move left
and right and forwards and backwards, there are two degrees of freedom. So that's what I'm saying that
the sphere is a 2-dimensional manifold even though it lives in 3-dimensional space.
Other examples of 2-dimensional manifolds would be the surface of the donut or this famous Klein
bottle, yes, if I live on it again I wouldn't be able to tell the difference from- from a plane. Just
like it's very hard to tell the difference between a plane and the the Earth, the sphere, from where we
are. To really tell them apart you would have to go far away and see that you come back. There are
also 1-dimensional manifolds like this circle right here, it's a 1-dimensional manifold. If we
only lived on this- on the circle then we would only have one direction in which to move in. And
then there are 3-dimensional manifolds like our ordinary space or the universe. Around every
point it looks like we have three dimensions to go into, we also have the bottom and top,
uh but we actually don't know the shape of the universe. It's a 3-dimensional manifold; it
could be hyperspace but it could be what's called a 3-dimensional hypersphere. Perhaps if we go
very high up then we would come back to where we are. Nobody has done that but we don't know
if it's possible or not, just like on the surface of the Earth when you go around you come back.
So we just know it's a three-dimensional manifold- - (Is there no way to test for that? It seems strange)
(that we could not know that even the one we're living in, the one we're living and breathing in.)
(Is the only way to test to walk forever and see if I come back?) - No there are other ways, people are
looking at cosmic microwave background radiation, and they they have put forward various theories
about what the shape of the universe is but there is no definite theory that everyone
accepts, that it's not yet known what shape it is - but there are some hints from the universe.
Okay so we have 3-dimensional objects and then we can think about 4-dimensionals and
so on. A good example is this hypersphere; so in dimension 1 we have the circle which
is a 1-dimensional manifold and it's the set of points at the given distance from the origin. It's
given by the equation x squared plus y squared equals 1. Similarly we have the sphere which
is, again, the set of points at distance 1 from a given point but it lives in 3-dimensional
space and it's given by the equation x squared plus y squared plus z squared equals 1. And
we can generalize this; we can talk about the n-dimensional hypersphere which is given by the
equation x1 squared + ... + xn squared plus another variable xn plus 1 squared equals
1. It lives in n+1-dimensional space and it's an n-dimensional manifold.
Now I'm getting to the crux of the matter: the exotic objects. What are exotic objects?
These are things that are topologically the same but are smoothly different. So there are two ways
of thinking about the formations in topology; I showed you before that a circle can be turned
into a square and in the process of course you have to make a corner. Now the circle is smooth
and you can turn it into other smooth curves - like maybe something like an ellipse - and the process
would be smooth. Now could it be that we have two smooth objects and you can deform one into
the other but the only way to do so is by making corners somewhere in the middle? Then those objects
would be topologically the same but smoothly different. And- I don't know, if you're more familiar
with one of the objects you would say that the second one is an exotic copy of the other one.
It's like if you live on the first object and you go to the second one it's like going to an exotic
vacation; some things are the same but some things are just different. Now how can this be? How can you
have two smooth things but to deform one into the other you need to make corners? Here is an example,
we have a figure eight and can I turn it into a circle? Well I can I can do so if I make one of
the loops smaller and smaller but at some point I have to make a corner and then it becomes a circle.
Now that's not actually a good example; because if I had a 3rd dimension, if I allow myself to just
lift this up, then it would become a circle just smoothly. So this is not a real example of exotic
objects - in fact I cannot show you exotic objects because they don't exist in dimensions 1, 2
and 3, in the dimensions that we can easily visualize. They only start existing in dimension
4 and higher. The first examples were found in 1956 by a mathematician called John Milnor and
they were found in dimension 7. They are exotic hyperspheres of 7 dimensions. So there is some
object, an exotic sphere, that you can deform into the ordinary 7-dimensional sphere but not
smoothly. An exotic sphere is a smooth object that I can deform into a sphere. But I cannot deform
it smoothly, I would have to make corners in the middle. Let me show you now the numbers of spheres
in each dimension: in dimension 1, 2, 3 there's just one, the standard one. The same goes in dimensions
5 and 6. Now in dimension 7 we actually have 28 spheres so there's the standard one and
then there are 27 different ones. Then we have two spheres in dimension 8; then eight then
six. Then in dimension 11 we have 992 spheres. In dimension 12 we have just one, then 2, then
16256 and so on. If you look at this sequence of numbers it looks strange, I grant
you that, but there are some patterns. For example in dimensions 7, 11, 15 - every four dimensions - you
have- you get a lot more exotic spheres than in the other nearby dimensions. One thing you can
ask is, in what dimensions do exotic spheres exist? And in what dimensions does just the standard one?
We see that there's just one sphere in dimensions 1, 2, 3, 5, 6, 12; and
also mathematicians figure out the same is true in dimensions 56 and 61. - (So does that mean if I)
(was an object living in a 61-dimensional universe it's a little bit more boring than some of the)
(other ones?) - That's right, yes, these are the boring dimensions. You don't have exotic spheres - although
you do have some other exotic objects in these dimensions just not spheres. - (These numbers seem so)
(arbitrary; I assume it's impossible for someone to easily explain why it's certain numbers that are)
(boring and other ones that are hyper exotic?) - These numbers are hard to compute, there's a whole story
involving Bernoulli numbers and homotopy groups of spheres. The most interesting thing about them
is that we can compute them in any dimension except in dimension 4. That's where I left
a question mark. That's the biggest mystery, it's one of the most famous open problems in topology;
it's called the 4-dimensional smooth Poincaré Conjecture which says that there are no
exotic spheres in dimension 4. - (But we don't know that for sure yet?) - We don't know. There are
examples of objects that are smooth and you can deform them into the ordinary 4-dimensional
sphere by introducing corners; but people don't know if you can do that smoothly or not.
I have also looked for exotic 4-dimensional sphere myself and I couldn't find any.
(If you did would you become a very famous mathematician?) - Absolutely. - (Yeah? Is that- is it- it's not)
(like a millennium problem is it or anything- or is it?) - No there was a similar conjecture in
dimension 3 - the Poincaré Conjecture - that was the one problem in topology among the
millennium problems. That was solved by Perelman in 2003; and I think now the most
famous remaining open problem in topology is this one about dimension 4. But it hasn't yet made
it to the list of millennium problems. - (Professor, don't waste your time making Numberphile videos,)
(you need to go upstairs and solve this problem!) - Numberphile videos are important too.
(So why is dimension 4 so difficult?) - Good question. Well it's more difficult than the lower dimensions -
1, 2, 3 - because in those dimensions we can see things and there are also geometric techniques
that mathematicians use to study shapes in these dimensions. What's interesting is that it's also
simpler than dimensions 5, 6 and greater; and the reason for that is a trick about separating
disks. To explain it, let me say one way in which topologists distinguish between shapes. If I have
a lasso and I throw it somewhere on the plane, or even on the surface of the Earth which is a
sphere, it would surround a region. So I could just contract it into a blob around a little point. But
if I have the same lasso and I would put it around a surface- around a hole in a donut
then I wouldn't be able to contract it without breaking the donut. And that's one way you can
tell the donut apart from the sphere. So to study these objects mathematicians are looking at loops
and how you contract them, and the contraction spans a disc. A disc is a 2-dimensional object.
And it's easier to study these objects if you can split apart the disks so that you can study each
complication one at a time. All right, how can you split apart disks? Well let me start with lines. If
I have two lines and they're on the plane and they intersect, if I move one a little bit I'm not going
to be able to take it off the other. But I can take it off if I allow myself to go into space. The
reason is that 3 is greater than 1+1. Each line is a 1-dimensional object, and 1+1
is 2 but I have one extra dimension, and that's why I can do it. Similarly, if I'm interested in
separating disks: each disk is a 2-dimensional object and 2+2 is 4 so I need one
extra dimension. So in dimensions 5 and higher I can separate disks and that just makes topological
objects easier to study. This is called the Whitney Trick and it led to the classification of smooth
manifolds in higher dimensions back in the 60s. - (And this is the problem with dimension 4; you can't-)
(in dimension 4 you haven't got the space what- to to break- to separate two discs?)
That's right, we don't have the geometric methods from lower dimensions; and we don't have this trick about
separating disks. And that makes dimension 4 the hardest to wrap our heads around. To give you an
example of something that's unique to dimension 4: instead of looking at exotic spheres, let's
look at the exotic hyperspaces. Remember we had the line and then the plane and then the space and in
n dimensions we have the hyperspace. So could you have an exotic hyperspace? Something that looks
like like a plane or a space and you can deform it into the ordinary one but only by making corners?
Here are the numbers of hyperspaces in each dimension. In any dimension except 4 there
is just one; there's just a standard one and there are no exotic hyperspaces. But in dimension 4
we have infinitely many. In fact, it's more than infinitely many, it's uncountably many. It's not
the kind of infinity that you can count - 1, 2, 3 and so on - it's as many exotic hyperspaces
as there are real numbers. And this doesn't happen in any other dimensions. If you have exotic objects
in some other dimensions, you know that there are always finitely many of them. There's finitely
many exotic spheres, as I showed you that list of numbers. But in dimension 4 you can have
infinitely many exotic objects. And, yeah, that's what makes dimension 4 so fascinating.
(Does this mean dimension 4 is something that you should be spending more time on because it's so special)
(or is this a sign that maybe this is a bit of an aberration that you should leave alone? It's)
(a place where you shouldn't play?) - That depends on your perspective. There are mathematicians
who work on the other dimensions, and there are mathematicians that work on dimension 4. I'm
in the latter category. - (Because quite often when I speak to number theorists and they talk)
(to me about prime numbers, and they're showing me some interesting property of prime numbers, they)
(very often discard 2. They say we're not going to use 2 because that's an aberration and it ruins)
(everything. It ruins all our laws and principles and that and they just throw it aside. It sounds)
(like the 4th dimension is your equivalent.) - That's pretty much it. In many theorems you say
this is happening in all dimensions except 4 and we understand what's happening, and we just don't
know what's happening in dimension 4. But I find that interesting and worthy of study.