WEBVTT
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My name is Yukari Ito and I am
in the School of Faculty of Science,
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but also in Graduate School of
Mathematics at Nagoya University.
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So, welcome to this lecture
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and I want to explain
the relation between them.
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I don't know --
It is the Port Tower in Kobe
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and it is some -- I don't know the name.
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Do you know the name?
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And a lot of you know this is
in front of the Nagoya Station.
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So, anyway,
I will introduce symmetry first.
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And symmetry part is very enjoyable,
I think.
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So, what is symmetry in mathematics?
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So, here is our red one and blue one
and it was drawn by my son
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and it's something like dinosaur,
but there are two pictures
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and it looks like in front of mirror,
but there's different colors.
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But, in mathematics, the symmetry
with a person in the mirror,
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but please forget everything
except the shape from now
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because we want to
consider geometric symmetry,
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so you don't have
to mind the colors, okay...
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and those mirror and so on,
so only the shape we see.
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So, what is geometric symmetry?
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So the definition of symmetry,
symmetry can be observed
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after we rotate around a point
or a line or something
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and after the rotation, we can see
the image in the mirror or something.
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So that first original shape
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and the final shape is similar,
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but there are some
transformations between them.
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So, I will show some examples.
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So, after the rotation,
the shape is same as the original one.
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And mathematically it's called group
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and group can be explained
as situation well mathematically,
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but what is situation?
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The situation of symmetry.
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So the group is not group of
the human being, not human being,
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but group here is mathematical one
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and it's just a set of something.
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So, this is the picture by my daughter,
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few years ago.
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And this is line symmetry.
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In Japanese, sen taisho.
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And there is a line here and there is --
sorry, reflection.
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So, it is the reflection symmetry here.
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And other symmetry, well, you know,
but it's a point symmetry.
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In Japanese, tentaisho.
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It is rotated 180 degree,
then you can get another one,
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but after rotation,
then you will get original one.
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So there is a similar symmetry here,
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but I want to show you
very, very naïve mathematics definition,
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but if you don't understand,
don't mind it,
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but just I want to write down.
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So, the group is the following.
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So, it is called gun in Japanese
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and let G be a set with operator,
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here operator is something like
summation or a multiplication,
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just {Foreign language}
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and there are some sets, there is
numbers or some shapes and so on, so on.
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Then we call the set G as a group if it
satisfies the following conditions.
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So the condition 0 is --
so if x, y is in G,
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then x*y is also in G.
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So it is first rule.
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So, for example, if x and y is integer,
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then x+y is also integer,
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then you can say...
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it satisfies the condition 0.
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But there are other three conditions,
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so the next one is associative law.
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In Japanese, {Foreign language}
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And if you have x, y, z in G,
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then you can operate (x*y)*z
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or x*(y*z),
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the result is the same.
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It is the first condition.
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And second condition is,
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there exists identity element in G,
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which means it satisfies
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so that identity element
is something like...
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for example, in the set of integer,
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and if the operation is summation,
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then x+0=0+x=x,
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then 0 is identity element
in the set of integers.
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But here, so more generally you can say
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the identity element
satisfy this condition,
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but you should have
the identity element in G.
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And third final condition is,
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there exists the inverse element of x
such that, this one,
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so if you have x, then there is,
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you have always identity element e,
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but there are some other element y,
which satisfies
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x*y=y*x=e.
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For example, if G is a set of integer,
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then x-x is always 0.
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Then, the inverse element of x
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is minus x in the set of integers.
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So the set of integers
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with operator, summation,
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then this satisfies four conditions,
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then the set of integer
with summation is a group.
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This is one example and more generally
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if the set G satisfies this condition,
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then we can say,
always we can say it is group.
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So, it's just a mathematical definition
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and we can't see any symmetry here...
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can you see?
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But we have to find some symmetry
in the following shapes, this one.
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This is several kinds of Japanese family
crest, it's called kamon in Japanese.
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And there are many,
many kinds of pictures.
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So, for example,
it's just dark, dark one,
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but there are some circles and kind of
flowers and birds and so on, so on.
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And we want to consider one of them.
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So, first one, so from now,
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I want to move each family crest...
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fixing the center part, means the following.
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We want to move the shape from now,
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but we want to connect with a group,
we're doing movement.
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The meaning of the definition of a group
in terms of movement of a family crest
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is the following.
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So number one rule
is just associative law,
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it's just rule for computation,
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but this computation is just
movement after movement.
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So, there are many movements,
so it's the summation of movement.
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And the second one is identity element,
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means, no move, do nothing.
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So, it happens always so it contains
in the movement of the family crest.
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And number three rule is inverse
element is opposite movement.
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So, if we rotate someway,
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then you can just
get the opposite movement.
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Then inverse element satisfies
the definition of groups,
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then the movement of the family crest
can be considered as a group.
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So, let's see some examples.
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This one.
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So, this is just for now
we consider rotation...
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so it's called cyclic group.
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In Japanese, junkai gun.
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And rotate suitably.
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So, you can rotate someway,
but in this case,
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it is a famous picture
on the face of Japanese drum
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and it's called mitsudomoe in Japanese.
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And there are three, something here...
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and if you rotate this 120 degrees,
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then it will coincide
with the original shape.
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And the next, and plus 120,
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so 240 degrees,
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then it also will also coincide.
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___ and sometimes it go back
to the original one,
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then it becomes
a cyclic group of order three.
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It is called cyclic group
of order three.
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It is just a rotation,
then it is a group.
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Okay.
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But next one is a little bit different.
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So, the second example
is called dihedral group.
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So it sounds something funny, but it's
the main diagram, two face they have.
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So it is a rotation and turning over,
so the upside down.
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Then, you can rotate this picture
for 72 degrees.
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It's just something like pentagon,
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but you can make it upside down or so,
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then you can -- the picture also
coincides with the original one.
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So I just go back to the last one,
so this one can rotate,
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but you cannot make it upside down.
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It is a different one.
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So, there is some direction
in the picture.
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So here, if you make it upside down,
then it is also the same picture.
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Then there are different movements.
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So, rotation and turning over,
then it is called dihedral group.
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And this one, this case is called
dihedral group of order five.
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Okay.
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So, there are two
different two movements.
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So from now, there is exercise time.
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Here, many family crests,
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so let's consider how you can move it.
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So, this is our flower Kiku
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and this is tree Kiri
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and this is Hanabishi, is a flower
and the shape is {Foreign language}
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I don't know the English name, sorry.
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Diamond.
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Diamond, okay.
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Flower diamond.
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And this is ___,
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there are two squares and also some --
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it's the same flower before.
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So from this one,
there are, I think 16 flowers...
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then 16 pieces,
so then you can rotate this picture
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and also make it upside down.
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So this is the dihedral group of order,
something.
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In this case,
this is a little bit different.
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You cannot rotate,
but you can make turnover...
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then it can turnover
and next one is the same, the original.
00:14:23.353 --> 00:14:24.721
So this is also the --
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you have no rotation on this sphere,
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but you have to turn, two movements,
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so it is also
the cyclic group of order, too.
00:14:43.935 --> 00:14:44.984
How about this?
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It is not a square, so it is diamond,
00:14:49.798 --> 00:14:54.262
so you can rotate 180 degrees
00:14:54.295 --> 00:14:58.376
and also you can turn it over.
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So there are two movements
and it is dihedral group of order two.
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Okay.
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How about this?
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It looks very beautiful
00:15:10.544 --> 00:15:15.453
and it is family crest
of Sakamoto Ryoma...
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if you see some historical drama,
00:15:20.294 --> 00:15:25.612
maybe you should see
what kind of family crest that he had.
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So it is two pictures,
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so there is a square and you can rotate,
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but inside there are five pieces
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and outside it's four corners.
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So then you cannot rotate, okay.
00:15:50.069 --> 00:15:55.335
And it cannot also turnover,
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upside down it's a different picture.
00:15:59.195 --> 00:16:01.402
Do you know, do you understand that?
00:16:01.435 --> 00:16:07.663
Then, the symmetry
has only identity element...
00:16:08.986 --> 00:16:11.178
so this is also symmetry.
00:16:11.211 --> 00:16:15.651
So this, no, you can't, there is some --
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how do you say, this one?
00:16:21.631 --> 00:16:22.726
Intertwined.
00:16:29.943 --> 00:16:31.030
Okay.
00:16:32.273 --> 00:16:39.203
So, then it has only identity symmetry.
00:16:39.236 --> 00:16:45.533
And when I give a lecture about
with these shapes in my lecture,
00:16:45.566 --> 00:16:48.470
the Sakamoto Ryoma,
do you know Sakamoto Ryoma well?
00:16:48.503 --> 00:16:54.095
So, he is very active person
and he met many person,
00:16:54.128 --> 00:16:57.549
but he doesn't belong to anyone,
any groups.
00:16:57.582 --> 00:17:01.937
So, it's something like Sakamoto Ryoma,
and also beautiful.
00:17:01.970 --> 00:17:06.196
So, anyway, it only has identity element
00:17:06.229 --> 00:17:09.292
and there are some several groups here.
00:17:09.325 --> 00:17:13.561
So then the next situation here.
00:17:15.137 --> 00:17:20.815
So, if you are designer of
the family crest, you will be very sad
00:17:20.848 --> 00:17:26.395
because there are two designs,
it is Kiri and this is Tachibana,
00:17:26.428 --> 00:17:28.309
so completely different picture...
00:17:29.423 --> 00:17:33.289
but in mathematics...
00:17:34.335 --> 00:17:41.103
if you consider the symmetry
of the picture, this is same.
00:17:42.747 --> 00:17:46.925
So, in the mathematical world,
in the group theory,
00:17:46.958 --> 00:17:52.762
the Kiri and Tachibana is same,
that the corresponding group is same.
00:17:54.010 --> 00:17:55.194
Okay.
00:17:57.836 --> 00:18:03.639
So, the corresponding group, as the
symmetry of the picture is the same,
00:18:03.672 --> 00:18:08.152
just cyclic order,
so it's just upside down.
00:18:11.846 --> 00:18:17.332
And we consider more difficult problems,
00:18:17.365 --> 00:18:21.895
so we consider rotation
and turning over,
00:18:21.928 --> 00:18:25.178
but more parallel displacement.
00:18:25.211 --> 00:18:27.292
It's heiko ido in Japanese.
00:18:28.308 --> 00:18:31.627
Then, you have some more pictures.
00:18:31.660 --> 00:18:36.353
So, for example, if you have many so,
00:18:36.386 --> 00:18:40.149
infinity, many mitsudomoe,
00:18:40.182 --> 00:18:45.078
then you have this kind of picture
and so on, so on.
00:18:45.111 --> 00:18:49.049
Then, it equals
two dimensional crystal groups.
00:18:49.082 --> 00:18:52.699
So, normally crystal
is a three dimension,
00:18:52.732 --> 00:18:57.950
but if it is a two dimension case,
then it is a two dimensional crystal.
00:18:57.983 --> 00:19:01.598
Then there is corresponding groups
00:19:01.631 --> 00:19:04.982
and which equals
two dimensional crystal groups.
00:19:05.015 --> 00:19:10.329
And number of the crystal group
in dimension two is only 17.
00:19:11.352 --> 00:19:15.316
So there are
several designs and wallpapers
00:19:15.349 --> 00:19:21.984
and tiles in Alhambra Palace in Spain,
but the pattern,
00:19:22.017 --> 00:19:26.984
the symmetry of the tiles
and also wallpapers
00:19:27.017 --> 00:19:29.255
is just 17.
00:19:29.288 --> 00:19:32.744
Is it many or few?
00:19:32.777 --> 00:19:34.448
How do you think?
00:19:34.481 --> 00:19:38.079
So, it is two dimensional case.
00:19:38.112 --> 00:19:42.823
So we want to consider
the three dimensional case for next,
00:19:42.856 --> 00:19:48.908
so you will see such pictures
in the textbook of chemistry...
00:19:50.077 --> 00:19:57.028
but it is three dimensional crystals
and it continues entirely.
00:19:57.061 --> 00:19:59.880
Then the number of
00:19:59.913 --> 00:20:05.598
the three dimensional crystal groups
are 217.
00:20:05.631 --> 00:20:07.354
Very many.
00:20:07.387 --> 00:20:12.662
And it is this number
what's given in mathematics,
00:20:12.695 --> 00:20:15.719
but the most of them are exist
in the nature.
00:20:17.459 --> 00:20:20.587
It's very surprising fact, I think.
00:20:20.620 --> 00:20:25.300
And mathematician is more crazy,
00:20:25.333 --> 00:20:28.646
they went to consider
four dimensional crystal group.
00:20:30.045 --> 00:20:35.954
Then you can compute mathematically
in four dimension or five dimension more
00:20:35.987 --> 00:20:41.565
and someone computed it,
then it was a big number,
00:20:41.598 --> 00:20:45.528
but it was proved in dimension four,
00:20:45.561 --> 00:20:49.794
but it's still an open problem
in five dimension.
00:20:49.827 --> 00:20:54.007
So, if you want to try it,
then you can do it, okay.
00:20:54.040 --> 00:21:00.021
But it is no meaning for
some real world, in our world,
00:21:00.054 --> 00:21:04.788
but it can be extended entirely,
00:21:04.821 --> 00:21:10.229
so four dimension, five dimension,
six dimension in mathematics, okay.
00:21:10.262 --> 00:21:15.801
So this is some
mathematical crystal groups.
00:21:15.834 --> 00:21:22.779
So, I want to explain some connection
with some natural sciences.
00:21:24.281 --> 00:21:28.121
So, symmetry in the nature
or natural science.
00:21:28.154 --> 00:21:32.177
So the group theory
is not only in mathematics,
00:21:32.210 --> 00:21:35.282
it is used in physics and chemistry.
00:21:35.315 --> 00:21:41.020
And we also saw
some picture of the crystals
00:21:41.053 --> 00:21:45.735
and also it is used to know
the structure of molecule.
00:21:45.768 --> 00:21:47.743
{Foreign language}
00:21:47.776 --> 00:21:52.399
Then, if you have something new,
00:21:52.432 --> 00:21:56.074
but if you want, can see
standard structure of the molecule,
00:21:56.107 --> 00:21:59.839
then sometimes you can make
new medicine and so on.
00:21:59.872 --> 00:22:05.139
So, it became good news
00:22:05.172 --> 00:22:09.307
for some people and so on and so on.
00:22:10.479 --> 00:22:13.242
And there are two symmetries
00:22:13.275 --> 00:22:20.237
which were regarded as Nobel Prize
in Nagoya University.
00:22:20.270 --> 00:22:26.716
So, I will show some
Japanese information here.
00:22:26.749 --> 00:22:30.295
And so 2001,
00:22:30.328 --> 00:22:34.257
the then Nobel Prize
for Professor Ryoji Noyori.
00:22:36.540 --> 00:22:42.896
So he artificially you can,
so if you want to make this,
00:22:42.929 --> 00:22:44.770
you always get another one,
00:22:44.803 --> 00:22:50.646
so mirror symmetries,
but he wanted to get one of them,
00:22:50.679 --> 00:22:55.433
then he found some way
to make one of the mirror symmetries
00:22:55.466 --> 00:22:59.668
and this is his prize paper,
00:22:59.701 --> 00:23:03.044
so it's a certification on his prize.
00:23:03.077 --> 00:23:07.848
And you can see here,
maybe some, all of them,
00:23:07.881 --> 00:23:10.832
so you can see his Nobel Prize.
00:23:10.865 --> 00:23:15.429
And symmetry in chemistry,
00:23:15.462 --> 00:23:18.605
but he didn't need the symmetry,
00:23:18.638 --> 00:23:22.483
he just want to break the symmetry.
00:23:22.516 --> 00:23:27.587
But there's another Nobel Prize,
in physics.
00:23:27.620 --> 00:23:31.595
So they also considered
some kind of symmetry,
00:23:31.628 --> 00:23:35.418
but they considered
some symmetry breaking,
00:23:35.451 --> 00:23:39.337
so they don't need the symmetry,
00:23:39.370 --> 00:23:43.026
but it was proved by some big --
00:23:46.238 --> 00:23:49.930
something, later after their papers,
00:23:49.963 --> 00:23:56.600
but you can see something
in the faculty of science.
00:23:57.996 --> 00:24:03.842
Now, I want to turn to the singularity.
00:24:03.875 --> 00:24:06.135
What is singularity?
00:24:06.168 --> 00:24:11.558
So, the singularity
is such kind of point,
00:24:11.591 --> 00:24:14.258
this one and this one,
00:24:14.291 --> 00:24:19.547
and if you ride on the jet coaster...
00:24:20.904 --> 00:24:26.344
if it is a ride of jet coaster,
can you go safely?
00:24:29.126 --> 00:24:31.525
Do you want to ride this jet coaster?
00:24:33.349 --> 00:24:35.260
Maybe you want.
00:24:36.400 --> 00:24:41.782
But if you want -- so, if it is a
jet coaster, then it's very safe.
00:24:41.815 --> 00:24:43.038
What is the difference?
00:24:49.198 --> 00:24:51.846
Of course, you cannot go away,
00:24:51.879 --> 00:24:55.138
but there is a singularity here,
so this part,
00:24:55.171 --> 00:25:00.062
if you have two jet coasters,
then you cannot pass here,
00:25:00.095 --> 00:25:03.434
but also you cannot pass here
very safely.
00:25:03.467 --> 00:25:06.319
But this one, you can go mostly.
00:25:06.352 --> 00:25:12.142
But if you have such a ride,
00:25:12.175 --> 00:25:17.114
so if it intersects, then you cannot,
00:25:17.147 --> 00:25:20.527
but you can imagine,
it is a different point here.
00:25:21.605 --> 00:25:24.919
There are two points,
so this one, here and there,
00:25:24.952 --> 00:25:27.525
there are two points, then it is okay.
00:25:27.558 --> 00:25:31.548
So, this kind of strange point
00:25:31.581 --> 00:25:34.838
is called singularity.
00:25:34.871 --> 00:25:37.282
And if you don't have singularity,
00:25:37.315 --> 00:25:42.856
you can't ride on jet coaster
very safely.
00:25:42.889 --> 00:25:43.961
Okay.
00:25:43.994 --> 00:25:47.015
That's the definition
of the singularity today.
00:25:47.048 --> 00:25:52.331
And I want to consider the group action
on the space,
00:25:52.364 --> 00:25:58.951
so we consider a group symmetry
until now,
00:25:58.984 --> 00:26:05.936
but for example,
if you have a paper, it is very smooth,
00:26:05.969 --> 00:26:07.968
but if you fold it...
00:26:09.555 --> 00:26:14.961
then the most part is two papers,
00:26:14.994 --> 00:26:17.207
but this part is only one paper.
00:26:19.001 --> 00:26:20.094
Can you see?
00:26:20.127 --> 00:26:24.324
So, you can bend some, of course, okay.
00:26:24.357 --> 00:26:26.958
So, if you have a wire like this...
00:26:28.524 --> 00:26:32.129
and bend it here, you cannot go ___...
00:26:33.430 --> 00:26:38.539
and other part is that to the point,
00:26:38.572 --> 00:26:40.454
but this is only one point.
00:26:40.487 --> 00:26:43.287
So, if you see the paper like this...
00:26:44.914 --> 00:26:48.164
so it is very strange point.
00:26:52.681 --> 00:26:55.316
So the folding paper is something...
00:26:58.675 --> 00:27:00.442
group action by two...
00:27:01.907 --> 00:27:06.842
so we can divide it two parts,
but there are strange points.
00:27:09.324 --> 00:27:10.459
Can you see?
00:27:10.492 --> 00:27:15.361
That this is upper part
and this is lower part,
00:27:15.394 --> 00:27:22.091
but it's not upper, not lower,
it's only one part, okay.
00:27:22.124 --> 00:27:24.190
So this is singularity.
00:27:24.223 --> 00:27:29.109
And we consider more complicated cases,
00:27:29.142 --> 00:27:32.268
but for example...
00:27:35.897 --> 00:27:39.974
we consider group action on space.
00:27:40.007 --> 00:27:44.034
So it is a little bit complicated,
I am sorry about it,
00:27:44.067 --> 00:27:48.506
but if you know some mathematics,
you can see.
00:27:48.539 --> 00:27:51.315
But it's just action,
00:27:51.348 --> 00:27:56.126
so we consider the group action.
00:27:56.159 --> 00:27:58.183
And what is the group action?
00:27:58.216 --> 00:28:02.539
It's just some movement on the space.
00:28:02.572 --> 00:28:08.313
So for x-axis ___...
00:28:09.687 --> 00:28:16.086
you can multiply
minus 1 after group action,
00:28:16.119 --> 00:28:22.327
so the group action on x-axis
is minus 1, times minus 1.
00:28:22.360 --> 00:28:25.856
Then, if it is x square,
00:28:25.889 --> 00:28:31.516
then x becomes minus x,
00:28:31.549 --> 00:28:38.440
but x square becomes minus x times
minus x, then it is x square.
00:28:38.473 --> 00:28:44.631
So, x square doesn't have a difference
after group action.
00:28:44.664 --> 00:28:47.794
And similarly on the y-axis,
00:28:47.827 --> 00:28:52.885
if it is multiplication by minus y
or minus 1,
00:28:52.918 --> 00:28:58.893
then y became minus y,
but y square became y square...
00:29:00.369 --> 00:29:06.372
and xy is x became minus x
00:29:06.405 --> 00:29:08.736
and y became minus y,
00:29:08.769 --> 00:29:15.734
but x times y becomes minus 1
and minus 1
00:29:15.767 --> 00:29:22.556
and xy becomes xy,
then it was called invariant monomials.
00:29:22.589 --> 00:29:25.776
So, just saying, you don't have to
consider what is monomials,
00:29:25.809 --> 00:29:29.818
but it is invariant {Foreign language}
00:29:29.851 --> 00:29:34.445
under the group action.
00:29:34.478 --> 00:29:40.637
Then, x square and y square and xy
doesn't change after the group action.
00:29:42.077 --> 00:29:47.867
And there are several monomials
and polynomials and so on,
00:29:47.900 --> 00:29:52.695
but it is minimal set of this action.
00:29:52.728 --> 00:29:58.840
And if you put x square as X,
now y square is Y
00:29:58.873 --> 00:30:01.101
and xy equals this,
00:30:01.134 --> 00:30:07.581
then there is the relation
between X times Y is Z square.
00:30:08.897 --> 00:30:15.850
And it is a little bit different style,
but more canonical formulation,
00:30:15.883 --> 00:30:22.831
then it can be written down,
X2 + Y2 + Z2 = 0.
00:30:22.864 --> 00:30:28.647
It is the definition of the space
after the group action.
00:30:30.343 --> 00:30:31.581
And what is this?
00:30:32.897 --> 00:30:34.246
Can you draw this picture?
00:30:37.202 --> 00:30:38.614
Maybe this one is more,
00:30:38.647 --> 00:30:44.491
X2 + Y2 equal to something is a circle.
00:30:45.596 --> 00:30:49.012
And I will show the pictures next.
00:30:50.815 --> 00:30:53.491
So, this is a picture of this one.
00:30:58.064 --> 00:31:01.471
But there is only one strange point,
00:31:01.504 --> 00:31:06.198
it is (0,0,0) part, is a singularity.
00:31:06.231 --> 00:31:08.107
So what is singularity here?
00:31:08.140 --> 00:31:13.178
So, if you want to do skating here,
00:31:13.211 --> 00:31:15.735
you can skate here
and you can skate here,
00:31:15.768 --> 00:31:17.278
but you cannot skate here.
00:31:23.346 --> 00:31:26.230
In algebraic geometry
00:31:26.263 --> 00:31:32.898
and I want to see more good shape,
00:31:32.931 --> 00:31:37.157
then we consider
resolution of singularities.
00:31:37.190 --> 00:31:41.629
So resolution of singularities means
remove singularities,
00:31:41.662 --> 00:31:46.999
but if you remove the singularity,
then you have some open part,
00:31:47.032 --> 00:31:49.140
so it will divide it two parts,
00:31:49.173 --> 00:31:53.850
but in spite of this point,
00:31:53.883 --> 00:31:57.589
we just make some circle,
00:31:57.622 --> 00:32:01.831
then you have cube here.
00:32:05.423 --> 00:32:10.225
This action is called
in mathematics,
00:32:10.258 --> 00:32:12.065
resolution of singularities.
00:32:12.098 --> 00:32:17.398
So, we have one circle after resolution
of singularities in this case.
00:32:17.431 --> 00:32:19.862
So, please remember this number,
00:32:19.895 --> 00:32:25.127
X2 + Y2 + Z2 = 0,
00:32:25.160 --> 00:32:31.889
then after resolution of singularities,
we have only one circle here.
00:32:34.204 --> 00:32:37.844
And you can see the picture here.
00:32:39.597 --> 00:32:44.868
In front of the Nagoya Station,
there is this singularities here
00:32:44.901 --> 00:32:51.443
and it is something like that,
but if you open it,
00:32:51.476 --> 00:32:56.172
then you can get
the Kobe Port Tower here.
00:32:57.426 --> 00:33:00.666
Maybe you need another one here.
00:33:00.699 --> 00:33:07.307
So this is lower part and we have
upper part and open this part,
00:33:07.340 --> 00:33:09.672
then you can get that,
00:33:09.705 --> 00:33:12.536
but it is only point,
00:33:12.569 --> 00:33:17.312
but this has one circle here.
00:33:17.345 --> 00:33:21.077
So, this is resolution of singularities.
00:33:21.110 --> 00:33:24.982
And I consider,
maybe I should go back...
00:33:26.302 --> 00:33:30.772
so this is our matrix
00:33:30.805 --> 00:33:37.273
but it is generator of the group
of order two
00:33:37.306 --> 00:33:42.756
means so that A square equals
to identity matrix,
00:33:42.789 --> 00:33:47.971
means there is only two elements,
A and identity.
00:33:48.004 --> 00:33:53.661
And A is some upside down or something
00:33:53.694 --> 00:33:57.302
turning over and original one.
00:33:59.109 --> 00:34:05.081
It is also same as cyclic group
of order two.
00:34:05.114 --> 00:34:09.801
And you have this equation
00:34:09.834 --> 00:34:12.865
after the action of order two,
00:34:12.898 --> 00:34:17.189
but if you consider
the order four case...
00:34:18.287 --> 00:34:21.577
then you have this number four here.
00:34:23.399 --> 00:34:27.038
And after the resolution
of singularities,
00:34:27.071 --> 00:34:29.362
I cannot draw the picture,
00:34:29.395 --> 00:34:35.266
but there are three circles
after the resolution of singularities.
00:34:35.299 --> 00:34:41.589
And more, if you consider
the cyclic group of order n+1...
00:34:47.720 --> 00:34:54.716
then the equation becomes
X2 + Y2 + Zn+1 = 0.
00:34:55.876 --> 00:34:59.186
And there is also singularity
at the origin (0,0,0)
00:34:59.219 --> 00:35:03.631
but it's a very strange point
in this picture.
00:35:03.664 --> 00:35:06.221
And after the resolution
of singularities,
00:35:06.254 --> 00:35:07.848
we have n curves --
00:35:10.080 --> 00:35:13.128
I didn't say it is curve, but n circles.
00:35:14.667 --> 00:35:19.269
And the number n is related to
with this part,
00:35:19.302 --> 00:35:24.786
but also it is same as number
of the group representations.
00:35:24.819 --> 00:35:27.371
I didn't define here,
00:35:27.404 --> 00:35:33.688
but some invariant number
00:35:33.721 --> 00:35:37.203
which was given only groups,
00:35:37.236 --> 00:35:39.545
so it is geometric part
00:35:39.578 --> 00:35:43.655
and the resolution of singularity
is in the geometry.
00:35:43.688 --> 00:35:49.754
But here, there is no geometry,
just if you have a group,
00:35:49.787 --> 00:35:54.603
then you can just compute,
you can compute automatically.
00:35:54.636 --> 00:35:57.446
There is no picture and so on, so on.
00:35:57.479 --> 00:36:01.252
But number n appears here and here
00:36:01.285 --> 00:36:06.730
and which is called McKay correspondence
in dimension two in mathematics.
00:36:07.920 --> 00:36:08.985
It is very...
00:36:15.656 --> 00:36:19.067
funny situation,
but it is proved mathematically.
00:36:20.348 --> 00:36:26.397
And as I told you that
if you have two dimensional crystal
00:36:26.430 --> 00:36:30.798
you want to consider three dimensional
or four dimensional crystal and so on,
00:36:30.831 --> 00:36:34.676
so people want to consider the higher
dimension McKay correspondence
00:36:34.709 --> 00:36:36.647
in mathematics.
00:36:36.680 --> 00:36:42.323
Then, there is higher
dimensional McKay correspondence.
00:36:42.356 --> 00:36:47.671
So I studied on some three
dimensional McKay correspondence,
00:36:47.704 --> 00:36:53.024
but there are many difficulties,
maybe you can imagine.
00:36:53.057 --> 00:36:57.463
So if you have two dimensional crystals,
00:36:57.496 --> 00:37:01.761
it is just a paper, wallpaper
and then so on, so on,
00:37:01.794 --> 00:37:05.653
but if you consider
three dimensional crystal,
00:37:05.686 --> 00:37:07.999
maybe you can see and you can imagine,
00:37:08.032 --> 00:37:10.606
but can you imagine
four dimensional crystals?
00:37:12.204 --> 00:37:14.676
Which maybe it's difficult.
00:37:17.869 --> 00:37:22.266
If you have a higher dimensional object,
00:37:22.299 --> 00:37:26.528
it's very difficult to consider
sometime for human being,
00:37:26.561 --> 00:37:30.448
but maybe it can be computed
by computer, something,
00:37:30.481 --> 00:37:34.809
but there are some
more mathematical tools for.
00:37:34.842 --> 00:37:39.152
But on the other hand, there is some
relation with results in physics
00:37:39.185 --> 00:37:41.489
for three dimensional
McKay correspondence,
00:37:41.522 --> 00:37:43.754
so I want to show you next.
00:37:49.094 --> 00:37:55.407
So, there are two famous
singularities in physics.
00:37:55.440 --> 00:37:57.510
So, one is black holes.
00:37:57.543 --> 00:38:01.633
Black hole, maybe you know the name,
00:38:01.666 --> 00:38:05.423
it has very heavy gravity,
00:38:05.456 --> 00:38:11.466
so everyone who goes to the black hole,
it goes down and so on, so on.
00:38:11.499 --> 00:38:16.022
And finally it goes to the one point
00:38:16.055 --> 00:38:21.352
and it maybe singularity
in the universe.
00:38:22.852 --> 00:38:29.214
And big bang
is beginning of the universe,
00:38:30.240 --> 00:38:37.223
so the physicists believed
that the original universe is one point.
00:38:39.598 --> 00:38:44.869
And it became larger and larger
00:38:44.902 --> 00:38:47.556
and that we live here
00:38:47.589 --> 00:38:49.971
and that the first part is big bang
00:38:50.004 --> 00:38:53.227
and it is also something
like singularities...
00:38:54.997 --> 00:38:57.526
I mean, historically singularities.
00:38:59.071 --> 00:39:02.412
So, it is a kind of a singularity,
00:39:02.445 --> 00:39:08.469
but it's not in relation
with mathematics at this moment,
00:39:08.502 --> 00:39:12.096
but my three dimensional
McKay correspondence
00:39:12.129 --> 00:39:17.899
is related with super string theory
and it is said that
00:39:17.932 --> 00:39:22.845
it will be some most strongest theory
00:39:22.878 --> 00:39:26.321
to explain the universe in physics,
00:39:26.354 --> 00:39:32.498
but they consider the dimension
of universe is ten...
00:39:33.857 --> 00:39:39.748
and space time, so space time is --
time is one dimension after big bang
00:39:39.781 --> 00:39:45.381
and also space is three dimension,
then the space time is four dimension...
00:39:46.607 --> 00:39:50.682
but you have another remaining part,
six dimension part
00:39:50.715 --> 00:39:53.336
and if ___ Calabi-Yau manifold
00:39:53.369 --> 00:39:57.821
and physicists
draw a picture of Calabi–Yau
00:39:57.854 --> 00:40:01.677
like something like that.
00:40:01.710 --> 00:40:06.554
I think mathematicians
don't draw this picture,
00:40:06.587 --> 00:40:10.810
but you can just imagine
Calabi–Yau part here.
00:40:10.843 --> 00:40:16.945
And this is six dimensional,
but I consider --
00:40:16.978 --> 00:40:23.206
I told you
three dimensional McKay correspondence
00:40:23.239 --> 00:40:26.055
and the two dimensional
McKay correspondence,
00:40:26.088 --> 00:40:33.045
so in mathematics, you can consider
also the real axis and so on,
00:40:33.078 --> 00:40:40.040
but usually
we consider the complex numbers.
00:40:40.073 --> 00:40:47.015
So, if you know the complex,
1 complex x + yi,
00:40:47.048 --> 00:40:50.932
there is a movement,
two dimensional movement x and y,
00:40:50.965 --> 00:40:53.296
it is two dimensional.
00:40:53.329 --> 00:40:58.339
Then, x can be two dimensional
and y can be two dimensional,
00:40:58.372 --> 00:41:01.006
z can be two dimensional
and so on, so on.
00:41:01.039 --> 00:41:05.399
Then, three dimensional
McKay correspondence
00:41:05.432 --> 00:41:09.216
considered
in the complex three dimension,
00:41:09.249 --> 00:41:16.078
means, there is six dimension, okay.
00:41:16.111 --> 00:41:22.036
Then, it corresponds
to the physicist Calabi-Yau space
00:41:22.069 --> 00:41:28.448
and they compute some invariant
or some numbers in physics here...
00:41:29.461 --> 00:41:32.061
in their study.
00:41:32.094 --> 00:41:38.448
And some of the numbers correspond
to some mathematical number
00:41:38.481 --> 00:41:42.361
and we considered
what is this number in mathematics.
00:41:43.697 --> 00:41:49.444
But nowadays we have
several universes in physics,
00:41:49.477 --> 00:41:53.001
so I wrote here the universe,
00:41:53.034 --> 00:41:58.223
but multiverse
is a mutli-version of universe.
00:41:58.256 --> 00:42:04.269
So, uni is one,
so multiverse is many universe.
00:42:04.302 --> 00:42:05.311
Okay.
00:42:05.344 --> 00:42:09.852
Then they considered
the several kinds of universe
00:42:09.885 --> 00:42:13.655
and we live in one of them,
they consider,
00:42:13.688 --> 00:42:16.828
it is not my theory but they consider,
00:42:16.861 --> 00:42:22.807
and each space time they have
a different Calabi-Yau, they consider.
00:42:23.896 --> 00:42:30.260
Then, they considered several
kinds of Calabi-Yau spaces.
00:42:31.330 --> 00:42:36.958
So it is an idea in physics,
but mathematically
00:42:36.991 --> 00:42:42.146
I consider the resolution of
singularities in dimension two,
00:42:42.179 --> 00:42:44.309
and as I told you...
00:42:45.830 --> 00:42:52.622
so we have three circles and so on,
00:42:52.655 --> 00:42:59.337
but in dimension two, the resolution
of singularity is unique...
00:43:00.715 --> 00:43:05.606
but if you consider
three dimensional case,
00:43:05.639 --> 00:43:12.040
the way of the resolution
of the singularity is not unique.
00:43:12.073 --> 00:43:16.525
There are several
types of singularities.
00:43:16.558 --> 00:43:19.180
So from one singularity,
00:43:19.213 --> 00:43:22.759
you can get several shapes
00:43:22.792 --> 00:43:26.505
after the resolution of singularities.
00:43:26.538 --> 00:43:31.780
Then here, this is one Calabi-Yau space,
00:43:31.813 --> 00:43:36.529
so maybe you can consider as one number
of resolution of singularities,
00:43:36.562 --> 00:43:41.709
but you can also consider the different
type of resolution of singularities.
00:43:44.839 --> 00:43:47.480
And in mathematics,
00:43:47.513 --> 00:43:54.374
we consider the whole space
of all resolution of singularities,
00:43:54.407 --> 00:44:01.227
then it is similar to the idea
of the physicist in multiverse.
00:44:01.260 --> 00:44:07.001
So, I don't know the idea of multiverse
is correct in physics
00:44:07.034 --> 00:44:09.160
or natural sciences,
00:44:09.193 --> 00:44:12.331
but mathematically it's very interesting
00:44:12.364 --> 00:44:19.039
and also we can consider some
relationships physics and mathematics.
00:44:21.216 --> 00:44:24.799
So, that's all my slides,
00:44:24.832 --> 00:44:28.918
maybe you can ask questions
and so on, so on.
00:44:28.951 --> 00:44:30.343
Okay, thank you very much.
00:44:31.748 --> 00:44:33.167
Thank you very much.
00:44:35.779 --> 00:44:37.103
Very elegant.
00:44:38.415 --> 00:44:39.509
Thank you.
00:44:41.168 --> 00:44:43.337
So, any questions from the audience?
00:44:44.875 --> 00:44:45.911
Hello.
00:44:45.944 --> 00:44:48.751
In the middle of the talk,
there was an equation
00:44:48.784 --> 00:44:54.594
which said X2 + Y2 + Z2 = 0, right?
00:44:54.627 --> 00:44:58.145
I think if you think naturally,
00:44:58.178 --> 00:45:02.370
I believe that the solution
is only (0,0,0),
00:45:02.403 --> 00:45:04.421
but I don't know why?
00:45:04.454 --> 00:45:05.852
No, no, no.
00:45:05.885 --> 00:45:07.813
You can consider imaginary number.
00:45:07.846 --> 00:45:08.936
-Imaginary number?
-Yes.
00:45:08.969 --> 00:45:12.081
So that if it is -- yes, good question.
00:45:12.114 --> 00:45:13.220
Thank you very much.
00:45:15.812 --> 00:45:20.490
So, this equation holds only for (0,0,0)
00:45:20.523 --> 00:45:23.735
in real numbers,
00:45:23.768 --> 00:45:30.025
but if you consider
the imaginary number or complex number,
00:45:30.058 --> 00:45:32.805
then you have some similar points.
00:45:33.993 --> 00:45:38.169
So, for example,
i and minus i and so on, so on.
00:45:38.202 --> 00:45:39.860
Yes?
00:45:41.305 --> 00:45:42.545
Specific of the…
00:45:42.578 --> 00:45:47.631
So, there are
four dimensional object,
00:45:47.664 --> 00:45:52.002
but I just draw
a three dimensional picture.
00:45:52.035 --> 00:45:54.082
I see.
00:45:54.115 --> 00:45:55.545
In the real analysis,
00:45:55.578 --> 00:46:00.170
usually the coefficient of
these squares should be minus 1…
00:46:00.203 --> 00:46:01.554
Yes, yes.
00:46:01.587 --> 00:46:05.150
Because we are thinking
about the complex numbers,
00:46:05.183 --> 00:46:07.383
we can write this way.
00:46:07.416 --> 00:46:09.272
-Thank you very much.
-Yes, that's right.
00:46:09.305 --> 00:46:10.453
You're welcome.
00:46:11.934 --> 00:46:15.206
Thank you for a good question.
00:46:15.239 --> 00:46:17.135
Anybody else?
00:46:17.168 --> 00:46:22.866
You said that it is ___,
00:46:22.899 --> 00:46:29.538
___
00:46:29.571 --> 00:46:31.215
but I think...
00:46:38.062 --> 00:46:39.127
Matrix?
00:46:39.160 --> 00:46:41.321
___ knowledge...
00:46:46.374 --> 00:46:53.231
so it is difficult to ___
00:46:53.264 --> 00:46:58.430
so, please speak ___
00:47:01.072 --> 00:47:04.352
Elementary, about what?
00:47:04.385 --> 00:47:10.246
So, he was saying that using matrixes...
00:47:10.279 --> 00:47:13.036
and he considers that university level,
00:47:13.069 --> 00:47:15.402
so he would like more ___ level.
00:47:15.435 --> 00:47:16.972
Okay.
00:47:22.954 --> 00:47:27.898
So, this matrix means
00:47:27.931 --> 00:47:32.963
a quotient of {Foreign language}
00:47:32.996 --> 00:47:36.749
There is x, y variable
00:47:36.782 --> 00:47:41.603
and minus x plus 0 times y
equals to something
00:47:41.636 --> 00:47:47.162
and 0 times x minus y
equals to something.
00:47:50.698 --> 00:47:57.126
Then, this is also considered
as action on x, y...
00:47:59.641 --> 00:48:04.263
then x will be multiplied by minus y
00:48:04.296 --> 00:48:07.338
and there's no action on z in this case.
00:48:07.371 --> 00:48:09.857
And this part, there is no action on x,
00:48:09.890 --> 00:48:15.381
but there is minus 1 times y.
00:48:15.414 --> 00:48:19.406
So this is meaning of matrix, okay.
00:48:20.616 --> 00:48:27.314
Then you consider the invariant
under the action on x and y here, okay.
00:48:31.437 --> 00:48:33.168
Maybe this was the difficult part.
00:48:36.222 --> 00:48:37.336
Anybody else?
00:48:40.762 --> 00:48:42.041
Can I ask you a question?
00:48:42.074 --> 00:48:43.641
Yes.
00:48:43.674 --> 00:48:46.332
For example, if you have like
00:48:46.365 --> 00:48:51.474
two dimensional system,
00:48:51.507 --> 00:48:58.421
is the number of singularities, ___,
only one or can there be more?
00:49:00.108 --> 00:49:02.450
-In dimension two?
-Yes.
00:49:02.483 --> 00:49:04.124
There's only one point.
00:49:05.278 --> 00:49:06.368
That's it.
00:49:06.401 --> 00:49:08.675
In this case.
00:49:08.708 --> 00:49:10.740
-In this case?
-In this case.
00:49:10.773 --> 00:49:15.575
And other cases also has
only one point at the origin,
00:49:15.608 --> 00:49:18.007
but in dimension three,
00:49:18.040 --> 00:49:22.969
the origin is always singular,
00:49:23.002 --> 00:49:28.335
but for example, like that,
so it is not a point,
00:49:28.368 --> 00:49:31.715
sometimes line and sometimes surface
and so on.
00:49:31.748 --> 00:49:35.753
So, singular part
is connected to the origin,
00:49:35.786 --> 00:49:42.318
but it's a little bit funny structure
like ___ and so on, yes.
00:49:42.351 --> 00:49:45.026
And if there are multiple singularities,
00:49:45.059 --> 00:49:49.829
is it still possible to resolve them
all at once or?
00:49:49.862 --> 00:49:52.283
Yes, in dimension two, it was solved,
00:49:52.316 --> 00:49:57.600
but in higher dimension
there are many open problems, yes.
00:50:02.767 --> 00:50:03.791
Thank you very much.
00:50:03.824 --> 00:50:05.236
Okay. Thank you very much.