our next speaker is Professor R bot uh
我们的下一位演讲者是 R 教授

who is Professor ameritus in the
谁是 ameritus 教授

mathematics department at Harvard
哈佛数学系

University R it's a pleasure to welcome
大学 R 欢迎大家，非常荣幸

you 你

[Applause] [掌声]

oh well it's quite a crowd
哎，人挺多的

here it's a great honor to be been
这里是一种极大的荣幸能被……

brought here all the way from
从远方带来

California on this very nice
加利福尼亚非常不错

occasion 场合

I came here of course 56 years
当然我来这里 56 年了

ago to a large extent I think my
以前在很大程度上我想我

benefactor who brought me here was
资助人把我带到这里的是

Herman 赫尔曼

Val 瓦尔德

the George Dyson 乔治·迪森

yesterday informed me that he found the
昨天通知我他发现了

paper where they were selecting the
论文中他们在进行选择

people for admission that
人们对于入学的要求

year and he found my name on the list
年份，他在名单上找到了我的名字

that of doubtful 可疑的那一个

candidates so I was a candidate
候选人 我是一名候选人

that had not been informed that they had
他们尚未被告知那件事

accepted but but they had not been
已接受但是他们尚未

informed that they had not been accepted
已通知他们未被接受

yet 仍然

so I decided that as usual between my
所以，我决定和往常一样，在和我

two friends I'm talking that I'll be on
两位朋友我说我会在

the lighter side of this
这一面的轻松愉快

subject and after the imposing stuff mik
主题以及之后的高大上玩意儿米克

was told I think you'll find my talk
我觉得你会发现我的谈话很有趣

much easier to understand
更容易理解

and coming back to heral again he was he
回到赫拉尔再次他曾是

liked to sit with us young people up in
喜欢和我们这些年轻人一起坐在一起

the in the little restaurant and then he
在小小的餐馆里 然后 他

would often say oh you poor people
常常会说哦，你们这些穷人

mathematics has become so
数学已经成为如此

complicated especially I remember he
复杂 尤其我记得他

once mentioned 曾经提到

Fritz's Brooks's reman Rock serum which
弗里茨·布鲁克斯的剩岩血清，其中

he thought was unbelievably complicated
他认为是难以置信的复杂

compared to the mathematics that he
与他的数学相比

himself 他自己

had done in his life and he would give
已在他一生中所做的，他会给予

examples and so the title of my talk is
例子以及因此我的演讲题目是

really what Maron Morse missed by not
真地，Maron Morse 错过的东西

talking to her a 与她说着话

m in a sense and it was my good fortune
在某种程度上，这是我好运气的体现

to to sort 对 对 排序

of to pick up on
从…中继续

there so 那里如此

the I I came here as an engineer and
我作为工程师来这里

topology was a pretty difficult subject
拓扑学是一门相当难学的课程

to understand 理解

but questions of this sort I was able to
但这类问题我能解决

understand 理解

so according to mors if you had a
所以，根据莫尔斯电码，如果你有一个

function on a 在上的函数

manifold you should look at its critical
多方面您应考虑其关键

points 点

and count the critical points by how
并且按如何计算关键点

many directions of steepest descent came
许多最速下降方向出现

from it now even as an engineer I knew
从它现在甚至作为工程师我也知道

what that was so for instance I learned
那个比如说是这样，我学到了

the following picture here is the two
以下图片中的是两个

sphere you take a point
球体取一点

outside and look at the distance
外面看看距离

function then of course it's easy to see
函数，当然很容易看出

that the critical points correspond to
那些关键点相对应

the places where the straight line lines
直线经过的地方

meet the thing at right
遇到右边的东西

angles and so the two critical points
角度和所以两个关键点

are here and 不存在

here here is a minimum and here's the
这里 这里是一个最小值，这里的是

maximum and here is a wonderful point
最大的是这里有一个非常好的观点

the center of the 中心的

circle you notice that these two points
圆形你注意到这两个点

are the Symmetry one takes into the
是取入对称性的

other and the 其他 和 的

this is the minimum this is the maximum
这是最小值 这是最大值

and this is what in the in the in
并列这个是什么在在在

Geometry is called the focal point of
几何学被称为……的焦点

this shape along this line and it's of
这个形状沿这条线，它属于

course means because all these normals
课程意味着因为所有这些法线

come together 聚在一起

here and so in the Morse counting of of
这里并且如此在莫尔斯计数中

this function you would write down a one
此功能您会写下单个

for 对于

this and a t^2 for this
这和这个的 t^2

because there are two this two two
因为有两个这两个

directions of steepest 最陡方向

descent independent directions and this
下落独立方向及此

would then be the 则将是

Morse counting function of this
摩斯计数功能此

function and the nice and to me
功能、美好并且对我

immediately I immediately love this
立即我立即爱上这个

theorem that his theorem was really that
定理其实是他定理的那个定理

if you do this in General on a
如果你在一般上这样做

manifold this function of course depends
多重此功能当然依赖于

on the original function the counting
在原始功能中的计数

function but it satisfies certain
函数但它满足某些

inequalities but more than that if
不平等，但不仅如此如果

you're 你是

lucky then they actually the
幸运的话，他们实际上就是

inequalities become 不平等加剧

equalities and in particular for
平等，特别是在以下方面

instance if all the exponents that you
实例 如果所有指数都是你

find are even then 找到甚至是那时

automatically this function is a perfect
这个功能自动是完美的

as we call it perfect M
我们称之为完美的 M

function well the 功能良好 的

this is what I would sort of call the
这是我可以称之为的那种

the 没有提供需要翻译的源文本，因此无法进行翻译。请提供需要翻译的内容

baby what we used to call the baby Morse
宝宝 我们曾经叫的宝宝摩斯

Theory what really fascinated me was
理论真正吸引我的是

that he 他那

had and that was of course
已有 当然

also to a large extent his Fame rested
在很大程度上，他的名声也建立在……上

on this theorem that for instance he
关于这个定理，例如他

proved that there are an infinite number
证明了存在无限个数

of De descs joining any two points on a
在上的两点之间

sphere 球体

and he 并且他

used the Morse Theory there in the
在该处应用了莫尔斯理论

following way so the problem
以下方式所以问题

was the theorem is that you can do the
该定理是你可以做的

same sort of thing if you study the
同类事物，如果你研究的话

function on the loop 循环上的函数

space of a 空间的一个

manifold that 多 MANIFOLD

means you take a manifold and
意味着你取一个多尺度体

M and you study all the paths that join
M 和你们研究所有相连的路径

a point P to 一个点 P 到

Q you take the energy along that path
你沿着那条路获取能量

and the extreme of that are the
极其是那个

geodesics and you can count the
测地线并且你可以计数

geodesics by they are critical
大地测线至关重要

points and they turn out to have the
点并且结果证明他们有

marvelous property that the number of
奇妙的属性，数量

negative directions is always
负面方向总是

finite and so 有限且所以

the you can formally write down the same
你可以正式写下相同的

sort of 有点

Series so let's for instance do that
系列，比如让我们这么做

for the two 为这两个

sphere 球体

again we start 再次开始

here and we choose a nearby Point
这里我们选择一个附近的点

q and we start to study the critical
q 我们开始研究关键

points of this 这个的要点

situation 情况

well you see the geodesic
好，您看，这是地理经纬线

segments here is of course the geodesic
这段当然是指测地线段

but that's not 但这不是

all you can also 所有您也能

go this 去这里

way you can of course also go this
当然，您也可以走这条路

way and so on so you see that there's an
方法和等等，所以你看这里有个

infinite number of critical points
无穷多个临界点

immediately and and you can write down
立即并且你可以写下

the more series very easily because it
更多系列非常容易因为它

turns out that what is the focal point
原来焦点是什么

there in this situation becomes the
在这种情况下变成的

conjugate points and the conjugate
共轭点和共轭

points are precisely the conjugate point
点恰好是共轭点

of p is precisely the antipode and P
p 的反点恰好是 P

itself and so instead of
它自身以及因此而是

looking Mora's very beautiful so-called
寻找莫拉的美貌所谓的

index theorem said that you to to count
指数定理说你来数

the number of negative directions you
你负向指令的数量

just have to count the number of
只需计算数量

conjugate points along each
每条线段上的共轭点

zodic so for instance if we do this you
zodic 所以例如如果我们这样做，你

will see on the two sphere you will see
将看到在两个球体上你会看到

that this thing here counts for
这东西很有价值

one this fellow 一位这位家伙

here 这里

where I don't see quite for yeah this
我在哪里看不到 quite for yeah this

this fellow here crosses the conjugate
这里这个人跨越了共轭

Point once so it adds it
点击一次，使其添加

T and then you'll find one that they add
T 然后你就会发现他们添加的其中一个

or one more each time we go around so
每次绕一圈再加一个

you get this 你得到这个

series now this series actually the way
系列 现在的这个系列 实际上 这种方式

it 它

stands would only satisfy
只能满足架子

inequalities actually if I I write
不平等实际上如果我要写

it it's 1 1 minus
它它是 1 减 1

t shorter way of writing it
t 更简短的写法

so here the we don't know whether this
所以这里我们不知道这是

is 是

really describes a b polinomial of the
真的描述了一个 b 多项式

loop space but actually one could show
循环空间但实际上可以显示

that it does but certainly if i' switch
但这肯定如果我能切换

to a higher sphere if I switch let's say
若我转换，比如说，到更高的层次

to the three 至三个

sphere then it's fairly easy to see that
球体然后相当容易看出

these multiplicities change and so then
这些多重性会改变，然后

you would have had for the three sphere
你会有三个球体

the corresponding 1 / 1us t^ 2
对应 1/1us t^2

this would be the 这将会是

answer and 答案及

the this would actually this would prove
这个这实际上会证明

that the 那个

chology on of the space of loops on a
心理学中关于环状空间的研究

sphere has this point R
球面有这一点 R

Series so I think that was a
系列 所以我认为那是

wonderful result and it it's interesting
奇妙的结果，而且很有趣

that when I was here in Princeton the
当时我在普林斯顿的时候

topologists pay no attention to this
拓扑学家对此不予关注

really you know I mean and never did I
确实你知道我的意思，我从未那样做过

hear on fine Hall anybody comment on
听到在漂亮的礼堂里有人评论

this fact which was completely
这一事实完全

ignored and this was of course in
被忽略了，这当然是

49 then of course Sarah appeared in 51
49 然后当然莎拉出现在 51

or so and suddenly Loop spaces and so on
或如此，突然出现环空间等等

became the bread and butter of all of us
成了我们大家的生计来源

topologists but in a quite different
拓扑学家但相当不同

different way not in the Morse way and
不同方式，不是莫尔斯码方式

before s nobody thought it was
在 S 之前，没有人认为它是

interesting that one could compute the
有趣的是，人们可以计算

chology of such interesting function
心理学中这样有趣的功能

Spaces by such easy geometric
空間由如此簡單的幾何形狀構成

computations all except me I must say I
计算，除了我必须说，我

I I enjoy this very much
我喜欢这个

so is there an ER eras
so is there an ER eras 如此有一个 ER 时代

here can I erace 这里可以删除

I guess 我想

not a bit 一点也不

terrifying oh oh 恐怖 哦 哦

thanks so when I 谢谢，所以我

left when I left the Institute went to
离开研究所时去了

Michigan and there I started to learn
密歇根州，我开始了学习

about leag 关于联赛

groups and uh as Michael was saying one
组和嗯，正如迈克尔所说的，一个

one loves one's first theorem so that's
一人深爱着自己的第一个定理 所以那

the one I'm going to talk
我要谈论的那一个

about which I discovered there in in the
关于我在那里发现的

in the about 52 or so so the as I
在大概 52 左右所以我是

learned about Le groups of course I came
当然，我学到了 Le 群，我来了

to appreciate this picture of the two
欣赏这张两张照片

spere in a different way the two sphere
在大球中寻找不同的空间

is the adjoined orbit of the SO3 group
是 SO3 群的相邻轨道

of 的

symmetries 对称性

and so when then I went home and I as I
然后我回家，我当然后面

say in Michigan I started to think about
在密歇根我开始思考

the next group so to speak
下一组，可以说是

su3 is a three 三是个三

sphere and then I saw right away that we
球体，然后我立刻就看到了我们

had already computed the point series of
已计算了点序列的

the three sphere so we were on the way
三个球体，所以我们正在路上

so sorry 很抱歉

su2 su2 really morse's way of computing
su2 su2 真的是摩尔斯计算方法

gave the coity of the loop space of the
给出了环空间的城

of the three 三个中的

sphere and so the next step was to look
球体，所以下一步是查看

at SU 在苏

three and 三和

now the marvelous thing is that you see
现在奇妙的是，你能看见

in the the orbit of of
在轨道中

su2 the two sphere intersects this line
su2 两个球面与这条直线相交

in just these 仅在这些

points well there's a corresponding
点上对应

phenomenon in all the groups has to do
所有组的现象都有关联

with the cartan sub algebra but in this
与 Cartan 子代数但在这一

example it has to do with that you have
例子它与此有关

a two Di dimensional plane and if you
一个二维平面，如果你

take the 取

orbit the orbit of anything in the
绕任何事物的轨道轨道

adjoint representation it will intersect
伴随表示它将相交

that two plane in a similar way in which
那两个飞机以类似的方式

this orbit intersected that line and in
这条轨道与那条线相交于

fact what you find then
事实你然后找到的

is that if you take the matrices
如果取这些矩阵

then 然后

3x3 matrices in su3 diagonal matrices
3x3 矩阵在 su3 对角矩阵中

and write down the 并写下

lines where two ion values are
行中存在两个离子值

equal you will find a picture of this
平等你会发现这张图片

sort and if you take the orbit of any of
对...进行排序，如果您选择任何行星的轨道

any point 任意一点

here The Marvelous thing is that you'll
这里奇妙的是你会

find this orbit is perpendicular to that
找到这个轨道与此垂直

two plane and it hits
两个飞机并撞击

it comes back 它回来了

once in each one of these symmetry
在每个这样的对称中一次

places so the orbit goes out and comes
地点使得轨道出去又回来

back so if you study the distance from a

general Point let's say to the
一般观点来说，咱们可以这样讲

orbit then you easily check that the
轨道然后你很容易检查

critical points of that functions are
那些函数的关键点为

precisely these 精确地这些

intersections 交叉点

so there we found only two here we we
这里我们只找到了两个

would find 将找到

six and more than 六和更多

that if 如果

we draw the 我們繪製了

lines just as here you remember we had
行就像这里你记得我们曾经有

the minimum when we didn't hit a focal
当未击中焦点时的最小值

point and the maximum there here you the
点，最大值在这里你

number of negative 负数个数

directions at the corresponding critical
对应的临界方向

point is just the number of times you
点是次数

cross these lines so that means that
跨过这些线就意味着

this thing would be 这东西会是

two this would be two that's
两个这将是两个那就是

four four and six so that the
四四加六所以是

mor of this adint orbit would be 2
此广告积分轨道的周期为 2

t^2 plus what am I doing here 2 t + 2 T
t^2 加我在这里做什么 2t + 2T

to 4th plus t to the 6 in L and behold
至第 4 加 t 到 L 的第 6，看哪

they were all even so this is the actual
他们都这么整齐，这才是实际的

topology of an orbit in
轨道的拓扑

su3 and now this thing here com
苏 3 以及现在这里这个玩意儿

is 是

the how should I put
我不知道该怎么说

it the marvelous thing is that this this
它奇妙之处在于这个这个

happens for all these compact Le groups
发生在这所有紧凑的李群中

if the picture is the same this is
如果图片相同，这就是

called the V group of course this is the
被称为 V 组当然这是

least you you have to look only at the
至少你要看的只是

cartan sub algebra you have the V group
卡坦子代数您有 V 群

The General orbit Hy orbit will now is
通用的轨道 Hy 轨道现在将是

actually parameterized by the elements
实际上由元素参数化

of the V group and these indices that I
V 组及其这些指数我所

have written down geometrically the
已将几何地写下

algebraist usually like to write down as
代数学家通常喜欢写成

the number of negative weights that have
负权重数量

been changed by an element of the V
已由 V 的一个元素所改变

group so this was this was a
组所以这是这是

I think I was the first to notice this
我觉得我是第一个注意到这个的

and it was great fun and
很愉快且

it and it started to make me think well
它让我开始思考

maybe we can do the same thing for the
也许我们可以为同样的东西做同样的事情

loop space of a Le
莱的环路空间

group 组

so and as it turned out you you can
结果证明你可以

indeed 的确

so this is where 所以这就是那里

this picture 这张图片

will um in this 将..

is unfortunately 很遗憾

the Blackboard is too small for
黑板太小了

me no but we we have enough room here
我没事但我们这里足够宽敞

now what is this remember I I just
现在这是什么 记得我 我只是

showed you in s SO3 in su3 in the Le
展示了您在 SO3 中的 s 在 su3 中的 Le

algebra in the matrixes of Trace zero
矩阵中迹为零的代数

3x3 su3 matrices the places where two
3x3 su3 矩阵的两个位置

Ang values were equal now you can
Ang 值现在相等了，你可以

actually look in the 实际上看看

group at the diagonal matrices
对角矩阵组

and draw on on the space of diagonal
从对角线空间中汲取

matrices which is a two two-dimensional
矩阵是一个二维的

Taurus all the places where two I values
金牛座所有两个 I 值的地方

are 是

equal and then you take the universal
平等然后你取普遍的

covering of that and you get obviously
覆盖那部分你显然得到

this 这

picture and this picture is has
图片和这张图片是有的

additional information namely if this is
附加信息即如果是

the 没有提供需要翻译的源文本，因此无法进行翻译。请提供需要翻译的内容

origin then here is the
起源 然后这里就是

latice that 格子

and so 并且所以

on the so I recover now on the plane I
so 我现在在飞机上恢复

recover the picture of 恢复图片

the this diagram as it's called of the
这张被称为的图

league group on the maximal Taurus and I
联赛集团在最大牛头犬和我

get I get a a picture of this sort we
我得到这种照片的图片

get a 获取

latice together with planes called the
晶格连同称为平面的

root planes but they now now form a
根平面但现在现在形成

Tessellation of the whole
整个镶嵌

plane now the beautiful thing is the
飞机现在美妙的所在是

geometry comes into the fact that in a
几何涉及到在的事实中

Le group if you start a geodesic in a
您在启动大地测量的组中

general direction at the origin on a
以原点为起点的总体方向

Taurus it will stay on the Taurus so
泰坦神将留在泰坦上

that the only 那是唯一的

geodesics that join two close by points
大地测线的两点相连

which are in general positions you can
这些通常是在一般位置的你可以

actually see them as straight lines on
实际上将它们视为直线在

this Universal covering 这通用覆盖

so if you for instance if this is the
因此，如果你比如说这是这个

origin and I'm looking at the space of
源头和我在观察空间

paths which go from here to
路径从这儿到

here then here is the first
这里，然后这里是第一个

geodesic but of course 球面测地但当然

here is another 这里又是另一个

one just like in that earlier picture
就像那幅早期图片中的一样

what I was drawing there you get a very
我在那里画的东西你看得非常清楚

beautiful linear picture of all the
所有美丽线性的画面

geodesics in question 问题中的测地线

and so you have to then only count again
所以您必须再次计算

try and find the directions of steepest
尽力找出最陡的方向

descent and you find just as I just as
下降并且你发现就像我现在这样

before as you might conjecture that if I
在您可能猜测的那样，如果我是

draw for instance this 绘制例如这个

geodesic I just have to count the number
球面坐标我只需要计算数量

of times it crosses planes and multiply
次数穿越平面并倍增

by two and that gives me the number of
乘以二，这就是我的数字

negative 负面

directions so so I have a
指示一般 我有一个

complete a complete picture of all the
完整展现全部的

theoretics joining two general position
理论上连接两个一般位置的

points on the lead group now you fiddle
领先组现在你瞎折腾

a bit about this there's an obvious the
关于这一点有一点明显的

Symmetry group now is larger than before
对称群现在比之前大

so I 所以，我

can I can now do the following thing
我现在可以做到以下事情

that I can try and keep track of this
我可以尝试并跟踪这个

situation more systematically
情况更系统

and this is called an
以及这被称为一个

Al a positive chamber in the in the

regroup 重组

industry so you choose a positive wild
行业所以你选择积极的野

chamber and then obviously every
chamber 然后 显然 每

geodesic can be rotated into this one by
地球仪可以通过以下方式旋转到这一位置

conjugation so I 对化所以我

can take these points which are spread
可以采取这些分散的点

all over and always rotate them back
遍及并始终将其旋转回来

into into the fundamental chamber
进入基本室

and beautiful thing is that I completely
这是美妙之处在于我完全

fill out this chamber so that I don't
填写这个房间以便我不

have to draw 得画

these cues so carefully anymore I can
这些提示现在不再那么小心翼翼了

indicate them by 指明它们

just so there Q here there's a q here
只是这里有 Q，那里也有 Q

there's a q here A
此处有一个“q”字母，A

Q so on so I just well the positive ra
Q 所以如此，我只是很好，正面的 ra

chway is here chway 在这里

and so to each one of these there is a
并且对于这些中的每一个都有

[Music] [音乐]

unique geodesic that corresponds to it
独特的球面几何形状，与之相对应

and I can read off its its um index now
并且我现在可以读出它的索引了

because it the index is given by
因为指数由给定

counting how many lines you cross let's
计算你穿越了多少行 让我们

do that here for instance so this is of
做这里，例如，这样这就是

course this this is called an alve in
课程这就是所谓的肺泡

the industry this alov represents
该行业代表的是 alov

zero here here the Q that was in here
零 这里 这里 的 Q 在这里

you you get to with Crossing just one
你只需穿越一次就能到达

line so here you put a
行所以这里放

two here you put a
两个这里放一个

four 四

four six and so 四六所以

on so for 对于所以如此

su3 you see that the fun series and now
遂有你看那趣味系列如今

only because they are even this already
仅仅因为它们已经是这样的了

actually computes the Co wary and so the
实际上计算了 Co wary 因此所以

answer is if you do a little counting
答案是你稍微计算一下

here this line gives you 1 / 1 -
这里这行给你 1 / 1 -

t^2 and this and then you can repeat
t^2 和 这 然后 你 可以重复

this once you have added all these
这已经添加了所有这些

together you 一起

can add these lines and there you notice
可以添加这些行，然后你就会注意到

that you always go up at four so it is m
您总是四点起床，因此是…

1us t 4

and that of course is the correct
当然是对的

answer and this way of getting at it
答案以及这种方法

proves that there's no torsion in the in
证明在内部的扭力为零

the space path on an E Group and that
在 E 组上的空间路径以及那个

was at the time 当时

unexpected 意外

because of course the leag groups the
因为当然，联盟分组了

higher Le groups do have torsion but
较高阶的 Le 群确实有挠率，但

they lose it all in the step to their
他们在这个过程中失去了所有

Loop 循环

space I don't know how much time I got
太空我不知道我还有多少时间

left so all right so
左边所以右边所以

the so this now you can see quick quite
所这现在你看快很

easily how it extends to the classical
轻易地它扩展到经典

groups these this sort of information
将这些这类信息分组

about a Le group is precisely the sort
关于勒氏集团这正是那一种

of information that is used in
用到的信息

classifying them and so it was very easy
将它们分类，所以非常容易

to write down the corresponding
写下相应的

formula let's actually for those of you
公式实际上是为了你们这些

who don't 谁不

know let's compute this way what the
不知道我们就这样计算一下这是什么的

loop space of G2 is you know G2 is a
G2 的环路空间是，你知道 G2 是

more sophisticated object which most
更复杂的物体，大多数

people don't know so well but I I know
人们不太了解但我知道

there are some people interested here I
此处有些人感兴趣

don't know do you know
不知道你知道

the series 系列

of well if you look at the so-called
当然，由于翻译任务的性质，没有提供具体的源文本，因此无法给出翻译结果。请提供具体的源文本，我才能进行翻译

diagram of 图表

G2 it's very easy to obtain it from from
G2 很容易获取它

the one of 这一

su3 I mean these pictorial methods of
梳 3 我意思是这些图画方法

course only work last for groups of rank
课程仅适用于等级分组

no greater than two but G2 than go rank
不大于两但 G2 比 GO 排名

two so all you have to do is add these
两个所以你只需要添加这些

lines 行

to I hope I 希望我

I'm missing a 我缺少一个

few 少数

uhoh maybe I have enough
哎呀，可能我足够了

now you see here here now
现在你看到这里 现在现在

the the wild chamber would be like this
原始文本：the the wild chamber would be like this 译文：这个野生的大厅会是这样

and I can and it's these little these
并且我能 这些小 的

it's these shapes that play the role of
这些形状扮演着角色

the 没有提供需要翻译的源文本，因此无法进行翻译。请提供需要翻译的内容

alcoves and so I start here with
凹室，所以我从这里开始

zero 2 零 2

4

6

8 and you'll find that the first time I
8 你会发现，第一次我

get something else is 找其他东西

10 here I have two 10
10 这里我有两个 10

join so the so the the the loop space of
加入这这样这样的循环空间

G2 is one over 1 - t^
G2 是 1 除以 1 - t^

2 1 - t to the
21- t 至

10th and on the so as far
第十及以后

as the homology of the Spheres that
作为球体的同构

underl they would look like a three
下是他们看起来像三个

sphere cross an 11 sphere and while this
球穿过一个 11 球体，同时这个

work great fun for me to find out
工作对我来说非常有趣，发现它很有趣

because I knew nothing about G2 at all
因为我对 G2 一无所知

but but the world at that time didn't
但那时的世界还没有

know anything about the topology of G2
了解 G2 的拓扑结构吗

and from this one could immediately
并且从此可以立即

decide that for instance Pi 3 was the
决定例如 Pi 3 是

infinite cyclic and that the next
无限循环以及下一个

homotopic group was was 11
同伦群是 11

dimensional well so this is this is to
维度井 因此这是 这是这

show you what I say mors missed if you
展示我说的 mors 丢失了如果你

had talked to her Val just a little bit
与她 Val 交谈了一会儿

right this is really it was all in his
没错，这真的就是全部都在他那里

completely in his grasp I mean he he
完全在他的掌控之中 我意思是说 他他

could have done this in a moment's time
瞬间就可以做到这件事

but we all unfortunately tend to miss
但我们都不幸地倾向于错过

these 这些

opportunities this at this stage I could
此阶段我能做的机遇

really 真的很

start a a long lecture
开始一个漫长的讲座

because because these diagrams of course
因为因为当然这些图表

you should think of them as
你应该把它们想成

little what sticks out in an iceberg
小冰山突显的部分

because since the period of I did this a
因为自从我做了这个时期

tremendous amount of information has
巨大的信息量已经

been found 被发现

out 出

about in particular about this situation
关于特别是关于这种情况

you notice that this to me it already
你注意到这对我来说已经

was clear this looks very similar to my
这看起来和我的一样清楚

original computer mutation in the Le
source computer 在 Le 中的原始突变

algebra situation that could never and
代数情景，永远无法

here you notice that this this Loop
这里注意到这个循环

space seems to be made out of many
太空似乎由很多组成

copies of what I found in the
副本我在那找到的

infinitesimal picture well all that has
无穷小图所有这一切

been completely cleared up in the
已完全解决在

meantime the this 与此同时，这个

Marvelous World of the cut smoothly
妙不可言的剪辑世界

algebras their 代数它们的

diagrams the topologies the
拓扑图定义

representation 表现

series that to me was a very exciting
系列对我而言非常激动人心

moment when when really the whole
瞬间当当整个

classical representation theory of the
经典表示理论

compact groups suddenly was lifted up
紧凑组突然被举起来了

and extended to the katsumi algebras and
并且扩展到 Katsumi 代数及

the representations of positive
正表示

energy these things came as an impetus
能源，这些事物起到了推动作用

from 来自

physics and of course 物理和当然

they've had the repercussions in not
他们已经尝到了后果的滋味

just in this context but in algebraic
只是在这个上下文中，但在代数上

geometry and 几何与

in and even in pic
在内以及在内

Theory but I think that's all I have to
理论，但我认为这就够了

say thank you 说谢谢

[Applause] [掌声]

thank you Ralph for beautiful talk and I
谢谢拉尔夫，您的谈话很美

think we have time for one perhaps two
认为我们还有时间，可能有一个或者两个

questions or 疑问或

comments I'm always very 评论 我总是非常

clear oh no no no
清晰 噢 不 不 不

no that trouble 无那麻烦

well Mar Morris was in some sense a sort
嗯，Morris 在某些意义上是一种类型

of he communicated only 他仅传达了信息

[Laughter] [笑声]

outward yeah yeah know I didn't see him
向外 嗯 嗯 我没看见他

ping around with any of the professors
和任何教授闲聊

very much but he was a
非常但他是

wonderful although he was domineering in
虽然他有时专制，但很棒

that sense he was at the same time
在那种意义上他是同时的

rather humble and and and
相当谦虚

you know for instance and very
您知道，例如，非常

informal on my 26th birthday he was
不正式的在我的 26 岁生日那天他

invited to a little party we had on the
被邀请参加我们举办的那个小型派对

thing and he immediately challenged me
东西 他立刻挑战我

to a to a race to the nearest
到最近的比赛去

tree 树

yeah but I I the only people that I saw
是啊，但我只看到了这些人

talking in a in an easy way to each
用简单的方式彼此交谈

other were really 其他真的很

foron and her foron 和她的

and really of the permanent members at
并且真的永久成员在

the time herova seemed the most at ease
她的時候赫羅瓦似乎最自在

here and was a wonderful
这里是个美好

presence hey thank you R again for a
存在 嘿 谢谢 R 再见 a

beautiful call 美丽呼唤

[Music] [音乐]