One job.

On.

The first of the

rest.

Of the Royal Society.

Of.
The.

Site.

For a.

Number.

Of.

Very high for

their.

Own.

Wonderful.

And.

An and.

That's all I'd like to thank.

Star Ross and other people responsible for

inviting me here is a great pleasure
to come it's my first time at.

Georgia Tech and it's been nice of a look
around seemed like a great campus.

I was coming from Nashville and
I thought I was coming south so

it will be coming to a warmer place and
this is what a.

Big call here than I had had predicted.

Right so.

One of the formalities of
my first to go through.

Let's just get straight on to talk.

So the title of the talk of
the sea is how quantum and

statistical mechanics gave rise to
a polynomial vary in the north so

this is an old story it goes back
a step of the states go back to

basically begins in about nine
hundred eighty or even before.

And the part of I'm really going
to talk about was over and

about nine hundred eighty five.

But it must be an oldie but goodie because
every time I give people a choice of

several pokes at the they might want
to hear this is the one they choose so

and I have to say I haven't yet
fit up with giving it so

hopefully it gives me a chance to
relive my glory days or something.

So that's what we that's for
them and talk about.

Another wave of.

Perhaps more.

Graphic way of saying what this talk
is about is from that and that.

To.

That and that So let me just explain some
of these pictures this this one here is

put possibly the most puzzling it
will actually this is probably.

Some group of mathematicians
my most famous picture.

OK as my piece of.

This was my attempt to.

Draw a certain for Norman algebra which is
something we're going to be talking about

and this was a better as accurate
description as I could get of it

this is supposed to be a laugh this two
dimensional lettuce on and on which

would live something like the easing
model from statistical mechanics but

some of them this one kind
of mechanical models and

this is supposed to be a not will
I look to say about knots and

the amazing one of the amazing end
points of this this very rapid voyage

was going from these very abstract and on.

Well that's the that's the best
picture I could draw of what I was

talking about through real life
in a biology in labs and so

on this is a picture of a D.N.A. molecule.

And not a D.N.A. molecules no doubt that.

The D.N.A. molecules get
themselves tied up in knots so

I probably won't have time to say
anything about that but it was.

A mind blowing trip to all of us and
a couple of years.

Right now the person who's really
responsible for it is this gentleman here.

This is this is John for Norman and

it was basically his idea is that I
was following and his inspiration and

I'm still following in my in my
mathematical and full physical Korea so

he was an amazing person I'm assuring that
you know about him I put him up twice for

what he wanted the difference.

Could there's lots of and it don't and
if you by any chance you're someone who

doesn't know about phenomena go look him
up right away you will be absolutely

staggered boy what
the mind of that person.

Right so I want to start my
talk I'm not entirely sure what

background the people who want
to try to turn you on to.

It's phenomenal the bus boy
asking you a very simple for

mathematicians question.

Which is that what is the dimension of.

C. squared two copies of the complex
numbers a Cutty's in product of two copies

of the complex numbers would
have pairs of complex numbers.

Right the would I mention will be
in Forth reduced to just deal him

what is dimension all the complex numbers.

And if your methods as you go you
use first reaction is what is he

talking about they must be kidding
because the answer is to write.

Two copies of the complex
number system Engine two.

But then well that was supposed to be
the answer I would have asked the question

right so now I'm going to give you another
answer which would be the next one you

would get the income which would be.

Four right.

Because you know the complex numbers
is two copies of the real numbers real

imaginary and the four for
those all together in C. squared OK.

Again I'm going to propose another are so
which would have pleased Norman and

will be the spirit of
full Norman in these so

called for
a moment of this is the other answer.

Hough.

OK.

Well what's this nonsense going to be
the mention how that's ridiculous Well

it's not ridiculous.

And here why do we get a half well
to figure that out how did we get

the answers to and for
well we thought of C.

squared as it's two copies of C.
It's the number of copies

of the field underlying thing on
which everything is represented

that you see in the in the space so
you see two copies of the complex and.

Well if you want to work out on the.

The reels you see four
copies of the reels so

I claim that you see one half a copy of.

The two by two major cities right because
the two were two major cities over

the complex numbers a two by two
matrix is something called A B. C. D..

Where all of these are complex numbers and
if you take half of that

split that in half then you get into.

So the lesson is that
the dimension of some space

in the snowy sense depends entirely on
what you're measuring with respect to

it's two copies of the complex
numbers focus of the real numbers and

exactly one half of a copy of the two but
to matrices in the two but

to make the C's act on the the.

The C. to the usual formula for
Matrix multiplication

matrices by vectors I'm afraid if
you don't know how that works.

There's a door because you know that's
supposed to be common to every educated

scientist these days is how matrices
act and how you multiply them Victus.

So I'm not going to go any lower
than that right so it's going on.

What's the now we can say or will we have.

The space of the complex numbers or reals
we know all the possible dimensions of

the spaces of their it's the integers one
two three four five six seven according to

how many copies of the scales
you see in the first place so

what's the corresponding answer for
the to go to major cities.

So in other words what is the set
All numbers which are the mentions.

Use a method to go to modules
over the tube or to major cities.

Places on which the two boys emergency say
well we've seen that certainly contains

a half from these guys and he can't get
any smaller than that the mattress is just

one act on anything when that So on the
line you can or you can take the tube or

two matrices acting on two copies of this
system over creation by themselves and

that will give you one right that's the
whole philosophy if you take the matrix

to go to majesty acting on themselves it
should be dimension one just as a complex

numbers they can sell should be dimension
one real number Second this is OK so

we're going to win one and
then if you just keep adding more and

more copies you get this on.

The set of all dimensions Victus
plays all modules over the two What

two matrices erode one half
one three have to have and

if you want to string infinitely many of
them together you get infinity as well.

Right so that's the.

Stuff for
now sort of hopefully motivated you all

got you on the road on which I want
you to travel for the next minute.

Because no one to replace to.

Boy another interview so
we would get to be a variable into and

with change the question to what is
the set of all the mentions of spaces

over the envoy in major
cities OK Well obviously

the smallest one is going
to be someone tell me.

Stories are Thank you this
is over the next month.

Or next month whatever it right and
then threw over and

three over and so
on OK it's not up there is no.

There it is zero one over into over and
through and

what's happening is in gets bigger and

bigger we're filling out the unit interval
right now what for Norman one of his.

Great discoveries Besides you know
how to make an atomic bomb work and

you know game theory and.

Or Thomas and both of those things one
of his great discoveries was that there

are these things that are some kind
of infinite general dimensional

generalization all of the envoy
in major cities these so-called

two on factors I mean it's a it's a funny
term but you get used to it because on

the set Hellboy in this talk he sings
call to talk to one factor this Him and

their role in life is that
if you take the dimensions.

Of.
Spaces over them.

You get all of the get between these
integer one over and get filled in and

you get the whole positive real
line in zero and infinity.

OK.

So you don't need to know what exactly
they are the definition Bubba but

what you need to retain is that they
said it will dimensions of these things

is continuous since
the whole real line OK and

this one is going to be one distinguished
one which is going to be M.

which is some kind of infinite
dimensional matrices acting on itself

that's going to have the mission one
remember to go to matrices on the two or

to make the series with the missing
one in billion matrices on the in by

matrices with the mentioned one or
just maybe.

The point nuff in the.

Previous one but there's always the hero
did I put this one here because of one

that is the sort of comical
thing acting on self so so

it is with the he's.

B. school too in fact this is the
dimension of him acting on self because

the one hearing is normalized as
a God given element of dimension one.

Or right now you know.

Just the other change so typographical
sayings are put in here is a very to H..

Zero point.

Zero zero.

H. and B.
that's because I just Hilbert space

we're going to talk a lot about
who this place is and these

so-called two hundred factors have
a lot to do with with who gets places.

There's a sort of logic we were into
the world of things being infinite

dimensional over the complex numbers
even if the fun of the mention of

measured bodies to one factor.

OK.

Norman was so taken by this idea.

That he wrote a book on these things and

called this continuous geometry and
this is the.

Book this pope peps not somehow
surprisingly not all that well known

these days but back in the you know
one thousand nine hundred fifty S.

this was a much more well
known field of study so

he studied.

The if you like the most loose possible
context in which the the theory

would work so he showed that you don't
really need to have a two in fact

to get a continuous jointly but
it's turned out the reason of this book is

a little bit obscure these days this
is the most interesting example but

if these two affect us.

OK.

Right now I want to make an important
technical or sawed I want to

explain to you just how when you
working with these things and

doing calculations how this
continuous don't vary.

Royce's have this continuous
dimension rises or

reminding you of how.

The dimension of a space.

Is naturally not.

In ordinary.

And you know the first winner
of the course you know the.

Like I say that when you're older
these days should be common ground

to all scientists anyone who dealing
with numbers in fact I've been one of

the people thank God no one's ever taking
me seriously perhaps but I've always been

arguing that we should teach linear
algebra before we teach calculus.

Because it's just you
know any discipline and

science is going to be dealing
with numbers statistics and so

on good enough linear algebra.

OK That in fact if you want to understand
calculus you got to know linear algebra so

for me it's all basic thing and
we should teach it first.

Actually I think the reason we don't
this is bloody boring to teach.

Anyway.

How do we actually get the stuff mentioned
Well what we do is we realise the space P.

we realise this these spaces
the images of projection so

we think of the space as being so it
gets projected on from some maybe higher

dimensional thing we this is a projection.

And projections.

A special kind of matrices they have
that piece squared equals piece

it will pay and this for
this property which I want to bring out

piece critical piece therapy which Eric
algebraically characterizes them but

what it means in sort of hands on terms
is that if you choose a basis approach

appropriately with a few diagonal as
your matrix you can always assume that

projection is all zeroes except
a few ones on the dying off and

the number of ones on the diagonal is
a dimension the space so the answer.

As the the dimension of the space
is the trace of a projection

in the trace at the diagonals so.

The dimension of V. is the trace of
a projection in some algebra matrices

well so it is with these projections of
that kind of operator the kind matrix and

for four more minutes
the same is true one fact is.

Have a trace just like that in by and
matrices so to matrices over him so

we can play exactly the same game and
we can realize from Norman's

continuously varying dimension by taking
the traces ending up to diagnose months

of projections so

I've introduced two things are going
to be key projections in the traces.

OK maybe saying well you know was that
have to do with not well you'll see.

That's the adventure we're on.

OK.

Now.

I want to remove the physicists in
the audience they probably is bored but

as a mathematicians have been
sofa I just want to remind

the audience of the basic mathematical
structure of quantum mechanics.

And this was most clearly understood
by from learning him selfe

wrote a major book on the subject.

And there's a picture this is
a relatively recent addition but

what I want I'm just going to talk about
vanilla stuff there's no no one will

dispute anything that I'm bout to say and
it's contained in some form than in any

serious text on quantum mechanics so
here it is this is the gospel.

This is the gospel of quantum mechanics.

And the so you have a quantum
system is the states of the quantum

system a given by one dimensional
subspaces of some complex who but by.

Yes I don't tell you what
it was places I sorry.

I can tell you they're complete and

fundamental complex vector space
was opposed to the definition for.

Several The pure States a recording system
a given by the one dimensional subspaces

of some complex Hilbert space the
observables of the system such as position

and momentum and spin and so
on a given by self or joint operators so

the things that halfway to
being predictions I will say.

Operators self join operate is on
the Hilbert space which may be banded

as in the case of the components of spin
or unbounded in the case of momentum or

position.

So in the now there's a great thing
if the connecting these things

these vectors in those places supposed
to be states operators are supposed

to be doable Z. and the connection
between them is this formula I think so

exhausted with what is supposed to mean is

that number member I guess I'm
really going to hold this place but

it was places always have these in
a product which are written with these.

You know with these pointed braces put
into brackets that number which is

space in a product is the average
value of repeated measurements of I

if the system is prepared in this in a
state given Boeing saw I saw one mechanics

is inherently known that a mistake and
this if you prepare if you

set up the system on an identical way you
don't get the identical measurements but

this average value of these measurements
of I was given by this product.

The repeated measurements of the.

Of the repeated measurements of the system
of prepared in the stadium like so on.

So if an observable or
committing family a verbal.

As given then this is wonderful
theorem called the spectral theorem

which is the infinite dimensional
version of diagonalization matrices and

what it says is that you can express
the Hilbert space as this so

this is square in the room square integral
functions from some space into some other

some other victim space the said this
is all right now these are called wave

functions you can't avoid this
if you do quantum mechanics.

And the given observables become
multiplication operators.

On the ill to space.

But that's the way it
is a said that in any.

Textbook read to beginnings
quantum mechanics one o one.

Where the scale is this X.
for instance is supposed to take care

of all of the spatial momentum observables
and then the sum of jury hope its place K.

which might take take out of some more.

More subtle I guess in things
called degrees of freedom such as I

said it's been it's been
one of the supposed to be.

I won't go into what they have but
physicists in the wings will sort of.

Be happy with that.

Right so that then is the gospel there's
a few boxes of the gospels a bit longer

observables A and B. can be similar
tenuously observe the tree accuracy only

if they commute a big was B.A.
itself or joint operators.

And.

Otherwise than this some.

Inherent limits on how well you
can observe these two things and

plenty Asli and.

The free analysis gives
the Heisenberg uncertainty principle

which restricts the standard
deviations of momentum and position.

Observables well others
not just momentum and and

position but this others and
the similar droughty and

operator language on what basis
expressed by this P Q minus Q.P.

is equal to the I square that's
the square root of minus one two and.

That's some fundamental
constant of due to Planck.

Constant which will play the role in this
torque as being a very important variable.

Those first be a joke.

If the Gospel goes on the pure state
can be superimposed but adding

in ending the unit vectors and this is so
the deep mystery of quantum mechanics and

no one really understands I mean I
think for I'm in the foot famous for

saying anyone who thinks they understand
quantum mechanics is a full and this is

where there's this like of understanding
because when you add together two things

the the states a given by the vectors
up to a phase but when you add them.

The phase counts you know doesn't have any
meaning for the for you know observing for

the values of observables this
is a source of great mystery and

confusion as a resulting state depends
on the phases of the representing

victims it gets in the way when you
read things on the other hand there's

a simpler notion which for a moment really
temp in the mix but make States and

their states that you'll call the convicts
combinations of pure states of observables

they give them by things called Trace
crossover it is and this formula in

force is the bottom trace of
either the minus H. and I.

Is the thing that gives every
devalues of observables for

the system in the state
even is not a pure state.

Going to a state for member with.

Vectors one dimensional prediction.

And predictions themselves are called.

If both a simple thing is the state and
observable and

they simply these question observables P.

is the system in a certain
state on this early pure.

Right.

To end the gospel one important
thing which is if you have

two quantum systems given by it was
places H. and K. if you want to get

the Hilbert space given by the joint
system you take the tense of product.

And the tense approach to what's places
and what is meant by a joint system

is not always so clear because these
things call for millions and both zones

with when you put two of them one with
itself you only get this symmetric or

any sort of trick parts
of the tense a product.

OK now they're into the gospel right so
everything I just said is done is done and

you know the first few weeks months
possibly of the really quantum mechanics

course and it's beyond argument right.

Anyone comes to argue with
duke it out afterwards so

now for Norman sort of went off in
a different direction from perhaps most of

from physicists and that he introduced
the sort of trying to get deeper

into the mathematical structure of what
was going on introduce these algebras

of operators on Hilbert spaces and
I closed under the stuff of ration

the Hilbert space is a joint This
is transposed like received.

Because observables are supposed to be so
joined so the two one factors

are examples of these things which
have this continuous dimension.

But you know as soon as we think want to
think about subsystems of quantum systems

we're smack in the phenomena of the land
and I'll say a couple of words on that.

Because the subsystem of the system is
going to be described by some phenomenal

run.

Of observables.

In the gospel of full Norman elder of
the whole system of I want to be ambitious

talk about the whole universe.

And it has its place of all
possible states of the universe and

the fundamental truth the whole thing is
all possible observables all possible

states of the universe which is maybe
you know more than we could possibly

contemplate just being living in
this tiny little corner of it.

So but so will it naturally be talking
about subsystems of the whole universe and

they will be given boy from Norman
algebras of observables corresponding

subsystem.

OK So while everything.

Is given by all operators
on this big open space.

That's too much for
our mortal Munns to get groups

around the mathematically with necessarily
going to restrict to swot systems

which are mathematically given by the
small algebras and phenomenal diverse and

that's why for Norman was so interested
as one of the reasons why he was so

interested in phenomenal but
now what he thought was.

What he hoped so hear this from Norman
himself thought and hope that these

two one fact is that he that he loved so
much we're going to be the right ones for

quantum mechanics because of this trace
because of this continuous dimensionality

and so on but
he turned out to be wrong and that.

And here I have a picture
that I don't know.

The interview will recognize this.

Anyone know what that is.

Right.

Then right this is
a David Rittenhouse laboratory

at University of Pennsylvania where.

The mathematics department is housed
there is not put it this is this up

was because I used to have in my
position a book of handwritten notes

from Norman called.

Quantum mechanics continuous geometries
with a transition probability quantum

mechanics something like that
it was one hundred page or

a notes were at the end of it he
realized he'd made a mistake OK.

We couldn't quite get to where we wanted
and I presume this was one of my treasured

possessions uses for Norman's handwriting
with for Newman's crossings out and so

on and one point when I was
a pen offices were burgled.

Very sad because I lost my.

Hand written from when the very
said thing actually to be fair

just about every last book that
I think I used to have and

I don't have any more so
it was taken from me.

Burglaries a pencil you know but
it probably was actually.

Anyway.

There's a saying for Norman was
wrong in that do the work of hug and

kiss let another system that to be not
the case it was a two one fact in this but

on the other hand he was right in the
sense to do the work of to me to talk so

I can call on it turns out
that in some strong and

usable sense all phenomenal
Jabbers including the ones that.

Wanted to show that you have to use
its so-called top three factors or

they could be reduced to the form
women's beloved put to one factors.

So in that picture that I
wear gave the beginning of

the thought that just black
circle there was my attempt.

To describe to you a picture of
what it to infect looks like.

OK actually more precisely what the
projections inside to infect look like.

So now moving right along.

I want to get to some fact this sort of
stuff was mentioned some fact is in the in

the talk I want to now talk about
to one fact is some fact is and

two in fact is so since a factor itself
was just this big blog with pretty had to

say what amazing what
the subject is like but.

Let's just say that a subject
of a two in fact are credible in

the big two in fact it is
him the small ones in.

And the small one is going to act on
the big one by multiplication just like

matrices act on major season of images
inside a magazine act they always act by

Mobic ation so therefore we have the right
to consider this number the dimension for

Norman's dimension measured by in of him
a member has a whole order the beginning

the mentions of things depend on what how
they're being acted on and if it was some

factor which is just a small factor
inside him we can look at this number.

Right so what is supposed to mean.

In some no evil perhaps less naive sense
is how many copies of India see and

saw it in what was our
dimensions the number of copies

of your base field you see inside
the thing that's acting on

it was how I started this
lecture the secret was to.

For half the pinning on what
would have been the OK so

now here we have an example you may
not know what a true infector is but

certainly going to be able to take
the two by two matrices over it and so

we have a first example of a some fact
that is the two boys who might receive

in M. as the two by two matrices of N.
OK so

what is the index in this case.

OK so the.
This is a question though and posing and

I'm not going to go on until
someone comes up with the answer.

How many copies of N. Do you see inside
the two by two major Cs over itself.

Go this a great audience for
because there's a B. C. D.

so forth so there's the answer for
so we can go on think of that so

now even though you have no idea what to
infect is you know what this dimension

of him of in is and we call this
the fancy index things in the X.

of the sun factor and
in that example it's four.

OK.

Sort of I didn't say this or.

The dimension measured by end
of the two by two major cities.

In.

Four.

Or eight so that is to vary as we are and
have a doing we see that we get all

squares of integers as I mentioned in the
season some factors of two one factus for

better or one for.

Nine sixteen twenty five and so on and

infinity we don't have zero
right one is the smallest So

whoever it was said with zeros is morally
wrong the smallest one is one but.

If we look at Galois theory we can
actually improve on that we import

Galois Theory into the picture looks like
a subfield groups on it before we get all

integers so we get one two three
four five six right and so on.

OK but

there's something wrong here because that
mention was according to full Norman.

Was supposed to be a real number variance
is going to be at least one because you

know you can't get this and
one copy within is contained in.

So we've only come up with integers and

we actually did quite a lot of work to
come up with and just took the matrices.

And we use Galois theory so
we have mediately.

Pointing go a theory risk it all
positive than the just so we immediately

pose a question but remember that it's
supposed to be a continuously very

real numbers what for one and
then of these then to effect the four and

we've only come up with
the positive integers.

So the question is what's the answer.

And here is the theorem that sort of got.

The got some factors going and.

The.

Connection with physics makes the answer
kind of interesting possibly for

physics so the answer is that if the index
is less than four then for Koso and

squared in for any groups three four five
six come of the well the hell is that for

doing the and I was wondering that myself
wouldn't realize these numbers converge to

four as intense infinity so for
those of you OK So if it's less than or

equal to four that is
one of those numbers.

So the something wrong here that.

It's less than a very top should be a
listener equal to and makes makes it makes

it exists then amazingly enough a norm
is continuous to mention turns on and

from four on it every
possible real number.

OK So this is a mathematical result for

prove that about nine hundred
eighty one or eighty two.

And here's a picture of these things these
little crosses the four course could

put N. OK like there's been States
of a sort of you know physics and

like the bound states of an atom and then
from four on you get the scattering States

with any possible real number as being
the value of in the X. for some factor.

So you notice that the first few values.

A one that's focused group of
the three this is focused.

Before and the first sort of exciting
one the four course quid pro five

which is also known as the square of
the golden ratio again in the three and

you go up to converge up to four and then
continuous for moments which is on and

you get all real numbers from then on.

OK So you know I could
Wexler work about this for

at least half an avid I never
get to where I want to get so.

So let me skip all that and
just say that I'm not going

to give any indication of the proof
except that in the course of the proof

you run up with us you run into a sequence
of projections why insisted on these

predictions these operators that this
property and I'm going to call them.

In so I did too in fact.

So whatever's going on is one
thing as an Ambien that's a trace

because from them and
said trace the mention move on and

these projections satisfy
these magical relations.

That.

Well you know it was the That's
very magical isn't this is

something missing this
should be you know I squid.

The first line should read.

Squid is.

And then these funny things you know
you know A plus A minus one equals some

real number which is actually
the inverse of the index.

So if he has a next door if you meet them
and that's what they said is four and

on the other end of the two
apart they commute.

So as observables a simple thing is the
observable a million they interact to feel

like if the next door and for
me this is ambient trace.

For Norman trace and that says
that the trace of if you take any

bunch of the first in of them take you.

Multiply them a multiple of the next one
then what happens is this number comes out

trace breaks.

This.

These relations should sort of evoke
something you know some stirrings of

neurons moving through most people
if they think about them for

a while this will be some
relation with Markov chains

This is some kind of independently of
the first in they have the next one and

something happens there but
other was things that the Independent or

lattice statistical mechanics
will let us you know

quantum chains of things we put these
you know as one next to each other and

is in the squad of spin chain something
happens the next door of the Was this

the nearest neighbors local
nearest neighbor interaction.

OK.

In fact you can say physically that
the you know if you're a quantum computer

the the eyes are chain of cube it would
say only in direct of the nearest

neighbors and
way back in the day papa actually show

the can be realized and what one would
say these days is a China would agree

to spend cubit by a sum certain.

For but for matrices.

In the Trice in that cute picture
the trace is given by the ordinary matrix

trace the to the end but to the end
matrices of the moneys be the age and

sex for some and will Tony and
and some in this temperature.

So all of this is what's
about a sort of quantum spin

systems quantum statistical mechanics.

OK.

Right now let's change gears
completely and check the times as.

Hopefully not going to
run out sixteen of the.

The good or sad thing about this
Beemer is that you always see

the little thing have you know this is
sixteen have twenty nine which is how

the works are getting on towards the end
the final goal and the penny on how you

are enjoying the thought you think God I
still got the thing these damn pages to

go where you think man is to
bed this is fun all right so

here's a no no it's just
a three dimensional clothes and

the nice thing about them is you can
represent them faithfully in two

dimensions with a picture
this picture of a knot.

And braid.

Is something very akin to a thought it's a
way of talking you have to horizontal buzz

these blue buzz toward the top ones to the
bottom ones with a rigid bits of string OK

so that's braid now braids
has the advantage of a knot.

And they form a group.

You can.

Concatenate them so.

I'll do that in the board in
case you don't want to do.

If you take a braid.

Here with the with
the ridge strings Alpha.

System and

you have another one with the same
number of strings then you can put the.

Second one on the bottom and
just join them up and you get to braid.

Remove that central bun to get another
braid so this is the composition of braces

is the group operation Alpha Beta
will be two alpha depending on.

Which hemisphere you come from or.

With your left handed or right and
something anyway the idea is that braids

have a group and this group is generated
by some very elementary braids these just

simple crossings just because if you see
any braid you can organize the crossings

one after another between adjacent
strings and so drawing the sigmas.

Don't signal one and sigma two here for
the three string very group.

And anyway what you find displayed
during pictures is that the braids.

Abstract very individual simple crossing
breaks between adjacent strings satisfy

the following relations sigmoid is like
my post once again it was Sigma plus one

hundred of us one the next door
to some kind of interaction.

And if they're not next door they commute
physics the simple tiniest observable

no interaction.

Right.

So what's going on here well you know.

The.

Gist of.

The literally ended up was talking about
these breaks at the same time though I

finished talking up about the zero
as when this remembers just quit.

OK.

It's that error has.

Repeated itself for the voices of
cutting and pasting obviously and.

Well if you look at those two things
together next to each other then you just

can't help thinking maybe there's some
relationship between these braids and

these index for some factors and
indeed what you find is that

if you choose T. carefully.

Then the and just right right don't
you find that these relations.

Employ these relations to define
sigmoid to be this particular

combination of E O I.

This is not a one cause this is
an I T.E.O. I mine is one want to see or

in this particular combination of.

Ers in the identity actually these
relations imply the braid relation so

what you find is the the break group is.

End up being represent.

Did these braids this group of brains
in that be represented in saw it

it to infect.

OK.

It's sort of amazing.

And here's a picture of the place
where I actually realized that

some of my one person
two people three people

certainly recognize this building
here this is Geneva and this is

where the mathematics the method it's
the problem the sort of stuck in a P.S.

somewhere in the it's really amazing place
to have a mathematics department and

people have been complaining about it for
at least fifty years but

no one's been every direction we do it the
methods from the still we're stuck between

a supermarket and
some banks some Swiss bank so.

This is in the people told me about the
breakthrough I didn't know anything about

them this place any way to get the news
the story will stop and even move on to.

Mexico City which is where I salute
the next step in this thing so

I discovered these braids in Geneva and
I thought boy that's really neat.

And so I just tried to show these
representations of the bridge group to

people and no one absolutely no one in the
entire mathematical world took any notice

whatsoever because there was no you know
no one was interested very instant braids

Fallis representations of the braid
group just wasn't a go but

when I was in Mexico City sort of
was a was a spouse my wife was

the official dignitary there and
I had some free time on my hands and

mucked around in the.

In the log or the.

University and I found the representation
interesting no one else did but

then I came across this
book by Cox of the.

Old Regular complex probably Topes.

And on the cover of that book
was this rather fascinating but

perhaps to complicate
overcomplicated picture but

then I looked inside the book and I saw
that cock to the fore this to describe

this picture was all about three minutes
to make or sees three major cities annoyed

already written down as
representations of the brake group.

On the certain number generators exactly
bankruptcy's which I knew were in

the course of the book so then I thought
well that's that's really interesting

my representation maybe everyone else is
wrong and I'm right these are interesting

representations of the bridge group
I couldn't get anyone interested and

then in a year later visited mathematician
John Berman There she is she was for

Mrs braid and.

I visited her Iran working on this
is Columbia which of which she was

annoyed showed her my representations and
I thought well I'm going to go to the top

she was no more interesting
than anyone else said to say

she in fact it was somewhat depressing
that you know now absolutely no one

not even the world's top exposing braids
was interested at all in my ribs and

patients so I went away but saddened but

fortunately Joan was very generous and
she did tell me all kinds of things

in particular about the rip the connection
details of the connection between

knots and braids and and so very quickly.

Here it is if you have a braid this
is the braid that we had before or

you can do is what you call closing
the braid in closing the braid is

tying the tops to the bottom of
the brain so you do that is the braid

of taking the top strings and I'm just
going around doing them to the bottom and

that Patton and then I remove the buzz
remove the blue buzz and what do you have.

You have.

A note or possibly a link a link is
a note with several components and

you might get a link I think in this case.

I think it's a no no
it's just one component.

Right.

So this is a well known
correspondence between braids and

knots and in fact it's known.

Boy Alexander this is not the Alexander
who destroyed notes this is

the Alexander who.

The American John Alexander who.

The lots were not in the early
twentieth century so

he showed among other things that you can
actually obtain all notes in this way that

you can give a not you can turn to a close
braid Injuns showed me had do that

remember through noise of.

So all links can be obtained in this
way I'm still gonna don't also told me

with result of I cough which
says exactly what different

braids give the same wink so given a wink
you can get it as a close braid and

lots and lots of different ways but
there's a theorem to mark off.

That.

Says those different ways then you know so

week after this after my
big disappointment that.

You know zero of the human beings compared
to Crete cared about my representation

a miracle it could.

And that was that this Marcos theorem.

Joy of exactly with this condition
that I had about the full

Normans trace right this thing trace
of axiom plus one OK which is so

far been missing and in fact what
Michael Steele showed you more or

less immediately but it was.

Another miracle we even
mention this we call this

trace a mark of condition because.

Those or a mark of.

Well actually it's not
the same a Michael of.

Not the same a Michael.

But the father and son.

Amazing so.

Anyway the point was putting
all the ingredients together

taking the trace of a braid you get a.

Invariant of knots and
links called which I call V L of T.

by the following procedure first or
you take a link.

Which remember a link was just an I would
very close could have a quick solution

you're represented by Alexander serum
as a cook the closure or braid L.

for in some braid group you represent that
braid in the sail through the eyes in

the one factor but a sending the good
right here three more goes to see so

obviously here I didn't cut and
paste this T.E.O.

I minus one month over there
remember that formula.

Until you remember just some version
of the index of the subject.

And then you take the trace of the brain
in the algebra multiply sort of

simple fudge factor involved you
have an invariant of knots and

links not to speak of examples So

this unfortunately in my first bus I knew
would be running out of time so I am.

I didn't include this in the.

So here's an example.

So here is a not this is the simplest
anyone know this is the thread for and on.

And at the end of the day the pulling this
not is the plastic you administered in

the fourth So if you go through
this long complicated business

this is the answer that comes out.

Right now this Alexander I've
been talking about.

He actually had another poem you'll
call the Alexander phone a meal and

his Pono meal for this not was T.
plus the minus one plus two.

In the US.

This is the only gun for.

It so now you know just wanted to
put yourself in my position for

a second after the meeting with Joan.

A week later so the US will bolt
upright in the moonlight and

realize that I had this pulling on me.

And I calculated for the trip for
by the next morning and

I had that answer.

And there's a Selig's on the phone in
the middle of the been around since one

thousand nine hundred eight.

So what would your reaction be.

First reaction or
made a horrible mistake rubbish and.

Well you know a calculated again again
again I think that all over the seem to

be OK on thing to be no reason well.

What's your next reaction was.

In there's not much difference between
these guys just rediscovered the Alexander

phone so that would have been
a disappointment in fact because

of the connection with statistical
mechanics or sort of completely.

Glossed over having talked about it when
the been to bed because I knew that if

this thing that I discovered was
the Alexander poem you know or

would have solved the pot small
this was a famous open problem and

still is in statistical mechanics so
it was a nice win win situation

anyway it turned out to be.

It turned out to be.

A new phone number.

So this all happened a long time ago in
the summer of ninety ninety four which was

bad year for well characters of course but
a great year for me it was very exciting

times this is a long long time
ago this is ancient history.

The physics connection was
there right from the start in

fact even before the start and.

If we use this many of Princeton papa
bidding of the you know is inside the to

go to Major season what we have is
a quantum spin chain Well Prince but

it was a quantum spin change
on the not the spins and

the not interactions at the crossing and
the.

Jones circle John's point was just
the trace the the every dryly of the.

Hamiltonian on the quantum spin chain and

writing this formula to
explicit formula for V.

For a picture of the knot gives
you the quantum spin Jane right so

that's ancient history now not having much
time for it being already over time I just

want to run very rapidly through
the recent developments.

Has been about.

Thirty years.

So we are going to have to spend all
this time on this old stuff going to

Iran else's work and best thirty seconds.

Is totally unfair so Calvin gave
a beautifully simple way to define and

calculate this polynomial
without using braids

many people discover many more polar
normals along the same lines or

unified by the thought of quantum
groups and their matrices.

Wouldn't show using quantum field theory
ideas interest in mid June so I mean sex

and its connection with conformal
field theory had to extend V.L.T.

from the links an arbitrary
three minute falls on all of

the developments since then this
is probably the the big one.

One hundred eighty eight including
Mt links and thus invariance and

three manifolds themselves.

With a silly if and just a new kind
of not invariant from the other

break topology of the space
of all knots and Berman and

so help was related to these pulling
the mills can save it showed

how these finite what became of finite
top invariance were related to the put

a bit of expansion in quantum field theory
not the only one but maybe do the best.

And now to say that two of the world's
main experts in these and other and

all the aspects are here to take in this
audience several SCO fully disentangling

So if you want more of this after we're
gone then you got no more thought you

can't get a more authoritative version and
then and now want to end of story in

this cover in a from ology which is
the most recent which is a ten year old

perhaps big new development
this well I want to end.

With.

In spite of all these sophisticated tools.

Or quantum groups and provide for
modes in spectral sequences and form and

theory and you know that goes on gauge
theory and so on his own without him there

are two very simple problems which are
open in one thousand nine hundred four and

remain open today so
I won't tell you what that first problem

does there exist a non-trivial not
who's John's problem it was trivial.

And it was does the job for me of the take
the unknown if you drew a picture

it's very hard to tell the four looking
at it whether you can until I don't can

calculate it John for no and we'll tell
you it's possible one that is open.

Feelings on the other hand for
lengths of more than one component

your answer has been known to be yes for
over ten is so little for it or

rather his computer discovered that
particular link on two components and

that link the problem is the same
as a pony of two component unlink.

Rising So it's a fifteen crossing link.

Not so
on the other hand thanks to stuff for us.

More Once computers now working I
believe on all twenty one crossing not

supposed to be done by Christmas.

And the computer the the I think
the game was the one one of them.

Billion right in the more than
a billion knots to be looked at and

none of them has turned up as having
trivial don't fall on them so

if you want to go on you
know solve the problem you.

It's not very hard to stay than don't look
at a not less than twenty one crossings.

And the second problem which is quite
related this is this representation

of the bridge group inside the.

One factor if you take your
place a breakthrough but

images which I describe how to get inside
the two in fact is that capture the entire

structure of the break
group that's also open.

But although it's true that if one wells
more general presentations it's come from

Poland from another poem Mills
the answer has been known to be yes for

over ten years Also as
with a big long Kraemer.

And the the last word hopefully is that.

The reason these problems are so
hard is that in spite of everything no

one has the foggiest notion
what this variable T.

or at least no useful notion
of what the variable T.

in the poem you actually means Alexander
it's quite easy to understand.

And.

Here are some of the answers
a cropped up and

different several different guises this
variable is the almost number of elements

in a finite field the Sigsbee exponential
of Planck's constant is in varies it's

an inverse temperature it's a number of
spin states it's also the the lambda and

cons classification of top three
one factors but sadly none of

the interpretations gives any hint as to
how the top logically would you metrically

manipulate nots to control on a mill.

So that's the end thanks for listening.

Yeah.

Yeah both using me.

But I think that we may regress even
though it really feels like most all.

Like really this is all this
is a person thing when you're.

Out.

Late you know when I was.

Into the verse a student.

I was asked this question but
I don't pick month who was actually

the this is a voice of one of the most
recent Fields Medal goes from Geneva.

And I've been thinking about
a sort of sense on and off and.

I think it.

Lets the stuff it was a little
bit run the major cities and

trying to solve that problem but.

For me personally it's pretty cunning.

And I don't think it's properly
understood mathematically even today.

A very good question.