My pleasure.

OK So this is doing go to great.

And Leslie that I saw somewhere here.

So all in the start of a big amount so.

We have already seen the independents
that pull in the mail but

let me just thirty finish.

The definition OK So what do we have
you have some underlying graph G.

and then we look at the over
independent thirds of

alum that do the size of the independents
that are here for example for

this graph here that
are seven in the pen and.

The first one has size zero so

it has weighed on these have size to have
weight so I could go on the square and

the independents at
the polynomial over the gain of.

Now would a lot of the problem the deal
going to be looking at so it's going

to be decelerated to me question
where are we to thank to make a deal.

That are labeled by a number Lumba And

if you feed the good half into
the good and they're labeled by law

then what it spits out the independents
pulling the meal over that good stuff.

So if you had to get in there and

there was to you she does get
off into it it will spit out.

Whatever nine plus.

Eight seventeen OK now.

You could ask for the answer to be
exact but then it turns out to that.

There is no damage interesting that
you can say OK so maybe you can

have a poll in the mail time I get it
I'm bummed that he would get out but

that's about it and everything is easy.

But any other type of good if
you look at the regular graph or

any kind of the problem will be difficult.

So then.

I'm going to be lax manufacturing.

And we are going to ask only for.

Good and there's that spit out
the answer which I didn't but

I fission and
the profusion is also part of the input.

So input now is a good half but
a season and you want to have independence

polynomial or they're going to do that but
I mean to with some precision.

People look at the before of course and
by now we understand the situation for

a completely OK so or almost
completely maybe it would be exception

of the critical critical point OK so
there is some critical threshold.

That depends on the maximum
degree of the graph and

if you look at the class of
the graph of make some degree Delta.

The problem is easy for London below
the threshold and it's N.P. hard.

Above that threshold and if.

They didn't work.

We have to change our market plan.

Let's see why would we be interested
in looking at this question OK So

well one motivation for studying.

This independence polynomial is
that this is the simplest or

one of the two simplest spin model
thought of graphical models so

this is probably the first thing to
try to build that understanding on.

Well and we have all the big gain from
understanding in the previous work.

The second motivation.

Could be from the.

Very end of loco look at Lemme.

Return the strongest known the very
end of the local local Lima OK so

what is the setting there we
have a collection of bad events.

We have some notion of a dependency
graph between these bed events and

I would like to understand
under what conditions

can we conclude that the probability
of no bad even happening is positive.

And that there is a condition that

looks at every possible subgraph
of the bending to get half.

The independence polynomial
a negative value of P.

in the condition is that all
have to be positive OK so.

The conclusion from this light should be
that it should be interesting to look

at negative values of alum
decibel they are interesting for

for example this application.

Now.

One more motivation OK.

Detail but that is the technology
developed by battery Norco and further by

pattern that act that allows to construct
approximation already tons of these.

Problems if you understand where they're.

Pulling on the L.

has complex heroes and
if you managed to find it he didn't write

the polynomial doesn't have complex then
you get a fish and I'll bet it I'm OK So

the conclusion from this light should be
that even looking at complex values of

is an interesting question OK so.

Now we made the problem more complex so
let's expand our mind to

complex numbers and let's look at
the problem with complex numbers in mind

OK Now our understanding from
video line is still there.

And we would like to understand a little
bit more and what will play out

all in their understanding is this region
that is a precise definition there but

not doing but it didn't identify
them by bottle and that ECT.

And.

There was some work on this and

what people managed to do is they
managed to get a little bit.

Why there's.

A line where we have all got it them and
they managed to figure out and

I'll get it when we have complex
values in the small that it circle.

And then there are some
negativities old for

the real line if you
have negative numbers we.

Had to N.P. hard and
if it is old previously.

So are the result OK so the result is we
are going to show that the problem is N.P.

hard everywhere outside of the city
didn't identified by Buttle in that act.

So I'm going to explain how
dirty didn't comes about.

Yes.

An algorithm in the circle.

Probably no but it's kind of difficult.

OK So let me.

Let me try to do the proof for
more or less in.

On a high level OK now.

As any is any how did this prove
we'll have to create gadgets OK now.

It's hard to search for gadgets or
to be about them when they're good but

it turns out that if we did expect to.

Know how to score a model you don't
have to think about other gadgets

that are going to be enough and
if she's got.

All of it be understood in two thousand
and six now probably not everybody

sort of telling me let me do a proof that
you only have to think about thirty OK so

here is a graph where the probability
that it takes there is occupied.

Uniformly to undermine the penance that So
let me assume that day is equal

to one now for simplicity so there were
seven independent sets two of them contain

the better tax so they are to be two over
seven now if I look at that like this

well it's a path if you wanted to compute
the probability that the Vatican is OK.

Counting independent sets on the like
looking at Fibonacci numbers so you

could figure I would go down for days and
it will turn out to be two over seven for

these are good enough so
the quantities are getting.

How to do this in general so this is due
to the whites two thousand and six OK So

we are going to look at the probability
of the better pictures occupied but

actually it will be easier to look at
the the ratio of the probability that you

is occupied divided by
the probability that it's not OK.

And then.

Just in pictures what
you can do is you can

take your graph and look at the all
the configurations where it's in

divide by the configurations out
of a sort of circle means that

I'm committing to the particular value OK
now you believe the better takes into.

It doesn't matter OK along with
I was equal to one so no change.

And the only important thing about but

it putting this about effects
in order node in is what

it does to the neighbors and here we
are doing the same thing to the neighbors.

Now once you have read it and
like this you can but

I think there's a telescoping product

now the next step is to understand
one term in the product OK so.

There is only one disagreement.

What does the disagreement do well here is
the better Texas in this one cannot be in.

Configurations of this type if this
one is empty then I have possibility

of taking this empty or
occupied I divide by numerator.

Getting means I look at the ratio OK so
you get equation like this

the ratio of the probability
of Occupy to only could bite

you can write as a product of.

Things.

What did we accomplish their objective is.

To be accomplished so that it would
be accomplished on the next flight OK

so we had a drink to accomplish did
whatever you can accomplish on the graph

you can accomplish on a thirty K.

So if you are looking for gadgets
the only thing that you have to look at.

Or the three who are coming in
now are coming in the clearly.

Trying to model this by three.

Reduce the size of the graph
by one better fix by induction

those can be done by three I put
those who demanded the better pics

if you apply the equality on
the thirty you will get difficulty.

And you are done.

Well has support if you send it I
mean how far would you have to go but

I believe the exponential
in the size of the graph.

But these are gadgets and they are going
to be constant size in the end.

But in principle as I search for the
gadget I only need to look for three so

at this point OK so the.

Bulb it vaguely have a day that is.

OK but in the sense or no.

OK now.

Back to having this part
of me to lumber that mean.

OK so if you want to have it then and
I don't know what it's X.

now with the ratio of the to the
probability that there were these occupied

but it doesn't matter OK.

If you want to figure out that the
probability that these occupied on the or

the ratio you can figure it out on
the thirty by days it occurs the formula

by taking their smaller teeth and
computing the value for them.

And there is some function F.

that does this computation for you.

The probability that you are OK by
done this study I think if that ratio.

If the ratio of Occupy to undercut by one
of these three if the ratio of occupied

by to these three that X.

two X.

is the for the last three and
then you are going to get better.

For them to three.

Now you can forget about you can
forget about the game will be now just

looking at this function and seeing what
you can do with it OK so the game now is.

Function you started with one dollar or
whatever and

you can never have you can plug in for
that but I mean to function those without

the number you have a new number you can
decide to a plug for some of the values

you still have the one you can put in
there you get in your mind number but

he did it be that he beat
What is it that you can get.

Yeah X.

is not a one for dollars that
there's just the initial value.

People I don't think they studied these
maps for complex value as much in.

Complex dynamics.

But they would study these univariate maps
where you have some national function and

then your map X.

using directional function and
try to see whether this process of growth.

If you ever do a Monday will be.

Such a thing on a computer you
are doing something like this.

OK so now we come back to where
did that he didn't come from.

So we had a going to look at this.

And then.

What happens if you eat I didn't map so.

I guess the behavior will
depend on the value of lumber.

And it the way this that
agent is defined is

well if you're outside that he didn't
all the fixed points of the repulsive.

Where if you are inside the city
didn't attract a fixed point.

What does that mean OK so
what that a fixed points will fix points

out of the things the function takes for
themselves OK

how do you find them well you multiply
you have a polynomial of degree D.

plus one and you have the roots
those are the fixed points OK Now

you take the fixed point a new
perturbed out of a little bit and

you started applying the map do fly
away or do go towards the fixed point or

what happens OK so formally what
you're going to look at is you have

to look at the Jacoby end of this map
at the fixed point and see where the.

Specs that although they do is less
than one that would correspond to

the text of fixed points or
it has spectral.

Bigger than one trip correspond
to people from fixed points.

Did.

I think that is unique that
is unique to fixed point.

Well there is only one fixed point
there that I did that are about to but

here are all the plus one repulsive and

here I think of them are repulsive and
one of them is that active.

It is depends on the value of London so
there are days I

love that one days upon a meter and
the Caribbean will have the lumber in it.

OK So the next plan is to
very publicly fixpoint.

What does that mean OK so we are going to
be using some tools from complex dynamics.

And smart enough to figure all
of these out on our own OK so

people that study these are national maps
developed quite a bit of technology.

And there are a few notions
that I'm going to need OK so

if you have this complex dynamics
that is the notion of to sit so

these out of the points that
travel the difference and

they keep the friends on a leash OK so
here is a formal definition.

So what this means is that.

If you start close enough to the point and
your reply did then I mix and

you go your friend is always
the standard most epsilon from you

as you go to the future so all your
travels your friend Bill stay with you.

Up to distance up your lawn.

For ever if you have trouble
remembering the return if.

It stands for friends I guess if
you're a member now I have trouble.

So example of it.

At that active fixed points so

if you are attacked if you keep
your friends on the leash.

Now the other course is this Julie S.

it so that the compliment to sit and
that means your friends

no matter how close they are going to
divert to eventually so example or.

Point to that is in the for
to set out of the repulsive fixpoint.

The right that's right there.

Now and don't ask me about the fine
details of a devout do you describe

can happen or not I don't know but
especially in complex than a mix.

OK.

To use if.

That says that if you

manage to get a dense set of
points in a open neighborhood.

Of a point that is in
the in the do list it OK so

if you manage to get your hands on it and
you get then set on the point.

In the Julia set then if you'll apply
the map and actually financially

many times then you're going to get
all the points in the complex plane.

With few exceptions but
that's not important OK So somehow.

Or other beings to gain from from this
complex dynamics that it's enough

to get close to a fixed point and
get a set of points around their.

Complex Numbers complex
numbers that infinity I guess.

We we played with this for
quite a while and

it's kind of difficult to get your
hands on a policy fixpoint OK So

that's the whole difficulty how do
gets to any policy fixed point.

And we tried all kinds of things and

the last ditch effort we
just said well what if we.

Look at the following the very end of
the map or the following way of dealing.

With the quantity the quantities that we
have OK So we started only with one OK but

then we take the ones and
we get to New Well you and

then we shift the value of the new
values in and applied the function F.

on one one one one and

then you value then we remove one of the
ones at the end and apply the function F.

get the new point get a new point and so
on OK So we have the sequence of point and

we always take the last of the of
them we apply the function F.

on them and then we get the new point
that he that he didn't beat OK so

here we are heavily using
the fact that we have this.

Multivariate access to the function F.

we had no dues in the universe it might
but we had it using as heavily as possible

the fact that we can plug in different
axes into the into the function F..

And the fact is that if you do this then

this process gets to the fixed point of
of it it gets to be the fixed point of

the universe at the recursion
where the proof Well the proof.

You set up a slightly different author.

Like this.

And you compete with that
I can answer it and then.

You argue that there is
held of this process

if you take that ratio of
constitutive terms then they are.

These are the current.

You can well understand

solutions of occurrences that I
do just figure out about the.

Transfer to make that exists on your
computer the largest eigenvalue and

you are going to figure out
where the process goes to.

And then there are only a little
thing that you need to check

well in here you need to check
the do never divide by zero.

And took a while a get in to figure
out how to argue that but luckily.

These are the case.

Dispose of the do said he if it's
actually counting independent sets or

independence polynomial
in a get off like this.

And.

If you look at the get up for a while you
see that it's a call for the graph so

it doesn't contain a subgraph like this
and if you have a Closer to get it's known

that independence that polynomial
cannot have complex roots real

don't have to worry about this possibility
that you will be dividing by zero and

if you have that then you.

Pretty much settled OK so
now what we have is we have that

this process gives us access to a.

Fixed point.

Now it actually goes to the fixed point
to be the smallest making a cute OK so

the sum in about isn't.

Actually no two fixed point of the same
magnitude if you have complex lumbered on

the system but proof OK if you
have fixed point it's not just

a question like this if you have another
fixed point equation like this if.

There are magnitudes about equal then
the norm of the of the inside would

be the same therefore the magnitude of
that identified would be the thing so

that means the points are equal distance
from zero that equal distance from one

that means they must be called to get but

if they were going to get there that
would have to be quite a bit to itself so

long there would have to be deal but
we are only dealing with the complex case

he had a lot of the deal was
dealt a bit better with papers.

Saw what do we have.

So we have a defect Milady we know how
to get to a fixed point now we will be.

Real need this real need to get that he
could add numbers in some interval OK

this is just a high level description of
what do do if you want to achieve and

then sort of points inside it if it.

Supposed that you had a set of
maps F one F two or three or

four the data from the set.

To some think outside this or
maybe slightly outside the set and

the madness contracting and all together
the map cover devoted you know.

If you have such I think then
you can increase prices and

you can create the density of points how
do you do that you have some targets that

you want to achieve if you look
at the map that maps to that but

the contracting for
the premier of the target is smaller.

Now you look right at the target lies and.

In each of the maps.

The target gets bigger target to get
the bigger target to get bigger you go beg

back at the end of the target is
sufficiently big We'll have a sufficient

lead then for the point here you will have
something in the bull's eye nearby and

if you forward the maps you
will get to put it to close

to the point that you want it so that it.

Gets you close to a fixed point now.

This.

OK so a little.

Bit close to a fixed point.

Will do.

That at least then sort of points out on
the fixed points once you have more that

at least at this fixed point you will
construct these maps that you saw here.

Moderately set of points.

Then set of points that you can achieve.

On the fixpoint.

Well but.

That only polls if fixpoint if you
are outside the cut of the region so

that the point you get to a fixed point
is it repulsive for a detective and

it's going to be repulsive because all of
them are repulsive if explained means.

If you get dense around it you're
going to get everything OK.

Yes Julia said.

Thank you now

how do we get moderately dense of
points around the fixed point.

So you have this map you wrote to
a fixed point of the of the universe.

Now.

You are almost at the fixed point but
lay off OK so you are off by.

A fixed point got off by
a small amount OK now

if you apply the map numbers
are small it's going to be linear

that everything is lean on a small
scale OK So we have a lean that a map.

It.

It expands.

A little bit if you had all coded
into the same if the preservation

of the same the map would actually
expand its policy fixpoint but.

If you made some of the corner there do
you know it would be actually contracting

OK so the only point here.

The map.

Fixpoint behaves really naturally.

And it multiplies by some quantity
which is smaller than one.

Quarter this is capital F.

of the of the of the left hand side.

And since it you can do addition
you can do multiplication

the P.D.P. the repeat you're going to be
able to do any polynomial that you wish OK

now if you can do any polynomial
that you wish well only polynomial

with integer coefficients but
do you have this bill in the mail and

you EVER late it at this point to
me to see which is which is the.

Multiple out of the Jacoby
under the fixpoint OK so

if you if you are able to
achieve this polynomials

then you are every thing polynomials
with integer coefficients and you think

polynomials with integer coefficients
ever learned it at complex numbers

with magnitude less than one you can
get that then sort of points OK So

this is how it this is how you get
the first initial set of points OK so

just high level you you go to one of the
fixed point you have a little bit distance

away but you are still know that the fixed
point now everything is a linear but

always gets multiplied by this but I mean
is a complex number of in magnitude less

than one you are going to build
polynomials out of these and

you're going to get evolutions of this
ball in a Mills and they're going to be

pretty dense because polynomials
integer to question the validity of

the complex number magnitude less than
one give you anything OK so now you have.

More that it sort of points
around the fixed point

you used those to build these maps F.

one F.
two F.

three F.
for that cover.

Some target circle and you just
zoom to any point that you wish OK.

This is the third part.

This is the third part yeah maybe
I should switch it into talk.

You know work yes.

Why are you trying to do the right so

you are trying to have a reduction that
I had a new in the introduction you.

Will see in a second.

You want to have an all kinds of values
for that because they do allow you to.

Do some kind of behind the research on
the number of independents that in a good

are somehow what do you need
to do is like this initially

there is some initial set up where you set
up these maps and this constant stuff.

That depends on lambda but

then you have these machinery did
allows you to zoom to any point.

In a clear many steps and that's for
that's for like so that you don't

have high complexity when you don't need
an actual input when you do deductions.

Or levels just.

OK so that some of the.

Higher level stuff in the proof.

All.

Right.

This is a constructive proof of getting
something care in the neighborhood

after you are done in the neighborhood
then you're going to use the hammer so

once you dance in the neighborhood you hit
the hammer new good you get everything.

Well the hammer just says that
if you have any disks then

after financially many steps
the disk maps to everything so

it has a bound on the leg how
many traditions you have to do

well I mean that I would the number of
iterations that you have to do the course

then depending on the ideas of the circle
but we don't care I mean less fixed and

then we got interested in complexity so

computer science but I guess we did all
these very old the oldest construction but

the question still is where the need to
get down there particularly of lumber that

exists right so these need to
get in there is they exist and

I guess the only way to do it is look at
the directions OK so let's assume that

we have a meat meat grinder that does the
computation for us and we are going to.

Create me to get in there
of industrial quality that

something something that we
know cannot be done OK so

we are going to let's say look
at the problem of Everly think.

Independence polynomial for
a five foot three to go to graft at five.

How do they did action you know
that you can do anything so

what you're going to do to go
to get you're going to take.

Attach a thirty.

To a very very picks that effectively
changes the Lumba you're going to change

lumber to five and now if you ever
late they're independents blooming all

Revelle you complex really lumber down
this one it will effectively computed

the value of the Independence polynomial
of eight over if you get five but

it had a problem therefore this
would be a hard problem with well.

Now we threw you out an old hippie weed.

OK So this is the meta good and

that is you construct if you're
out of the happy with this one.

And they're not good enough
to begin before at least for

it being required your own dignity to the
and maybe one child to how to Nashville

you have to do that action
slightly differently OK so

now we are going to do that action when we
are going through exactly computer number

of independent sets how do we do it and
action like that well here to what

you do is you take your graph age and you
attach that to the to ever have attics.

And know what kind of independent
indies get off eight.

To be decent dition Well I do
take this about it takes but

then you kind of take anything from this
graph or you don't think it is about

a text and then you count the number
of independents that in this graph.

And now we should do by and that if.

You are going to figure out when
this is both at the fort and

they get the If so you're going to exactly
cound a number of independent thought

in this order general graph H.

OK now is cheating a little bit we have
a high degree of error things here

that's no deal out so what do you
have to do is you have to do some

reduction with the diggity How do the
direction we did you construct a little

gadget allows you to copy
information OK so we're

going to have a gadget where the number of
quarter the weight of configurations of.

Points.

Is the door but if they agree it
has the same weight you can do it

by the full length of the with carefully
chosen weights for the one of the three.

And I think that's it.

Connected I am also like M M M was
like the bind that I searched but

I meter so M M Was that ratio of two
on it could bite on the study and

that allows you to figure out
how many independents that.

May be negative for them and

if they're going to be a negative
value and you're going to.

Will.

No no no no these only covers it for
a complex a lot of.

It it covers it for
that's right now if order for

a positive one do you have to
do whatever Alan was doing for

modalists that is there is no cheap way
out OK so if you look at the picture.

Everywhere we have Shot of the hardness
but on this line we have N.P.

hardness and some shut behind this
if you can get out of behind this

they do usually easier to do and
be hot and I think so so the.

Line I think by nature has to
be more difficult to prove.

Well there either is not going to be so

I would be how they think and
the complexity theory.

That I'd.