>> David Wilson: Our next speaker in the afternoon session is going to be
professor Mario Bonk from the University of Michigan. As with many of the
people we've been hearing from, Mario is one of Oded's coauthors. They worked
on hyperbolic geometry. Mario's going to talk to us about transboundary
extremal length.
>> Mario Bonk: Yeah. Thanks. Well, we all enjoyed these little anecdotes about
Oded, so let me add just another one. So when I first came to Seattle visiting
here, Oded had moved from the Weizmann Institute to Microsoft I asked him,
well, so is there big change in your life now? What are you doing? What's
different? And he said oh, well, nothing much has changed. I have a boss now.
And sometimes I have to pretend that I under what my Mike Freedman is talking
about. [laughter].
So yeah, it's -- it was already mentioned I have done one joint paper with Oded,
and our collaboration actually also started during a hike. So it was a [inaudible]
conference, and well, I think this was one of the features about Oded that I liked
very much. So you give him a couple of definitions, and if he gets interested in
the problem, he immediately starts thinking about it. And this is also a what we
did in our collaboration. So it was about metric geometry, which is a little bit off
his main area interest. But you know, I told him a couple basic things and then
well immediately started working.
Today I don't want to talk about our joint work. I rather want to talk about
something that I consider one of the most important original ideas of Oded in the
field of conformal geometry and classical uniformalization, namely this concept of
transboundary length.
So I should remind you that before Oded switched to probability, he wrote a
sequence of very important papers. Well, in circle packings, we've heard this.
But also in, well, conformal geometry and classical uniformalization. In particular
there was a series of very important papers jointly with He, and I think it's fair to
say that these papers I would say are the most important contributions to this
general area that's in the last 50 years or so. Okay? So there are many papers.
And of course I don't have the time to talk about them all. So I just picked one
little gem that I think is the most important original idea of Oded in this area. And
that's the concept of what he called transboundary extremal length. So this
appeared in 1995 in a paper published in the Israel Journal.
The concept is a little technical. So in order to give you full appreciation of this
very simple but clever idea I first have to talk about classical modules in extremal
length. So let me start with some basic definitions. So we on the Riemann's
sphere we have a couple of paths, so think of these paths as wires like this. And
what you would like to do is you would like to put a mass distribution on the
Riemann sphere in such a way that whenever you integrate along one of these
paths the path integral over this distribution picks up at least length one. Okay?
So if this is the case for some non negative density, then I want to call it
admissible for the path family.
So the modules of the path family is now simply the infimum of the L-2 norms
over all admissible densities. Right? So somehow you want to spread out this
density row in an efficient way so that whenever you travel along curve, the curve
picks up length one, row length one but you want to do it in such a way that the
L-2 norm, the total L-2 norm of the density is as small as possible. And that's
called the modulus of the curve family.
Well, there's a little bit of controversy here in this area. What should consider this
quantity the modulus rather it's reciprocal and work with that. So the reciprocal is
called the extremal length here. So actually Oded liked extremal length, but I
consider this as a little bit of a heretic view, so I rather work with modulus and I'll
have to convince you in a second that actually modulus is slightly better concept
even though extremal length has a somewhat more geometric touch to it. Okay?
So the most important property of modulus is its conformal invariance. So if you
have a -- if you have a region -- so if your curves sit in some region omega and
you have a conformal map applied to this region well, then you get some image
curve family that I call just F of gamma, so F of gamma just consists of all
compositions of your conformal map with the paths in the family right, I mean
these parent trust paths. And, well, when can show simply by essentially
applying the transformation formula that the modulus of the image curve family is
equal to the modulus of the original curve family.
So this conformal invariance of modulus is very useful to explain certain
properties that we all know. For example, one of the things that you all know
from basic complex analysis is that if you have two annuli, say one with inner
radius one, the other one with inner radius R, and then one with inner radius one,
outer radius R prime, well, then, the only conformal equivalent if R is equal to R
prime. Okay?
This can very nicely be explained by this conformal invariance modulus, namely
if you look at the curve family that connects the inner to the outer boundary, then
this modulus of this curve family gamma is just, well, two pi over log are. Right?
So because of conformal invariance of the modulus, if you have the situation well
then necessarily R has to be equal to R prime.
So this quantity allows us to immediately explain such effect. Another effect of
the same type is if you have two rectangles and you have a conformal map
between two rectangles so that vertices go to vertices, well then we all know that
this can only work if the side ratio is the same, right? So if you have a conformal
map like this, with a property that a vertices go to vertices, well, then necessarily
A prime over B prime is equal to A over B, and this again can be explained per
modulus because you can interpret this quantity A over B here just as the
modulus of the curve family that connects the sites in the rectangle like this.
Right? So if you call this curve family gamma, then you get the modulus of this
curve family.
Well, the reason why I think modulus is slightly better than extremal length is this
subadditivity problem that I wrote down here. So if you have a countable
collection of curves and you all throw them together, then some of the modulus
behaves like an outer measure on the family of curves, and of course you can
write down the same quantity with extremal length but then the interpretation is a
little more contrived, right? So I think this is the main reason one -- why should
one -- why one should actually work with modulus and not with extremal length.
>>: [inaudible].
>> Mario Bonk: Well, there are various monotonicity properties. And I didn't
write them down. Right. Maybe with extremal length you can some of them can
be interpreted in a better way. One intuition that you should have about modulus
is that the modulus of a curve family is a little bit like electrical conductivity. If you
consider the paths in your family as wires.
So then some of the monotonicity properties that you can easily prove are very
obvious in this interpretation mainly if you're very, very long wires, well then that
the conductivity should be small and that the same with modulus. So if you have
long curves in your path family then typically the modulus is very small. If you
have many curves in your path family, well, then the modulus becomes large,
right, exactly the same as with conductivity. And if you think of this interpretation
then some of these equations that are in the board here become very intuitive.
So remember if I have a rectangle like this and I look at this path family that
connects opposite sides like this, well, then, I told you that the modulus, this
curve family is A over B, and this ratio can easily be explained by this
interpretation, right, because if you make A large you get more curves and this is
why A is numerator here. We make B large and the curves become longer and
this is why B is in the denominator.
Okay. There's a special feature of modulus in the plane that is different from
modulus in high dimensions. The definition of modulus actually works in all
dimensions, even in general metric spaces if you have some base underlying
base measure. But this property that I call reciprocity here is a special feature of
the plane and there are various versions how to formulate it. So I formulated it
somehow in the simplest possible context, namely for moduli that come from
quadrilaterals. Similar way as this modulus that comes from a rectangle. So
quadrilaterals by definition just -- well, a close Jordan region with four marked
vertices on the boundary and well, this gives you subdivision of this Jordan
curve. And let me just denote the sides by A, B, C, and D. Okay?
And what you can do then, you can consider two of path families similar, the one,
the path family that connects A to C and the one that connects B to D.
And it turns out that these moduli are related, namely one is the reciprocal of the
other.
The way to prove this is actually -- to map the quadrilateral here to a rectangle
like this and for the rectangle case it's obvious by this equation, right? If you take
the other -- the other modulus that you get from this rectangle L, then you get
exactly the opposite ratio here.
Well, if you want to apply this concept of modulus in conformal geometry, then of
course you have to relate it to geometric properties, right? This is simply -- this is
roughly the game that you play. So you have a map between two regions, a
conformal map, and somehow you wanted to say something about the geometric
properties of this map, and you know that this modulus is preserved quantity. So
if you can relate it somehow to geometric properties of some let's say a continua
in the region, then you get some information on the map.
And the nice thing about modulus is that it relates to geometric quantities. So,
well, let me introduce some notation here. So we have a region omega. And in
this region omega we have two continua, so two contact connected sets
consisting of more than one point. And we would like to look at this curve family
that connected these two continua inside the region. This curve family I call
gamma -- well, E, F, omega.
So turns out that modulus is related to what I call the relative distance of these
two continua. So the relatively distance of two continua is just the distance of
them rescaled by the smaller diameter. And one can actually show that you
always get an upper bound of this type here. So the modulus of this curve
family, well, I guess there's a typo here, it should be gamma of EF. This is
controlled by the reciprocal of the relative distance up to function. So this
notation here means up to a monotonic function.
Unfortunately that causes a big problem in this control mapping problems. You
don't get an inequality in the other direction. So you cannot guarantee that the
modulus is uniformly large if you have some control -- some control on the
relative distance of two continua. This is through if the region actually is the
whole Riemann sphere. But for general regions it's not true. And intuitively this
is kind of clear because think of the general region, you will have a bunch of
holes here, and if you have these holes then you are forced to go around the
obstacles. So you get less and less paths in your family, so it's clear that you
cannot expect any kind of uniform bound. Okay? So that's a big problem. So
modulus is unfortunately not bounded below by a quantity that just depends on
the relative distance.
Transboundary extremal length solves this problem. So transboundary extremal
length or transboundary modulus is a quantity where you do get this uniform
lower bound and that is why this concept is so important and useful. Okay?
But before I come to the definition of transboundary modulus, let me tell you a
little bit about extremal densities, okay? So what is an extremal density? Well, in
the definition of modulus we will this concept of and admissible density and they
call indensity extremal, well, if it realizes the smallest possible L-2 norm that I can
get for an admissible density. Okay? So you can prove the existence of such
extremal densities in fairly general situations. And you also get uniqueness
because it's easy to see that if you have a bunch of admissible densities for a
given curve family well then any convex combinations of them is also admissible
and this L-2 function is strictly convex, right, from this you immediately get some
uniqueness statement for extremal densities if they exist. So more precisely you
get uniqueness almost everywhere, up to sets of measure zero.
So let me tell you two instances about these -- two situations where you can say
something about extremal densities. The first one is again has to do with these
quadrilaterals that I talked a moment ago. So if you have a quadrilateral with
these marked sides A, B, C, D, and you look at this curve family that connects A
to C, well, then the extremal density is actually a derivative of the absolute value
of the derivative of a conformal map that maps the quadrilateral to a rectangle
well with the right normalization, I guess the right normalization is that this side
length should be one and of course corners should go to corners. Okay?
Similar results also holds in the multi-connected situation. So suppose we have
the same picture but now we have a bunch of holes here. And been you can ask
yourself, well, what's the extremal density for the path family that stays inside the
region but avoids the holds. And this is actually very interesting. It turns out that
the extremal density comes from a conformal map of this quadrilateral to a
rectangle with slits removed. So the extremal density is again just the absolute
value of the derivative of conformal map. Well, maybe I should push this a little
higher. That way you have slits like this removed.
And that's actually very useful if you want to map an arbitrary region into some
kind of a normal domain. And, well, there's a classical theorem that actually goes
back more than 100 years to the times of Hilbert. So he had an argument to
show that every reach, whatever the connectivity can be connected to what we
now call a slit region. And the region where slits are removed in a similar way
like this. And from what I just told you it's very easy to understand why this is not
so hard to prove because you have an natural extremal problem whose solution
actually is related to the conformal map.
All right. So one way to prove this effect about uniformizations of arbitrary
domains by slit domains is just to look at your domain, introduce a picture like
this, then, you know, consider the modulus of this path family here, look at the
extremal role, and then, well, try to cook up the map actually from the role.
That's one venue. Well, there's a famous problem that is more than 100 years
old in this area that is still unsolved, and that's Koebe's
Kreisnormieurungsproblem. And, well, it uses as normalizing regions the
so-called circle domains. So by definition the circle domain is just a domain in
the Riemann sphere whose commentary components are round disks. And
Koebe in 1908 asked whether every region in the plane is actually conformally
equivalent to one of these circle domains. So a domain where you have holes
that are round disks.
And he actually proved this in 1920 for the finitely connected case. You know,
that's a very important and useful uniformalization theorem. If you have infinitely
connected domains, if you have only have finitely many holes in your domains, if
you have a situation like this well, then there's a conformal map on to something
that looks like this, where all these boundaries here are just circles. One big
advance in this area is a theorem to He and Oded from the early '90s. And when
they showed this in the case that you actually have countably many components,
and that was a big progress in the area. And the original proof actually was by
using ideas from circle packings. But later on Oded actually found a different
proof by using well, this transboundary extremal transboundary length that I will
introduce in a moment.
But before I do this, let us think why this problem is so hard. Well, the reason for
this is that there's some unknown natural extremal problem that you can solve
into solution, but some will be related to conformal map on to such a circle
domain. So I think the main observation that Oded made is that if you change
the geometry of these holes to slightly different shapes, actually then you do get
some extremal problem which is very natural. The shapes that occur in these
pictures actually are squares. Okay?
And that has to do now with this what I call the transboundary modulus. So if you
see the definition for the first time it's a little technical, but let me try to explain
and maybe here on this other board. So again you have a domain, and you have
a -- you have a bunch of holes. So I guess I call them KJ. It could be even
countably many, but let's not make things too complicated here.
So we have a domain omega and these complementary components and you
have a path family. And the path family is actually allowed to travel through
these holes here. What you would like to do is you would again like to associate
a quantity to this path family that is very similar to classical modulus. And again
you will have some continuous density here, this role that actually lives on the
region, but then you have extra turns and that's the main point, these extra turns
actually live on the holes. So you assign to each of these holes here discrete
quantity row 1, row 2, and row 3, and so on. And now the admissibility condition
is essentially the obvious thing. So you integrate your serve here along this
continuous part of the density. This is this turn here.
In addition to this, you pick up a row from each of the holes that you meet okay?
And if this sum is greater than or equal to one, then you call well this density
consisting of a continuous inter-discrete part admissible. In some sense this is -some sense the curves you can travel through a hole by paying a flat rate. So
flat taxi rate doesn't matter, you know, whether you hit the hole several times. If
you use it once, well you pay your row I and, well, it goes into this admissibility
condition like this.
And well, then again you try to look at a certain infimum and it's kind of clear what
you want to infimize, you want to infimize the analog of the L-2 norm here, right,
so just the L-2 norm of the continuous part plus, well, the natural sum that comes
out of the discrete part.
The funny thing is, this quantity, even though it uses things that are not in the
domain is still conformally invariant on the maps of the domain. That makes it
actually so useful. So this is this conformal invariant.
Actually that formulated it a little differently. So he -- well, he introduced some
[inaudible] space where he actually punched the hole to one point. But let me
not do this like this, let me do it a little simpler. So this conformal invariance here
says that if you have any homeomorphism on the sphere that is conformal on the
domain. So no requirement whatsoever on the holes, then you actually do get
this conformal invariance of transboundary modulus.
The proof is almost a triviality so for the continuous part you do again the same
transformation formula as you did for classical modulus. And while the other
thing you use is somehow in any conformal map these holes are somehow
shuffled around, even though the map is not defined, right? I mean, to make this
rigorous you can talk about the end compactification or something like this. But
it's essentially clear that you just shuffle around these discrete -- these discrete
weights and well then you get this invariance.
Well, I'm running out of time, so let me quickly say what you can do with this.
And somehow you get some intuition of what is going on if you just look at
extremal densities, namely if you again have some quadrilateral, some corners,
you have some holes here, and you look at the path family that connects the
sides of the quadrilateral, they are allowed to actually hit the holes, then the
extremal density here of transboundary density has a continuous part which is
again just the derivative -- the absolute value of the derivative of a conformal
map on to a rectangle with these squares removed. And the discrete parts are
just the side lengths of the squares. All right. So the extremal rows J here
correspond to the side length of the Jth of square. Okay. And once you see this,
I mean, then you know that this is a very useful for some conformal mapping
problems.
Let me just give you one quick theorem that Oded proved. Using this concept
that has to do with what he called fat sets and co-fat regions. So what is a fat
set? So fat set is a set with a property that if you place any point in the set and
you look at both centered at this point of a radius whose diameter is not too
large, let's say, well, less than the diameter of the set, then the percentage that
you see here, this is eaten by the intersection of the ball and the set is fixed for
every such ball. Right? So this is what I call tau. So if you look at the ratio of
the area of the interconnection to the area of the whole ball, well, then, you get
this fixed percentage tau. And well, a co-fat region is just a region who is
complementary components are fat in the sensor just single points. Okay. So
this is what Oded called a co-fat region.
And the important thing about transboundary modulus now is that in contrast to
classical modulus, you do get uniform lower bounds for moduli of path families
connecting to continua, just depending on the relative distance of this continua.
And this somehow allows you to play the usual game. I mean, you use the
conformal invariance. You have lower bounds of the modulus for path families in
your source domain. Well, if upper bounds in your target and you can use this to
your advantage to for example prove this theorem that is due to Oded, if you
have any co-fat domain, then you can actually conformally map it to a circle
domain.
That is a very nice theorem. So what this essentially says is that, well, if you
want to soft curve as conjecture, it didn't really matter whether you map to circle
domain or for example to domain whose complementary components are
squares, right, because squares are also uniformly fat. So you can essentially
choose your favorite domain and solve the problem there. So I think I'll stop
here.
[applause].
>> David Wilson: Questions?
>>: What's the obstacle [inaudible] methods to use the [inaudible].
>> Mario Bonk: Well, that's a difficult story. So essentially somehow you lose all
control. So if you had some unit, uniform continuity in the map, all right, then you
could just go from the finite case by some limiting argument to the finite case, I
mean the case of finitely many connected components but unfortunately you
don't have this kind of control. So the whole game really is to get this type of
control. And you can get such control in the case we have countably many
connected components. They are this kind of naive approach works. But in the
general case unfortunately doesn't.
>> David Wilson: Other questions?
Okay. Please join me in thanking -[applause].
>> David Wilson: It's a great pleasure to introduce Greg Lawler from the
University of Chicago. Greg is a long-time and very distinguished contributor to
probability theory. And in the context of his conference of course some of the -some of the Oded's most famous and most ground breaking work was done in
collaboration with Greg and with Wendelin. Greg is going to speak to us today
about something different from what's in the program, it's path properties of
Schramm-Loewner evolution.
>> Gregory Lawler: Okay. Thank you very much. I'm very happy to have an
invitation here. Like many people in this room, my personal research has been
changed greatly by the insights of Oded Schramm, and I'd like to focus on some
of the idea that Oded have done that particularly influenced me, and that is this
beautiful object which we now call the Schramm-Loewner evolution.
We've already had an introduction to this in Wendelin's talk, so I'm assuming that
talk is a prerequisite to this talk and we're going to build from there. So if you
recall what Oded did was looking at scaling limits of various things, particularly
loop erase walk at the time. And what he did was he said -- thought suppose
there's a conformally invariant limit that satisfies a certain Markov property we
know which is satisfied. What would it have to be? And as we've now learned,
he then recovered the Loewner equation, wrote down the Loewner equation,
then figured out basically that it would have to be the Loewner equation with a
Brownian motion input. And that's part of his original paper in the -- from the
Israel Journal that Wendelin talked about.
So if there's a limit you have an equation like this. I did this a little bit differently
than normally done. I'm in the upper half plane. So Z's in the upper half plane. I
didn't write that there. I should say that there was also not a [inaudible] of Oded
that for a lot of the properties, although most of the Loewner equation is written in
the disk or the component of the disk and the complex variables literature is
actually easier for a lot of the properties to actually look at it in the upper half
plane. And in particular the Poisson kernel, which basically drives the Loewner
equation is easier in the upper half plane.
So you basically had this equation in the upper half plane where GT is a function.
UT is a function on the really line in this case, a continuous function on the really
line. And A is the parameter we can play with. Now, the way Oded and most
people do it, they make A equal two and they do more variance -- variation here.
But just because the way -- I tend to make this a Brownian motion of variance
one and vary this. It doesn't make -- it's just a constant linear time change.
Doesn't make any difference.
But if you see A -- wherever you see As in this talk, A is equal to two over kappa,
and that's consistent. So you just see -- if you ever see it there.
But the question is now that you know it's got to satisfy this, well, does -- are
there things that satisfy this? Well, if I stick in any continuous function here, then
this equation has a well defined solution up to a certain time, and it's not too
difficult, a little bit of work, to show that in fact if for any time T the solution of this
will actually be a conformal map from the -- from some domain of some
unbounded subdomain of the complex plane to the complex plane with a certain
boundary behavior at infinity. It looks like the identity of infinity. The next term,
the constant term is zero. And then there's another term which is one over Z.
That A is the same as this A. It comes from the rate here. Then you have
corrections. [inaudible]. Well, there should be a T. That's type. But there
should be A times T. Thank you. There should be.
But the question is -- but just because we had this unbounded domain HT, that
doesn't necessarily mean that it comes from a curve. So the questions that I'm
going to deal with is -- the basic questions, does there exist a curve, gamma,
such that HT is the unbounded component of H? So remember that Oded
originally was looking at random curves. So this should be coming from the
curve. But it's not obvious to see what's happening with that curve. Is the curve
simple, IE, non-self-intersecting? Does it stay in the upper half plane? Which is
really -- does it turn out to be essentially equivalent questions? And finally what's
the Hausdorff dimension of this curve? Is there actually going to be fractal
curves?
The main paper of Oded's I've got to worked on is actually a paper of Oded and
Steffen, and I actually thank Steffen for not speaking about this, which means I
can speak about this.
But the main thing is basic properties of SLE. Now, it was finally published in
2005. There are earlier versions of this. Actually once talked to Steffen and told
him how much I liked the paper but I really was struggling with the basic
properties of SLE was kind of the wrong title.
But now that I think about it now, I think that it's the right title and these really are
the basic properties of SLE, so I even agree with the title of this paper. So let's
talk about exist of the path. And I didn't draw any pictures here. I just have a
simple, simple picture here. So this is what we're thinking the curve should look
like. So let's assume it was a curve. GT is mapping the curve back to the upper
half plane. And this point gamma T should be mapping to this point use of T,
which is the use of T in the equation.
So if I want an equation for gamma T, since that should be mapping there, so UT
should just be GT inverse or if I write FT for GT inverse, FT of UT should be
gamma T. So there we go. That's the curve. Gamma T equals GT inverse.
The only problem is that when you have rough boundary for conformal maps,
these thing may not exist, these things off hand could be very, very rough. And
in fact, one can actually find out that there are the solutions to the Loewner
equation might not come from curves. Let me just go back to the equation.
Here's the equation up there. This is dying. Okay. Well, we'll get the equation
up there.
The behavior that equation does depend on use of T. And there's various work
in the deterministic world. I mean Tom Marshall, Steffen Rohde, Joe Lind and
some other people as well, I'm sorry if I've -- but if you -- what happens
with this equation, if you look at it, if you've never seen it, very quickly what
happens is this slit here tends to grow in time, in time T this slit tends to grow
about delta T, what's called Brownian scaling.
So if I just fixed a growing point there, this would grow at about delta T. So what
keeps us from being a curve is that this starts moving faster than it's growing in
this direction. When this thing moves square root delta T or more in a certain
direction, then you might be in trouble. And that tells you to some extent that the
natural borderline for this would be UTs that are Holder continuous of order
one-half.
And in fact, if UT up here were Holder continuous of order a half, then you can
actually show that if A is large enough, this actually gives you a simple curve.
That's actual what they show in various points is what they showed. But their
examples or Holder continuous functions order one-half and small As such as
you don't have them.
Now, a Brownian motion path U sub-T is not Holder continuous of order one-half.
But in some sense it's randomly Holder continuous of order one-half. It's like it's
Holder continuous of order one-half, but the constant is random on different
scales. So you can sort of thing of it as -- so it's not clear there whether or not it's
going to exist, but if it's going and you like to do that, so how do we try to prove
this? Well, we just approximate we'll let gamma N of T approximate it by instead
of evaluating UT and UT plus I times N, that's well defined, just move it up a little
bit here. It's certainly defined there. And then just take the lemma. Okay.
So the question is can you show this limit exists, and then once you show the
limit exists, can you give bounds on gamma S minus gamma T, just shows that
you actually get a continuous curve. Well, you just think about gamma T minus
gamma NT is just less. If I go along here, it take its image over here and get the
light there, all I have to do is look at the derivative of the map. So, thank you.
That's a good chairman. Okay. So to get this, to be more precise what I should
say is that if this integral is finite, this integral is finite, then I'll be able to define
gamma T such that that's true. If this integral is infinite then I may not be able to
define limiting value gamma T. But you actually get this thing. So there's the
quantity to start estimating. So if one can show that this derivative is such that
it's bounded say less than Y to the delta minus one for some delta bigger than
zero, then in fact we get this.
Actually, this is almost true deterministically. Deterministically you basically get Y
to the minus one. That's the worst that you can get. However, Y to the minus
one is not good enough to make this integral converge. So you need something
better than that.
Okay. Now, there's a technical difficulty if you're dealing with very wild use of U
sub-Ts that you would have to -- you have to do this for every possible value of
T. But for Brownian motion we have this approximate Holder continuity and
basically with the Holder continuity that says I don't have to take every T, I have
to take just maybe if I'm taking N here, then I only have to do N squared values
there.
Ten years ago I knew very little about say complex variables two. When it comes
to conformal maps now, I found out how much you can do with just a couple of
tools. So know the distortion theorem, know the one, Koebe one-quarter
theorem, know the [inaudible] estimates. You hardly need to know anything else.
That's, you know but you need to know those.
Okay. So that's at one T. And now what happens if you do other Ts? Well, now
we just basically do here on triangle inequality. So if I'm doing -- so FS hat is just
-- is, okay, you can just go up here. So that's the approximation, at S the
approximation at T, and we just split it into two pieces because we have S as
appearing two places. So we just take one of the Ss and one of the other Ss.
And as it turns out, if I -- so if I fix a point, and I want to vary S and T, that's no
problem. This difference in this derivative is of the order of this. For S and T
small. I'm not going to go through proof and say this is just deterministically from
the Loewner equation, from the differential equation. Holds for every path. And
now if I want to compare FTI epsilon plus USI epsilon plus UT, so this is with the
same function but two different points, that's where the distortion estimates are -and when you do that, and now if I use the Holder continuity I get this.
So basically this sort of -- this difference is okay, and so we get sort of this, only
just so -- so if I'm taking gamma S minus gamma T, there are -- well, there are
three terms here but basically the gamma NS minus gamma NT turns out not to
take too much work. So it's mainly the gamma S minus gamma NS, gamma NT
minus gamma T. Those are the ones that have to be handled.
And to handle this by this is basically can I handle the derivative? So how do I
get estimates of this derivative? And now, so so far we haven't used any
probability at all except for the modulus of continuity of Brownian motion. So now
what I need to do is look at this function and also be in the Rohde and Schramm
paper they showed, and this is actually not difficult, use the fact that just
deterministically this function, which is an inverse, has the same distribution as a
Loewner equation with reverse time or Loewner equation in the reverse direction
so this is the same equation we had before, except we have a minus sign in the
denominator.
So if I want to take moments of this, it's the same thing as taking moments of
this. So how do I do this? Well, there's a few different equivalent ways of finding
moments. You can either -- you talk about writing second order differential
equations or things like that. I prefer now to talk about it in terms of what's the
appropriate martingale or local martingale.
So in this particular case, I take MT of Z, the HT prime [inaudible] lambda, that's
just the imaginary part of the image, sign to minus sign, don't think too much
about this, but it's just a computation that if I choose these parameters correctly
with respect to each other, I could choose R to be whatever I want, choose the -then in fact this is a martingale. And this is Ito's formula. It's just an application
of Ito's formula. And therefore the expected value of this let's just take Z to be I.
The expected value of this is one.
Well, if I didn't worry about correlations here, this is the reverse flow, this -- the
image of this flows like T to the one-half. And if things don't skew very much and
I stay to the middle and don't hug the -- and don't hug the boundary, that should
be of order one, so this guy should be order one times T to the zeta over two
times that, and you actually get this estimate. And that's actually giving you the
moment that you want.
Let me just say this isn't true always. You have to put some restrictions on R.
And this calculation -- but you want to go through the calculations, but I'll just say
once you have this, this estimate is good enough to prove existence of the curve
for kappa not equal to eight. And this is essentially the proof in the
Rohde-Schramm paper.
Inform kappa equals eight, what I've done here is not good enough to do things.
This is the curve follows separate from work, this is a joint paper, but like many of
the results in joint papers this really should have the S circled on it as a Schramm
result. So if you talk about for kappa equals eight, the -- how to prove that SLE
kappa really approaches SLE eight or the uniform spanning tree approaches SLE
eight sufficiently closely to derive this is okay. Now we know it's true for kappa
equals eight.
I do -- okay. Well, actually when you do this, you also get a guess for what the
Holder continuity is. And in fact, that's now a theorem and I like to say in fact you
can actually find what the Holder continuity of gamma is. Let's say for T away
from zero, T equals zero is different. And there's the formula. It actually turns
out to be not particularly nice. But in some sense that's because this capacity
parameterization of the curve is not very natural. But there is one thing if you just
want to plug it in to note for yourself is that you have a -- it is Holder continuous
with a positive number if kappa's not equal to eight. If kappa's equal to eight, in
fact it's now a theorem that the curve is not Holder continuous of any alpha
bigger zero. Now, an open problem I think still open is to find a different proof for
kappa equals eight which follows this line but is much more careful, it handles all
the logarithms correctly. But we'll talk about that. I mean, okay. But that gives
the curve. Okay. So that's the existence of the curve.
Now I want to talk about the dimension of the curve. And this is just a notation.
I've define it so I'm going to define conformal radius actually just for convenience
I do one-half conformal radius. So by the simply connected domain D, and a
point W in it, this will be one-half F prime of zero where F is the map from the unit
disk into D from taking zero to W. So factor one-half is just -- I like gamma of the
half plane and I to be one. That's actually an upsilon, sorry, it's upsilon.
But the reason to put this is a conformal radius turns out to be something more
interesting or easier to deal with than this distance from W to the boundary.
They're actually comparable quantities. So in many ways I'd really want to look
about the distance from a point to the boundary, but it's comparable to this
conformal radius. That's the Koebe one-quarter theorem.
I'm also going to give some formulas. I just want to give some notation. So if I
have a simply connected domain and I have two boundary points, I could talk
about the argument of a point W with respect on the inside with respect to those
two points. And it's just the argument that you would get if you map it to the
upper half plane. And I'm actually S will be the sign of the argument. So if it's a
-- if it's in the upper half plane and I've got a point, I'm just looking at the sign of
the angle of the point. And for other domains the conformally version is -- okay.
So that's just notation.
Okay. Actually well maybe I'll do this. Okay. We first might ask the question do
we hit W. Okay. Does this curve actually hit W? Well, what you do is in that
case whether something hits W, you take the Loewner equation but you change
the parameterization so that it's parameterized by this conformal radius which
decreases. Or it's -- we can call this the radial parameterization. It's actually the
parameterization of radial SLE. And if you look at it and see what happens to
this angle, you get an equation like this.
So this is the angle at W some point on the inside, and now as we see what
happens, this is now a fairly simple differential quo equation, co-tie attached in
theta until it looks like more or less one over theta tilde. The question is what
happens -- does this reach the origin if I -- if I'm going to hit a point then as I get
close to that point, the angle keeps fluctuating. However, if I miss a point, then
this thing disappears and it turns out using this you can see that for kappa bigger
than or equal to eight, the SLE curve actually hits points. For SLE, for CPA less
than eight, SLE does not hit points. And this equation also, and this actually
goes back to Oded's first paper, for kappa less than eight you can actually use
this equation to determine whether the curve goes to the right or the left of W.
I won't prove it here, but something for kappa less or equal to four, you actually
get that the paths are simple for kappa between four and eight they're not simple.
But I won't do that, because I want to discuss a little bit about the fractal
dimension.
So how do you determine a dimension of a curve? I mean, the -- when I started
working on SLE, and this was very knew, it was very hard work. I had
techniques to work with Brownian motions. But SLE, what you're doing is you're
-- you're looking at the evolution of a curve by measuring its effect on points
away from the curve. Not by the curve itself. There's not an equation on the
curve itself. So how do you final like what's the fractal dimension of the curve?
Well, if I want to find out what the fractal dimension you might ask well, how
many balls of radius epsilon are needed to cover a curve? If I want to know how
many balls of radius epsilon are needed what I can do is I take a particular point,
W, and ask what's the probability that the curve gets within distance epsilon of
W? And at least heuristically in terms of expectations that should behave like
epsilon to the two minus D where D here denotes fractal dimension of the curve.
Since this is distance, I could replace the distance, and this is [inaudible] with the
conformal radius. What's the probability the conformal radius looks like that?
So how do you compute this? Well, let's think of it as suppose there were a
function which I think it was a Green's function, think of this roughly as being
what's the probability that the -- that an SLE in the domain D from Z1 to Z2 gets
close to W?
So the probability that the distance is less than epsilon should look like epsilon
the two minus D times this thing, this sort of Green's function which has to do
with the domain.
Well, there's a certain scaling rule but just because the way we have this epsilon
here, if I take a conformal transformation, this epsilon to the two minus D means
that this thing should conform like this, transform like this. And the other fact
which is basically very simple idea if I'm looking at this point right here, and I say
just think, what's the probability I get to that point, well, the -- you ask what's the
probability I get to this point given I've seen the curve so far? Well, as I moved
this curve, the probability gained there given the curve so far should be a
martingale, or at least a local martingale? I'm not going to worry about that
distinction. So -- and what's the probability -- and once I've gotten this far, well,
here now my new domain is D minus gamma zero. This really should be -- this is
right if gamma's a simple curve. A little more complicated if gamma's not a
simple curve.
Basically look at the domain I'm in right now. We're now going from gamma T to
Z2. What's this? And that need to be a local martingale. So that plugs in to Ito's
formula. And if you just plug in to Ito's formula, you actually get -- and this is
done in Rohde-Schramm, that the dimension has to be one plus kappa over
eight. And you actually get the Green's function as well. They use that term, this
function can be found in their paper. And it's just the formal radius to a power
times the sign door power. Signed or positive power.
So this actually you find the beautiful result is that for dimensions between one
and two Hausdorff's dimensions strictly between one and two there's only one
value of kappa that corresponds to it. So it's a one-to-one relationship between
dimension in kappa up until we saturated kappa equals eight.
Okay. Maybe I'm running agent short on time. I say you can approve on this and
actually show that the probability of that, but maybe I will skip that. So I want to
say a little bit about this slide. So one beautiful thing but disadvantage of this is
that focusing on one point, it's giving one point estimates. There's another
question about second moment estimates. So for those who know about
dimension, what Steffen an Oded showed in their paper was they gave an upper
bound for the Hausdorff dimension and gave good heuristics for the lower bound
but did not prove the lower bound.
To prove the lower bound, there's a standard technique for proving the lower
bound which in this case requires a certain second moment estimate. This was
not done in -- by Steffen and Oded but it was done later by Vincent Beffara.
Now, I will say that this is much harder to prove. And it's kind of unnatural. And
well, I'll just say there were some mis-- there were some problems with early
versions of this paper Vincent, and in order just to give a little credit, I do think
Oded was one of the people responsible in helping to get this thing to -- but it's
now published and it certainly is correct.
There's another alternative proof which can be given using the reverse Loewner
flow, for which the second -- so we actually have that the Hausdorff dimension is
like that. And actually maybe I'll -- maybe -- actually maybe I will skip that,
because I'm nearing the end.
Let me just emphasize again here how what was very unnatural to me 10 years
-- when Oded was first doing this, and by the time Oded had shown this to me or
to Wendelin and me, Oded already had some experience with this, how you
show something about a curve by looking at what happens away from the curve.
In the -- when I've given colloquially on SLE, the question I always get at the end
is what about three dimensions? [inaudible]. So what you can do in this curve,
this equation really has two things in it. One is true for three dimensions, and
that is that if you -- if you want to observe a curve you can actually look at a point
away from it and see the effect of that point away from it and make it a
differentiable smooth function of that. That's doable. However, what's also true
in here is in two dimensions is you have conformal invariance of all these
harmonic functions. And that's not true in three dimension. So that this really
combines both of them when you do this.
Okay. I think I've used up my time here.
[applause].
>> David Wilson: Thank you, Greg. We have time for questions.
>>: What about four dimensions? [laughter].
>>: [inaudible].
>>: So one problem with three dimensions is -- I mean, I found the longer I work
on this, the harder I find it [inaudible] I mean in two dimensions even the curve
that you discussed is even the non empty set. So in high dimensions I think it is
possible you work out [inaudible] and two dimensions that create nothing in
higher dimensions. So is there a flow that is exactly the analog of the Loewner
equation so things flow down in [inaudible].
>>: It's not a [inaudible].
>>: Well, of course not. So it's a flow -- it's sort of the natural thing [inaudible].
>>: Isn't [inaudible] in three dimensions?
>> Gregory Lawler: If I may comment on your comment. Just to explain my final
comment. What I called the Loewner equation or the -- is a weaker thing than
what you were probably saying. So like for two dimensions the Loewner
equation is something that -- the flow of conformal maps. But you can think of it
as two steps. One is a flow on the potential that's going on, and the other is the
piece that makes it a conformal map. And if you're only looking at the part on the
potential without mapping it anyplace, that's actually what I was talking about
when a Loewner -- the hitting probability from a particular point.
>> David Wilson: Okay. I think we're -- I think coffee is here, so please join me
in thanking both speakers from the session.
[applause]