>> Jennifer Chayes: So, as you've all just said, we're here to learn a little bit
about Oded's mathematics, to celebrate his mathematics. As I'm sure most of
you know, he was an amazing mathematician who managed with his
mathematics to unify fields that people had never realized had such a significant
and deep, deep unification.
And as we go through the sessions today, we will see some of the different areas
in which he worked and some of the areas that he really -- he really created with
his mathematics.
So our first speaker is one of his principle collaborators, Wendelin Werner, who
won the Field's medal in large part due to their beautiful joint work on SLE. So
Wendelin.
>> Wendelin Werner: Thank you. So I'm -- it's a very a big honor and emotional
moment for me to be the first speaker at this conference. Can you hear me?
Okay. And I'm -- before leaving, you know, I said to my friends, I'm going to this
conference honoring Oded's mathematics. And people said oh, that will be
tough, you know, one year after and sad and so on. And my answer was no, I
mean, you know, when you talk about Oded mathematics, when you open his
papers, each time it's just a pleasure and you -- you are happy.
And I'm sure that these two days will be just happy moment that we'll be able to
share by just, you know, looking at remembering his ideas and some tricks he
learned us and he teached to us, sorry, and so he -- and for those of us
somehow who were not, you know, in everyday life contact with him, we can all -I mean his absence is very much, you know, felt but nevertheless his
mathematics is really very vibrant and present all the time.
And so before actually starting my lecture today, you know, the -- I spent some
time thinking about, you know, what should I tell to this sort of audience that we
have today. We have some specialists. We know very well everything about, for
instance, SLE, which is the topic of my talk and some non experts or some
people who don't know at all what SLE is about. So how can I, you know, do sort
of give a talk that will be okay for everybody simultaneously? And okay, Yuval
and David gave us the instruction that we should, you know, focus on some
specific things, so little parts try to show the beauty of some arguments, and
somehow I failed -- I said that personally, you know, at those moments during
this past year where I felt, you know, Oded's absence very much, what I did was
actually that I played Sokoban, which is one of Oded's favorite games at the time
when I met him in 1999. And it's this type of game where you have to push these
boxes and if you don't -- the goal is to put one of these boxes on each of the red
dots. And sometimes it's very tricky to find the route to do it, because if the box
is say in a corner, say, then you're stuck, you're not able to push it anymore,
because you have to be behind the box to push it out.
And I remember very well sort of the way Oded was looking. So Oded would
look at this and just find a way without just by knowing -- find the trick and we you
know amateurs, we just start pushing, you know, move it around and start
pushing and see how this works if we do -- if we are able to work our way out.
And if you would see us find the trick, then he would smile and say okay, you're
good, that's nice. But it's a very nice feeling when you find the trick yourself.
And I think when we did mathematics with him, it was a little bit similar so that he
would -- he would know, you know, he has some sitting above you and looking
you play and then sometimes he gives you the time to find the thing by yourself
or -- and I recommend that you can download these things for free on the
Internet and play this on the plain back. It's very -- it's fun.
Okay. Anyway. So let me now sort of start with my actual slides. And I suppose
SLEs, which is basically an invention of Oded's I mean arguably one of his really
most important contributions to science and to our scientific life today. And you
know, students in graduate school will just hear by suppose that Oded
Schramm's SLEs.
And so the goal of this talk basically is not going to be obviously an instruction to
SLE nor a description for description of SLE as those of us who had to teach this
know that it's not -- you know, takes time to actually -- so it takes more than one
lecture to actually give a real introduction, mathematical full picture of SLE and
the arguments used. And actually it's a great fun when you -- well, fun, you can
actually watch, you know, the awesome recorded lectures by Oded himself
where he describes this in his own words and own style so there is this MSRI
meeting where he -- which is really nice to listen because he -- you know,
everything is very clear in his head put when explains it somehow, it probably
those who were present in that meeting maybe remember that his presentation of
SLE was not the most transparent one. And that was really -- but actually if you
listen now when you know it and a you listen to everything he says, of course
everything is transparent. But -- and the ICN meeting in 2006, for instance, his
talk has been recorded and you can hear him sort of recent -- more recent times,
2006 and as I said last year you can -- once you switch it on, you hear his you
know singing voice, special tone, and you feel very much -- well, it's very
emotional.
And also he wrote several duty full introductional surveys about SLE itself, so the
goal of this talk is not going to be just to try to do this, but I'd rather focus on
actually, you know, Yuval mentioned that, you know, there were plenty of gems
all around the place and that all the planted but I would say that one -- obviously
one pig diamond in the entire mathematical production is S -- the paper that has
been later published and is a really jumble of mathematics in 2000. That is sort
of put on the archive in April, '99, where he basically explains what SLE is and
maybe -- so the entire talk more or less will be devoted to trying to show you that
how much in that paper was already there or that the global picture was already
the frame for everything that happened later was already set and also I should
comment that Oded did not write so many papers single author papers. You
know, he loved to share and to write joint papers with people. And but this one is
a single authored paper. And in his single authored paper somehow he puts
more of himself than in the joint papers. So when you read it, you feel like, you
know, he puts more things like I believe, I think that, this is nice, and so on. And
this type of statement is not so nice, is not so often present in the joint papers,
obviously.
And always when he gives, you know, some -- this type of statement you, you
know, in retrospect 10 years later you can feel how, you know, perfectly right he
was and try to show you such -- some parts of that. And of course I mean for
me, this paper was -- when the preprint came out or more or less actually one
month before when he sent it out said hi, guys, here is a preprint, please send
me comments or -- at some point then I failed -- oh, now, this is really important,
and actually I in some way I felt ashamed because when I met first Oded, that
was when he came to visit Rick in Orsay some months before and actually
almost one year before, I think, and where he gave actually a lecture on SLE and
the relation to loop-erased random walk. And at that time of course I mean I
guess Rick was one of his main -- main people he wanted to discuss with
because Rick had worked out some computations that enabled then in that paper
Oded to actually figure out what the parameter, right parameter was in order to
understand the scaling limit of loop-erased random walks. And I remember very
well that the two of them, you know, they were in very excited discussions and
somehow I didn't really understand what was going on, you know, was outsider
and when he gave this lecture on SLE, I didn't -- okay, I said good, good. Didn't
understand Loewner's equation basically, and then like most of us it takes some
time -- coming from a probability theory takes some time to actually understand
how this thing works.
And then basically in that mail when he says that's also that something -- okay,
and you will see that -- I mean, he explains also that something important is
going on here in this paper. And so I miss completely basically the prehistory of
how SLE was actually created in his brain and how the procedure went. And I
guess Steffen, Itai, David and Rick are probably the -- those who saw it, you
know, on -- by there, and discussed with Oded during that pre or SLE formation
period. And I'm sure they have anecdotes to tell about the genesis of SLE.
Actually Steffen wrote this -- where is Steffen? This memorial paper you know
about -- and if you will of anecdotes, some of which I stole from his paper to put
in my talk. And I recommend strongly that you read it, because it's great.
So when you see something slanted on the slide, it means that it's just taken out
of one of Oded's e-mails or of the paper. And basically in the start of the
introduction so he sets the frame of you know what is the scaling limit of a lattice
model? Of course this, you know, was already discussed earlier by Aizenman
and other people. And one just one little thing I want to mention here is this last
sentence here where he says actually it's important to discuss what the scaling
limit is. In his last sentence he says actually I have a somewhat different
definition that I will propose in the forth coming paper. And I think the forth
coming paper in question is still somewhere in Strauss's computer because it has
transformed into a joint paper of Strauss and Oded. But I remember vividly that
when I visited Oded a few months later he told me something at the time I didn't
understand either like you know actually the elegant way to looking at scaling
limits of percolation would be to, you know, the set of quadrilaterals that actually
cross my path, you know, there is some topological structure and these things
are nice and so this was already somehow present there. Of course then
combined with the fact that sort of -- so this was a frame for proving some scaling
limit and now the frame is also filled by Strauss's proof of conforming variance of
percolation. So that's why it's sort of he waited to -- before showing this.
And here there's also another very interesting sentence that explains that
basically the first task which is actually fundamental is when you try to
understand the scaling limit of a discrete model, lattice model on the large scale.
Oh, I see that you don't see. Is that you have to answer the following two
questions. What kind of object is the scaling limit? What does it mean to be the
scaling limit? And then -- and there's this thighs explanation that actually there's
more than one right answer, that they are -- it's not that there is one good way to
look at scaling limit and one bad way but actually that there are several different
ways to look at things and that all of them give -- shed sort of different lights on
this.
And so now maybe I start immediately with the sort of the description of -- I mean
wouldn't make sense to still to despite everything I said before to give a lecture -I mean to have these two days without at some point spending five minutes
about what SLE is. So of course this will be just five minutes. But nevertheless
since SLE would be you know discussed at -- appear in many of the talks during
these two days it makes sense to say this.
And so in that paper, what he explains to us or in an extremely clear way is
basically the -- how SLE shows up naturally in the context of loop-erased random
walks.
So basically the idea is, okay, what is a loop-erased random walk? So imagine
that you have a very fine grid here and in the very large disk and you start a
random walk at zero. So simple random walk you start from it the center of the
disk, and you condition it to get out -- yes?
>>: I just thought darker pen.
>> Wendelin Werner: A darker pen. So I'll take black and green is okay. So
when you condition random walk to go out at say this point here, one, so here I
take a very small fine mesh grid and so this is a unit disk and you have a random
walk, a random N that moves on your disk. And it moves around. And you force
-- and you condition it to get out of the disk here. Which is not such a big deal
because it has to get out somewhere. You can always sort of more or less rotate
the disk to make sure that -- that's what you see.
So basically -- and then you have this random walk and then this object here
gives a path that goes from zero to one, which is the trajectory of the random
walk, and you can loop erase it. So along the way each time you create a loop,
you erase the loop and in this way basically you are going to create -- okay, now
I'm -- some sort of self-avoiding curve which joins zero to one, which is the
loop-erased random walk from zero to one in the unit disk which is a random
curve that goes from zero to one.
And there's a nice statement in Oded's paper that actually that, you know, this
model has been invented or created or much studied by Greg Lawler as a
substitute for self avoiding walk which was more difficult to study. And then
Oded says I think actually that loop-erased random walk is as interesting as self
avoiding walk because of the connection to random walks and uniform spanning
trees and all these things. So this is the type of statement that are not so present
in the -- I mean, you know, joint -- when he says I think actually that -- and
loop-erased random walk as was sort of pointed out to us by, okay, many people
actually -- I mean, okay. Has been studied -- had been studied before. David
Aldous, Robin Pemantle, Greg Lawler, David Wilson, and there's one property of
the loop-erased random walk that can be pointed out immediately. So imagine
that this -- so basically you have a random -- this is loop-erased random walk
from zero to one in the unit disk. Or discrete version of the unit disk. I could
have another, you know, shape in the unit disk and all the points than zero and
one you could adapt a definition easily. And the important property of
loop-erased random walk that he exploited was then that basically -- the
following, that you look at the time reversal, so the loop-erased random walk has
a path going from one to zero. So you take the path loop erase from zero to one
but then you condition -- you say, okay, I know that the curve is going to finish
like this. You already know the curve is going to finish like this. So that means
that the remaining path -- I mean the thing you don't know is basically the path of
the path that goes from here to that point.
And the key observation for loop-erased random walk was that actually if you
condition on this path of the curve, the law of the remaining guy is just the law of
the loop-erased random walk from zero to that point in the domain in the slit
domain like when you take the disk and when you cut this out here.
So in some sense, there was something preserved, press vacation of some
property when you were growing -- instead of looking at this path going from zero
to one as you would now usually do, you look at it backwards, and if you start
growing -- discovering the information by growing the slit backwards, some law is
preserved.
So this is what is usual called the Markovian property of loop-erased random
walk or something like that. Okay. So that was the first observation that was
around before.
And the second observation is Loewner's situation. So Loewner's is the basic
tooling in complex analysis that has been used a lot in the context of the
[inaudible] conjecture so important complex analysis question. So the idea is the
following. It goes as follows: Imagine now that I have a curve, gamma that gross
from in the unit disk from here to there. It's a simple curve. So at time zero I'm
here, at time T I'm here, so I'm not -- the idea you should think of it as you're
cutting open the disk like this going from here to zero. And the curve is then
going to be parameterized in such a way that -- so this is just, you know, coming
from Riemann's mapping theorem that basically if you define the conformal map
from the unit disk on to the slit domain, so basically where here you have the unit
disk like this, and here basic this unit disk is smaller circles, you know I'll
transform it to something like that. When you have this conformal map FT that
you normalize the -- I mean parameterize the time on the curve in such a way
that F prime T of zero where FT is such that FT of zero is zero and derivative at
zero is horizontal and that this is E to the minus T. And Loewner's equation it just
says that if you have this picture like this, here you have something like -- which
is zeta of T which is FT minus one the preimage of this tip here, that for each
fixed point you're in the unit disk, the motion when time evolves with which the FT
of zed moves is given by sort of a smooth or ordinary differential equation. F
prime T of that means the derivative with respect to space.
So I don't want to explain you Loewner's equation there, but basically this is
already written in the abstract. It just says that basically there's a nice equation
satisfied by growing solicit like this that can be encoded by sort of as a -- by a
differential equation in terms of a conformal map.
And the key observation by Oded was to say well, actually if you think of this and
think of that together, then it's very natural you get fairly quickly to the following
sequence. If you assume the existence and conformal invariance of scaling limit
of loop-erased random walk. So that means you let delta go to zero, the mesh
size go to zero here. So this discrete random walk converges may be in load to
some random continuous function. And that if this is true and if the continuous
function is conformally invariant, which everybody believed was true and which
can Rick partially proved to be true at the time, then the law of this scaling limit
should be that of the curve gamma that you obtained there by plugging in zeta to
be a Brownian motion moving on the boundary of the circle at a certain speed.
So there's two remarks that one should make at that point which is the first one
sort of historical remark and it might be surprising and is surprising for most of us
at some point that to use Loewner's equation to produce this, because this curve
is a very rough and fractile type curve. It's a scaling limit of something, so you
would expect it to be fractile type curve. And on the other hand, what you have
over there is a smooth -- you know, it's a smooth ordinary differential equation so
how can it be that some of the smooth differential equation you know gives rise
to a rough fractal curve? And in a way this type -- of course, Steffen of course
knows the story much better than I do. In a way, also, in the complex analysis
community usually Loewner's equation was really used for smooth curves
because in the context of a [inaudible] conjecture, they were only using, you
know, it was enough to use smooth curves. Nevertheless sort of in the '60s Paul
Miranker [phonetic] and others had observed that you could construct rough
curves using Loewner's equation.
And actually he comments at the end of so conjecture 1.2 in his paper is that
actually the conditions here are true. And he says actually it's plausible that
perhaps soon there would be a proof of conjecture 1.2. And of course he was
right, because a couple of years later the conjecture turned out to be okay. And
so he predicted that this was no big actually conceptual obstacle somehow to be
able to prove -- to fill in the gap that -- to make this conditional statement into an
actual statement.
Okay. So now there's something I learned, you know, if you are a probabilist and
you were trying to understand the scaling limit of these curves and you see the
picture and somehow this is what I thought, you know, for a long time. And last
month reading seven's paper I realized that I was completely wrong because I
thought okay, you will hear the background of Oded in complex analysis, so
obviously, you know, he had all this -- he knew all of this perfectly because that's
the background you have in complex analysis and he was -- you know, he
interacted a couple of years with Itai, and Itai, you know, persuaded him that
actually probability theory was fun and that there was a lot to do there and that
there was a lot to do there and that look at this and let's do. And Itai can be very
persuasive. And so Oded started to look at probability theory questions. And so
he ended up pretty quickly on those parts of the probability theory where that -we're touching a little bit too much complex analysis and that of course he made
the connection between what he knew from are complex analysis and what he
knew there.
And this is completely wrong because this is an e-mail he sent -- Oded sent three
years ago to Steffen and Yuval when they tried to understand, you know, how
SLE was created at all.
And so what Oded explains in that e-mail that basically before working on SLE
itself he did not really know what Loewner's equation was. He knew vaguely that
it had to do with -- that it was important, that it had to do with the coefficient
problems and that it involved something with the differentiation in terms. But
then he says,well, I kind of rediscovered Loewner's equation the context of SLE
and made the connection.
So and if you know really Oded when he says that it's -- you know, you can be
sure it's true, right? He's not claiming something. So it's obvious that actually he
refound basically all these formulation. Actually there was this other Loewner
equation, the half plane, and that basically in the process of creating SLE, he
went all by himself through the procedure to actually recover basically this type of
theory that we have here.
And I remember this paper of '66 by Paul Miranker that he quoted and that we
quote then because it was -- turned out in the joint paper. He said well, actually,
okay, we need to reprove it and by the way let me write the proof because I have
a -- I think my proof is maybe, you know, safer and so then he reput his own way
of looking at this characterization of what -- of Loewner chains in a geometric
way in that paper.
And I think this is a very important thing about Oded that it's not, you know -when we say, you know, he merged together different fields or something, it's not
-- he did not just plug in, you know, the missing thing because he actually
constructed or reconstructed all by himself both sides as well see also on
probability theory it's the same.
And in the same paper then he continues. And this I love this sentence. So the
one that is going to come. Then he says actually if instead of putting a Brownian
motion, I'd run that speed two times T on the unit circle, you'd put something
else. And something funny happens that may be when Brownian motion on the
unit circle moves too fast, then the thing you construct is not the simple curve
anymore. You know, instead of constructing some curve that grows here like this
in a simple way, cutting something open like simple, this is not exactly true any
more when kappa is too large and that paper actually proves that point. Kappa is
larger than four, indeed it's not so the face transition at kappa before was so he
says when kappa's larger than four, it's not a simple curve any more, here's the
proof. And I conjecture actually that this is actually kappa equal four's, the right
is actually the correct face transition.
And then this sentence. Given some kappa, even when kappa is not -- so that
means even when this is not a simple curve, the process is quite interesting. So
I don't know if there's something, you know, in Hebrew that makes this translation
you know, what does it mean to be quite interesting? You know, because
everything that happened later had precisely to do with the -- this quite
interesting object he creates. Of course, you know, if you invent something, you
are not going to say this is very interesting. But you -- Oded would not do that.
But he says it's quite interesting. And I remember then -- you know, I was
remembered by this last year, you know, the last e-mail I got from Oded is the
following: That was after a conference we organized in Moriel. And he said
Chuck, John, and Wendelin, we were the three who organized it, thanks once
again for organizing this work shop. I think it went very well, and it was quite
worthwhile. [laughter]. So then I thought, well, okay, well then quite worthwhile,
know, especially that -- you know, that thought, it was quite worthwhile to make a
trip to a conference and I very much remember last year when I sent it, quite
worthwhile. I've seen this quite before already in Oded's -- and actually sort of a
-- okay. And then comes what sort of made me tilt in some sort of -- you know,
when I said okay, now, this is really going to be important because then he says
in fact, I plan to show in a subsequent work, assuming a conformal invariance
conjecture analogous to conjecture 1.2, that actually a similar process described
the scaling limit of the outer boundaries of percolation clusters.
And he also -- I mean he will -- so he will describe it, everything. And actually in
the paper, he actually already sort of explains everything more or less what he
plans to do at the time. And that this sort of implied many things, would imply
sort of at least explain many things that appeared in the physics literature having
to do with the scaling limit of percolation.
And so for those of you who don't know percolation, this is one of the pictures
that Oded created and that is used all over the world basically which is this
interface, is this scaling limit that is going to converge to one of these -- one of
his SLE processes, which is basically you have this hexagons, you toss a coin,
each one is colored in gray or in white, and then you follow the boundary of a big
island or connected island of black hexagons, you follow that boundary, and this
is this black curve that I've been also -- okay. That's the percolation boundary
which -- and understanding sort of the large scale property of this curve was a
big problem that the physicist gave us to bite on and that we had to at that time
look to pretty stuck. We were very stuck.
Okay. And just a comment on what Oded was sort of aiming to prove at the time.
I mean, what he was promising to put in his later paper and somehow the later
paper, you know, got delayed because you know then Greg and I sort of showed
up and say let's prove this because with your thing you can, you know, you can
solve our problem. And then Strauss came with his proof of conformal invariants
of the discrete model and then sort of the conditional thing was okay, and so it
opportunity make sense then to write that paper. We promised. But we just
wanted to say that how did he knew that kappa equals six was the right
parameter? Basically there's the following thing is just the following algebra. If
you have some symmetric square like this, also some symmetric object like this,
if you look at SLE sort of any parameter going from here to here for -- when it's
SLE that bounces on the boundary, basically it will have probability but simply
because of symmetry it will have probability one-half to hit this path before that
path. You know, it's just black and white are symmetric in the previous picture.
So if you start with symmetric pictures has probability one-half to go more to the
right than to the left or vice versa.
So this is okay. But then if you do the -- what Oded already computed at the time
was something like okay, suppose that now that I take my SLE that goes from
here to here instead of going from here to here. I take the curve that goes from
here to here. Remember the curve is defined but you have to decide a priori
where the starting point is, where the end point is.
And so he did the computation basically in that context of what's the probability
first, go hit this guy before that guy. And there's something very special about
percolation which is that basically you don't -- the fact that this guy's covered in
white or black or this one's covered in white or black is not going to influence the
curve that you are going to draw inside the domain. And so therefore that -- for
percolation or percolation scale limit the probability that the SLE from here to
there also has probability one-half to hit that before that. This is only happens
only for kappa equals six.
So basically you could, you know, change slightly. You can move the target
point you are aiming at, and certain probabilities would still be preserved under
this moving of the end point you look at. And so basically this is for those who
know the story then that came later. This is basically that he actually already had
some pre locality statement just looking at Cardy's formula that he computed for
the various values of kappa.
And so that it was no surprise that when we said oh, actually can you prove
locality of, he said yeah, I know this will be -- this is doable. Right? And he
already had this type of argument in that paper, also. And at the end of the
introduction of that paper what he explains us is that -- the following, that -- well,
the big picture. The picture is that different values of kappa in the differential
equation that we wrote there produced different parts which are scaling limited of
naturally designed processes and that these paths can be space filling or simple
paths or neither depending on the parameter kappa. So space filling is this -has to do with the uniform spanning tree model that I don't discuss here. Simple
path is kappas more than 4. And then you have this intermediate phase which is
this neither corresponds to kappa between 4 and 8.
So he has already set the entire you know -- now, the goal will be to prove all
these different things and the -- and now I want to emphasize one thing. I mean
experience of Oded sort of also present in that -- from that paper which has to do
with his relation to probability theory and to stochastic calculus. And so in the
proof of that paper, where he proves that basically the -- it's not a simple curve as
soon as kappa is larger than 4, for this procedure does define actually objects
that are not simple curves but are still probably quite interesting.
What he writes is the following: We now show that a certain function satisfies a
second order differential equation inside E in an interval, using Ito's formula. And
then he gives the following thing: The reader, unfamiliar with stochastic calculus,
can have a look at -- and then he gives a reference, for example, or he could try
-- so if you are the reader unfamiliar to stochastic calculus, can just try to derive it
directly. You can just, you know, invent stochastic calculus on your own. Why
not? And then he explains the latter is a bit tricky but can be done. Right? So
basically what he did at the time was that -- well, he just probably did on his own,
you know, the entire understanding of how you get the second order differential
equation. And actually I remember now that since I'm at Microsoft I can say that
when I discuss with him, you know, a priori, we were saying well, if you're
working at Microsoft or if you were going to work at Microsoft how to do, because
there's no library. And at that time, you know, digitalized things were not so
present, 1999. And obviously you know, not having a library was not such a
problem for Oded because he would just, you know -- the library was just more
there for, you know, to cite the appropriate reference and give credit to the
appropriate people but he would, you know, always derive things first,
understand things first by himself. And doesn't look like he would read them out
from a book.
And actually this funny thing then that continues on a page -- related page which
is so he completes the proof of this equation here that this should satisfy this,
and so the way he does it is assume that the function is C2, and then apply Ito's
formula and that's just for those of you who know stochastic calculus and then
you end up immediately with that equation.
And then he said well, actually there's a problem here if I apply Ito's formula, you
know, if I don't derive it directly, that I haven't proved that this function that is
defined by being a probability of a certain event is actually C2. And then he says
well, there should be a reference, so that's in this version one of his preprint,
implying this, but we have not located one. Okay? So how does he do it? Well,
to deal with this, here's the way -- so I'm sorry I didn't find the appropriate
reference but here is why it is C2, right? And so he explains the thing that well
you don't need to know that it's C2 because you just once you know the function
and you put it in, you know the solution to the equation is C2, you plug it in, then
you verify that indeed by using the stopping -- and that's actually the way -standard way is proof.
So again, sort of he opportunity know where to find this but he said well, that's
the way to do it, actually, better way. And actually I very much remember with
when with Greg at some point, well, actually that's just a funny -- funny thing. We
just discussed on the parking lot of Homestead Studio's with Greg this morning
and we were sort of saying well, actually sort of when one tries to discuss you
know or our joint papers as you know the things where you have to prove
technically things, somehow it was not so interesting to look at those because
you know, the proofs were improved later on, and simplification were found and
so indeed those technical aspects. But actually in this paper basically nothing
that you -- there's nothing that you want to in this first paper everything is still like
it should be. There's no real update except of fours other things were proved
later on, but you don't want to really replace the arguments that he produced.
And in one of our technical papers later there was a tricky question involving
precisely Ito's formula but you know differentiating whether you could apply these
things uniformly when you had a family of Ito things and differentiating with, you
know -- and sort of, you know, I was scratching my head and saying well I don't
know any reference for that, is this true or whatever? And Oded said you know,
of course it's true. You know. Of course this is going to be true because
uniformly you can differentiate okay and then I was supposed to be the one, you
know, who grew up in stochastic calculus and was supposed to be able to swim
or you know speak the language fluently and then okay then okay we managed
to say find a reference actually that saved us of having to reapprove it. But so he
really was -- had this personal understanding which he understood it. And then
just to finish maybe I'm actually too fast in my lecture, but again sort of when he
says it's a bit tricky but can be done, it's very interesting because in one e-mail
exchange that he had with Steffen and that Steffen forwarded me, he shows
basically how he thinks about the -- it shows -- it's not that -- you know, the
content of what I'm going to show you is important, it's more the way sort of he
understands it and the style with which both the style mathematical and
gentlemen type of a style with which he explains it in e-mail, and I'm sure all of
you who interacted with him had this sort of e-mail exchange where he gently,
you know, explains to you on the e-mail an argument. And so basically they
were discussing the relation between the generator and you know, the differential
equations that pop out and the stochastic differential equation that you had
before, which is hard to do with this, which is then, you know, turned out to be
these equations I wrote later.
And Steffen says, well, is it easy to see? All right? So you say -- and here's the
answer of Oded. So you'd probably look up some book on the relations between
PDE, SDE and operator semigroup because my background in this stuff is not as
strong as I would like.
So that's something very funny because I always felt that he had this -- he would
have liked to be really, you know -- and he always felt these guys, you know,
doing stochastic analysis and probability theory maybe they know something I
don't or, you know, they -- so he would have liked to have a strong background
there. But I'll give you my understanding for the little that it's worth. And then he
gives two ways to look at this. So -- and that's the way so that he was looking at
this type of objects in the time where he was -- that's not the -- you know, it's just
at the same moment where he was, you know, working out all these things by
himself. So of course, later on he would have explained it in a completely
different way. But it more shows how, you know, he -- the way he created these
things.
So you don't have to read this, but then he says here's one way. And so he
explains this, and he says you know, you have this density, total mass is
conserved, you differentiate here. And assume it's smooth. You can
differentiate. You substitute. And you have your hand waving argument and
that's what goes out of this. In a way, I would say this is more in the stochastic
calculus version. And then so much for this approach. Now another one. And
then basically he completely says well, anyway, this problem is a linear problem
so we can, you know, you know, divide it, you know, sort of step by step. So let's
first look at what happens if, you know, there's a uniform density to start with.
Let's first, you know, look at what happens if I have a drift to the left on the one
side of the domain -- on R plus and then a drift to the left on R minus. Well, then
the single [inaudible] and this will have to do with the derivative. And so he
basically explains all the Ito calculus just by, you know, it's linear and all the
terms are coming out just by looking at them one by one. And that's what it is.
So that's the reason for that term for instance.
And I like very much the third one, also, which is actually I have the third way -that's everything that's his answer, right? So sometimes you got answer which is
very -- which are -- to think of it through approximating the motion you know,
which diffusion of a random walk on the grid which was not evenly spaced, but I
don't think it would be helpful for you if I tried to recall that. So of course for
those of you who know -- so he had all this picture. And everything, it's there that
he doesn't -- he works with his entire -- you know, his own object. So simple
random walks, I understand, I do this. So he always had some feeling for the
object that he plays with.
And okay. Then of course, you know, gentlemen touch, sort of I hope this was at
least a bit helpful. The type of thing when he explains you something, obviously
it took him I don't know, 40 minutes to tied this whole thing trying to explain you
gently, and he did this, you know could you currently in -- for to probably all of
you who interacted with him. And always, you know, just saying this is the way
trying to explain, but maybe I didn't succeed. Sorry if I didn't. And actually -okay.
And so I hope -- so that's -- there's really one thing important thing that I
understood in a way I felt it when I interacted with him and then when basically I
was understood that even Loewner's equation he basically recrafted on his own
and now of course I mean -- okay. I should not say that. I mean, you know he -the fact he used Loewner's equation in the context of DLA and other context was
you know in the air there was Carlson and Makarov and so but basically the way
he wanted to under it was just by free understanding or reproving everything
himself. And that this was not just, you know, I saw it on my -- with my eyes that
this is the way he proceeded in probability theory and stochastic calculus but
looks like, you know, that was like the same -- was the same way in actually all
areas that he touched or many areas that he touched.
And again so I want to emphasize really that it's just a pleasure to read his paper
and this type of sentences that I showed you is quite interesting or I believe that
this is worthwhile or something. You always smile when you read these things.
When you know Oded and when you knew Oded and you read his papers and
you smile. And actually sometimes you laugh. So the -- I told you, you know, I
said for instance to my family I'm going to Oded's memorial conference, and I
have to prepare actually my talk so -- and so I was just sitting in the sofa of my
living room reading his papers and the very same sofa that I was -- you know
where last year devastated when I learned that Oded fell from the -- in the
mountains and I was just reading this and I was just laughing, you know. I was
just smiling and laughing. And then my daughters came along and said well,
what is it that is so funny? They say we want you tell us, you know. I said
actually I'm reading just a paper of Oded and it's just nice. And they were looking
at me like, you know, something is wrong. You know, he tells us he's going to a
memorial conference, this looks okay, sad moment and he's just laughing or
smiling by reading his papers. But really, I mean, it's just -- I just recommend to
all of you sort of to when you fly home or that you just print out the Israel Journal
of Math paper or rather the version one of the preprint on the archive. And or
actually Oded's sort of ICM paper is also a very nice thing to read because he
just gives plenty of problem to the community. Some of them have been solved
by now because that was 2006 and now it's 2009. But so if he, you know, just
wants to give problems and also explaining things in a gentle and simple and
transparent way, not hiding anything, you know, there's not -- there will never be
something that, you know, you have the impression that he didn't want to share.
And also really if you want to feel a little bit like Oded probably felt every day
when he proved or found the tricks because it was -- that's one other thing when
you interacted within that Steffen writes which is that that you can just see how
fast, you know, he found the tricks or went around. You know, there was let's try
this way, doesn't work. Okay. And the rest of us would just feel, okay, this proof
breaks down completely, we don't know how do we -- how -- we're stuck and you
know two hours later actually here's another way, you know, or maybe we could,
you know, go around the mountain and climb the other.
And so the speed with which he would, you know, find a trick and solve the
concrete problems that we were working on. And I -- you know, if you take one
of these Sokoban things, sort of that we have here, and if you move a little, play
with it around, and then you find the trick and say oh, you have first to push the
boxes in that way to -- and that's -- and then the moment when you find this then
I guess you feel a little bit like Oded and probably felt, you know, you mile, you're
happy and that's it, you don't make a fuss out of it just because you are in front of
your, you know -- and you are happy to actually also maybe then share it with
somebody who had went through the same thing and I guess that's -- okay. My
recommendation you download it and you play it on the plane back and -actually it's interesting because Sokoban in a way, if you start playing with this,
it's not completely unrelated to the entire Loewner story and so on because, you
know, once the boxes or the other [inaudible] problem or something, because
you know once the boxes on the boundaries somehow you can't push it out of
the boundary anymore and, you know, something -- it's not completely unrelated.
And maybe the time you know Oded spent playing like this, Sokoban was very
useful for all of us in the way he crafted in SLE. Okay. And thank you very
much.
[applause].
>>: I'm not sure that if there are questions or comments maybe comments or if
you -- or if Rick or David or Steffen or Itai want to say something about ->>: I want to say something. So about your first comment. So in '97, a bunch of
us were going to conference in Italy, and this is certainly after we work with
Oded, Itai and Russ Lyons on uniform spanning trees, and then we -- Oded
consult me, he said well I want to talk about something a bit speculative. And
this is '97. And he presented a large part, not everything, put a large part of what
was eventually his Israel journal paper. This is 1997. But then he [inaudible]
describe it to me, and I said this sounds like total science fiction. [laughter]. Talk
about something more secure. And he didn't accept that advice then put that's -[laughter].
And the picture and the other thing that released to the stochastic analysis story
is that some point, this is between '97 and '99, I get you know from Oded, he was
asking, you know, actually pretty basic things about Brownian, about Brownian
motion and its characterizations, and only we had characterizations of them then
Oded just -- you know, if he didn't know this, this is [inaudible] recovered
[inaudible] no. Because they are arrive exactly in the story when Wendelin was
saying this and then he asked what's the reference for this property of [inaudible]
so it's not that he combined these things, he realized okay, we need
characterizations of Brownian order in order to prove [inaudible].
>>: Any other comments or questions?
>> Wendelin Werner: Maybe I'll -- just one other comment. Somehow -- okay.
My first experience with Oded was sort of when we had this hike with Rick in
fountain blue, and maybe that was one of my first experience of mathematician
that would actually love to do mathematics on a hike, right? So that they were -the two of them were just -- it was a bit frustrating because I -- you know, I didn't
understand what was going on. And so they were discussing math during the
hike in a very -- and I remember sort of also that probably Greg also remembers
that somehow the proof of loop-erased random walk convergence in the end was
just took place at this MSRI conference during the walk when we decided let's
walk down you know from MSRI down to Berkeley, and when we started on the
top, we started discussing this and down towards okay looks like it's doable, we
knew nothing is preventing us from anything. So he loved, and I guess he had to
talk really mathematics also, you know, while walking, or by, you know, being
sitting you know really like in his chair or -- well, in his sort of low sofa, I guess, at
Microsoft you have plenty of those. We were just you know like hands in his
pockets and then said well, actually, I don't think that -- and then providing you
the argument by sorry for my handwriting is not good but I will try to sketch it on
the blackboard and then yeah. So things went always pretty quickly. It was not
like he was -- the ideas were not -- you know, took hours for them to show up
with like interaction and discussion which was also ignition for his creativity.
>>: Let's thank Wendelin.
[applause]