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LAWRENCE GUTH: Hey, everyone!

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Today and this
week, we are going

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to talk about the connection
between homogeneous dynamics

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and projection theory.

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Today is, by the way, the first
day of the Simons lectures.

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Maryna Viazovska is
giving them this year.

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And she's going to be
talking about sphere packing.

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And I got interested
in this two years ago

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when Elon Lindenstrauss came
to give the Simons lectures.

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And he was talking about
homogeneous dynamics.

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And he explained a little
bit this connection

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with projection theory.

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And I have been trying to
understand it since then.

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And it's a pretty
geometric visual thing.

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So I brought some tools with me.

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The goal of the class is to
try to understand, visually,

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geometrically, how
homogeneous dynamics is

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related to projection theory.

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So let's first say a little bit
what is homogeneous dynamics.

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So we're going to be
working with spaces

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that have a lot of symmetries.

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And so the spaces will
come from a Lie group.

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So G will be a Lie
group, for example,

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S ln(R) And inside of G,
there'll be a discrete subgroup.

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So e.g.

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S ln(Z) And then we're going to
be studying the quotient space,

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G mod gamma.

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So this is called a
homogeneous space.

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So we'll see it has
a lot of symmetries.

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And so the homogeneous
and homogeneous dynamics

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is reacting on this
homogeneous space.

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Now, what does dynamics mean?

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Suppose that H is
a subgroup of G.

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And then if x is a
point in G mod gamma,

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then we can look at H dot x.

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So H acts on G mod gamma.

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So we can talk about H dot x.

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That's called an orbit.

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And the question is to
describe the orbits.

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So let's make a picture.

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So in actual Lie group G--

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group G looks like this--

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there might be
some group element.

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And then you could
take a coset H dot g,

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and h might be non-compact.

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So that might go on forever.

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But somehow, it's
not that complicated.

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Yeah.

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AUDIENCE: If we're taking the
quotient of G mod gamma, what

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are the conditions for gamma
to be a normal subgroup

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to allow this question?

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LAWRENCE GUTH: So this
object is not a group.

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It's just a space.

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AUDIENCE: A space, OK.

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LAWRENCE GUTH: So gamma is a--

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so the question was about
whether gamma is normal.

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And the answer is gamma
doesn't have to be normal.

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So this is not a normal
subgroup of this.

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That means that this is not a
group, but it's still a space.

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So G mod gamma--

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oh, and I'll put something else.

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So G mod gamma, we'll
assume that the volume--

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we'll be interested in cases
where the volume of G mod gamma

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is finite.

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So the volume of G is infinite,
but the volume of G mod gamma

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is finite.

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So G mod gamma looks
sort of like this.

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And so then a point x in there,
and the orbit goes through x,

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but then it wraps around.

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The picture looks a lot
more complicated already.

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And we want to understand
what does this do.

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Is it going to
wrap back on itself

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and just go around over and
over again, the same thing?

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Or is it going to
evenly distribute

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all over G mod gamma
or something else?

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So to make it concrete, I'd
like to begin with a very simple

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warm up, which is with
a commutative group.

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So it's simpler than that.

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So I think the simplest example
of this is that G could be R2,

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and then gamma could be Z2.

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So G mod gamma is R2 mod Z2.

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It's a torus.

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And H is going to
be a subgroup of G.

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And let's say it's like
a one-parameter subgroup.

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We'll do a couple of examples.

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H could be the subgroup t, 2t.

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t in R. There's
a subgroup of R2.

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And if we do this, what happens?

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So I'll visualize G mod gamma as
a square with sides identified.

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And if I were to take--

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so let's say x0 is
just gamma, which

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is an element of G mod gamma.

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It's like you take the
identity in G and push it down.

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So what is the orbit
through x0 look like?

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Well, it's this line of
slope 2 looks like that.

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And then it comes
back where it started.

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And it just keeps
going like that.

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And actually, it wouldn't
be-- it's a little harder

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to draw accurately, but it
wouldn't be different if you

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started at any other point.

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So x, if you start
at any point x,

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then if I were to
add 1 comma 2, that

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would be equal to
x in R2 mod Z2.

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And so, actually, every
orbit is periodic.

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So the conclusion
in example one is

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that H dot x is
periodic for every x.

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Example two, I'll take a
different one-parameter

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subgroup, t comma square root of
2t t in R. So now what happens?

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Well, if I, again, start
at x0, it's the identity.

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And now I'm going up at slope
square root of 2 at the top.

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And keep going at
slope square root of 2.

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And this time, this
will never close up

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because the slope is irrational.

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And in fact, this orbit will be
dense and evenly distributed.

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So now what happens is that
H dot x is dense for every x.

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I won't do a proof right
now, but I'll make a remark.

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So there are several
proofs, but there's

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a nice proof using Fourier
analysis that we might talk

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about later if we feel like it.

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Questions or comments so far?

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So we've seen two
things that can happen.

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The orbit could be periodic.

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And the orbit could be dense.

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And we haven't yet
seen other things,

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but we might wonder about
whether other things could

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happen.

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And now we can shift to a more
complicated noncommutative

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group, like SL2(R).

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So now let's say G is
SL2(R) and gamma is SL2(Z).

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There are multiple reasons for
looking at these quotients, G

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mod gamma.

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But one of them that comes up
for this particular example

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is that these
parametrize lattices.

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So G mod gamma is
the space of lattices

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in R2 with the property lattices
lambda in R2 with the property

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that the area of
R2 mod lambda is 1.

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So let's do a proof sketch.

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Any such lattice, we
can write as g times

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the lattice Z squared,
where G is in SL2(R).

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Make any lattice by
starting with Z squared,

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and applying a linear
transformation.

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However, there is
more than one G

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that gives you the same lambda.

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So if H is in SL2(Z),
then hZ2 is Z2.

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And therefore, G times
hZ2 is g times Z2.

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And these guys give
you the same lattice.

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So the space of lattices
is g modulo SL2(Z).

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Now, this space of lattices,
G mod gamma, is not compact.

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And I will give you an example
of a sequence of lattices

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that don't converge
to anything--

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don't have a subsequence
that converge to anything.

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So for example, we could
take lambda epsilon

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to be epsilon Z direct
sum epsilon inverse.

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So this is a lattice, which is
very skinny in one direction

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and very tall in
the other direction.

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It has covolume one.

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And as epsilon goes
to 0, this is not

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going to converge to anything.

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So the space of
lattices is not compact.

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And this is the only reason
that the space of lattices

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is not compact.

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So a definition, K epsilon
is a subset of G mod gamma.

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K epsilon is defined to
be the set of lattices

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so that the minimum over nonzero
v in our lattice of the size

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of v is at least epsilon--

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is greater than or equal
to, let's say, epsilon.

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And there's a proposition,
which is called the Mahler

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criterion, that says for
every epsilon bigger than 0,

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K epsilon is compact.

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So a picture of G mod gamma
is something like this.

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G mod gamma is
three-dimensional,

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so it's not super easy to draw.

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But over here, so this
would be G mod gamma.

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There's this noncompact part.

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And we'll make a
cut off maybe here.

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And this stuff is K epsilon.

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So I won't prove this in detail.

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This is not super difficult. If
you think about two vectors that

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generate lambda, if
we're in K epsilon,

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the vectors definitely
cannot be too close to 0.

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And you can also
always find two vectors

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that are not too giant, because
the area of the thing is one.

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And this gives you a compact set
of choices of these two vectors.

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So that gives a little bit of
intuition about what the space G

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mod gamma looks like.

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Now, inside of there, we have
some interesting one-parameter

00:12:55.780 --> 00:13:01.690 align:middle line:90%
subgroups of G of SL2(R).

00:13:01.690 --> 00:13:04.980 align:middle line:90%


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So one of them is the
unipotent subgroup.

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So that's the set of
1, t, 0, 1, t in R.

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And the other one is
the diagonal subgroup.

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So I'll call this--

00:13:16.580 --> 00:13:18.040 align:middle line:90%
so it's the set of UT.

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So this matrix is U sub t.

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And there's the diagonal
subgroup, e to the r,

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e to the minus r, 0, 0,
little r in the reals.

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And this I'll call
the set of a sub r.

00:13:31.900 --> 00:13:34.380 align:middle line:84%
So this matrix here
is called a sub r.

00:13:34.380 --> 00:13:36.033 align:middle line:84%
So those are
one-parameter subgroups.

00:13:36.033 --> 00:13:38.700 align:middle line:84%
And now we can ask our question
about what the orbits look like.

00:13:38.700 --> 00:13:47.826 align:middle line:90%


00:13:47.826 --> 00:13:52.880 align:middle line:84%
So there's a theorem that
goes back to the 1930s,

00:13:52.880 --> 00:13:56.600 align:middle line:90%
Hedlund in the 1930s.

00:13:56.600 --> 00:14:07.400 align:middle line:84%
He said that a U orbit is
either periodic or dense.

00:14:07.400 --> 00:14:10.800 align:middle line:90%
But in contrast, there's a--

00:14:10.800 --> 00:14:13.120 align:middle line:84%
I guess I'll also
call it a theorem--

00:14:13.120 --> 00:14:16.840 align:middle line:84%
I don't know who
proved this one--

00:14:16.840 --> 00:14:22.400 align:middle line:84%
that D orbit does
not have to be maybe

00:14:22.400 --> 00:14:26.330 align:middle line:90%
neither periodic nor dense.

00:14:26.330 --> 00:14:29.902 align:middle line:90%


00:14:29.902 --> 00:14:31.360 align:middle line:84%
Let me say something
more than that

00:14:31.360 --> 00:14:35.030 align:middle line:84%
about how these D orbits can
be really quite complicated.

00:14:35.030 --> 00:14:37.150 align:middle line:90%
So here's G mod gamma.

00:14:37.150 --> 00:14:39.470 align:middle line:84%
And inside of G
mod gamma, pick out

00:14:39.470 --> 00:14:44.110 align:middle line:84%
three small balls that are
not too close together.

00:14:44.110 --> 00:14:47.880 align:middle line:90%
So B1, B2, B3 are small balls.

00:14:47.880 --> 00:14:51.630 align:middle line:90%


00:14:51.630 --> 00:14:58.270 align:middle line:84%
Then for any sequence
of 1's and 2's, so

00:14:58.270 --> 00:15:07.590 align:middle line:84%
for example, the sequence
1121212221, any sequence,

00:15:07.590 --> 00:15:18.430 align:middle line:84%
there exists an x in G mod gamma
so that the D orbit of x never

00:15:18.430 --> 00:15:20.910 align:middle line:90%
hits B3.

00:15:20.910 --> 00:15:29.180 align:middle line:84%
And the D orbit of x hits B1 and
B2 many times in this sequence.

00:15:29.180 --> 00:15:34.170 align:middle line:90%


00:15:34.170 --> 00:15:35.950 align:middle line:84%
So what must this
orbit look like?

00:15:35.950 --> 00:15:40.490 align:middle line:90%


00:15:40.490 --> 00:15:41.070 align:middle line:90%
Let's see.

00:15:41.070 --> 00:15:43.770 align:middle line:90%
So first, it hits B1.

00:15:43.770 --> 00:15:45.410 align:middle line:90%
It goes through there.

00:15:45.410 --> 00:15:50.650 align:middle line:84%
Then it goes around, and the
next thing it hits is also B1.

00:15:50.650 --> 00:15:52.330 align:middle line:90%
And it goes around.

00:15:52.330 --> 00:15:55.010 align:middle line:84%
And the next thing
it hits is B2.

00:15:55.010 --> 00:15:56.030 align:middle line:90%
And it keeps going.

00:15:56.030 --> 00:15:59.970 align:middle line:84%
And the next thing it
hits is B1, and so on.

00:15:59.970 --> 00:16:01.180 align:middle line:90%
And it never touches B3.

00:16:01.180 --> 00:16:04.850 align:middle line:90%


00:16:04.850 --> 00:16:08.290 align:middle line:84%
So this orbit,
apparently, is not dense,

00:16:08.290 --> 00:16:10.650 align:middle line:90%
because it never intersects B3.

00:16:10.650 --> 00:16:14.850 align:middle line:84%
And it is not periodic if
this sequence is not periodic.

00:16:14.850 --> 00:16:15.990 align:middle line:90%
So it's something else.

00:16:15.990 --> 00:16:20.370 align:middle line:90%


00:16:20.370 --> 00:16:26.970 align:middle line:84%
So D orbits can do this somewhat
crazy thing, somewhat chaotic

00:16:26.970 --> 00:16:29.650 align:middle line:90%
thing.

00:16:29.650 --> 00:16:31.750 align:middle line:90%
But U orbits can't.

00:16:31.750 --> 00:16:34.750 align:middle line:90%
U orbits can be periodic.

00:16:34.750 --> 00:16:43.260 align:middle line:84%
So if you were to take U
of x0, this is periodic.

00:16:43.260 --> 00:16:45.990 align:middle line:90%


00:16:45.990 --> 00:16:47.950 align:middle line:90%
And the reason is--

00:16:47.950 --> 00:16:49.870 align:middle line:90%
so if you take--

00:16:49.870 --> 00:16:59.030 align:middle line:84%
so x0 corresponds to the
matrix 1, 1 times SL2(Z).

00:16:59.030 --> 00:17:03.070 align:middle line:84%
So now if I were to
take 1, 1, t, x0,

00:17:03.070 --> 00:17:09.420 align:middle line:84%
that's equal to x0 in G mod
gamma if t is an integer.

00:17:09.420 --> 00:17:11.990 align:middle line:90%


00:17:11.990 --> 00:17:13.410 align:middle line:90%
So that's a periodic orbit.

00:17:13.410 --> 00:17:18.670 align:middle line:84%
It makes a closed
circle in G mod gamma.

00:17:18.670 --> 00:17:24.829 align:middle line:84%
But if you were to start with
a kind of weird-looking choice

00:17:24.829 --> 00:17:27.550 align:middle line:84%
of x0, and you were
to do this, then

00:17:27.550 --> 00:17:35.510 align:middle line:84%
there won't be any obvious
periodicity, so for most x.

00:17:35.510 --> 00:17:43.830 align:middle line:84%
So let's suppose now x is
root 2, 0, 0, 1 over root 2.

00:17:43.830 --> 00:17:46.290 align:middle line:90%
So now what happens when we--

00:17:46.290 --> 00:17:48.930 align:middle line:90%
and then times SL2(Z).

00:17:48.930 --> 00:17:53.690 align:middle line:90%
So now what is UT of x?

00:17:53.690 --> 00:17:58.240 align:middle line:84%
That's 1, 1, t, root
2, 0, 1 over root 2.

00:17:58.240 --> 00:18:01.450 align:middle line:90%


00:18:01.450 --> 00:18:04.810 align:middle line:90%
So that will be root 2.

00:18:04.810 --> 00:18:10.190 align:middle line:90%
That will be root 2.

00:18:10.190 --> 00:18:14.530 align:middle line:90%
That will be t over root 2.

00:18:14.530 --> 00:18:16.730 align:middle line:90%
This will be 0.

00:18:16.730 --> 00:18:18.670 align:middle line:90%
This will be 1 over root 2.

00:18:18.670 --> 00:18:25.850 align:middle line:90%


00:18:25.850 --> 00:18:27.170 align:middle line:90%
Oh, that one might be periodic.

00:18:27.170 --> 00:18:28.628 align:middle line:84%
I think I picked
the wrong example.

00:18:28.628 --> 00:18:32.840 align:middle line:90%


00:18:32.840 --> 00:18:33.400 align:middle line:90%
Yeah, yeah.

00:18:33.400 --> 00:18:34.760 align:middle line:90%
OK, sorry.

00:18:34.760 --> 00:18:35.980 align:middle line:90%
That was unhelpful.

00:18:35.980 --> 00:18:45.800 align:middle line:90%


00:18:45.800 --> 00:18:47.740 align:middle line:84%
So to see why they're
not all periodic,

00:18:47.740 --> 00:18:49.920 align:middle line:84%
let's instead try
to visualize what

00:18:49.920 --> 00:18:52.920 align:middle line:90%
is happening to the lattice.

00:18:52.920 --> 00:19:03.400 align:middle line:84%
So a point x corresponds
to a lattice lambda x.

00:19:03.400 --> 00:19:08.280 align:middle line:84%
And if you think about what
is happening if you take gx,

00:19:08.280 --> 00:19:14.520 align:middle line:84%
that corresponds to the
lattice g of lambda of x.

00:19:14.520 --> 00:19:16.480 align:middle line:90%
So g is in SL2(R).

00:19:16.480 --> 00:19:17.460 align:middle line:90%
It acts on R2.

00:19:17.460 --> 00:19:19.200 align:middle line:84%
It moves the lattice
to another lattice.

00:19:19.200 --> 00:19:21.840 align:middle line:90%
And that's what the action is.

00:19:21.840 --> 00:19:27.460 align:middle line:90%
And so now g is going to be ut.

00:19:27.460 --> 00:19:32.780 align:middle line:84%
And the way ut acts
on R2 is by shearing.

00:19:32.780 --> 00:19:38.460 align:middle line:90%
So ut acts on R2 by shearing.

00:19:38.460 --> 00:19:40.600 align:middle line:90%
So I'll do it this way.

00:19:40.600 --> 00:19:45.620 align:middle line:90%


00:19:45.620 --> 00:19:49.020 align:middle line:90%
So I'll draw some points in R2.

00:19:49.020 --> 00:19:51.060 align:middle line:84%
And perhaps,
coincidentally, these points

00:19:51.060 --> 00:19:53.620 align:middle line:90%
come from a lattice.

00:19:53.620 --> 00:19:58.860 align:middle line:84%
And then I'll
illustrate what ut does.

00:19:58.860 --> 00:20:07.420 align:middle line:84%
So if this is the point p,
then this over here is ut of p.

00:20:07.420 --> 00:20:09.100 align:middle line:90%
It's a shearing motion.

00:20:09.100 --> 00:20:12.600 align:middle line:84%
And so the shearing motion
moves everybody horizontally.

00:20:12.600 --> 00:20:14.500 align:middle line:84%
And it moves you
horizontally by an amount

00:20:14.500 --> 00:20:16.220 align:middle line:90%
proportional to your height.

00:20:16.220 --> 00:20:18.440 align:middle line:84%
So this one moves
horizontally, but a bit less.

00:20:18.440 --> 00:20:22.100 align:middle line:84%
And this one moves
horizontally a bit more.

00:20:22.100 --> 00:20:24.560 align:middle line:90%
Now, what would it mean?

00:20:24.560 --> 00:20:27.440 align:middle line:90%
So you start with a lattice.

00:20:27.440 --> 00:20:30.200 align:middle line:84%
And when we apply this
unipotent orbit to the lattice,

00:20:30.200 --> 00:20:34.000 align:middle line:84%
every point in the
lattice is shearing.

00:20:34.000 --> 00:20:36.820 align:middle line:84%
What would it mean if
the orbit was periodic?

00:20:36.820 --> 00:20:39.760 align:middle line:84%
Well, it would mean that this
point, this lattice point

00:20:39.760 --> 00:20:41.600 align:middle line:84%
is shearing along
and eventually it

00:20:41.600 --> 00:20:45.200 align:middle line:90%
reaches another lattice point?

00:20:45.200 --> 00:20:54.640 align:middle line:84%
So if ux is periodic,
that implies

00:20:54.640 --> 00:21:03.080 align:middle line:84%
that each lattice point
is at the same height

00:21:03.080 --> 00:21:09.160 align:middle line:90%
as another lattice point.

00:21:09.160 --> 00:21:14.720 align:middle line:84%
So there must be another lattice
point somewhere over here.

00:21:14.720 --> 00:21:16.560 align:middle line:90%
But now, think about this.

00:21:16.560 --> 00:21:18.320 align:middle line:84%
This difference is
also in our lattice,

00:21:18.320 --> 00:21:21.260 align:middle line:84%
because the lattice
is a subgroup.

00:21:21.260 --> 00:21:23.130 align:middle line:90%
And this distance is horizontal.

00:21:23.130 --> 00:21:30.430 align:middle line:84%
So this is if and
only if there exists--

00:21:30.430 --> 00:21:36.110 align:middle line:84%
if and only if lambda of x
contains a vector of the form

00:21:36.110 --> 00:21:37.340 align:middle line:90%
x1 comma 0.

00:21:37.340 --> 00:21:41.990 align:middle line:90%


00:21:41.990 --> 00:21:45.070 align:middle line:84%
So most lattices don't contain
a vector on the x-axis.

00:21:45.070 --> 00:21:48.790 align:middle line:84%
So u times x is
usually not periodic.

00:21:48.790 --> 00:21:52.190 align:middle line:84%
And then the difficult
theorem is in that

00:21:52.190 --> 00:21:54.200 align:middle line:90%
situation u times x is dense.

00:21:54.200 --> 00:22:02.870 align:middle line:90%


00:22:02.870 --> 00:22:06.190 align:middle line:90%
So this is the situation in SL2.

00:22:06.190 --> 00:22:11.110 align:middle line:84%
The situation in other Lie
groups, say SLn(R) for larger n,

00:22:11.110 --> 00:22:13.268 align:middle line:84%
is more difficult
and more recent,

00:22:13.268 --> 00:22:14.560 align:middle line:90%
but there are similar theorems.

00:22:14.560 --> 00:22:20.070 align:middle line:90%


00:22:20.070 --> 00:22:25.010 align:middle line:84%
For SL3(R), Margulis and Dani
in the 1980s proved an analog

00:22:25.010 --> 00:22:26.730 align:middle line:90%
of this theorem.

00:22:26.730 --> 00:22:30.410 align:middle line:84%
And then for all the SLn(R)
, and all the groups, Ratner,

00:22:30.410 --> 00:22:33.150 align:middle line:84%
around 1990 proved an
analog of this theorem.

00:22:33.150 --> 00:22:43.450 align:middle line:90%


00:22:43.450 --> 00:22:47.130 align:middle line:84%
So that's a little summary
of some kinds of theorems.

00:22:47.130 --> 00:22:49.190 align:middle line:84%
There's some different
motivations for this.

00:22:49.190 --> 00:22:51.320 align:middle line:84%
There's some
motivation from-- yeah.

00:22:51.320 --> 00:22:54.690 align:middle line:84%
AUDIENCE: I have just a
question about the link

00:22:54.690 --> 00:22:59.610 align:middle line:84%
between the lattices and the
elements of the quotient group.

00:22:59.610 --> 00:23:02.150 align:middle line:84%
So up there in the
definition of K epsilon,

00:23:02.150 --> 00:23:05.030 align:middle line:84%
there's kind of what looks
like some sort of norm,

00:23:05.030 --> 00:23:09.110 align:middle line:84%
the minimum size of v, or the
nonzero vector v in the lattice.

00:23:09.110 --> 00:23:10.770 align:middle line:84%
Does that generate
the same topology

00:23:10.770 --> 00:23:14.410 align:middle line:84%
as the quotient
topology from SL2(R)?

00:23:14.410 --> 00:23:20.935 align:middle line:90%


00:23:20.935 --> 00:23:22.310 align:middle line:84%
LAWRENCE GUTH: So
the question is

00:23:22.310 --> 00:23:24.870 align:middle line:84%
about the topology
of G mod gamma

00:23:24.870 --> 00:23:27.510 align:middle line:84%
and how the topology
of G mod gamma

00:23:27.510 --> 00:23:29.880 align:middle line:84%
plays with this
definition of K epsilon.

00:23:29.880 --> 00:23:32.790 align:middle line:90%


00:23:32.790 --> 00:23:35.770 align:middle line:84%
In a moment, we'll put
a metric on G mod gamma.

00:23:35.770 --> 00:23:37.210 align:middle line:84%
And that's what
this topology is.

00:23:37.210 --> 00:23:38.170 align:middle line:90%
It's a manifold.

00:23:38.170 --> 00:23:42.470 align:middle line:90%
It has a topology manifold.

00:23:42.470 --> 00:23:47.470 align:middle line:84%
And this thing here is
a continuous function

00:23:47.470 --> 00:23:54.430 align:middle line:90%
on our space of lattices.

00:23:54.430 --> 00:23:56.510 align:middle line:84%
Another way to say
what the topology is,

00:23:56.510 --> 00:23:59.790 align:middle line:84%
if you describe a lattice as the
span of two particular vectors,

00:23:59.790 --> 00:24:02.270 align:middle line:84%
if you slightly perturb those
particular vectors, that

00:24:02.270 --> 00:24:05.880 align:middle line:84%
will be a slight change
in the space of lattices.

00:24:05.880 --> 00:24:11.950 align:middle line:90%


00:24:11.950 --> 00:24:13.830 align:middle line:84%
AUDIENCE: Is that
definition similar

00:24:13.830 --> 00:24:15.965 align:middle line:90%
in any way to have a minimum?

00:24:15.965 --> 00:24:17.090 align:middle line:90%
Yeah, I guess you're right.

00:24:17.090 --> 00:24:19.420 align:middle line:90%
It's not quite-- thanks.

00:24:19.420 --> 00:24:27.860 align:middle line:90%


00:24:27.860 --> 00:24:30.920 align:middle line:84%
LAWRENCE GUTH: So when I
introduced homogeneous dynamics,

00:24:30.920 --> 00:24:35.300 align:middle line:84%
I mentioned some application
in number theory connection

00:24:35.300 --> 00:24:38.040 align:middle line:84%
to the Oppenheim-Davenport
conjecture,

00:24:38.040 --> 00:24:40.370 align:middle line:84%
which is about the values
of quadratic forms.

00:24:40.370 --> 00:24:43.740 align:middle line:90%


00:24:43.740 --> 00:24:45.080 align:middle line:90%
That's a nice connection.

00:24:45.080 --> 00:24:46.840 align:middle line:84%
I feel like we already
talked about it a little bit.

00:24:46.840 --> 00:24:48.298 align:middle line:84%
And if you wanted
to read about it,

00:24:48.298 --> 00:24:50.540 align:middle line:90%
it's also easy to read about it.

00:24:50.540 --> 00:24:53.200 align:middle line:84%
So I wanted to spend
the class time on--

00:24:53.200 --> 00:24:56.682 align:middle line:90%


00:24:56.682 --> 00:24:58.140 align:middle line:84%
so I wanted to
spend the class time

00:24:58.140 --> 00:25:00.860 align:middle line:84%
on trying to get intuition
for why something like this

00:25:00.860 --> 00:25:03.780 align:middle line:84%
would be true, and also
maybe why something like this

00:25:03.780 --> 00:25:05.740 align:middle line:90%
would be false.

00:25:05.740 --> 00:25:08.460 align:middle line:84%
And there are many
different ways

00:25:08.460 --> 00:25:10.480 align:middle line:84%
of thinking about
homogeneous dynamics.

00:25:10.480 --> 00:25:12.780 align:middle line:84%
There's a big
literature, but I want

00:25:12.780 --> 00:25:16.160 align:middle line:84%
to highlight the
way of connecting it

00:25:16.160 --> 00:25:18.280 align:middle line:90%
to projection theory.

00:25:18.280 --> 00:25:20.260 align:middle line:84%
And that's not one of
the classical ways.

00:25:20.260 --> 00:25:24.826 align:middle line:84%
It's actually not super easy
to find in the literature.

00:25:24.826 --> 00:25:32.360 align:middle line:90%


00:25:32.360 --> 00:25:34.040 align:middle line:84%
So it appears in
these recent works

00:25:34.040 --> 00:25:36.940 align:middle line:84%
of Lindenstrauss and Mohammadi
and their collaborators,

00:25:36.940 --> 00:25:38.920 align:middle line:84%
and they're trying--
they're doing the cutting

00:25:38.920 --> 00:25:40.957 align:middle line:90%
edge of homogeneous dynamics.

00:25:40.957 --> 00:25:43.040 align:middle line:84%
And the particular problem
they're trying to solve

00:25:43.040 --> 00:25:44.540 align:middle line:84%
is to do, first of
all, these higher

00:25:44.540 --> 00:25:48.000 align:middle line:84%
dimensional things that Margulis
and Ratner and Dani were doing.

00:25:48.000 --> 00:25:49.960 align:middle line:84%
And second of all,
instead of having

00:25:49.960 --> 00:25:53.760 align:middle line:84%
a qualitative theorem like this,
to have a quantitative theorem

00:25:53.760 --> 00:25:55.680 align:middle line:84%
that says, if you're
not approximately

00:25:55.680 --> 00:26:03.120 align:middle line:84%
periodic for some length, then
you're quantitatively dense.

00:26:03.120 --> 00:26:05.920 align:middle line:84%
And at that point, they
ran into some things

00:26:05.920 --> 00:26:09.400 align:middle line:84%
that many interesting
classical techniques could not

00:26:09.400 --> 00:26:11.432 align:middle line:90%
do, or could not obviously do.

00:26:11.432 --> 00:26:13.640 align:middle line:84%
And they invented this new
technique, this connection

00:26:13.640 --> 00:26:15.540 align:middle line:90%
to projection theory.

00:26:15.540 --> 00:26:17.620 align:middle line:84%
But in hindsight, you
could also look back

00:26:17.620 --> 00:26:21.380 align:middle line:84%
at the oldest theorems, the
oldest theorems like this one,

00:26:21.380 --> 00:26:24.900 align:middle line:84%
and you could approach it
using projection theory.

00:26:24.900 --> 00:26:27.272 align:middle line:84%
And it may not be the best
approach to this theorem.

00:26:27.272 --> 00:26:29.480 align:middle line:84%
It's certainly not the only
approach to this theorem.

00:26:29.480 --> 00:26:33.580 align:middle line:84%
But it's a way of looking
at it and quite a visual way

00:26:33.580 --> 00:26:35.763 align:middle line:90%
of looking at what's going on.

00:26:35.763 --> 00:26:37.180 align:middle line:84%
So the goal of the
class is to try

00:26:37.180 --> 00:26:45.180 align:middle line:84%
to explain how projection
theory could help to understand

00:26:45.180 --> 00:26:46.080 align:middle line:90%
something like this.

00:26:46.080 --> 00:26:51.540 align:middle line:90%


00:26:51.540 --> 00:26:53.040 align:middle line:84%
So we're going to
do some geometry.

00:26:53.040 --> 00:26:56.260 align:middle line:84%
So next, we need to think about
the geometry on the space G mod

00:26:56.260 --> 00:27:03.530 align:middle line:84%
gamma and how that geometry
interacts with the action.

00:27:03.530 --> 00:27:31.200 align:middle line:90%


00:27:31.200 --> 00:27:44.520 align:middle line:84%
So geometry of G mod
gamma, so first, we'll

00:27:44.520 --> 00:27:50.920 align:middle line:84%
talk about the right
action of G on itself.

00:27:50.920 --> 00:27:54.800 align:middle line:84%
So if G is in G, there's
a right action, which

00:27:54.800 --> 00:27:56.880 align:middle line:90%
is a map from G to itself.

00:27:56.880 --> 00:27:59.980 align:middle line:84%
And the right
action G of h is h.

00:27:59.980 --> 00:28:02.410 align:middle line:84%
And you multiply by
G on the right side.

00:28:02.410 --> 00:28:03.660 align:middle line:90%
So these groups don't commute.

00:28:03.660 --> 00:28:06.420 align:middle line:84%
So it matters which
side you put it on.

00:28:06.420 --> 00:28:10.750 align:middle line:84%
And actually, you have to be
quite careful in this story.

00:28:10.750 --> 00:28:13.260 align:middle line:84%
So that's a map
from G to itself.

00:28:13.260 --> 00:28:17.110 align:middle line:90%


00:28:17.110 --> 00:28:19.350 align:middle line:84%
Now, we're going to
use that to start

00:28:19.350 --> 00:28:21.670 align:middle line:84%
to put a geometry on
G. We're going to have

00:28:21.670 --> 00:28:29.130 align:middle line:90%
a right invariant metric m.

00:28:29.130 --> 00:28:32.310 align:middle line:90%


00:28:32.310 --> 00:28:36.990 align:middle line:90%
So how does this work?

00:28:36.990 --> 00:28:39.090 align:middle line:90%
So what does that mean?

00:28:39.090 --> 00:28:45.990 align:middle line:84%
So m is going to be a
Riemannian metric on G.

00:28:45.990 --> 00:28:50.270 align:middle line:84%
And for every g in G, we
have this right action G. So

00:28:50.270 --> 00:28:55.910 align:middle line:84%
that's a map from our group
with its metric to itself.

00:28:55.910 --> 00:28:59.990 align:middle line:84%
And this is-- for every g
in G, this is an isometry.

00:28:59.990 --> 00:29:03.350 align:middle line:84%
That's what it means to have
a right invariant metric.

00:29:03.350 --> 00:29:05.110 align:middle line:90%
Where do they come from?

00:29:05.110 --> 00:29:08.350 align:middle line:84%
The idea is that we'll put
a metric at the origin,

00:29:08.350 --> 00:29:12.170 align:middle line:84%
and then we'll use R sub g
to help define the metric

00:29:12.170 --> 00:29:13.270 align:middle line:90%
at every other place.

00:29:13.270 --> 00:29:21.970 align:middle line:90%


00:29:21.970 --> 00:29:25.610 align:middle line:84%
So a lot of what we'll say
here works for any group.

00:29:25.610 --> 00:29:29.050 align:middle line:84%
But to be concrete, I'm going
to stick to the group SL2(R).

00:29:29.050 --> 00:29:31.670 align:middle line:90%


00:29:31.670 --> 00:29:36.883 align:middle line:84%
The tangent space at the
origin, so e is the identity.

00:29:36.883 --> 00:29:38.050 align:middle line:90%
I was using the word origin.

00:29:38.050 --> 00:29:40.670 align:middle line:84%
I should say identity. e is
the identity in the group.

00:29:40.670 --> 00:29:43.610 align:middle line:84%
The tangent space in the
identity of this group

00:29:43.610 --> 00:29:48.830 align:middle line:84%
is the space of matrices A, B,
C, D with a plus d equal to 0.

00:29:48.830 --> 00:29:52.490 align:middle line:90%


00:29:52.490 --> 00:29:53.930 align:middle line:90%
So this is not a deep factor.

00:29:53.930 --> 00:30:00.570 align:middle line:84%
So SL2(R) is matrices A, B, C,
D, where ad minus bc equals 1.

00:30:00.570 --> 00:30:04.850 align:middle line:84%
And the identity is a matrix
that's sitting in there in R4.

00:30:04.850 --> 00:30:08.010 align:middle line:84%
So the group is a
3-dimensional manifold in R4.

00:30:08.010 --> 00:30:11.830 align:middle line:84%
And this is just its tangent
space in the usual sense.

00:30:11.830 --> 00:30:13.230 align:middle line:90%
So that's a tangent space.

00:30:13.230 --> 00:30:22.630 align:middle line:84%
So now, m0 is going to
be a metric on T e G.

00:30:22.630 --> 00:30:25.710 align:middle line:84%
And I can specify
it by specifying who

00:30:25.710 --> 00:30:27.390 align:middle line:90%
on an orthonormal basis.

00:30:27.390 --> 00:30:30.310 align:middle line:84%
And this choice doesn't
matter super a lot,

00:30:30.310 --> 00:30:32.830 align:middle line:90%
but there's a convenient choice.

00:30:32.830 --> 00:30:41.790 align:middle line:84%
So my orthonormal basis of
m0 is going to be 0, 1, 0, 0,

00:30:41.790 --> 00:30:43.990 align:middle line:90%
which I'll call u.

00:30:43.990 --> 00:30:47.430 align:middle line:84%
0, 0, 1, 0, which
I'll call u tilde.

00:30:47.430 --> 00:30:52.630 align:middle line:84%
And then 1 over root 2,
0, 0, minus 1 over 2,

00:30:52.630 --> 00:30:53.730 align:middle line:90%
which I'll call d.

00:30:53.730 --> 00:30:56.670 align:middle line:90%


00:30:56.670 --> 00:30:58.730 align:middle line:84%
These root 2's are not
actually important.

00:30:58.730 --> 00:31:00.590 align:middle line:84%
But this orthonormal
basis happens

00:31:00.590 --> 00:31:03.470 align:middle line:84%
to be the orthonormal basis
that corresponds to the metric

00:31:03.470 --> 00:31:05.150 align:middle line:84%
where you just take
the metric on R4

00:31:05.150 --> 00:31:10.300 align:middle line:84%
and restrict it
to this subspace.

00:31:10.300 --> 00:31:16.540 align:middle line:84%
So now how will we define
the metric elsewhere?

00:31:16.540 --> 00:31:21.140 align:middle line:84%
So now suppose g
is in our group.

00:31:21.140 --> 00:31:26.140 align:middle line:90%
R sub g of the identity is g.

00:31:26.140 --> 00:31:30.380 align:middle line:84%
It's easy to check from
the definition of R sub g.

00:31:30.380 --> 00:31:36.060 align:middle line:84%
And therefore, R sub g defines
a map from tangent space e of G

00:31:36.060 --> 00:31:41.460 align:middle line:84%
to the tangent space g of G.
And this map is an isomorphism.

00:31:41.460 --> 00:31:43.660 align:middle line:84%
If we have a metric
over here, we

00:31:43.660 --> 00:31:48.460 align:middle line:84%
use this map to define
a metric over there.

00:31:48.460 --> 00:31:56.900 align:middle line:84%
So now we define mg, which
is a metric or inner product,

00:31:56.900 --> 00:32:07.560 align:middle line:84%
mg on TgG so that Rg
goes from TeG comma m0

00:32:07.560 --> 00:32:12.700 align:middle line:84%
to TgG comma mg so that
that's an isometry.

00:32:12.700 --> 00:32:16.760 align:middle line:90%


00:32:16.760 --> 00:32:21.800 align:middle line:84%
And then it's an exercise to
check that this thing we just

00:32:21.800 --> 00:32:26.480 align:middle line:90%
defined is right invariant.

00:32:26.480 --> 00:32:29.909 align:middle line:84%
So that all of these maps
are isometries everywhere.

00:32:29.909 --> 00:32:53.240 align:middle line:90%


00:32:53.240 --> 00:33:08.320 align:middle line:84%
So now, because m is right
invariant, we get a metric.

00:33:08.320 --> 00:33:11.830 align:middle line:84%
It extends to a metric
m on G mod gamma.

00:33:11.830 --> 00:33:16.980 align:middle line:90%


00:33:16.980 --> 00:33:19.540 align:middle line:90%
So how does that work?

00:33:19.540 --> 00:33:23.580 align:middle line:84%
So we have all of G. We
have a projection map

00:33:23.580 --> 00:33:26.260 align:middle line:90%
pi to G mod gamma.

00:33:26.260 --> 00:33:28.580 align:middle line:84%
We have a point
x in G mod gamma.

00:33:28.580 --> 00:33:31.460 align:middle line:90%
And x has many preimages.

00:33:31.460 --> 00:33:36.260 align:middle line:84%
So x is equal to, say,
little g times little g--

00:33:36.260 --> 00:33:38.010 align:middle line:90%
little h times gamma.

00:33:38.010 --> 00:33:53.780 align:middle line:90%


00:33:53.780 --> 00:33:56.980 align:middle line:84%
So the preimage of x is
these many different points.

00:33:56.980 --> 00:33:58.680 align:middle line:90%
And I have a metric upstairs.

00:33:58.680 --> 00:34:01.290 align:middle line:84%
And I'd like to try to
define a metric downstairs

00:34:01.290 --> 00:34:02.610 align:middle line:90%
in a natural way.

00:34:02.610 --> 00:34:05.690 align:middle line:84%
So I like to define
a metric here.

00:34:05.690 --> 00:34:07.850 align:middle line:84%
So for each of these
lifts upstairs,

00:34:07.850 --> 00:34:09.270 align:middle line:90%
I can take the metric there.

00:34:09.270 --> 00:34:10.770 align:middle line:90%
And I can use the map pi.

00:34:10.770 --> 00:34:13.889 align:middle line:84%
And I can produce
a metric over here.

00:34:13.889 --> 00:34:15.489 align:middle line:84%
There's no canonical
choice of which

00:34:15.489 --> 00:34:18.570 align:middle line:84%
of these purple points in
the preimage I should use.

00:34:18.570 --> 00:34:21.070 align:middle line:84%
But because our metric
upstairs is right invariant,

00:34:21.070 --> 00:34:22.590 align:middle line:90%
they all give the same answer.

00:34:22.590 --> 00:34:26.850 align:middle line:84%
And so there's a
coherent choice.

00:34:26.850 --> 00:34:30.750 align:middle line:84%
So that gives us a
metric on G mod gamma.

00:34:30.750 --> 00:34:35.650 align:middle line:90%


00:34:35.650 --> 00:34:46.210 align:middle line:84%
So now we have also a
left action of G on G.

00:34:46.210 --> 00:34:52.370 align:middle line:84%
So if g is in G, the left
action of g is also a map from G

00:34:52.370 --> 00:34:53.570 align:middle line:90%
to itself.

00:34:53.570 --> 00:34:58.990 align:middle line:84%
Left action g of h is g
inverse on the left h.

00:34:58.990 --> 00:35:01.590 align:middle line:84%
So the word left
in left action is

00:35:01.590 --> 00:35:05.583 align:middle line:84%
because the thing that
comes from g is over here.

00:35:05.583 --> 00:35:07.750 align:middle line:84%
The inverse is not super
important in our discussion

00:35:07.750 --> 00:35:10.270 align:middle line:84%
today, but it causes this
to behave correctly when you

00:35:10.270 --> 00:35:12.030 align:middle line:90%
compose elements in the group.

00:35:12.030 --> 00:35:13.690 align:middle line:84%
And so it's
traditional to do this.

00:35:13.690 --> 00:35:23.030 align:middle line:90%


00:35:23.030 --> 00:35:34.310 align:middle line:84%
Lg is not an isometry of our
group with our right invariant

00:35:34.310 --> 00:35:36.510 align:middle line:90%
metric.

00:35:36.510 --> 00:35:38.238 align:middle line:90%
And that's very important.

00:35:38.238 --> 00:35:40.030 align:middle line:84%
It's very important to
our whole discussion

00:35:40.030 --> 00:35:47.150 align:middle line:84%
to understand, geometrically,
how this operation affects

00:35:47.150 --> 00:35:47.800 align:middle line:90%
this space.

00:35:47.800 --> 00:35:54.270 align:middle line:90%


00:35:54.270 --> 00:35:55.700 align:middle line:90%
So let's see what happens.

00:35:55.700 --> 00:36:01.770 align:middle line:90%


00:36:01.770 --> 00:36:05.850 align:middle line:84%
So it's easiest to see what
happens around the identity.

00:36:05.850 --> 00:36:10.210 align:middle line:84%
And you can do it starting
at other points, too.

00:36:10.210 --> 00:36:12.130 align:middle line:90%
So here's the identity.

00:36:12.130 --> 00:36:14.490 align:middle line:90%
And here's some g.

00:36:14.490 --> 00:36:17.530 align:middle line:84%
And I have a map,
the left action,

00:36:17.530 --> 00:36:21.970 align:middle line:84%
by g inverse that takes
the identity over here.

00:36:21.970 --> 00:36:25.170 align:middle line:84%
And now, this action,
it's not clear from what

00:36:25.170 --> 00:36:27.190 align:middle line:84%
we've said whether it
preserves the metric.

00:36:27.190 --> 00:36:30.410 align:middle line:90%
In fact, it distorts the metric.

00:36:30.410 --> 00:36:33.610 align:middle line:84%
But how can we figure
out what this mapping

00:36:33.610 --> 00:36:36.130 align:middle line:90%
is doing to the metric?

00:36:36.130 --> 00:36:40.290 align:middle line:84%
Well, our metric was defined
to be right invariant.

00:36:40.290 --> 00:36:43.130 align:middle line:84%
So there's a map going
the other direction,

00:36:43.130 --> 00:36:46.970 align:middle line:90%
the right action by g inverse.

00:36:46.970 --> 00:36:48.950 align:middle line:84%
That takes g back
to the identity.

00:36:48.950 --> 00:36:51.590 align:middle line:84%
And this one we do understand
what it does to the metric.

00:36:51.590 --> 00:36:53.890 align:middle line:90%
It's an isometry.

00:36:53.890 --> 00:36:57.870 align:middle line:84%
So if I want to understand what
this map did to the metric,

00:36:57.870 --> 00:37:02.750 align:middle line:84%
it's a reasonable idea to
look at this, and then this.

00:37:02.750 --> 00:37:04.250 align:middle line:90%
That part is an isometry.

00:37:04.250 --> 00:37:08.750 align:middle line:84%
So all of the distortion of
the metric came from this part.

00:37:08.750 --> 00:37:11.270 align:middle line:84%
And this composition is
going to be a little easier

00:37:11.270 --> 00:37:13.830 align:middle line:90%
to understand.

00:37:13.830 --> 00:37:20.950 align:middle line:84%
So what happens if I do this,
if I take a left action by g

00:37:20.950 --> 00:37:28.230 align:middle line:84%
inverse and a right action by
g inverse and I apply it to h?

00:37:28.230 --> 00:37:30.598 align:middle line:84%
These guys commute
with each other.

00:37:30.598 --> 00:37:31.890 align:middle line:90%
We'll see when we write it out.

00:37:31.890 --> 00:37:35.190 align:middle line:84%
So I don't have to be super
careful what order to write them

00:37:35.190 --> 00:37:35.710 align:middle line:90%
in.

00:37:35.710 --> 00:37:36.210 align:middle line:90%
Yeah.

00:37:36.210 --> 00:37:39.350 align:middle line:90%


00:37:39.350 --> 00:37:43.070 align:middle line:84%
AUDIENCE: Is one of
these [INAUDIBLE]?

00:37:43.070 --> 00:37:44.204 align:middle line:90%
LAWRENCE GUTH: What's that?

00:37:44.204 --> 00:37:45.246 align:middle line:90%
AUDIENCE: Oh, never mind.

00:37:45.246 --> 00:37:48.957 align:middle line:90%


00:37:48.957 --> 00:37:49.790 align:middle line:90%
LAWRENCE GUTH: Yeah.

00:37:49.790 --> 00:37:51.810 align:middle line:84%
I guess I should really
write it in the other order.

00:37:51.810 --> 00:37:52.977 align:middle line:90%
Although, it doesn't matter.

00:37:52.977 --> 00:37:56.283 align:middle line:90%


00:37:56.283 --> 00:37:57.700 align:middle line:84%
So I want you to
imagine I started

00:37:57.700 --> 00:38:00.180 align:middle line:84%
with e, or maybe
somebody else near e,

00:38:00.180 --> 00:38:04.380 align:middle line:84%
I applied left g inverse, and
then I applied right g inverse.

00:38:04.380 --> 00:38:07.460 align:middle line:84%
And we're going to
see what happens.

00:38:07.460 --> 00:38:10.340 align:middle line:84%
So just from the formulas,
this is right action

00:38:10.340 --> 00:38:14.580 align:middle line:90%
of g inverse of gh.

00:38:14.580 --> 00:38:17.740 align:middle line:90%
And that's g h inverse.

00:38:17.740 --> 00:38:20.040 align:middle line:84%
So actually, it's something
natural and important.

00:38:20.040 --> 00:38:22.540 align:middle line:90%
It's conjugating.

00:38:22.540 --> 00:38:28.090 align:middle line:84%
So I'll call this C sub g
of h, conjugating g by h.

00:38:28.090 --> 00:38:32.100 align:middle line:90%


00:38:32.100 --> 00:38:36.980 align:middle line:84%
So conjugating by h
then induces a map

00:38:36.980 --> 00:38:41.740 align:middle line:84%
from the tangent
space of e to itself.

00:38:41.740 --> 00:38:46.380 align:middle line:84%
So conjugating by G is a map
from the group to itself.

00:38:46.380 --> 00:38:49.032 align:middle line:84%
And it sends the
identity to the identity.

00:38:49.032 --> 00:38:50.740 align:middle line:84%
And therefore, it
sends the tangent space

00:38:50.740 --> 00:38:53.400 align:middle line:90%
to the tangent space.

00:38:53.400 --> 00:38:56.260 align:middle line:90%
And we can see what--

00:38:56.260 --> 00:38:57.400 align:middle line:90%
we can write this down.

00:38:57.400 --> 00:38:59.680 align:middle line:90%
And we can see what it does.

00:38:59.680 --> 00:39:05.660 align:middle line:84%
And if this thing is
distorting the geometry,

00:39:05.660 --> 00:39:07.920 align:middle line:84%
it means that this thing
was distorting the geometry

00:39:07.920 --> 00:39:11.960 align:middle line:90%
in exactly the same way.

00:39:11.960 --> 00:39:18.360 align:middle line:84%
So we could say more than
this, but the singular values

00:39:18.360 --> 00:39:26.190 align:middle line:84%
of Lg inverse are equal to
the singular values of Cg.

00:39:26.190 --> 00:39:32.480 align:middle line:90%


00:39:32.480 --> 00:39:35.130 align:middle line:90%
So what they are depends on g.

00:39:35.130 --> 00:39:36.380 align:middle line:90%
So I'm going to pick a nice g.

00:39:36.380 --> 00:39:37.838 align:middle line:84%
And we're actually
going to compute

00:39:37.838 --> 00:39:40.540 align:middle line:84%
them to see how the
geometry is being distorted.

00:39:40.540 --> 00:40:10.560 align:middle line:90%


00:40:10.560 --> 00:40:13.740 align:middle line:84%
So the interesting
g's are going to be

00:40:13.740 --> 00:40:16.140 align:middle line:84%
from those one-parameter
subgroups up there.

00:40:16.140 --> 00:40:18.540 align:middle line:84%
And the one that
looks the nicest

00:40:18.540 --> 00:40:20.540 align:middle line:84%
and where this calculation
is the most important

00:40:20.540 --> 00:40:23.300 align:middle line:90%
is for that diagonal matrix ar.

00:40:23.300 --> 00:40:26.820 align:middle line:84%
So we're going to
compute what that is.

00:40:26.820 --> 00:40:30.780 align:middle line:90%
So I'm going to look at Car.

00:40:30.780 --> 00:40:36.220 align:middle line:84%
That goes from the
tangent space to itself.

00:40:36.220 --> 00:40:40.960 align:middle line:84%
So the tangent space is the
set of matrices a, b, c,

00:40:40.960 --> 00:40:44.140 align:middle line:90%
d with a plus b equals 0.

00:40:44.140 --> 00:40:52.490 align:middle line:84%
And let's call this
matrix M. And so Car of M

00:40:52.490 --> 00:40:57.730 align:middle line:90%
is ar M ar inverse.

00:40:57.730 --> 00:41:03.010 align:middle line:84%
That's e to the r, e to
the minus r a, b, c, d,

00:41:03.010 --> 00:41:06.270 align:middle line:90%
e to the minus r, e to the r.

00:41:06.270 --> 00:41:07.590 align:middle line:90%
So this is a sub r.

00:41:07.590 --> 00:41:08.870 align:middle line:90%
That's just a definition.

00:41:08.870 --> 00:41:16.530 align:middle line:84%
This is a sub r inverse,
so I switch the minus signs

00:41:16.530 --> 00:41:19.930 align:middle line:90%
I did this very carefully.

00:41:19.930 --> 00:41:22.590 align:middle line:84%
It's not super instructive to
watch me do it very carefully.

00:41:22.590 --> 00:41:25.610 align:middle line:84%
So I'm just going to
write down the answer.

00:41:25.610 --> 00:41:33.850 align:middle line:84%
It's a, d, e to the 2r
b, e to the minus 2r c.

00:41:33.850 --> 00:41:35.730 align:middle line:84%
So this conjugation,
what it does

00:41:35.730 --> 00:41:38.050 align:middle line:84%
is it doesn't
change the diagonal.

00:41:38.050 --> 00:41:42.690 align:middle line:90%
It magnifies this component.

00:41:42.690 --> 00:41:47.150 align:middle line:90%
And d magnifies that component.

00:41:47.150 --> 00:41:48.670 align:middle line:84%
That happens to be
something we can

00:41:48.670 --> 00:41:51.740 align:middle line:84%
describe really nicely in the
orthonormal basis that I picked.

00:41:51.740 --> 00:41:55.550 align:middle line:90%


00:41:55.550 --> 00:42:06.550 align:middle line:84%
So Car of u, so it
is e to the 2r u.

00:42:06.550 --> 00:42:15.190 align:middle line:84%
Car of u tilde is e to the minus
2r u tilde, and Car of d is d.

00:42:15.190 --> 00:42:19.170 align:middle line:84%
This conjugation behaves
really nicely in this basis,

00:42:19.170 --> 00:42:21.790 align:middle line:84%
but you can see that it does
not preserve the metric.

00:42:21.790 --> 00:42:23.590 align:middle line:90%
It stretches that one.

00:42:23.590 --> 00:42:25.430 align:middle line:90%
It compresses that one.

00:42:25.430 --> 00:42:29.390 align:middle line:90%
And it preserves that one.

00:42:29.390 --> 00:42:42.350 align:middle line:84%
So the conclusion from this
is that L ar inverse of u,

00:42:42.350 --> 00:42:49.330 align:middle line:84%
the length of that is e to the
2r L ar inverse of u tilde has

00:42:49.330 --> 00:42:54.090 align:middle line:84%
length e to the minus
2r L ar inverse.

00:42:54.090 --> 00:42:56.490 align:middle line:90%
d has length of 1.

00:42:56.490 --> 00:43:00.230 align:middle line:90%
And it also preserved angles.

00:43:00.230 --> 00:43:03.410 align:middle line:84%
So we just stretched things,
but we didn't tilt anything.

00:43:03.410 --> 00:43:11.950 align:middle line:84%
So L ar inverse of u of u
tilde and of d are orthogonal.

00:43:11.950 --> 00:43:14.890 align:middle line:90%


00:43:14.890 --> 00:43:19.010 align:middle line:84%
So this describes how the
left action by this matrix

00:43:19.010 --> 00:43:21.200 align:middle line:84%
is distorting the
geometry of the space.

00:43:21.200 --> 00:43:24.170 align:middle line:90%


00:43:24.170 --> 00:43:26.720 align:middle line:84%
And we also can make
some kind of picture.

00:43:26.720 --> 00:43:29.530 align:middle line:90%


00:43:29.530 --> 00:43:34.970 align:middle line:84%
So suppose this is a,
quote, unquote, "cube" in G,

00:43:34.970 --> 00:43:39.190 align:middle line:90%
where this is the u direction.

00:43:39.190 --> 00:43:41.270 align:middle line:90%
This is the u tilde direction.

00:43:41.270 --> 00:43:44.560 align:middle line:84%
And the direction into the
board is the d direction.

00:43:44.560 --> 00:43:48.320 align:middle line:84%
So what will happen
to it when we do this?

00:43:48.320 --> 00:43:50.440 align:middle line:84%
The u direction
gets really long.

00:43:50.440 --> 00:43:52.780 align:middle line:84%
The u tilde direction
gets really short.

00:43:52.780 --> 00:43:55.440 align:middle line:84%
The other direction
is preserved.

00:43:55.440 --> 00:43:57.350 align:middle line:90%
So it looks kind of like that.

00:43:57.350 --> 00:44:04.920 align:middle line:90%


00:44:04.920 --> 00:44:06.840 align:middle line:84%
One thing that you can
notice from the picture

00:44:06.840 --> 00:44:12.920 align:middle line:84%
and from the calculation is
that while L ar inverse distorts

00:44:12.920 --> 00:44:18.040 align:middle line:84%
lengths, distorts the
metric, it nevertheless

00:44:18.040 --> 00:44:19.710 align:middle line:90%
preserves the volume.

00:44:19.710 --> 00:44:36.520 align:middle line:90%


00:44:36.520 --> 00:44:39.730 align:middle line:84%
So that's how the left
action distorts the geometry.

00:44:39.730 --> 00:44:53.300 align:middle line:90%


00:44:53.300 --> 00:45:03.170 align:middle line:84%
And we also have a left
action of G on G mod gamma.

00:45:03.170 --> 00:45:06.740 align:middle line:90%


00:45:06.740 --> 00:45:09.460 align:middle line:90%
And so how does that work?

00:45:09.460 --> 00:45:17.300 align:middle line:84%
Well, Lg of h gamma is
just g inverse h gamma.

00:45:17.300 --> 00:45:18.860 align:middle line:84%
Because the gamma
is on the right,

00:45:18.860 --> 00:45:22.124 align:middle line:84%
and this guy is on
the left, they sort of

00:45:22.124 --> 00:45:23.840 align:middle line:90%
don't interfere with each other.

00:45:23.840 --> 00:45:28.140 align:middle line:84%
And so this is an
actual group action.

00:45:28.140 --> 00:45:33.780 align:middle line:84%
So Lg maps G mod
gamma to itself.

00:45:33.780 --> 00:45:38.580 align:middle line:90%
And it is a bijection.

00:45:38.580 --> 00:45:40.410 align:middle line:90%
It distorts the metric.

00:45:40.410 --> 00:45:43.280 align:middle line:90%


00:45:43.280 --> 00:45:46.580 align:middle line:90%
But it preserves the volume.

00:45:46.580 --> 00:45:50.680 align:middle line:90%


00:45:50.680 --> 00:45:53.440 align:middle line:84%
And maybe I should
say, we just computed

00:45:53.440 --> 00:45:55.360 align:middle line:84%
this carefully for
a particular element

00:45:55.360 --> 00:45:58.240 align:middle line:84%
g, which is these
diagonal elements.

00:45:58.240 --> 00:45:59.660 align:middle line:90%
You could take any g.

00:45:59.660 --> 00:46:01.400 align:middle line:90%
It doesn't have to be diagonal.

00:46:01.400 --> 00:46:04.440 align:middle line:84%
It will distort the metric
in some different way.

00:46:04.440 --> 00:46:07.120 align:middle line:84%
Every g is different, but
they all preserve the volume.

00:46:07.120 --> 00:46:09.620 align:middle line:84%
We didn't really prove that,
because I just did one example,

00:46:09.620 --> 00:46:12.320 align:middle line:90%
but it's true.

00:46:12.320 --> 00:46:12.880 align:middle line:90%
Yeah.

00:46:12.880 --> 00:46:15.640 align:middle line:84%
AUDIENCE: Are you able
to show that by showing

00:46:15.640 --> 00:46:20.080 align:middle line:84%
that the second u of t is
a one-parameter subgroup?

00:46:20.080 --> 00:46:21.540 align:middle line:90%
It doesn't change the volume.

00:46:21.540 --> 00:46:25.767 align:middle line:84%
And do those two one-parameter
subgroups generate a G of gamma?

00:46:25.767 --> 00:46:26.600 align:middle line:90%
LAWRENCE GUTH: Yeah.

00:46:26.600 --> 00:46:28.320 align:middle line:84%
So the question
was, if we wanted

00:46:28.320 --> 00:46:30.800 align:middle line:84%
to check that it preserved--
that every g preserves

00:46:30.800 --> 00:46:34.160 align:middle line:84%
the volume, it would be enough
to check a few g's that generate

00:46:34.160 --> 00:46:35.200 align:middle line:90%
it.

00:46:35.200 --> 00:46:39.260 align:middle line:84%
And yeah, so if we
did the computation

00:46:39.260 --> 00:46:43.140 align:middle line:84%
with the unipotent 1, that
would basically finish the job.

00:46:43.140 --> 00:46:44.980 align:middle line:90%
So it's not that difficult.

00:46:44.980 --> 00:46:52.100 align:middle line:84%
We also could just describe
the g as a, b, c, d, and then

00:46:52.100 --> 00:46:53.520 align:middle line:84%
the computation
is a little messy,

00:46:53.520 --> 00:46:55.810 align:middle line:84%
but we could just
check once and for all.

00:46:55.810 --> 00:46:57.560 align:middle line:84%
There's probably a
better way of doing it,

00:46:57.560 --> 00:46:59.080 align:middle line:90%
but that's not my part of math.

00:46:59.080 --> 00:47:09.660 align:middle line:90%


00:47:09.660 --> 00:47:12.800 align:middle line:90%
So now we can say--

00:47:12.800 --> 00:47:17.220 align:middle line:90%


00:47:17.220 --> 00:47:20.940 align:middle line:84%
so now, an important and
interesting part of this story

00:47:20.940 --> 00:47:30.660 align:middle line:84%
is what is the difference
between the unitary ones

00:47:30.660 --> 00:47:33.660 align:middle line:90%
and the diagonal ones.

00:47:33.660 --> 00:47:37.170 align:middle line:84%
And there are many
different answers to this.

00:47:37.170 --> 00:47:40.730 align:middle line:84%
We could come back
to it multiple times,

00:47:40.730 --> 00:47:42.890 align:middle line:84%
but I'm going to mention
one way of looking

00:47:42.890 --> 00:47:45.400 align:middle line:84%
at it, which will start to get
us to the projection theory.

00:47:45.400 --> 00:47:48.690 align:middle line:90%


00:47:48.690 --> 00:47:52.370 align:middle line:84%
So there's a special
feature of unitary matrices

00:47:52.370 --> 00:47:54.870 align:middle line:84%
that we saw also
maybe by coincidence,

00:47:54.870 --> 00:47:57.090 align:middle line:84%
but it also came up when
we were talking about SL2

00:47:57.090 --> 00:48:01.010 align:middle line:84%
of Fp, which what we saw there
basically is that something cool

00:48:01.010 --> 00:48:04.050 align:middle line:84%
happens when you
conjugate a unitary matrix

00:48:04.050 --> 00:48:06.250 align:middle line:90%
by a diagonal matrix.

00:48:06.250 --> 00:48:09.350 align:middle line:84%
And that is the special
feature of unitary matrices.

00:48:09.350 --> 00:48:11.190 align:middle line:84%
And it really helps
to understand them.

00:48:11.190 --> 00:48:32.975 align:middle line:90%


00:48:32.975 --> 00:48:41.756 align:middle line:84%
So what is special
about unitary matrices?

00:48:41.756 --> 00:48:46.830 align:middle line:90%


00:48:46.830 --> 00:48:51.950 align:middle line:84%
So there's a computation that
if I take a unitary matrix,

00:48:51.950 --> 00:48:56.230 align:middle line:84%
and I conjugate it
by a diagonal matrix.

00:48:56.230 --> 00:48:58.970 align:middle line:84%
So I have e to r,
e to the minus r.

00:48:58.970 --> 00:49:01.820 align:middle line:90%


00:49:01.820 --> 00:49:04.670 align:middle line:84%
AUDIENCE: Should
it be unipotent?

00:49:04.670 --> 00:49:06.390 align:middle line:90%
LAWRENCE GUTH: Unipotent, yeah.

00:49:06.390 --> 00:49:10.740 align:middle line:84%
Thanks Yes, thank
you, unipotent.

00:49:10.740 --> 00:49:15.030 align:middle line:90%


00:49:15.030 --> 00:49:22.270 align:middle line:84%
So if you multiply
this out, you get this.

00:49:22.270 --> 00:49:25.670 align:middle line:84%
So if you're interested in
a unipotent matrix ut, where

00:49:25.670 --> 00:49:30.350 align:middle line:84%
this t is really large, you can
get at it by starting with u1,

00:49:30.350 --> 00:49:33.360 align:middle line:84%
and conjugating it by one
of these diagonal matrices.

00:49:33.360 --> 00:49:36.850 align:middle line:90%


00:49:36.850 --> 00:49:42.190 align:middle line:84%
So let's say u0t, this is
a set of unitary matrices.

00:49:42.190 --> 00:49:44.210 align:middle line:90%
This is ut.

00:49:44.210 --> 00:49:46.650 align:middle line:90%
t goes from 0 to t.

00:49:46.650 --> 00:49:52.010 align:middle line:84%
And so to understand
an orbit, our goal

00:49:52.010 --> 00:49:59.830 align:middle line:84%
is to understand u0 t
of x, where t is large.

00:49:59.830 --> 00:50:03.450 align:middle line:90%


00:50:03.450 --> 00:50:11.930 align:middle line:84%
And because of this slick little
formula, we can write u0t x.

00:50:11.930 --> 00:50:27.930 align:middle line:84%
So let's say we can write
it as ar u01 a minus R of x,

00:50:27.930 --> 00:50:34.560 align:middle line:90%
where e to the 2r is T.

00:50:34.560 --> 00:50:36.320 align:middle line:90%
Why is this helpful?

00:50:36.320 --> 00:50:42.242 align:middle line:84%
So this here is just some
point in G mod gamma.

00:50:42.242 --> 00:50:43.700 align:middle line:84%
It's not the point
we started with.

00:50:43.700 --> 00:50:47.760 align:middle line:84%
It's some other point,
but it's just a point.

00:50:47.760 --> 00:50:50.370 align:middle line:84%
This here is a
short unitary orbit.

00:50:50.370 --> 00:50:55.815 align:middle line:90%


00:50:55.815 --> 00:51:00.200 align:middle line:84%
A short unitary orbit isn't
so intimidating to draw.

00:51:00.200 --> 00:51:02.720 align:middle line:90%
And then all the action is here.

00:51:02.720 --> 00:51:06.600 align:middle line:84%
And the question is,
what does this map

00:51:06.600 --> 00:51:08.390 align:middle line:90%
do to a short unitary orbit?

00:51:08.390 --> 00:51:11.200 align:middle line:90%


00:51:11.200 --> 00:51:17.640 align:middle line:84%
So if we make a picture,
here's our G mod gamma.

00:51:17.640 --> 00:51:22.800 align:middle line:84%
Here is our new starting
point, a minus R x.

00:51:22.800 --> 00:51:24.360 align:middle line:90%
Here is our orbit.

00:51:24.360 --> 00:51:25.540 align:middle line:90%
It's not very long.

00:51:25.540 --> 00:51:29.180 align:middle line:84%
It doesn't wrap around yet,
so it's not that intimidating.

00:51:29.180 --> 00:51:34.860 align:middle line:84%
That's this guy,
short unitary orbit.

00:51:34.860 --> 00:51:39.000 align:middle line:84%
And now we're going to take this
map of G mod gamma to itself.

00:51:39.000 --> 00:51:40.400 align:middle line:90%
And we're going to apply it.

00:51:40.400 --> 00:51:42.480 align:middle line:84%
We want to see what
happens to this blue thing.

00:51:42.480 --> 00:51:45.140 align:middle line:90%


00:51:45.140 --> 00:51:47.620 align:middle line:90%
Again, unipotent, fix.

00:51:47.620 --> 00:51:51.280 align:middle line:90%
Short, unipotent.

00:51:51.280 --> 00:51:51.780 align:middle line:90%
Thanks.

00:51:51.780 --> 00:51:55.460 align:middle line:90%


00:51:55.460 --> 00:51:57.820 align:middle line:90%
And we can also make a remark.

00:51:57.820 --> 00:51:59.160 align:middle line:90%
aR is a subgroup.

00:51:59.160 --> 00:52:03.500 align:middle line:84%
So aR is like a1 to
the R over a little r

00:52:03.500 --> 00:52:06.980 align:middle line:90%
to the big R over little r.

00:52:06.980 --> 00:52:09.420 align:middle line:84%
Big R, little r is
a natural number.

00:52:09.420 --> 00:52:11.980 align:middle line:84%
So we don't have to
do this all at once.

00:52:11.980 --> 00:52:16.920 align:middle line:84%
We could take a less dramatic
map from G mod gamma to itself,

00:52:16.920 --> 00:52:18.800 align:middle line:84%
and we repeat that
over and over again.

00:52:18.800 --> 00:52:20.680 align:middle line:84%
We want to see what
happens to this thing.

00:52:20.680 --> 00:52:26.580 align:middle line:90%


00:52:26.580 --> 00:52:30.600 align:middle line:90%
So we are led to try--

00:52:30.600 --> 00:52:33.560 align:middle line:84%
so we are led to
a goal, which is

00:52:33.560 --> 00:52:40.280 align:middle line:84%
to try to understand
and visualize

00:52:40.280 --> 00:52:46.200 align:middle line:84%
the action of a little
r on G mod gamma.

00:52:46.200 --> 00:52:48.028 align:middle line:84%
And if we understand
that really well,

00:52:48.028 --> 00:52:49.820 align:middle line:84%
we should be able to
understand this orbit.

00:52:49.820 --> 00:53:35.330 align:middle line:90%


00:53:35.330 --> 00:53:39.230 align:middle line:84%
All right, so how might we
try to get a handle on this?

00:53:39.230 --> 00:53:41.530 align:middle line:84%
Let's look at a
fundamental domain.

00:53:41.530 --> 00:53:45.990 align:middle line:84%
So F is a fundamental
domain of G mod gamma.

00:53:45.990 --> 00:53:51.230 align:middle line:84%
F is a subset of G. And
I'll draw it as a cube.

00:53:51.230 --> 00:53:54.606 align:middle line:90%
It's not literally a cube.

00:53:54.606 --> 00:53:56.770 align:middle line:90%
G mod gamma is not compact.

00:53:56.770 --> 00:53:58.850 align:middle line:90%
So F cannot be compact.

00:53:58.850 --> 00:54:02.190 align:middle line:84%
So it should have a little tail
that's going off to infinity.

00:54:02.190 --> 00:54:05.928 align:middle line:84%
But for what I'm about to say,
that's not super important.

00:54:05.928 --> 00:54:07.470 align:middle line:84%
Actually, by the
way, the whole story

00:54:07.470 --> 00:54:10.270 align:middle line:84%
also makes sense for
gammas that are cocompact.

00:54:10.270 --> 00:54:14.390 align:middle line:84%
And then this picture
would be better.

00:54:14.390 --> 00:54:18.310 align:middle line:84%
So now, we'd like to
understand what a does.

00:54:18.310 --> 00:54:23.470 align:middle line:84%
So the first step is
I'm going to apply aR.

00:54:23.470 --> 00:54:26.750 align:middle line:84%
And we just thought about
what aR does, geometrically,

00:54:26.750 --> 00:54:28.890 align:middle line:90%
how it distorts the geometry.

00:54:28.890 --> 00:54:31.390 align:middle line:84%
We were looking at what
it does to the metric,

00:54:31.390 --> 00:54:35.810 align:middle line:84%
but it's very similar to
what it does to a unit cube.

00:54:35.810 --> 00:54:38.840 align:middle line:90%
So this thing is going to be--

00:54:38.840 --> 00:54:44.890 align:middle line:90%


00:54:44.890 --> 00:54:51.650 align:middle line:84%
so this is aR of F, which is
still sitting in the group

00:54:51.650 --> 00:54:54.343 align:middle line:84%
G. It got longer in one
way, shorter in another way.

00:54:54.343 --> 00:54:56.510 align:middle line:84%
The real thing would have
more dimensions than this,

00:54:56.510 --> 00:54:59.130 align:middle line:90%
but just to give the idea.

00:54:59.130 --> 00:55:02.450 align:middle line:84%
Now, the next thing
that happens is

00:55:02.450 --> 00:55:04.930 align:middle line:84%
we have to apply
the quotient map.

00:55:04.930 --> 00:55:08.160 align:middle line:84%
So pi is the map from
G to G mod gamma.

00:55:08.160 --> 00:55:10.850 align:middle line:90%


00:55:10.850 --> 00:55:14.170 align:middle line:84%
And when we do that, I'll
visualize G mod gamma, again,

00:55:14.170 --> 00:55:15.752 align:middle line:90%
as this fundamental domain.

00:55:15.752 --> 00:55:17.210 align:middle line:84%
And what happens
is that this thing

00:55:17.210 --> 00:55:21.450 align:middle line:84%
is going to wrap around
the fundamental domain.

00:55:21.450 --> 00:55:26.550 align:middle line:84%
So it will look
something like this.

00:55:26.550 --> 00:55:27.970 align:middle line:90%
There, it'll go in there.

00:55:27.970 --> 00:55:30.590 align:middle line:90%


00:55:30.590 --> 00:55:34.910 align:middle line:84%
It's very difficult
to draw this well.

00:55:34.910 --> 00:55:37.070 align:middle line:84%
This map from the
fundamental domain to itself

00:55:37.070 --> 00:55:39.223 align:middle line:90%
is going to be a bijection.

00:55:39.223 --> 00:55:40.890 align:middle line:84%
So somehow this is
going to wrap around.

00:55:40.890 --> 00:55:44.470 align:middle line:84%
It's going to cover
every point exactly once.

00:55:44.470 --> 00:55:48.000 align:middle line:84%
It's like some sort of a
beautiful jigsaw puzzle.

00:55:48.000 --> 00:55:49.250 align:middle line:90%
I don't know how to draw that.

00:55:49.250 --> 00:55:51.470 align:middle line:90%
So I'll just do--

00:55:51.470 --> 00:55:52.910 align:middle line:84%
I'll just have
this come out here,

00:55:52.910 --> 00:55:56.230 align:middle line:90%
and I'll do a dot, dot, dot.

00:55:56.230 --> 00:55:57.750 align:middle line:90%
And then I don't know.

00:55:57.750 --> 00:56:00.533 align:middle line:84%
If we were visualizing, if
we had something of interest,

00:56:00.533 --> 00:56:02.450 align:middle line:84%
eventually, it would be
some unipotent orbits,

00:56:02.450 --> 00:56:04.510 align:middle line:84%
but maybe, initially,
we're interested in what

00:56:04.510 --> 00:56:05.830 align:middle line:90%
happens to that.

00:56:05.830 --> 00:56:10.390 align:middle line:84%
So in the first
step, I did this.

00:56:10.390 --> 00:56:17.310 align:middle line:84%
And then in the next step,
we see this and this.

00:56:17.310 --> 00:56:18.970 align:middle line:84%
Somewhere over
here, we see this.

00:56:18.970 --> 00:56:25.290 align:middle line:90%


00:56:25.290 --> 00:56:28.110 align:middle line:90%
Cool.

00:56:28.110 --> 00:56:29.585 align:middle line:90%
So yeah.

00:56:29.585 --> 00:56:31.710 align:middle line:84%
AUDIENCE: Why does the math
have to be a bijection?

00:56:31.710 --> 00:56:36.497 align:middle line:84%
Why can't the projection
map not be injective?

00:56:36.497 --> 00:56:37.330 align:middle line:90%
LAWRENCE GUTH: Yeah.

00:56:37.330 --> 00:56:43.810 align:middle line:84%
So the question is, why
is this map a bijection?

00:56:43.810 --> 00:56:45.450 align:middle line:90%
So we were saying--

00:56:45.450 --> 00:56:47.950 align:middle line:84%
so this-- I guess
it's actually there.

00:56:47.950 --> 00:56:53.610 align:middle line:90%


00:56:53.610 --> 00:57:00.050 align:middle line:84%
So we said earlier that the
left action of G, G mod gamma

00:57:00.050 --> 00:57:04.010 align:middle line:90%
goes to itself is a bijection.

00:57:04.010 --> 00:57:05.893 align:middle line:84%
And if we believe
that, then if we

00:57:05.893 --> 00:57:07.310 align:middle line:84%
start with the
fundamental domain,

00:57:07.310 --> 00:57:10.290 align:middle line:84%
we have one representative
of each point in G mod gamma.

00:57:10.290 --> 00:57:12.433 align:middle line:84%
And then we end up
back in G mod gamma.

00:57:12.433 --> 00:57:13.850 align:middle line:84%
We should cover
everything exactly

00:57:13.850 --> 00:57:17.010 align:middle line:90%
once if this is a bijection.

00:57:17.010 --> 00:57:19.360 align:middle line:90%
Now, why was this a bijection?

00:57:19.360 --> 00:57:21.160 align:middle line:90%
So what is this map?

00:57:21.160 --> 00:57:27.640 align:middle line:84%
It's the map h gamma goes
to g inverse h gamma.

00:57:27.640 --> 00:57:29.380 align:middle line:90%
So why is this a bijection?

00:57:29.380 --> 00:57:40.820 align:middle line:90%


00:57:40.820 --> 00:57:44.120 align:middle line:90%
So it's injective.

00:57:44.120 --> 00:57:51.520 align:middle line:84%
That means that if h1
g inverse h1 gamma is

00:57:51.520 --> 00:57:59.060 align:middle line:84%
equal to g inverse h2 gamma,
then h1 gamma equals h2 gamma.

00:57:59.060 --> 00:58:02.880 align:middle line:84%
That's what it means
for it to be injective.

00:58:02.880 --> 00:58:05.840 align:middle line:90%
So is this clear?

00:58:05.840 --> 00:58:10.120 align:middle line:84%
Yeah, you multiply
on both sides by g.

00:58:10.120 --> 00:58:30.500 align:middle line:84%
Surjective, that means
that for every h2 in G,

00:58:30.500 --> 00:58:39.660 align:middle line:84%
there exists h1 in G, so that
g inverse h1 gamma is h2 gamma.

00:58:39.660 --> 00:58:43.262 align:middle line:84%
So yeah, we set these
equal, and solve for g.

00:58:43.262 --> 00:58:45.220 align:middle line:84%
AUDIENCE: I guess that
causes the invertibility

00:58:45.220 --> 00:58:46.707 align:middle line:90%
of the [INAUDIBLE].

00:58:46.707 --> 00:58:47.540 align:middle line:90%
LAWRENCE GUTH: Yeah.

00:58:47.540 --> 00:58:48.040 align:middle line:90%
Yeah.

00:58:48.040 --> 00:58:57.500 align:middle line:90%


00:58:57.500 --> 00:58:59.260 align:middle line:84%
I have to say, I went
through this, too.

00:58:59.260 --> 00:59:01.280 align:middle line:84%
When I drew these
pictures, I asked myself,

00:59:01.280 --> 00:59:03.620 align:middle line:84%
is this actually going to
fold up and cover everything

00:59:03.620 --> 00:59:04.900 align:middle line:90%
exactly once?

00:59:04.900 --> 00:59:08.592 align:middle line:84%
And then I was like,
is it really bijective?

00:59:08.592 --> 00:59:09.550 align:middle line:90%
Yeah, it should happen.

00:59:09.550 --> 00:59:19.680 align:middle line:90%


00:59:19.680 --> 00:59:23.140 align:middle line:84%
I guess you can make easier fun
examples that are more linear.

00:59:23.140 --> 00:59:27.240 align:middle line:84%
So you just have a matrix
acting on R2 mod Z2,

00:59:27.240 --> 00:59:29.280 align:middle line:90%
or something like that.

00:59:29.280 --> 00:59:31.180 align:middle line:84%
And then the fundamental
domain is a square.

00:59:31.180 --> 00:59:32.680 align:middle line:84%
And you can draw
what happens to it.

00:59:32.680 --> 00:59:37.300 align:middle line:84%
And you can look up-- so this
was inspired by that example.

00:59:37.300 --> 00:59:39.260 align:middle line:84%
They called that the cat
map, a particular map.

00:59:39.260 --> 00:59:41.180 align:middle line:84%
And instead of a smiley
face, they draw a cat.

00:59:41.180 --> 00:59:42.888 align:middle line:84%
They have better
artistic skills than me.

00:59:42.888 --> 00:59:47.760 align:middle line:84%
You can google that, and see
some nice pictures like this.

00:59:47.760 --> 00:59:49.720 align:middle line:90%
OK, cool.

00:59:49.720 --> 00:59:57.580 align:middle line:84%
So here, this thing, Lg Ll
with this diagonal action,

00:59:57.580 --> 00:59:59.680 align:middle line:90%
you've broken it into two steps.

00:59:59.680 --> 01:00:04.440 align:middle line:84%
And out of these two steps, I
find this one quite confusing.

01:00:04.440 --> 01:00:07.540 align:middle line:84%
It's difficult to write this
down sufficiently accurately,

01:00:07.540 --> 01:00:09.240 align:middle line:84%
and then to say
what happened here.

01:00:09.240 --> 01:00:11.875 align:middle line:84%
And did this come
back over here,

01:00:11.875 --> 01:00:13.000 align:middle line:90%
over there when it wrapped?

01:00:13.000 --> 01:00:14.630 align:middle line:90%
Where did it wrap?

01:00:14.630 --> 01:00:16.930 align:middle line:84%
All of that seems very
intimidating to me,

01:00:16.930 --> 01:00:19.150 align:middle line:84%
so very difficult to
write down in a clean way.

01:00:19.150 --> 01:00:24.510 align:middle line:84%
But this first
step is not so bad.

01:00:24.510 --> 01:00:27.910 align:middle line:84%
And so we're going to try to
understand the first step very

01:00:27.910 --> 01:00:31.350 align:middle line:84%
well and use that to see
how much we can figure out

01:00:31.350 --> 01:00:36.400 align:middle line:84%
about this map Lg acting
on our homogeneous space.

01:00:36.400 --> 01:00:39.110 align:middle line:90%


01:00:39.110 --> 01:00:43.070 align:middle line:84%
Now, the object that
we want to follow

01:00:43.070 --> 01:00:45.600 align:middle line:84%
is not a smiley face,
but a unipotent orbit.

01:00:45.600 --> 01:00:52.030 align:middle line:90%


01:00:52.030 --> 01:00:53.100 align:middle line:90%
Let's see here.

01:00:53.100 --> 01:01:02.670 align:middle line:90%


01:01:02.670 --> 01:01:04.410 align:middle line:90%
I guess we can erase all of it.

01:01:04.410 --> 01:01:25.210 align:middle line:90%


01:01:25.210 --> 01:01:29.010 align:middle line:84%
So I erased something
that is useful for us.

01:01:29.010 --> 01:01:34.810 align:middle line:84%
So recall, we worked out that
the left map by ar inverse,

01:01:34.810 --> 01:01:38.050 align:middle line:84%
we found its singular
values and singular vectors.

01:01:38.050 --> 01:01:44.970 align:middle line:84%
So in the u direction, the
length of this is e to the 2r.

01:01:44.970 --> 01:01:52.250 align:middle line:84%
And in the u tilde direction,
the length is e to the minus 2r.

01:01:52.250 --> 01:01:55.630 align:middle line:84%
And in the diagonal
direction, the length is 1.

01:01:55.630 --> 01:02:00.690 align:middle line:90%


01:02:00.690 --> 01:02:07.050 align:middle line:84%
So this is if we're thinking of
L ar inverse from TeG to TarG.

01:02:07.050 --> 01:02:10.910 align:middle line:90%


01:02:10.910 --> 01:02:18.830 align:middle line:84%
And this little u, this is
also the tangent direction

01:02:18.830 --> 01:02:20.750 align:middle line:90%
of our group u.

01:02:20.750 --> 01:02:23.383 align:middle line:90%


01:02:23.383 --> 01:02:25.050 align:middle line:84%
So that group is going
to get stretched,

01:02:25.050 --> 01:02:27.610 align:middle line:84%
which is the same thing we saw
in the computation over there.

01:02:27.610 --> 01:02:37.670 align:middle line:90%


01:02:37.670 --> 01:02:43.750 align:middle line:84%
Step one, so if we start with
a single unipotent orbit,

01:02:43.750 --> 01:02:47.950 align:middle line:84%
it starts at some
point like this.

01:02:47.950 --> 01:02:52.710 align:middle line:84%
So that's u 0, 1
of some point x.

01:02:52.710 --> 01:02:54.500 align:middle line:90%
What happens when we apply Lar?

01:02:54.500 --> 01:03:00.270 align:middle line:90%


01:03:00.270 --> 01:03:05.400 align:middle line:84%
Well, this thing is going
to get tall and skinny.

01:03:05.400 --> 01:03:09.420 align:middle line:90%


01:03:09.420 --> 01:03:11.660 align:middle line:84%
And the direction
that got stretched

01:03:11.660 --> 01:03:13.660 align:middle line:90%
is the direction of this orbit.

01:03:13.660 --> 01:03:16.900 align:middle line:84%
So now we're going
to have a long orbit.

01:03:16.900 --> 01:03:21.460 align:middle line:84%
And the height of this
thing is e to the 2r.

01:03:21.460 --> 01:03:24.620 align:middle line:90%
Then we're getting a quotient.

01:03:24.620 --> 01:03:26.180 align:middle line:84%
So when we quotient,
I don't know

01:03:26.180 --> 01:03:27.770 align:middle line:90%
exactly where it's going to go.

01:03:27.770 --> 01:03:31.140 align:middle line:90%


01:03:31.140 --> 01:03:36.300 align:middle line:84%
But it will go along--
it'll go out the top

01:03:36.300 --> 01:03:38.540 align:middle line:84%
and come in the
bottom somewhere.

01:03:38.540 --> 01:03:41.580 align:middle line:84%
And then we'll have
a bunch of these.

01:03:41.580 --> 01:03:48.520 align:middle line:84%
So we'll have e to the
2r u orbits of length 1.

01:03:48.520 --> 01:03:51.620 align:middle line:90%


01:03:51.620 --> 01:03:53.242 align:middle line:84%
And I have no idea
where they are.

01:03:53.242 --> 01:03:54.700 align:middle line:84%
So we haven't made
any progress yet

01:03:54.700 --> 01:03:57.325 align:middle line:84%
on our main question, which is
to see if these are distributing

01:03:57.325 --> 01:03:59.220 align:middle line:90%
kind of evenly.

01:03:59.220 --> 01:04:01.500 align:middle line:90%
But now we can start again.

01:04:01.500 --> 01:04:02.810 align:middle line:90%
So let me move this over here.

01:04:02.810 --> 01:04:07.880 align:middle line:90%


01:04:07.880 --> 01:04:12.520 align:middle line:84%
So now we have a bunch
of unipotent orbits.

01:04:12.520 --> 01:04:14.740 align:middle line:84%
And we're going to
stretch them out.

01:04:14.740 --> 01:04:18.120 align:middle line:90%


01:04:18.120 --> 01:04:20.660 align:middle line:84%
Now, what happens when we
stretch them out this time?

01:04:20.660 --> 01:04:30.580 align:middle line:90%


01:04:30.580 --> 01:04:32.140 align:middle line:90%
So each one of them got longer.

01:04:32.140 --> 01:04:36.240 align:middle line:84%
It's going to go from
the bottom to the top.

01:04:36.240 --> 01:04:41.560 align:middle line:84%
But also, there's also
some compression, which

01:04:41.560 --> 01:04:43.880 align:middle line:90%
is kind of in this direction.

01:04:43.880 --> 01:04:46.040 align:middle line:84%
And that could potentially
cause some orbits

01:04:46.040 --> 01:04:49.560 align:middle line:90%
to get closer together, or not.

01:04:49.560 --> 01:04:51.280 align:middle line:84%
And this is the
part that we have

01:04:51.280 --> 01:04:52.940 align:middle line:90%
to understand really carefully.

01:04:52.940 --> 01:04:57.160 align:middle line:90%


01:04:57.160 --> 01:05:02.167 align:middle line:84%
AUDIENCE: So this box is
some small box [INAUDIBLE]?

01:05:02.167 --> 01:05:03.000 align:middle line:90%
LAWRENCE GUTH: Yeah.

01:05:03.000 --> 01:05:04.280 align:middle line:90%
So this--

01:05:04.280 --> 01:05:07.167 align:middle line:90%
AUDIENCE: [INAUDIBLE]

01:05:07.167 --> 01:05:08.000 align:middle line:90%
LAWRENCE GUTH: Yeah.

01:05:08.000 --> 01:05:08.640 align:middle line:90%
That's right.

01:05:08.640 --> 01:05:09.440 align:middle line:90%
That's right.

01:05:09.440 --> 01:05:13.820 align:middle line:84%
So this box here is
a fundamental domain.

01:05:13.820 --> 01:05:17.220 align:middle line:90%
So it's scale 1.

01:05:17.220 --> 01:05:19.540 align:middle line:84%
And sure, it could be
around the identity.

01:05:19.540 --> 01:05:22.220 align:middle line:84%
The thing I most want
to convey in this class

01:05:22.220 --> 01:05:28.580 align:middle line:84%
is a feeling for how this map
is like twisting and distorting

01:05:28.580 --> 01:05:30.220 align:middle line:90%
these curves.

01:05:30.220 --> 01:05:33.500 align:middle line:84%
And so I'm going to try to
make a really big picture of it

01:05:33.500 --> 01:05:34.040 align:middle line:90%
there.

01:05:34.040 --> 01:05:36.390 align:middle line:84%
I also brought
with me some props.

01:05:36.390 --> 01:05:58.020 align:middle line:90%


01:05:58.020 --> 01:06:02.140 align:middle line:84%
So I'm going to visualize our
fundamental domain or maybe

01:06:02.140 --> 01:06:04.920 align:middle line:90%
a piece of it as a cylinder.

01:06:04.920 --> 01:06:06.960 align:middle line:84%
And I'm going to draw
some unipotent orbits

01:06:06.960 --> 01:06:08.480 align:middle line:90%
in different colors.

01:06:08.480 --> 01:06:11.290 align:middle line:84%
And they all go from
the bottom to the top.

01:06:11.290 --> 01:06:21.560 align:middle line:90%


01:06:21.560 --> 01:06:23.020 align:middle line:90%
Maybe I'll draw three of them.

01:06:23.020 --> 01:06:31.480 align:middle line:90%


01:06:31.480 --> 01:06:34.640 align:middle line:84%
Now, what happens
to this picture?

01:06:34.640 --> 01:06:37.280 align:middle line:84%
So first of all, it's
going to stretch that way.

01:06:37.280 --> 01:06:38.960 align:middle line:84%
That part's not that
hard to imagine.

01:06:38.960 --> 01:06:40.060 align:middle line:90%
It would go off the board.

01:06:40.060 --> 01:06:42.240 align:middle line:84%
So I'm not going
to emphasize that.

01:06:42.240 --> 01:06:45.320 align:middle line:84%
Then each one of
these slices, so we

01:06:45.320 --> 01:06:49.380 align:middle line:84%
have some slices that say this
is t equals 0 up to t equals 1--

01:06:49.380 --> 01:06:50.680 align:middle line:90%
other t's in the middle.

01:06:50.680 --> 01:06:53.520 align:middle line:84%
Each one of these
slices, one direction

01:06:53.520 --> 01:06:55.880 align:middle line:84%
is going to be squished,
and the other direction

01:06:55.880 --> 01:06:58.720 align:middle line:90%
is going to stay the same.

01:06:58.720 --> 01:07:04.530 align:middle line:84%
So let's say that at t equals 1
cross-section of this picture,

01:07:04.530 --> 01:07:09.000 align:middle line:84%
I see a triangle
that looks like this.

01:07:09.000 --> 01:07:11.910 align:middle line:90%


01:07:11.910 --> 01:07:16.790 align:middle line:84%
And there's going to be some
direction that gets compressed.

01:07:16.790 --> 01:07:21.770 align:middle line:84%
When I apply Lar inverse, that
direction gets compressed.

01:07:21.770 --> 01:07:24.990 align:middle line:84%
The output is going
to look like so.

01:07:24.990 --> 01:07:27.630 align:middle line:84%
And that has been
actually worked

01:07:27.630 --> 01:07:32.550 align:middle line:84%
out so that this blue and yellow
get kind of squished together.

01:07:32.550 --> 01:07:35.950 align:middle line:90%
And orange is still there.

01:07:35.950 --> 01:07:41.030 align:middle line:84%
And at t equals 0, if I
look at the bottom disk,

01:07:41.030 --> 01:07:42.200 align:middle line:90%
I have the same picture.

01:07:42.200 --> 01:07:47.750 align:middle line:90%


01:07:47.750 --> 01:07:51.670 align:middle line:84%
And there's some direction
that's going to be compressed.

01:07:51.670 --> 01:07:55.310 align:middle line:90%
And it's not the same direction.

01:07:55.310 --> 01:07:57.290 align:middle line:84%
I do somewhat
arbitrarily that one.

01:07:57.290 --> 01:08:05.566 align:middle line:90%


01:08:05.566 --> 01:08:10.250 align:middle line:84%
If you compress that way, that
will look sort of like so.

01:08:10.250 --> 01:08:13.810 align:middle line:84%
And the blue point
will be over here.

01:08:13.810 --> 01:08:17.019 align:middle line:84%
And the yellow and orange point
will be somewhat close together.

01:08:17.019 --> 01:08:19.529 align:middle line:90%


01:08:19.529 --> 01:08:22.090 align:middle line:84%
And as you look at all
the intermediate heights,

01:08:22.090 --> 01:08:23.510 align:middle line:90%
you'll have a similar picture.

01:08:23.510 --> 01:08:25.569 align:middle line:84%
But this compression
direction is

01:08:25.569 --> 01:08:28.910 align:middle line:84%
going to be changing
continuously as you go up.

01:08:28.910 --> 01:08:46.029 align:middle line:90%


01:08:46.029 --> 01:08:47.750 align:middle line:84%
So I'm not sure if
this will be helpful,

01:08:47.750 --> 01:08:49.770 align:middle line:84%
but I brought several
props to try to illustrate

01:08:49.770 --> 01:08:51.569 align:middle line:90%
this important process.

01:08:51.569 --> 01:08:55.609 align:middle line:84%
So if you imagine that you have
a bunch of unipotent orbits

01:08:55.609 --> 01:08:58.870 align:middle line:84%
in your homogeneous
space like so,

01:08:58.870 --> 01:09:04.670 align:middle line:84%
and we apply a, first of all,
they will stretch this way.

01:09:04.670 --> 01:09:07.050 align:middle line:84%
And second of all, there
will be some compression.

01:09:07.050 --> 01:09:10.750 align:middle line:84%
So maybe at the top, they might
get smooshed together this way.

01:09:10.750 --> 01:09:12.229 align:middle line:84%
And at the bottom,
they might get

01:09:12.229 --> 01:09:16.299 align:middle line:84%
smooshed together a different
way, sort of like this.

01:09:16.299 --> 01:09:22.630 align:middle line:90%


01:09:22.630 --> 01:09:29.310 align:middle line:84%
Or if this piece of Play-Doh
is this cylinder here,

01:09:29.310 --> 01:09:32.390 align:middle line:84%
what's going to happen to
it when we apply the map?

01:09:32.390 --> 01:09:35.870 align:middle line:90%
Well, it's going to get taller.

01:09:35.870 --> 01:09:37.970 align:middle line:90%
And it's going to get squished.

01:09:37.970 --> 01:09:41.750 align:middle line:84%
And maybe the top will
get squished that way.

01:09:41.750 --> 01:09:45.149 align:middle line:84%
But the bottom will
get squished that way.

01:09:45.149 --> 01:09:46.910 align:middle line:84%
And in the middle,
it'll get squished

01:09:46.910 --> 01:09:49.149 align:middle line:90%
some direction in the middle.

01:09:49.149 --> 01:09:52.750 align:middle line:84%
And that will-- if I
had done it carefully.

01:09:52.750 --> 01:09:54.350 align:middle line:90%
I practiced it at home.

01:09:54.350 --> 01:09:56.490 align:middle line:84%
It produces something
that's sort of like a helix.

01:09:56.490 --> 01:10:12.480 align:middle line:90%


01:10:12.480 --> 01:10:15.570 align:middle line:84%
So let me say briefly
why this is relevant.

01:10:15.570 --> 01:10:20.060 align:middle line:90%


01:10:20.060 --> 01:10:22.200 align:middle line:84%
And then if we have
time, if we feel like it,

01:10:22.200 --> 01:10:24.400 align:middle line:84%
we could try to do some
computations with matrices

01:10:24.400 --> 01:10:24.900 align:middle line:90%
to check.

01:10:24.900 --> 01:10:27.640 align:middle line:90%


01:10:27.640 --> 01:10:29.380 align:middle line:90%
So why is this relevant?

01:10:29.380 --> 01:10:34.030 align:middle line:84%
Well, if you had a
situation like this, then--

01:10:34.030 --> 01:10:38.620 align:middle line:90%


01:10:38.620 --> 01:10:42.380 align:middle line:90%
yeah, OK, let's go back up here.

01:10:42.380 --> 01:10:44.500 align:middle line:90%
So this guy is really tall.

01:10:44.500 --> 01:10:47.300 align:middle line:84%
We're going to cut it
into a sequence of pieces

01:10:47.300 --> 01:10:49.420 align:middle line:90%
that each have height 1.

01:10:49.420 --> 01:10:52.360 align:middle line:84%
And each one of these pieces
is going to fold back in.

01:10:52.360 --> 01:10:59.940 align:middle line:90%


01:10:59.940 --> 01:11:05.160 align:middle line:84%
So over here, there
is an unused color.

01:11:05.160 --> 01:11:07.930 align:middle line:84%
There's a purple
piece over here.

01:11:07.930 --> 01:11:11.560 align:middle line:90%


01:11:11.560 --> 01:11:15.820 align:middle line:84%
That purple piece is going
to fold back in over here.

01:11:15.820 --> 01:11:21.120 align:middle line:90%


01:11:21.120 --> 01:11:23.570 align:middle line:90%
Can you see that?

01:11:23.570 --> 01:11:25.320 align:middle line:84%
And then there are a
bunch of other pieces

01:11:25.320 --> 01:11:26.153 align:middle line:90%
of different colors.

01:11:26.153 --> 01:11:28.758 align:middle line:84%
Those other pieces will somehow
get slotted in in an order

01:11:28.758 --> 01:11:31.300 align:middle line:84%
that I don't understand at all,
but they'll all go somewhere.

01:11:31.300 --> 01:11:33.820 align:middle line:90%


01:11:33.820 --> 01:11:36.880 align:middle line:84%
Now, inside of this purple
piece, so over here,

01:11:36.880 --> 01:11:42.940 align:middle line:84%
we have a yellow orbit, a blue
orbit, and an orange orbit.

01:11:42.940 --> 01:11:44.840 align:middle line:90%
Let's just focus on those three.

01:11:44.840 --> 01:11:47.680 align:middle line:84%
They are all going
to pass through here.

01:11:47.680 --> 01:11:54.620 align:middle line:84%
And we could have the blue
orbit and the orange orbit

01:11:54.620 --> 01:11:58.300 align:middle line:84%
and the yellow orbit all kind
of spread out from each other.

01:11:58.300 --> 01:12:00.100 align:middle line:84%
Or it could be that
they are smooshed

01:12:00.100 --> 01:12:02.700 align:middle line:90%
together close to each other.

01:12:02.700 --> 01:12:04.740 align:middle line:84%
Anyway, whatever
happens here, this

01:12:04.740 --> 01:12:06.437 align:middle line:90%
is just slotted in over here.

01:12:06.437 --> 01:12:08.520 align:middle line:84%
So that will tell us how
they're spaced over here.

01:12:08.520 --> 01:12:15.260 align:middle line:90%


01:12:15.260 --> 01:12:18.300 align:middle line:84%
When we have
something like this,

01:12:18.300 --> 01:12:22.300 align:middle line:84%
that will produce two
unit length unipotent

01:12:22.300 --> 01:12:24.220 align:middle line:84%
orbits over here in
this purple thing that

01:12:24.220 --> 01:12:26.540 align:middle line:90%
are really close to each other.

01:12:26.540 --> 01:12:30.380 align:middle line:84%
And that phenomenon is the enemy
of the unipotent orbit spreading

01:12:30.380 --> 01:12:31.300 align:middle line:90%
out.

01:12:31.300 --> 01:12:33.580 align:middle line:84%
Unipotent orbit is definitely
going to be really long,

01:12:33.580 --> 01:12:35.020 align:middle line:84%
but it may not
spread out because it

01:12:35.020 --> 01:12:37.478 align:middle line:84%
may have many segments that
are really close to each other.

01:12:37.478 --> 01:12:43.220 align:middle line:90%


01:12:43.220 --> 01:12:45.180 align:middle line:90%
Let me scribe.

01:12:45.180 --> 01:12:54.290 align:middle line:84%
So to prove that u0t
of x spreads out,

01:12:54.290 --> 01:13:06.250 align:middle line:84%
we want to prove that
coincidences of this type

01:13:06.250 --> 01:13:06.950 align:middle line:90%
are rare.

01:13:06.950 --> 01:13:11.570 align:middle line:90%


01:13:11.570 --> 01:13:14.450 align:middle line:90%
And why are they rare?

01:13:14.450 --> 01:13:18.450 align:middle line:84%
Well, they're rare because this
compression angle is rotating

01:13:18.450 --> 01:13:21.130 align:middle line:90%
smoothly as we vary t.

01:13:21.130 --> 01:13:24.530 align:middle line:84%
And so even though there
may be one projection that

01:13:24.530 --> 01:13:27.300 align:middle line:84%
smooshes a lot of
things together,

01:13:27.300 --> 01:13:29.050 align:middle line:84%
we have learned in
this course that if you

01:13:29.050 --> 01:13:31.470 align:middle line:84%
look at the projection in
all different directions,

01:13:31.470 --> 01:13:35.530 align:middle line:84%
most of them do not smoosh
things together very much.

01:13:35.530 --> 01:13:38.910 align:middle line:84%
And that forces this
unipotent orbit to spread out.

01:13:38.910 --> 01:14:02.337 align:middle line:90%


01:14:02.337 --> 01:14:03.170 align:middle line:90%
AUDIENCE: I'm sorry.

01:14:03.170 --> 01:14:05.490 align:middle line:90%
I'm a little confused.

01:14:05.490 --> 01:14:10.950 align:middle line:84%
Why doesn't the direction
get compressed and changed?

01:14:10.950 --> 01:14:16.590 align:middle line:84%
From that picture, after
we applied LaR inverse,

01:14:16.590 --> 01:14:22.190 align:middle line:90%
it's always long.

01:14:22.190 --> 01:14:25.490 align:middle line:90%
[INAUDIBLE] another direction.

01:14:25.490 --> 01:14:28.630 align:middle line:90%


01:14:28.630 --> 01:14:30.770 align:middle line:84%
LAWRENCE GUTH:
Yeah, OK, all right.

01:14:30.770 --> 01:14:33.310 align:middle line:84%
So the question is,
why does this happen?

01:14:33.310 --> 01:14:35.430 align:middle line:84%
And in fact, there was
a point that the picture

01:14:35.430 --> 01:14:37.350 align:middle line:84%
that I drew up
there sort of makes

01:14:37.350 --> 01:14:39.360 align:middle line:90%
it look like it doesn't happen.

01:14:39.360 --> 01:14:39.860 align:middle line:90%
Correct?

01:14:39.860 --> 01:14:48.370 align:middle line:90%


01:14:48.370 --> 01:14:52.590 align:middle line:84%
So in my picture, this
is not exactly accurate,

01:14:52.590 --> 01:14:55.570 align:middle line:84%
but I drew it as though
this map was just

01:14:55.570 --> 01:14:59.010 align:middle line:84%
a linear map that takes a
cube to a rectangular solid.

01:14:59.010 --> 01:15:01.270 align:middle line:90%
And I drew it as though--

01:15:01.270 --> 01:15:02.910 align:middle line:84%
well, I tried to
show that this--

01:15:02.910 --> 01:15:07.150 align:middle line:84%
so these unipotent orbits
are more or less vertical.

01:15:07.150 --> 01:15:08.630 align:middle line:90%
So here's what would work.

01:15:08.630 --> 01:15:11.470 align:middle line:84%
So if the unipotent orbits
were literally vertical

01:15:11.470 --> 01:15:14.410 align:middle line:84%
and if this map was
literally a linear map,

01:15:14.410 --> 01:15:16.570 align:middle line:90%
then this would not happen.

01:15:16.570 --> 01:15:19.070 align:middle line:84%
The compression direction would
be the same at every height.

01:15:19.070 --> 01:15:21.770 align:middle line:90%


01:15:21.770 --> 01:15:24.523 align:middle line:90%
So I think you have to pick--

01:15:24.523 --> 01:15:29.270 align:middle line:90%


01:15:29.270 --> 01:15:32.550 align:middle line:84%
this Lie group is like
it's a submanifold of R4.

01:15:32.550 --> 01:15:34.930 align:middle line:84%
But if you want to
represent it in coordinates

01:15:34.930 --> 01:15:37.610 align:middle line:84%
in three dimensions, you
have to make a choice.

01:15:37.610 --> 01:15:41.370 align:middle line:84%
There's not a completely
canonical choice.

01:15:41.370 --> 01:15:47.270 align:middle line:84%
So we could make the
choice that the unipotent

01:15:47.270 --> 01:15:50.230 align:middle line:90%
orbits were literally vertical.

01:15:50.230 --> 01:15:55.390 align:middle line:84%
And if we did that, this picture
would not be right anymore.

01:15:55.390 --> 01:15:59.010 align:middle line:84%
If we did that, then this
would have a bit of a twist.

01:15:59.010 --> 01:16:00.310 align:middle line:90%
This would be more--

01:16:00.310 --> 01:16:01.970 align:middle line:84%
this would be like
a helix thing.

01:16:01.970 --> 01:16:05.030 align:middle line:90%


01:16:05.030 --> 01:16:06.690 align:middle line:84%
Or another way of
looking at it is

01:16:06.690 --> 01:16:10.190 align:middle line:84%
I think we could choose
coordinates in such a way

01:16:10.190 --> 01:16:15.670 align:middle line:84%
that this map was pretty
close to a linear map,

01:16:15.670 --> 01:16:19.470 align:middle line:84%
but then these unipotent
orbits would not be vertical.

01:16:19.470 --> 01:16:22.710 align:middle line:84%
And instead, they would
kind of spiral a little bit

01:16:22.710 --> 01:16:24.910 align:middle line:90%
as you went up.

01:16:24.910 --> 01:16:28.010 align:middle line:84%
And so the angle between
two unipotent guys

01:16:28.010 --> 01:16:28.760 align:middle line:90%
would be changing.

01:16:28.760 --> 01:16:33.750 align:middle line:90%


01:16:33.750 --> 01:16:40.070 align:middle line:84%
Now, at some point, why
do you believe that?

01:16:40.070 --> 01:16:42.982 align:middle line:84%
So at some point, we
should do a computation.

01:16:42.982 --> 01:16:45.660 align:middle line:90%


01:16:45.660 --> 01:16:47.680 align:middle line:84%
I don't think I'll do it
in the last 2 minutes,

01:16:47.680 --> 01:16:49.222 align:middle line:84%
but I'll just say
what you should do.

01:16:49.222 --> 01:16:52.500 align:middle line:90%


01:16:52.500 --> 01:16:59.267 align:middle line:84%
So we figured out which
direction is compressing.

01:16:59.267 --> 01:17:01.600 align:middle line:84%
We could have written it down
a little bit more clearly.

01:17:01.600 --> 01:17:03.142 align:middle line:84%
But if you trace
through our thoughts

01:17:03.142 --> 01:17:07.300 align:middle line:84%
today, at any point in the
homogeneous space, if you

01:17:07.300 --> 01:17:09.140 align:middle line:84%
ask which is the
tangent direction that's

01:17:09.140 --> 01:17:12.300 align:middle line:84%
going to get compressed
when you apply Lar,

01:17:12.300 --> 01:17:15.260 align:middle line:90%
we have a formula for it.

01:17:15.260 --> 01:17:19.060 align:middle line:84%
And then we can also say if
you have two nearby unipotent

01:17:19.060 --> 01:17:22.140 align:middle line:84%
orbits, and down
here at the bottom

01:17:22.140 --> 01:17:28.700 align:middle line:84%
there's a little vector between
them, as you go up the orbit,

01:17:28.700 --> 01:17:30.600 align:middle line:90%
what happens to that vector?

01:17:30.600 --> 01:17:32.980 align:middle line:90%
There's a formula for that, too.

01:17:32.980 --> 01:17:35.180 align:middle line:84%
If you compare
these two formulas,

01:17:35.180 --> 01:17:37.560 align:middle line:84%
you will see that
they don't match.

01:17:37.560 --> 01:17:42.380 align:middle line:84%
So if this direction initially
was the compression direction,

01:17:42.380 --> 01:17:45.400 align:middle line:84%
compression direction,
if you follow it up,

01:17:45.400 --> 01:17:48.480 align:middle line:84%
you don't get the compression
direction up here.

01:17:48.480 --> 01:17:50.990 align:middle line:84%
And you can compute
it all with matrices.

01:17:50.990 --> 01:17:54.680 align:middle line:90%


01:17:54.680 --> 01:17:55.200 align:middle line:90%
Yeah.

01:17:55.200 --> 01:17:56.783 align:middle line:84%
AUDIENCE: Is this
just kind of saying,

01:17:56.783 --> 01:18:01.880 align:middle line:84%
if there's a curvature tensor
that's non-zero from the,

01:18:01.880 --> 01:18:06.407 align:middle line:84%
I guess, one of the two
one-parameter groups?

01:18:06.407 --> 01:18:07.240 align:middle line:90%
LAWRENCE GUTH: Yeah.

01:18:07.240 --> 01:18:09.360 align:middle line:84%
The question is, is
this saying that there's

01:18:09.360 --> 01:18:12.600 align:middle line:84%
some kind of curvature
tensor that doesn't vanish?

01:18:12.600 --> 01:18:13.160 align:middle line:90%
Maybe.

01:18:13.160 --> 01:18:15.720 align:middle line:84%
It's not the Riemann
curvature tensor,

01:18:15.720 --> 01:18:19.320 align:middle line:84%
but it might be
something in that spirit.

01:18:19.320 --> 01:18:21.440 align:middle line:90%
Yeah.

01:18:21.440 --> 01:18:24.760 align:middle line:84%
AUDIENCE: The theorem
before is really two cases.

01:18:24.760 --> 01:18:28.200 align:middle line:84%
One, this periodic
number, the picture

01:18:28.200 --> 01:18:31.960 align:middle line:84%
kind of makes it seem
like it wasn't structured.

01:18:31.960 --> 01:18:33.120 align:middle line:90%
It's not periodic.

01:18:33.120 --> 01:18:35.200 align:middle line:90%
Is there something I'm missing?

01:18:35.200 --> 01:18:36.700 align:middle line:84%
LAWRENCE GUTH: Yeah,
great question.

01:18:36.700 --> 01:18:40.300 align:middle line:84%
The question was, this
appears to be a proof sketch

01:18:40.300 --> 01:18:43.060 align:middle line:84%
that every orbit of
the unipotent group

01:18:43.060 --> 01:18:45.500 align:middle line:90%
is evenly distributed.

01:18:45.500 --> 01:18:48.900 align:middle line:84%
And we saw at the beginning
that that is not true.

01:18:48.900 --> 01:18:50.800 align:middle line:84%
So what did what
did we do wrong?

01:18:50.800 --> 01:19:02.520 align:middle line:90%


01:19:02.520 --> 01:19:08.040 align:middle line:84%
All right, so this orbit,
we can write this way.

01:19:08.040 --> 01:19:09.540 align:middle line:84%
Actually, so let
me say the shortest

01:19:09.540 --> 01:19:10.800 align:middle line:90%
answer to your question.

01:19:10.800 --> 01:19:15.580 align:middle line:84%
The key thing is that the
cusp of the homogeneous space

01:19:15.580 --> 01:19:17.060 align:middle line:90%
is really important here.

01:19:17.060 --> 01:19:19.900 align:middle line:90%
And we were neglecting the cusp.

01:19:19.900 --> 01:19:25.180 align:middle line:84%
So the cusp produces a piece of
the fundamental domain that's

01:19:25.180 --> 01:19:26.260 align:middle line:90%
really skinny.

01:19:26.260 --> 01:19:28.240 align:middle line:84%
And it doesn't look
like the rest of it.

01:19:28.240 --> 01:19:32.180 align:middle line:84%
And when something's in
the cusp, it doesn't apply.

01:19:32.180 --> 01:19:34.840 align:middle line:84%
And so, actually, Hedlund
proved another theorem,

01:19:34.840 --> 01:19:38.730 align:middle line:84%
which is that if you take a
cocompact subgroup gamma, then

01:19:38.730 --> 01:19:42.890 align:middle line:84%
in G mod gamma, every
unipotent orbit is dense.

01:19:42.890 --> 01:19:46.170 align:middle line:84%
And this is a sketch of
the proof of that theorem.

01:19:46.170 --> 01:19:53.690 align:middle line:84%
But now what goes wrong is
that this a to the minus Rx

01:19:53.690 --> 01:19:57.690 align:middle line:84%
could be very far in the cusp,
and then our whole analysis

01:19:57.690 --> 01:19:58.890 align:middle line:90%
is wrong.

01:19:58.890 --> 01:20:02.330 align:middle line:84%
So what we've roughly shown
is that either our orbit

01:20:02.330 --> 01:20:07.690 align:middle line:84%
is very well distributed, or a
minus Rx is deep in the cusp.

01:20:07.690 --> 01:20:10.890 align:middle line:84%
And if you unwind what
that means about x,

01:20:10.890 --> 01:20:14.020 align:middle line:84%
it means that it's very close
to being a periodic orbit.

01:20:14.020 --> 01:20:23.090 align:middle line:90%


01:20:23.090 --> 01:20:24.530 align:middle line:90%
Cool.

01:20:24.530 --> 01:20:27.890 align:middle line:84%
So we'll talk a little more
about homogeneous dynamics

01:20:27.890 --> 01:20:28.490 align:middle line:90%
on Thursday.

01:20:28.490 --> 01:20:29.990 align:middle line:84%
I'm not sure exactly
what we'll say.

01:20:29.990 --> 01:20:32.400 align:middle line:90%
Maybe we'll fill in some things.

01:20:32.400 --> 01:20:38.000 align:middle line:90%