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Week of...
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Links
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Fall Semester
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1
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Sep 10
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About, Tue, Thu
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2
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Sep 17
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Tue, HW1, Thu
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3
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Sep 24
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Tue, Photo, Thu
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4
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Oct 1
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Questionnaire, Tue, HW2, Thu
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5
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Oct 8
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Thanksgiving, Tue, Thu
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6
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Oct 15
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Tue, HW3, Thu
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7
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Oct 22
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Tue, Thu
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8
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Oct 29
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Tue, HW4, Thu, Hilbert sphere
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9
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Nov 5
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Tue,Thu, TE1
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10
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Nov 12
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Tue, Thu
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11
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Nov 19
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Tue, Thu, HW5
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12
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Nov 26
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Tue, Thu
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13
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Dec 3
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Tue, Thu, HW6
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Spring Semester
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14
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Jan 7
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Tue, Thu, HW7
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15
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Jan 14
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Tue, Thu
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16
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Jan 21
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Tue, Thu, HW8
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17
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Jan 28
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Tue, Thu
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18
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Feb 4
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Tue
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19
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Feb 11
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TE2, Tue, HW9, Thu, Feb 17: last chance to drop class
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R
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Feb 18
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Reading week
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20
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Feb 25
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Tue, Thu, HW10
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21
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Mar 3
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Tue, Thu
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22
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Mar 10
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Tue, Thu, HW11
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23
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Mar 17
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Tue, Thu
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24
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Mar 24
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Tue, HW12, Thu
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25
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Mar 31
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Referendum,Tue, Thu
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26
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Apr 7
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Tue, Thu
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R
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Apr 14
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Office hours
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R
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Apr 21
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Office hours
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F
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Apr 28
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Office hours, Final (Fri, May 2)
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Register of Good Deeds
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Errata to Bredon's Book
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Announcements go here
Dror's Computer Program for C+
C
Our handout today is a printout of a Mathematica notebook that computes the measure of the projection of
in a direction
(where
is the standard Cantor set). Here's the notebook, and here's a PDF version. Also, here's the main picture on that notebook:
Typed Class Notes
The notes below are by the students and for the students. Hopefully they are useful, but they come with no guarantee of any kind.
First Hour
Today's Agenda:
Proof of Sard's Theorem. That is, for
being smooth, the measure
(critical values of f) = 0.
Comments regarding last class
1) In our counterexample to Sard's Theorem for the case of
functions it was emphasized that there are functions f from a Cantor set C' to the Cantor set C. We then let
and hence the critical values are
as was shown last time. The sketch of such an f was the same as last class.
Furthermore, in general we can find a
such function where we make the "bumps" in f smoother as needed and so
where
is a "very thin" Cantor set. But now let
which will have an image of
an interval.
2) The code, and what the program does, for Dror's program (above) was described. It is impractical to describe it here in detail and so I will only comment that it computes the measure of
for various alpha and that the methodology relied on the 2nd method of proof regarding C+C done last class.
Proof of Sard's Theorem
Firstly, it is enough to argue locally.
Further, the technical assumption about manifolds that until now has been largely ignored is that our M must be second countable. Recall that this means that there is a countable basis for the topology on M.
As a counterexample to Sard's Theorem when M is NOT second countable consider the real line with the discrete topology, a zero dimensional manifold, mapping via the identity onto the real line with the normal topology. Every point in the real line is thus a critical point and the real line has non zero measure.
We can restrict our neighborhoods so that we can assume
and
The general idea here is that if we consider a function g=f' that is nonzero at p but that f is zero at p, the inverse image is (in some chart) a straight line (a manifold). As such, we will inductively reduce the dimension from m down to zero. For m=0 there is nothing to prove. Hence we assume true for m-1.
Now, set
all partial derivities of f of order
for
Set
is not onto }. This is just the critical points.
Clearly
We will show by backwards induction that
Comment:
We have not actually defined the measure
. We use it merely to denote that
has measure zero, a concept that we DID define.
Second Hour
Claim 1
has measure 0.
Proof
W.L.O.G (without loss of generality) we can assume n=1. Intuitively this is reasonable as in lower dimensions the theorem is harder to prove; indeed, a set of size
in 1D becomes a smaller set of size
in 2D etc. More precisely, for
,
. Applying the proposition that if A is of measure zero in
then
is measure zero in
we now see that assuming n=1 is justified.
Reminder
Taylor's Theorem: for smooth enough
and some
then
for some t between x and
.
For
all but the last term vanishes and so we can conclude that f(x) is bounded by a constant times
.
Now let us consider a box B in
containing a section of
. We divide B into
boxes of side
.
By Taylor's Theorem,
of an interval of length
where the constant
is determined by Taylor's Theorem. Call this interval
Hence,
But
which tends to zero as
tends to infinity.
Q.E.D for Claim 1
Claim 2
has measure zero for
. We just proved this for k=m.
Now, W.L.O.G.
is the empty set. If not, just consider
which is still a manifold as
is closed (as it is determined by the "closed" condition that a determinant equals zero)
So, there is some kth derivative g of f such that
.
So
but
is at least locally a manifold of dimension 1 less. So,
which has measure zero due to our induction hypothesis.
Q.E.D for Claim 2
Claim 3
is of measure zero.
Recall
is defined differently from the
and so requires a different technique to prove.
W.L.O.G. lets assume that
is the empty set. So, some derivative of f is not zero. W.L.O.G.
is non zero near some point p. We can simply move to a coordinate system where this is true.
The idea here is to prove that the intersection with any "slice" has measure zero where we will then invoke a theorem that will claim everything has measure zero.
So, let U be an open neighborhood of a point
. Consider
and let
be onto. Using our previous theorem for the local structure of such a submersion W.L.O.G. let us assume
via
. That is,
.
Our differential df then is just the matrix whose first row consists of
. Then df is onto if the submatrix consisting of all but the first row and first column is invertible.
For notational convenience let us say
.
Now define
Also lets denote "critical points of f by CP(f) and "critical values of f" by CV(f)
Claim 4
The
.
But
has measure zero by our induction.
Lemma 1
If
is closed and has
then
.
Proof
Note: We prove this significantly differently then in Bredon
Sublemma
If
for an open U cover
for a closed A then
such that
Indeed, let
be
then d is a continuous function of a compact set and so obtains a minimum and since d>0 then
. But this
works for the claim. Q.E.D
The rest of the proof of Lemma 1, and of Sard's Theorem will be left until next class