
Differential Topology – Lecture 1 by John W. Milnor (1965) [video] - espeed
https://www.youtube.com/watch?v=1LwkljjLBns&list=PLelIK3uylPMFHC6Xny11XFXgwwtv9_PO3&index=1
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cplease
It would have been nice to list the source of these lectures where they are
freely available rather than a pirated Youtube link:

[https://www.simonsfoundation.org/science_lives_video/profess...](https://www.simonsfoundation.org/science_lives_video/professor-
john-w-milnor/)

Published sequel here:
[http://www.ams.org/notices/201106/rtx110600804p.pdf](http://www.ams.org/notices/201106/rtx110600804p.pdf)

~~~
jordigh
It's not freely available if they are not free to copy... and I really wonder
who could possibly be harmed by copying these videos around. I can't imagine
that Milnor would object to having his work widely disseminated. Do you think
if we copy something he made 50 years ago (which really should be in the
public domain by now), he'll be less motivated to generate new lectures at his
84 years of age?

Your usage of the term "pirated" seems to indicate more indignation than is in
order.

~~~
wodenokoto
Yes, it is called "free as in beer".

~~~
jordigh
It's not even free beer. If I go to an event and get a free beer, I'm not
forbidden from giving my free beer to anyone else. It is ridiculous that we
are forbidden from sharing a video that was created 50 years ago just because
Disney wants to make sure Mickey Mouse never falls out of copyright.

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graycat
Gee, I can like that!

For the set of real numbers R, he starts with a function

f: R --> R

such that f(x) is zero for x <= 0, strictly positive otherwise, and infinitely
differentiable. There is such a function, the same or similar, in an exercise
in Rudin, _Principles of Mathematical Analysis_.

At one time I used that function in Rudin to show that for positive integer n
and closed subset C of R^n there exists function

f: R^n --> R

so that f(x) = 0 for all x in C, strictly positive otherwise, and infinitely
differentiable. That result is comparable with the classic Whitney extension
theorem -- Whitney assumed a little more and got a little more.

I discovered this result for and used it in some work in the constraint
qualifications of the Kuhn-Tucker conditions: I constructed a counterexample
that showed that the Zangwill and Kuhn-Tucker constraint qualifications are
independent.

So, back to Milnor's lecture!

I want to see if he drags out the inverse and implicit function theorems
(essentially the local nonlinear version of what is standard from Gauss
elimination in systems of linear equations!).

~~~
SamReidHughes
Neat. Did you do it by making such a function for the open ball, and then,
given an open set defined as a countable union of open balls, define a
sequence of such functions scaled appropriately so that their derivatives all
converge uniformly?

Edit: Looks like you can make that work, according to the answer at
[https://math.stackexchange.com/questions/791248/every-
closed...](https://math.stackexchange.com/questions/791248/every-closed-
subset-e-subseteq-mathbbrn-is-the-zero-point-set-of-a-smooth)

~~~
graycat
I took a countable dense set in R^n - C, x_j, j = 1, 2, ..., and for each j
took the closed circle with center x_j and radius d(x_j, C) (distance from x_j
to C which is well defined due to compactness). Then I defined a function g_j:
R^n --> R 0 outside the circle and on the circle with values on the radii much
as in the function Milnor, Rudin, and I mentioned. So, g_j is infinitely
differentiable, strictly positive on the interior of the circle at x_j, and 0
otherwise. Then I added the functions g_j in a convergent way, again using the
exponential trick. The resulting function was as desired, 0 on C, strictly
positive otherwise, and infinitely differentiable.

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delish
Long ago I read a quote from a post about mathematicians who are good
communicators. I'd love to come across it again.

Of John Milnor, this post proclaimed, "When he speaks, you understand."

I try to notice rare compliments, like that one. Feynman received one like
that:

A female engineer says about him: _" Yes, [Feynman's sexism] really annoys
me," she said. "On the other hand, he is the only one who ever explained
quantum mechanics to me as if I could understand it."_

The only one!

[http://longnow.org/essays/richard-feynman-connection-
machine...](http://longnow.org/essays/richard-feynman-connection-machine/)

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tgb
What causes the halos in this film? I notice whenever there is an unusually
light or dark spot, it gets a halo around it of the opposite lightness. See
his hand as he starts writing shortly after this time stop to see what I mean:
[https://youtu.be/1LwkljjLBns?list=PLelIK3uylPMFHC6Xny11XFXgw...](https://youtu.be/1LwkljjLBns?list=PLelIK3uylPMFHC6Xny11XFXgwwtv9_PO3&t=831)

~~~
ddumas
It's not a filter, as others have suggested, but rather a standard artifact of
vacuum-tube-based television cameras that were in use at the time. (I don't
know why this lecture wasn't filmed, but the extensive dark halo effect makes
it clear this was shot with a TV camera. This was the early era for magnetic
video tape, but I assume that's how the recording was preserved.)

Anyway, the point is that these TV cameras are based on the fact that incoming
light will dislodge electrons from a thin plate in a vaccum tube in an amount
proportional to brightness. A very bright spot in the image produces a shower
of electrons that is more powerful than the rest of the tube (the part that
detects the electrons) can deal with. The net result of this "splash" of
electrons is a mild desensitization of the detection apparatus around the
bright spot. This makes the nearby stuff appear darker.

You mentioned that dark spots also seem to have a bright halo, but I don't see
that in the video, and it isn't consistent with the usual artifacts of these
cameras. Are you sure?

~~~
tgb
Very interesting, thanks.

There do seem to be subtle halos around dark spots, though. See here:
[https://youtu.be/1LwkljjLBns?list=PLelIK3uylPMFHC6Xny11XFXgw...](https://youtu.be/1LwkljjLBns?list=PLelIK3uylPMFHC6Xny11XFXgwwtv9_PO3&t=850)
It's not very noticeable in a still but is pretty clear when his hand shadow
moves.

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figure8
I really appreciate how he writes and draws the slides as he speaks. With math
lectures, this always encourages me to think deeply about the details and
implications of each line. I find it better than pre-written slides, now so
easily created and presented using Power Point.

~~~
jordigh
I think this style of presentation is still very common in maths departments.
Mathematicians tend to favour low-tech presentations. Blackboards are still in
vogue!

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mrcactu5
does anyone else dance to the music in the beginning?

