23 ms·

Differential Topology – Lecture 1 by John W. Milnor (1965) [video]

63 points by espeed 11y ago

tgb 11y ago
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=PLelIK3uylPMFHC6Xny11XFXgwwtv9_PO3&t=831 https://youtu.be/1LwkljjLBns?list=PLelIK3uylPMFHC6Xny11XFXgw...
- semi-extrinsic 11y ago
  Looks like someone has applied an unsharp mask. Example: https://upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Usm-unsharp-mask.png/320px-Usm-unsharp-mask.png https://upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Us...
  - theoh 11y ago
    Could also be a high dynamic range operator: they do the same kind of thing to preserve contrast across large ranges of intensity. See Larson's paper http://gaia.lbl.gov/btech/papers/39882.pdf http://gaia.lbl.gov/btech/papers/39882.pdf etc.
- ddumas 11y ago
  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 11y ago
    Very interesting, thanks. There do seem to be subtle halos around dark spots, though. See here: https://youtu.be/1LwkljjLBns?list=PLelIK3uylPMFHC6Xny11XFXgwwtv9_PO3&t=850 https://youtu.be/1LwkljjLBns?list=PLelIK3uylPMFHC6Xny11XFXgw... It's not very noticeable in a still but is pretty clear when his hand shadow moves.
delish 11y ago
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...
cplease 11y ago
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/professor-john-w-milnor/ https://www.simonsfoundation.org/science_lives_video/profess... Published sequel here: http://www.ams.org/notices/201106/rtx110600804p.pdf http://www.ams.org/notices/201106/rtx110600804p.pdf
- nairboon 11y ago
  So how do you pirate a Youtube link?
  - cplease 11y ago
    "Link to an unauthorized copy on Youtube." Feel better?
    - cft 11y ago
      Unauthorized by whom? You should contact the copyright owner and alert them of the infringement then!
- jordigh 11y ago
  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 11y ago
    Yes, it is called "free as in beer".
    - 11y ago
graycat 11y ago
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!).
- analognoise 11y ago
  Isn't f(x) = 0 in C a trivial solution?
  - graycat 11y ago
    No. The domain of f is all of R^n so that f(x) has to be defined for all x in R^n. Then there is the requirement that f be strictly positive otherwise, that is, in the open set outside C. The result is a bit amazing: It says that any closed set can be the level set for an infinitely differentiable function. Can add some interest when consider how bizarre some closed set are: So, in the plane can take C as a sample path of Brownian motion. So, there's an infinitely differentiable function zero on that sample path and positive otherwise. Next, the Mandelbrot set is closed. So, there's an infinitely differentiable function positive everywhere but zero on the Mandelbrot set. Next consider the Cantor set or Cantor sets of positive measure. Same story. So, a really smooth, infinitely differentiable, function can have for a level set anything at all considering that any level set of a continuous function is closed. There's a famous paper in mathematical economics by Arrow, Hurwicz, and Uzawa with a question with no answer, and my work answers the question. Yes, Arrow and Hurwicz won prizes in economics -- IIRC so far poor Uzawa has not won such a prize! For something intuitive, consider an infinitely differentiable mountain range, and pour in some water to make a lake or many lakes (assume a porous mountain) all with the same altitude. Then, the lake boundary is a level set of an infinitely differentiable function. As we know from Mandelbrot, commonly, roughly lake boundaries are fractals. So, we're fine; everything holds together: My result shows that for Mandelbrot's fractal lakes, the ground can be smooth, infinitely differentiable. Did Mandelbrot know that? For rational p/q for positive whole numbers p and q with p/q in lowest terms, consider in R^2 a ray from the origin of length 1/q at angle p radians. Do this for all such countably infinitely many rationals. Then the set of all these rays is closed but is a spiny urchin, really bizarre. That's the closed set I used for my counterexample. Can also apply this to zero day detection of anomalies in server farms and networks: So, suppose collect 20 times a second numerical data on 10 variables. Then that vector of 10 variables has a probability distribution. Suppose it also has a continuous density. Then pour in water to make lakes -- all the same altitude. Get a point in a lake, then declare an anomaly. The volume of land under the lakes is essentially the false alarm rate. Pour in more water and get a higher false alarm rate but also a higher detection rate. Here the set where we declare an anomaly has the highest possible area for the given false alarm rate, and there's another sense in which the detector has optimal combinations for false alarm rate and detection rate. How 'bout that!
figure8 11y ago
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 11y ago
  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!
mrcactu5 11y ago
does anyone else dance to the music in the beginning?

63 points by espeed 11y ago

tgb 11y ago

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=PLelIK3uylPMFHC6Xny11XFXgwwtv9_PO3&t=831 https://youtu.be/1LwkljjLBns?list=PLelIK3uylPMFHC6Xny11XFXgw...

semi-extrinsic 11y ago
Looks like someone has applied an unsharp mask. Example: https://upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Usm-unsharp-mask.png/320px-Usm-unsharp-mask.png https://upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Us...
- theoh 11y ago
  Could also be a high dynamic range operator: they do the same kind of thing to preserve contrast across large ranges of intensity. See Larson's paper http://gaia.lbl.gov/btech/papers/39882.pdf http://gaia.lbl.gov/btech/papers/39882.pdf etc.
ddumas 11y ago
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 11y ago
  Very interesting, thanks. There do seem to be subtle halos around dark spots, though. See here: https://youtu.be/1LwkljjLBns?list=PLelIK3uylPMFHC6Xny11XFXgwwtv9_PO3&t=850 https://youtu.be/1LwkljjLBns?list=PLelIK3uylPMFHC6Xny11XFXgw... It's not very noticeable in a still but is pretty clear when his hand shadow moves.

delish 11y ago

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...

cplease 11y ago

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/professor-john-w-milnor/ https://www.simonsfoundation.org/science_lives_video/profess... Published sequel here: http://www.ams.org/notices/201106/rtx110600804p.pdf http://www.ams.org/notices/201106/rtx110600804p.pdf

nairboon 11y ago
So how do you pirate a Youtube link?
- cplease 11y ago
  "Link to an unauthorized copy on Youtube." Feel better?
  - cft 11y ago
    Unauthorized by whom? You should contact the copyright owner and alert them of the infringement then!
jordigh 11y ago
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 11y ago
  Yes, it is called "free as in beer".
  - 11y ago

graycat 11y ago

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!).

analognoise 11y ago
Isn't f(x) = 0 in C a trivial solution?
- graycat 11y ago
  No. The domain of f is all of R^n so that f(x) has to be defined for all x in R^n. Then there is the requirement that f be strictly positive otherwise, that is, in the open set outside C. The result is a bit amazing: It says that any closed set can be the level set for an infinitely differentiable function. Can add some interest when consider how bizarre some closed set are: So, in the plane can take C as a sample path of Brownian motion. So, there's an infinitely differentiable function zero on that sample path and positive otherwise. Next, the Mandelbrot set is closed. So, there's an infinitely differentiable function positive everywhere but zero on the Mandelbrot set. Next consider the Cantor set or Cantor sets of positive measure. Same story. So, a really smooth, infinitely differentiable, function can have for a level set anything at all considering that any level set of a continuous function is closed. There's a famous paper in mathematical economics by Arrow, Hurwicz, and Uzawa with a question with no answer, and my work answers the question. Yes, Arrow and Hurwicz won prizes in economics -- IIRC so far poor Uzawa has not won such a prize! For something intuitive, consider an infinitely differentiable mountain range, and pour in some water to make a lake or many lakes (assume a porous mountain) all with the same altitude. Then, the lake boundary is a level set of an infinitely differentiable function. As we know from Mandelbrot, commonly, roughly lake boundaries are fractals. So, we're fine; everything holds together: My result shows that for Mandelbrot's fractal lakes, the ground can be smooth, infinitely differentiable. Did Mandelbrot know that? For rational p/q for positive whole numbers p and q with p/q in lowest terms, consider in R^2 a ray from the origin of length 1/q at angle p radians. Do this for all such countably infinitely many rationals. Then the set of all these rays is closed but is a spiny urchin, really bizarre. That's the closed set I used for my counterexample. Can also apply this to zero day detection of anomalies in server farms and networks: So, suppose collect 20 times a second numerical data on 10 variables. Then that vector of 10 variables has a probability distribution. Suppose it also has a continuous density. Then pour in water to make lakes -- all the same altitude. Get a point in a lake, then declare an anomaly. The volume of land under the lakes is essentially the false alarm rate. Pour in more water and get a higher false alarm rate but also a higher detection rate. Here the set where we declare an anomaly has the highest possible area for the given false alarm rate, and there's another sense in which the detector has optimal combinations for false alarm rate and detection rate. How 'bout that!

figure8 11y ago

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 11y ago
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!

mrcactu5 11y ago

does anyone else dance to the music in the beginning?