Differentiable structures definition and classification - Lec 07 - Frederic Schuller

[@zidanzidan1653]

It simply blow my mind away to know that 4 dimensions case is that much singled out!

[@mrnarason]

It's like nature is hinting at something about our physical world

[@LordVysh]

@@mrnarason It's kind of the opposite. The reason that exceptions like this or pi being irrational, etc. is that we are imposing our view of the world on the natural world. What is exceptional to us, is natural in the natural world. In the natural world it is our imposition that is exceptional.

[@smalljbug]

@@LordVysh well we evolved here so our imposition ought to be pretty fine tuned.

[@zfengjoe]

At 12:18, maybe the essential reason why holomorphicity on C is much stronger than differentiability on R^2 is NOT the approaching direction is arbitrary. Because the differentiability already requires this. The key reason behind the holomorphicity is that we can represent the Jacobian as complex numbers (while not every 2-d Real matrix can do this) which is exactly the C-R equation.

[@jasperday9020]

I love this explanation!

[@tim-701cca]

13:31-17:00 whitney theorem 46:40 structures 54:36 smooth structures up to diffeo. 58:00 surgery

[@TheNonHiddenSingularity]

23:30 I don't get what he is talking about : the function x: a --> a^(1/3) is differentiable at 0 it's just that it gives 0 no ?

[@theflaggeddragon9472]

At 11:43, I believe f:C --> C not C^2 --> C^2. Correct? edit: nevermind I just kept watching >_

[@millerfour2071]

11:41, 13:34, 24:10, 30:00, 36:56 (is it possible to have charts mapping open sets to different dimensions of R for the same manifold), 38:53, 40:22, 41:07, 45:20 (Bijection (Set)->Continuity (Topological space)->Differentiability (Smooth Mfd)), 48:44, 55:06, 1:02:03, 1:07:18, 1:11:04

[@SekerliRaki]

Late but I think if you have it any other dimension you may lose the bijectiveness and other properties. Not sure how embedding comes into this, it probably works with higher dimensions but not lower dimensions if it works at all.

[@KipIngram]

24:20 - What? I don't get it. How is there anything infinite in the inverse of either of those transitions? The inverse of the identity is the identity, and the inverse of x^(1/3) is x^3. Both are perfectly well-behaved.

[@nono-bt8gy]

The derivative of x^1/3 is not defined in 0

[@mappingtheshit]

I guess more clarification is needed for your question. Here, in the union of two atlases we consider two transition maps. First is id after inverse(x), which you correctly found as x^3 and second transition map is not identity to identity, but the x after inverse(id) that gives us x^(1/3), and this x^(1/3) has no derivative in the 0

[@SebastianBorszcz-to5zl]

I wonder why he then said that he chart containing x -> x^(1/3) by itself is C-infinity, as the derivative is not defined everywhere? So I even believe this chart on its own is only C-0..., not only the union of these two atlasses.

[@jaydevsinghrao7350]

@@SebastianBorszcz-to5zl maybe because there's no other chart to test the C-infinity compatibility with in the atlas?

[@SebastianBorszcz-to5zl]

@@jaydevsinghrao7350 I don't think so. I guess x -> x(1/5) is compatible with x -> x^(1/3), as it is also differentiable everywhere except 0. (same for x -> x^(1/7) and so on...).

[@gaiolobez]

59:10 THEY DID SURGERY ON A SPHERE

[@wasimrehman5816]

Great lecture course, Schuller's explanations are very clear. Thank you.

[@yumingbai2023]

Really like the professor whose lectures are always clear and precise.

[@juliangarcia3416]

What a beast this guy

[@scollyer.tuition]

At 49:00, Schuller's "stack of structures" seems to allow an arrow from the set layer to the diff'able structure layer via the group layer, avoiding the topology layer - this looks odd; we can't have diff'ability without a structure providing continuity. I guess this is an error?

[@fredxu9826]

the audience is so serious. It would be a blast in an U.S university when he says "went nuts"

[@McRingil]

the swears are mmuch less impactful in a second language

[@jelmar35]

I did not even notice it, that's how desensitized I am to English swearing

[@zfengjoe]

At 12:16, I think maybe the real definition of a complex manifold is more restrictive to just require the transition map to be holomorphic. (this is just the definition of almost-complex manifold? )

[@alpistein]

Yay, the missing lecture! Thanks so much Dr. Schuller!

[@sebastianmarquez9292]

How come can we apply the results of surgery theory to the light cone if it doesn't qualify as honest topological manifold in its entirety? I remember him saying that the light cone was strictly speaking not a top. manifold since the points of the two cones make it ill-denied.

[@wroanee]

At 29:56, shouldn't u be in Om and v be in On instead of M and N respectively?

[@kockarthur7976]

Well, any open set is a subset of the entire set - by definition - so what he wrote is true, but also any open set is "in" (a member/element of) the topology - also by definition. So both you and him are correct, but I think writing that U and V are subsets of the manifold is more heuristically transparent.

[@rounak5106]

Prof. Schuller said that you should know the difference between C-inf and C-w from your analysis course. Can someone tell which level of analysis course is he talking about? Baby Rudin doesn't mention anything about this. Is this supposed to be a course on measure theory or functional analysis or complex analysis?

[@chasebender7473]

Maybe it wasn't stated to you in this context, but any study of Taylor series requires a C-w function. Smooth (C-inf) does not imply analytic (C-w). If you are having trouble seeing this, try to construct a smooth function on [0,inf) where all the derivatives are zero at zero. Then a piecewise extension that is constantly zero on (-inf,0) will be smooth on R but not analytic at zero.

[@piercingspear2922]

It was taught in my university in the real analysis course. C-w simply means that f is analytical, meaning that its Taylor expansion equals f at any point on the domain. Obviously, f should be C-inf if f is analytical unless you would not able to define the Taylor expansion of f. But f being C-inf alone doesn't imply C-w, because f could be non-analytical.

[@jasperday9020]

@@chasebender7473The classic example of this is defined as f(x) = 0, x < 0, f(x) = e^(-1/x), x >= 0, where all derivatives are 0 at x=0 but the function is not everywhere 0.

[@shokrynada6439]

in 26.13 ........are C-infinity manifolds (not C-k manifolds)

[@reinerwilhelms-tricarico344]

Flower compatible?

[@rewtnode]

I really dig this lecture series. I am beginning to get this but there is a certain nagging feeling here, as if it’s always talking talking around something else out there (in German, um den heißen Brei geredet) . In this case, the properties of that map Phi between manifolds. Now here all these charts are introduced and things like differentiability are explained indirectly by means of these charts. Been there, done that, and remember some of that from undergraduate studies 40 years ago. But in the end, when this is applied, all this gets usually forgotten, or it’s simply assumed to be always there somehow in the background. Mainly the point of charts seems to be to somehow get to have coordinates - namely by projections into some nice regular space. But it’s often because of coordinates that everything becomes so awfully complicated in differential geometry. Makes me wish that there was more ways to forget about coordinates and have them only when you need to actually compute something. I mean there isn’t really anything like coordinates in nature, and if then they aren’t reals but rather integer or rationals based on something’s doing the counting and labelling. Also, it seems weird to always just have the usual Cartesian coordinates as examples. Is it just me being utterly confused? (Sorry, I’m just venting a feeling of “Unmut” here, so I can later revisit it. Please be kind. ;-)

[@bendonahoo8563]

rewtnode I think that’s just the beauty of coordinate-free geometry! You have the charts in case a calculation is needed, but the geometric objects themselves exist independently of any such choice of chart, and enjoy all their properties and relationships with other objects entirely in the abstract :)

[@synaestheziac]

I love the complex structure stacking diagram around 52 min... does anyone know if someone has made one of these that includes all the major mathematical structures?

[@tzimmermann]

I would definitely buy the poster !

[@mohamedmoussa9635]

Great lecture.

[@WayneKimRecords]

Then, any atlases with one charts are all C^infty atlases? I am confused...

[@blackflan]

No. All C^1 atlases (Atlases for which all transition maps are C^1 in the real analysis sense) contains a C^infty atlas. That C^infty atlas must be constructed by removing some charts from the C^1 atlas that DOESN'T have C^infty transition maps in the real analysis sense. We remove all of them until we only have charts for which their chart transition map is C^infty. What guarantees that there will be left some of those? Well, the theorem he showed in the lecture.

[@jamma246]

Wayne was correct. If you only have one chart then the only transition function is the identity, which is smooth. In this case your manifold is just an open subspace of R^n, with the same induced differentiable structure.

[@mehmetsolgun6816]

I think the atlas A_2 he defined at 21.02 is not C^(infty). even not C^1 since the derivative of the function x^(1/3) is not continuous at 0..

[@loadedices4543]

An atlas A is C^k if any two charts in A are C^k-compatible. Since A_2 has only one chart which is invertible, A_2 is C^(infty).

[@mehmetsolgun6816]

The C^(infty) issue is not just about invertibility. as I said, the map x^(1/3) is not evet C^1 at zero. So, there is some mistake.. But the lecture is still very helpful..

[@aguelmame]

A C^(infty) atlas should verify by definition : for any two charts that overlap, x \circ y^-1 is C^(infty) (this is on top of the chart map being a homeomorphism). For A2, this is clearly the case : (R,x^{1/3}) overlaps with itself, and the chart transition map is Id_R which is C^(infty)

[@kockarthur7976]

I think the confusing point in that example is that he chose the underlying manifold to be the Euclidean real line R. That specific chart he chose is indeed not even C^1 (when viewed as a map from R->R), but this is not what the notion of C^k compatibility is about. C^k compatibility is defined between different charts through their transition maps. If he had chosen some other abstract manifold and defined his charts more abstractly (where the only necessary ingredient for these charts is that they are homeos) then there wouldn't be this confusion.

[@andreshombriamate745]

Because you are doing precisely what it´s forbidden, namely , comparing it with the identity map. But if you take this coordinate, say, t, and build new maps employing x´(p)=f(t(p)) with the f maps beeing smooth, the whole structure will become a smooth manifold, of course non isomorphic to the one we are accostumed to.

[@neelmodi5791]

Isn't there some ambiguity with the definition presented at 30:00? Suppose (M, Om, Am) and (N, On, An) are differentiable manifolds but one is C^10 and the other is C^20, then for a map from one to the other do we regard them as both being C^10? Couldn't we even regard them as being C^9? What would happen if the transition map were C^9 but not C^10?

[@kockarthur7976]

I don't think there is any ambiguity. Beware, the "C^k" part of the term "C^k manifold" refers to the differentiability of the transition maps between overlapping charts. This notion is characteristic to a particular manifold, and does not have anything to do with relating two different manifolds, e.g. mappings between manifolds. A mapping between two manifolds could, in principle, be as crazy as we want it to be. It is only when the representation of this mapping on the charts/patches is C^q differentiable that we deem it a "C^q differentiable mapping between manifolds".

[@Nthgdu19]

So hard for me to get any intuition on Whitney's theorem, I cannot see how a C_inf atlas always exists when there exist a C_k atlas

[@kyubey3166]

It's far from obvious or intuitive. I guess one would need a few courses on differential geometry to see some light on why this is true. I mean, if you have a bag full of C_k compatible charts, somehow you can throw away some of them and boom, all the remaining are C_inf compatible. At least that's what I understood.

[@stonechen4820]

a C_k+1 chart is C_k as well, if we have C_k maximal atlas, then we have all charts that are C_k compatible and that includes C_k+1

[@RBanerj]

Fantastic.

[@siddarthsrinivasan5520]

59:13 they did surgery on a grape

[@UnforsakenXII]

:D Not HD but still good.

[@vincentpicaud5664]

Great video! remark: at 55: the cited theorem is Radó - Moise, and not Radon - Moise. Further details en.wikipedia.org/wiki/Differential_structure

[@sidddddddddddddd]

Don't know Cauchy-Riemann equations? Like seriously?

[@TimTeatro]

You'd be shocked. Even though they're a focal point of any complex variables/analysis course, a lot of people (I'm thinking specifically of engineers) forget them because they're not connected to the material that motivated those students to take the course in the first place! They don't really think about them again until graduate school when they need it to do something real, interesting and not a sanitized textbook problem.

[@synaestheziac]

I love the complex structure stacking diagram around 52 min... does anyone know if someone has made one of these that includes all the major mathematical structures?