[FRIAM] Manifold Enthusiasts

[Steven A Smith]

Glen -

Well corrected.   I think it IS important to expand the common-sense
notion of "closure" here, to the more general ideal of "closure under an
operation"  such as "closure under taking a step".   I think the basic
idea of "having no holes to fall through" was (mostly) good enough for
everyday thinking.

- Steve

On 3/11/19 10:42 AM, uǝlƃ ☣ wrote:
> I *almost* violated my standing directive to unplug on the weekends because of all the little beeps and buzzes from my phone.  It's fantastic to see so much traffic.
>
> I only have one comment on "closure", as used here.  I think it's a bit misleading to talk about turning a shroud into a balloon/sphere as "closing" it.  I think the only closure needed for a manifold is closure under particular operations (like the normal ones, +, -, *, /), where the point being operated on and the result are both *inside* the space.  So, while it's reasonable to think of a coastline and (imagine walking along the beach), if "step" is the operation, then you're on the coast before you step and still on the coast after you step.  While it makes intuitive sense to loop the coast around like Steve suggests to ensure that you're always still on the coast after taking a step, a manifold need not be a cycle in that way.  You might have, say, an infinitely long coastline and as long as you're stepping along the coast, you're still on it, it's closed under stepping, but it's not a cycle.
>
> To be clear, I don't think anyone said anything wrong. I just wanted to distinguish cycle from closure.
>
>
> On 3/9/19 3:17 PM, Steven A Smith wrote:
>> Nick -
>>
>>> All I can say is, for a man in excruciating pain, you sure write good.  Your response was just what I needed. 
>>>
>> Something got crossed in the e-mails.   *I'*m not in excruciating pain... that would be (only/mainly/specifically) Frank, I think.  But thanks for the thought!
>>
>> Any excruciating pain I might be in would be more like existential angst or something... but even that I have dulled with a Saturday afternoon Spring sunshine, an a cocktail of loud rock music, cynicism, anecdotal nostalgia, and over-intellectualism.  Oh and the paint fumes (latex only) I've been huffing while doing some touch-up/finish work in my sunroom on a sunny day is also a good dulling agent.
>>
>>> Now, when I think of a manifold, my leetle former-english-major brain thinks shroud, and the major thing about a shroud is that it /covers/ something.  Now I suspect that this is an example of irrelevant surplus meaning to a mathematician, right?  A mathematician doesn’t give a fig for the corpse, only for the properties of the shroud.  But is there a mathematics of the relation between the shroud and the corpse?  And what is THAT called. 
>>>
>> Hmm... I don't know if I can answer this fully/properly but as usual, I'll give it a go:
>>
>> I think the Baez paper Carl linked to has some help for this in that.  I just tripped over an elaboration of a topological boundary/graph duality which might have been in that paper.    But to be as direct as I can for you, I think the two properties of /shroud/ that *are* relevant is *continuity* with a surplus but not always irrelevant meaning of *smooth*.  In another (sub?)thread about /Convex Hulls/, we encounter inferring a continuous surface *from* a finite point-set.   A physical analogy for algorithmically building that /Convex Hull/ from a point set would be to create a physical model of the points and then drape or pull or shrink a continuous surface (shroud) over it.    Manifolds needn't be smooth (differentiable) at every point, but the ones we usually think of generally are.  
>>
>>>  So, imagine the coast of Maine with all its bays, rivers and fjords.  Imagine now a map of infinite resolution of that coastline, etched in ink.  I assume that this is a manifold of sorts.
>>>
>> In the abstract, I think that coastline (projected onto a plane) IS a 1d fractal surface (line).  To become a manifold, it needs to be *closed* which would imply continuing on around the entire mainland of the western hemisphere (unless we artificially use the non-ocean political boundaries of Maine to "close" it).
>>>  Now gradually back off the resolution of the map until you get the kind of coastline map you would get if you stopped at the Maine Turnpike booth on your way into the state and picked a tourist brochure.  Now that also is a manifold of sorts, right?  In my example, both are representations of the coastline, but I take it that in the mathematical conception the potential representational function of a “manifold” is not of interest?
>>>
>> I think the "smoothing" caused by rendering the coastline in ink the width of the nib on your pen (or the 300dpi printer you are using?) yields a continuous (1d) surface (line) which is also smooth (differentiable at all points)... if you *close* it (say, take the coastline of an island or the entire continental western hemisphere (ignoring the penetration of the panama canal and excluding all of the other canals between bodies of water, etc. then you DO have a 1D (and smooth!) manifold.
>>
>> If you zoom out and take the surface of the earth (crust, bodies of liquid water, etc), then you have another manifold which is topologically a "sphere" until you include any and all natural bridges, arches, caves with multiple openings.  If you "shrink wrap" it  (cuz I know you want to) it becomes smooth down to the dimension of say "a neutrino".   To a neutrino, however, the earth is just a dense "vapor" that it can pass right through with very little chance of intersection... though a "neutrino proof" shroud (made of neutrino-onium?) would not allow it I suppose.
>>
>> This may be one of the many places Frank (and Plato) and I (and Aristotle) might diverge...   while I enjoy thinking about manifolds in the abstract,  I don't think they have any "reality" beyond being a useful archetype/abstraction for the myriad physically instantiated objects I can interact with.  Of course, the earth is too large for me to apprehend directly except maybe by standing way back and seeing how it reflects the sunlight or maybe dropping into such a deep and perceptive meditative state that I can experience directly the gravitational pull on every one of the molecules in my body by every molecule in the earth (though that is probably not only absurd, but also physically out of scale... meaning that body-as-collection-of-atoms might not represent my own body and that of the earth and I think the Schroedinger equation for the system circumscribing my body and the earth is a tad too complex to begin to solve any other way than just "exisiting" as I do at this
>> location at this time on this earth.)
>>
>> If you haven't fallen far enough down a (fractal dimensioned?) rabbit hole then I offer you:
>>
>>     https://math.stackexchange.com/questions/1340973/can-a-fractal-be-a-manifold-if-so-will-its-boundary-if-exists-be-strictly-on
>>
>> Which to my reading does not answer the question, but kicks the (imperfectly formed, partially corroded, etc.) can on down the  (not quite perfectly straight/smooth) road, but DOES provide some more arcane verbage to decorate any attempt to explain it more deeply?
>>
>> - Steve
>>
>> PS.  To Frank or anyone else here with a more acutely mathematical mind/practice, I may have fumbled some details here...  feel free to correct them if it helps.
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