An Old Conjecture Falls, Making Spheres a Lot More Complicated

[e12e]

I enjoy this way more pretending it's the prelude to a Philip K Dick or H P Lovecraft story - than trying to actually grasp the math :)

[alan-crowe]

If you enjoy the intersection of H P Lovecraft and mathematics you may enjoy https://www.hulver.com/scoop/story/2009/1/15/182727/390

[billfruit]

The gossipy narrative style of the article is kind of jarring for an article on a topic like this. It took several paragraphs before it touched on the matter.

[topaz0]

I dunno about gossipy, but the narrative style is standard at Quanta. It's written for the subscriber who is reading for leisure, and wants a good story as well as some amount of technical depth, not for the HN reader who wants to quickly judge whether figuring this thing out is worth their time, and will abandon it if not.

[delocalized]

I always wonder what a popular science/math magazine would look like if it were oriented towards hackers. In this I mean people who have little background in the field but also the type of person who is used to bluntness and knows to RTFM.

I would subscribe to one. Journal articles are often opaque to people who aren't already in the field, and popular science falls too often into the storytelling trap seen here.

[jll29]

I read Quanta mag because of the narrative, and loved this article.

What a nice 65th birthday present to finally close off the last dangling piece of your almost-complete research agenda!

[ykonstant]

The following slides contain a more concrete description of the conjecture, its motivation and consequences: https://people.math.rochester.edu/faculty/doug/Talks/Glasgow...

[codeflo]

I hate slide decks like these, where every page in the PDF contains one more bullet point than the last one. Maybe I'm particularly bad at this, but I spend way too long scanning each page for the new information. Is it impossible to configure LaTeX to only produce the final animation step as a completed page and skip the intermediate ones?

[xeckr]

>(think of a 100-dimensional sphere)

Cheeky.

[unnah]

As the old joke goes: it's not hard at all, just think of an n-dimensional sphere and let n equal 100.

[quickthrower2]

I though of a 100 dimensional vector in python with buzzing numbers

[Nevermark]

So … convex?

[xtajv]

I'm terrible - one might even go so far as to say that the telescope conjecture has collapsed.

[datavirtue]

The first sentence should have been the ball-is-equal-to-egg explanation with mention of topology. Before that I had no idea what they were talking about.

P.s. I have to assume the rules forbid shapes with surfaces of zero thickness. Otherwise I can just smash a ball into an inner-tube. If the shapes have thickness mandated, what is it? Are the thickness of the surfaces a consideration when morphing from one shape to another? Is the surface thickness negative or positive from zero? All of these questions stem from my experience in 3D modeling where these parameters must be defined.

[brianpan]

Have you heard the quip that in physics a cow and a point are equivalent? This is because the physicist cares only about the motion of the thing.

In topology, a doughnut and a coffee mug are equivalent (a mug has exactly one hole, in the handle where your fingers grab). Because the mathematician doesn't care about how hard, thick, or breakable it is; they only care about how complex the shape is. So throw out thickness, size, elasticity, etc.

[dullcrisp]

There is no thickness (or it’s zero if you like). The deformations have to be continuous mathematical functions, so punching a hole isn’t possible.

The study is about the properties of (higher dimensional) shapes rather than concrete objects. It’s like asking what’s the thickness of a circle.

[agrounds]

To add on to what dullcrisp said, which is all correct, even spheres with thickness are “the same as” spheres of zero thickness from the perspective of homotopy theory. “Sameness” here means homotopy equivalence [1]. In fact the thin sphere is a deformation retract [2] of the thick one. The deformation pushes each point of the thick sphere along radial lines towards the thin sphere. Being a deformation retract implies the two spaces are homotopy equivalent.

[1] https://en.wikipedia.org/wiki/Homotopy#Homotopy_equivalence

[2] https://en.wikipedia.org/wiki/Retraction_(topology)

[topaz0]

You don't actually have to assume anything. You could ask instead, or read some background.

[bryanrasmussen]

>Infinitely more maps from spheres to telescopes means infinitely more maps between spheres themselves. The number of such maps is finite for any difference in dimension, but the new proof shows that the number grows quickly and inexorably.

is it actually infinitely - or just a lot?

[r0uv3n]

I think the article means that over all differences in dimension, the total number of missed maps is infinite.

[pbhjpbhj]

Lost me at the end, but, don't inner-tubes have 2 holes (genus 2), topologically: one for inflation and one for the wheel to fit in. This makes them distinct from a torus (genus 1) and no homotopy exists between them.

Clearly IANAM.

[superhuzza]

There is a distinction between a torus and a solid torus

A torus is like an inner tube - an inner void and a big hole in the middle.

A solid torus just has a big hole in the middle, like a donut.

https://en.wikipedia.org/wiki/Solid_torus

[toth]

You are right, but I am pretty sure they just meant the inner tube ignoring the puncture for inflation.

Usually people just use donut/bagel for an illustration of a torus. Not sure why they used inner tube here - maybe to make it clear it is just the surface?

[dclowd9901]

> It means that in very high dimensions (think of a 100-dimensional sphere)

Sure, no problem, author.

[tomcam]

I don’t know about you, but I start to get a little tentative at about 12 dimensions

[wwarner]

What a great read. Exotic spheres are the dark energy of mathematics.

[iraqmtpizza]

do they call them all spheres just to pretend that their work is relevant? I've heard from captain beyond that everything's a circle, but this is one step too far. a 100-dimensional non-uniform egg is not a sphere in any possible way. why is it not called an n-manifold or something like that

[Sniffnoy]

I'm not sure what you're talking about. These are, in fact, n-dimensional spheres -- the set of points at unit distance from the origin in n+1 dimensions. (It's n+1 because, e.g., a sphere in 3 dimensions is intrinsically 2-dimensional.)

An n-manifold would just mean any n-dimensional manifold. These are very particular n-dimensional manifolds, namely, spheres.

Now of course, this is topology, so our equivalences are broad; but the thing these are all equivalent (homeomorphic) to is a sphere. Sure, you can take a more complicated shape that's equivalent to a sphere, but that complexity is incidental; the broad equivalences of topology let us ignore them. (Although, alternatively, they also let us turn the sphere into, say, a cube, if that's easier to think about, which often it is.)

[bmitc]

An n-sphere is an n-dimensional manifold that can be translated, via various means such as diffeomorphisms, to other shapes like an n-dimensional egg. Since it's the "base" shape, that's what it's called. Note that an n-sphere has no holes.

[myownpetard]

Wait until you hear about hypercubes.

[Eduard]

what is this good for?

please no knee-jerk 'this is pure mathematics, it doesn't need applicability' answers.

[kmill]

This is about understanding something about what goes on as you go rightward on the table at https://en.m.wikipedia.org/wiki/Homotopy_groups_of_spheres (under General Theory)

There's an applications section in the Wikipedia article, but it's all to other parts of pure math. It's hard to summarize, but they've got to do with obstructions to untangling, unwrapping, or otherwise solving things to do with spaces.

[navels]

When I was working in the field it sure as heck wasn't because of practical applications. The mathematics involved is beautiful.

[quickthrower2]

Mathematics has a habit throughout history of coming up with useless (to layman, at first) looking ideas that become insanely valuable. Such as zero, negative numbers, imaginary numbers, group theory and so on. Modern life and progress needs this.

[plaguepilled]

Proving other theorems, which may themselves either prove further theorems or lead to direct applications. That's how the questions of "what to prove" often materialise.