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Telescope Conjecture Disproved (Mathematics)

Gord_in_Toronto

Penultimate Amazing
Gord_in_Toronto
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Jul 22, 2006
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If you are interested in HomotopyWP.

An Old Conjecture Falls, Making Spheres a Lot More Complicated

The telescope conjecture gave mathematicians a handle on ways to map one sphere to another. Now that it has been disproved, the universe of shapes has exploded.
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Honestly worth a read. Mathematical topology marches on.

There are different types of progress in math and science. One kind brings order to chaos. But another intensifies the chaos by dispelling hopeful assumptions that weren’t true. The disproof of the telescope conjecture is like that. It deepens the complexity of geometry and raises the odds that many generations of grandchildren will come and go before anyone fully understands maps between spheres.

“Every major advance in the subject seems to tell us the answer is a lot more complicated than we thought before,” Ravenel said.
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Gord_in_Toronto said:
Give it a try. The concepts involved are quite simple.
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Mmmm... nope. I made it slightly more than halfway through the article before my eyes glazed over so hard I needed a diamond cutter to free them. My mathematical brain is capable of basic mental arithmetic and simple algebra (which is really all that's required for real life and D&D games) and then it loses the thread. I once started running through the Khan Academy courses on mathematics starting from 1+1=2 and getting up to lowest common factors before I lost it.

This does look fascinating, though. I get the basic homotopy discussion, but once it careens off into higher dimensions and telescopes, I stop being able to visualise what it's talking about.

Don't get me wrong - I love pure mathematics. Or rather, I love the ideas of pure mathematics, and the fact that ingenious humans can comprehend these ideas is fascinating to me. But after 53 years of life I have found myself incapable of understanding on anything other than a purely abstract level.
 
Gord_in_Toronto said:
Give it a try. The concepts involved are quite simple.
Click to expand...


Morava E-theory, algebraic K-theory, and the exotic spheres discovered by John Milnor may be quite simple for you, but they are not quite simple for me.
That's despite having taken PhD-level courses in point set topology, set theoretic topology, and algebraic topology. Guess I should have gone to a better school.
;)
 
W.D.Clinger said:
Morava E-theory, algebraic K-theory, and the exotic spheres discovered by John Milnor may be quite simple for you, but they are not quite simple for me.
That's despite having taken PhD-level courses in point set topology, set theoretic topology, and algebraic topology. Guess I should have gone to a better school.
;)
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It depends on the level of detail of understanding. ;)

The concepts are simple. The basic idea of mapping one 3-D object onto another I do truly understand. I can visualize doing it. Extension to higher dimensions I take as an article of faith.

Now String Theory. :confused:*n
 
Gord_in_Toronto said:
It depends on the level of detail of understanding. ;)

The concepts are simple. The basic idea of mapping one 3-D object onto another I do truly understand. I can visualize doing it. Extension to higher dimensions I take as an article of faith.

Now String Theory. :confused:*n
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The concepts aren't that simple, e.g. the article manages to imply that infinity=1.

"start with a circle. Now map it onto the two-dimensional sphere, which is the surface of a ball. There are infinitely many ways of doing this."

"And if the space you’re mapping from is lower-dimensional than the space you’re mapping to (as in our example of the one-dimensional circle mapped onto a two-dimensional sphere), there is always only one map."
 
In the interest of saving you a click:

1. Topologists like to know how many unique mappings their are, of an object of more dimensions onto an object of fewer dimensions.

2. To make this question easier to answer/more computationally accessible, topologists have invented an object called a "telescope".

3. The question then becomes, how many unique mappings are there, of a higher-dimensional object onto a telescope?

4. The answer, up until recently, has been "Morava E-theory should tell you". Morava E-theory is a mathematical tool that tells you whether a mapping of a higher-dimensional object onto a telescope is unique, or if it is topologically identical to another mapping you have already made.

5. The Telescope Conjecture is that Morava E-theory will find every possible unique mapping. It's a conjecture because there is no mathematical proof that it must be true.

6. Up until recently, it looked like the conjecture was pretty consistent, and likely true.

7. However, the Telescope Conjecture has now been disproven. A new mathematical tool, called algebraic K-theory, is able to find unique maps that Morava E-theory does not.

8. Even worse (better?), algebraic K-theory finds an infinite number of unique mappings.

9. This is a huge discovery, since up until now, it was assumed that the number of unique mappings from a higher-dimensional object onto a telescope was always finite.

10. Don't worry, though: Your Mercator projection still works just fine.
 
sphenisc said:
The concepts aren't that simple, e.g. the article manages to imply that infinity=1.

"start with a circle. Now map it onto the two-dimensional sphere, which is the surface of a ball. There are infinitely many ways of doing this."

"And if the space you’re mapping from is lower-dimensional than the space you’re mapping to (as in our example of the one-dimensional circle mapped onto a two-dimensional sphere), there is always only one map."
Click to expand...

No, it implies no such thing.

The article says, correctly, that there are infinitely many places on a sphere, that you can project a map of a circle.

The article also says, also correctly, that no matter where on the sphere you project your circle, all the resulting maps are topologically identical.
 
theprestige said:
In the interest of saving you a click:

1. Topologists like to know how many unique mappings their are, of an object of more dimensions onto an object of fewer dimensions.

2. To make this question easier to answer/more computationally accessible, topologists have invented an object called a "telescope".

3. The question then becomes, how many unique mappings are there, of a higher-dimensional object onto a telescope?

4. The answer, up until recently, has been "Morava E-theory should tell you". Morava E-theory is a mathematical tool that tells you whether a mapping of a higher-dimensional object onto a telescope is unique, or if it is topologically identical to another mapping you have already made.

5. The Telescope Conjecture is that Morava E-theory will find every possible unique mapping. It's a conjecture because there is no mathematical proof that it must be true.

6. Up until recently, it looked like the conjecture was pretty consistent, and likely true.

7. However, the Telescope Conjecture has now been disproven. A new mathematical tool, called algebraic K-theory, is able to find unique maps that Morava E-theory does not.

8. Even worse (better?), algebraic K-theory finds an infinite number of unique mappings.

9. This is a huge discovery, since up until now, it was assumed that the number of unique mappings from a higher-dimensional object onto a telescope was always finite.

10. Don't worry, though: Your Mercator projection still works just fine.
Click to expand...

Exactly! Just like the article said. :thumbsup:
 
I'm confused, because it says, as sphenisc quoted "And if the space you’re mapping from is lower-dimensional than the space you’re mapping to (as in our example of the one-dimensional circle mapped onto a two-dimensional sphere), there is always only one map."

but it also says, about mapping a circle onto a toroid, that there are (at least) 2 distinct mappings.
 
zooterkin said:
I'm confused, because it says, as sphenisc quoted "And if the space you’re mapping from is lower-dimensional than the space you’re mapping to (as in our example of the one-dimensional circle mapped onto a two-dimensional sphere), there is always only one map."

but it also says, about mapping a circle onto a toroid, that there are (at least) 2 distinct mappings.
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A toroid like a doughnut, isn't a sphere. You can't deform a sphere into a doughnut without making a hole.

You can cut a sphere anywhere and get a circle, 1 map.

You can cut a doughnut in most ways to get a circle, but you can also cut it in another way to get something that's not a circle, more than one map.
 
I think I'm getting a little of it. It's possible that if the prestige and P0lka keep at it long enough, I'll get a little closer. I'll forget it all tomorrow, but it's interesting today.
 
p0lka said:
A toroid like a doughnut, isn't a sphere. You can't deform a sphere into a doughnut without making a hole.

You can cut a sphere anywhere and get a circle, 1 map.

You can cut a doughnut in most ways to get a circle, but you can also cut it in another way to get something that's not a circle, more than one map.
Click to expand...

Yes, I understand that part. It's the "And if the space you’re mapping from is lower-dimensional than the space you’re mapping to (as in our example of the one-dimensional circle mapped onto a two-dimensional sphere), there is always only one map." part which seems to disagree with there being 2 maps.
 
zooterkin said:
Yes, I understand that part. It's the "And if the space you’re mapping from is lower-dimensional than the space you’re mapping to (as in our example of the one-dimensional circle mapped onto a two-dimensional sphere), there is always only one map." part which seems to disagree with there being 2 maps.
Click to expand...
The whole statement was "If two spheres have the same dimension, there are always infinitely many maps between them. And if the space you’re mapping from is lower-dimensional than the space you’re mapping to (as in our example of the one-dimensional circle mapped onto a two-dimensional sphere), there is always only one map.",
the bit you quoted was referring to the highlighted i think, not in a general case of all shapes.
 
theprestige said:
This implies that, unlike the surface of a sphere, the surface of a torus is not two dimensional. But I am not topologist enough to know if this is true.
Click to expand...

Unless there is more than one definition of "two dimensional", the Math Department at Brown University says a torus is two-dimensional.

Slicing Doughnuts and Bagels

The sphere is called "two-dimensional" because we can identify any point (other than the north and south poles) uniquely by giving two numbers, the latitude showing the position of the point on its semicircle, and the longitude indicating how far the semicircle has been rotated. A torus is a two-dimensional surface in the same sense. We can give latitude and longitude coordinates for each point on the torus, where now the latitude shows the position anywhere on the vertical circle. Each point on the torus of revolution is specified uniquely by two coordinates. There are no "special points" like the north and south poles on the sphere.
Click to expand...

However. I'm not a registered topologist, so what do I know.
 
theprestige said:
In the interest of saving you a click:

1. Topologists like to know how many unique mappings their are, of an object of more dimensions onto an object of fewer dimensions.

2. To make this question easier to answer/more computationally accessible, topologists have invented an object called a "telescope".

3. The question then becomes, how many unique mappings are there, of a higher-dimensional object onto a telescope?

4. The answer, up until recently, has been "Morava E-theory should tell you". Morava E-theory is a mathematical tool that tells you whether a mapping of a higher-dimensional object onto a telescope is unique, or if it is topologically identical to another mapping you have already made.

5. The Telescope Conjecture is that Morava E-theory will find every possible unique mapping. It's a conjecture because there is no mathematical proof that it must be true.

6. Up until recently, it looked like the conjecture was pretty consistent, and likely true.

7. However, the Telescope Conjecture has now been disproven. A new mathematical tool, called algebraic K-theory, is able to find unique maps that Morava E-theory does not.

8. Even worse (better?), algebraic K-theory finds an infinite number of unique mappings.

9. This is a huge discovery, since up until now, it was assumed that the number of unique mappings from a higher-dimensional object onto a telescope was always finite.

10. Don't worry, though: Your Mercator projection still works just fine.
Click to expand...
I am glad that what I got from reading the article basically conforms with this elegant summary. Thank you.
 
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