A cubic surface, fibered by elliptic curves.pic.twitter.com/fbKhRYO9ef
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Replying to @stevejtrettel
Physicist here.. what does it mean to 'fiber' a curve?
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Replying to @InertialObservr
Here it’s like a fiber bundle (which is like a vector bundle but for more general types of spaces) except there can be some “singularities” or points where some of the spaces are bad. That happens three times in this example - when circles break off / recombine or collapse.
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Replying to @stevejtrettel
i see .. i still am not sure what a Fiber bundle is with 100% confidence .. wanna ELi5 for good measure (no pun intended)?
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Replying to @InertialObservr
For sure! Say you got a bunch (maybe...a bundle?) of sticks or wheat all tied together, and you want to describe to me. You could break the description into two pieces: 1) the shape of a piece of wheat, and 2) the shape of the cross section of your bundlepic.twitter.com/d46uiwuciM
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Replying to @stevejtrettel @InertialObservr
By telling me you've stacked a bunch of wheat together so the cross section looks like a disk, I could develop a sort of picture for what you are talking about (the bundle of wheat, a 3d thing) in terms of simpler data: a single fiber of wheat (1d) and the cross section (2d)
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Replying to @stevejtrettel @InertialObservr
A fiber bundle is a space that you can describe this way locally: that means you can chop it into pieces so that each piece looks just like a bundle of wheat: a bunch of parallel spaces stacked right next to one another. The spaces you're stacking together are called the fibers
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Replying to @stevejtrettel @InertialObservr
And the "cross section" is called the base. We already know and love lots of fiber bundles! The cylinder is a fiber bundle: with fiber a line and base a circle, corresponding to the description " take a vertical line and rotate it around the z-axis to get a cylinder"
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Replying to @stevejtrettel
would you call a bunch of parallel cylinders packed tightly also a fibre bundle?
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Replying to @InertialObservr @stevejtrettel
nvm this comment your reply made it clearer to me
5:56 PM - 18 Feb 2020
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