This is really hard for me to visualize. What does it look like for a 2-sphere to wrap around the core multiple times? Also, I would have expected it to be able to wrap around in multiple ways since there are more dimensions here, leading to pi^2(b^3 \ {0}) = Z^2. How would one even prove that this isn't the case?
Now, there are many ways of proving that there is only one way (up to homotopy) of wrapping a 2-sphere n times around another 2-sphere, but all of them are fairly involved. The simplest proof comes from an analysis of the Hopf fibration, which roughly describes a relation between the 1-sphere, the 2-sphere and the 3-sphere [1]. Other than this, it follows from the theory of degrees for continuous mappings, or from the Freudenthal suspension theorem and some basic homological computations.
[0] https://en.wikipedia.org/wiki/Homotopy_groups_of_spheres#/me...
You can have “two” spheres wrapped within the Earth with the following parametrization. Using a first coordinate r to denote the distance to the Earth’s center, so that (1,t,z) denotes the points in the sphere of radius 1:
(a,b)-> (1+cos(b)/2, a,b), for a,b in the interval [0,2pi].
Those are not proper spheres (the radius changes) but the surface so parametrized is homotopic to a sphere “counted two times”.
It is not possible to have a warped sphere which does not cross itself, as far as I can tell (but I might be wrong).
The wikipedia image linked by a sibling comment did not help me…
ETA: the issue is not the dimension (2) of your spheres but the codimension (1) inside the object, and the fact that you have only removed the center of the main sphere. I think (caveat emptor) that if you remove 2 points form the solid sphere, you get Z^2. Similar to the case of surfaces and holes.
A Sphere is a Loop of Loops (Visualizing Homotopy Groups)
The homotopy hypothesis has something mystical about it.
I completely understand where he is coming from. While it's true that "we don't know what we don't know", I would appreciate hearing more about his (opinionated) thoughts regarding the topics discussed during the interview.
How he got famous is such a mystery…
This differs in other podcasts. For example, Sean Carroll, a theoretical physicist, conducts interviews with colleagues, who are also theoretical physicists. This enables him to engage in a meaningful conversation with the person being interviewed. When both parties strive to use language that a wider audience can understand, it truly becomes enjoyable.
I wonder if a potential application of LLMS could be: have two experts have a really interesting but dense conversation with each other, and then translate the conversation into simpler language with interjections for explanations.
It may not be enjoyable for the most general audience, but it would scratch an itch for some of us.
Pure math just doesn't work like that. Literally no one has any idea when XYZ problem will be proven - there's no way to know! It's not like building a stadium or a road or even a company , where you can see or anticipate the coming of a proof before it arrives. It could happen tomorrow or it could happen never. For example, of Hilbert's original 23 problems posed in 1900, three are still open.
And then there are other problems for which progress has been made over the past decades. At times, a weaker version may have been proved. Occasionally, new ideas emerge in the field regarding how to advance effectively. An upper or lower bound may be identified and gradually improved over time.
Of course, while no one can predict the future, making an educated guess is of course interesting. I would appreciate it if Terry could share his thoughts on such estimates.
Even if that's true that's a lot of avoidable drama.
These kinds of conversation works great for pushing addictive engagement but I doubt it does a lot of good.
