[riemannzeta]

I think it's interesting that in physics, different global symmetries (topological manifolds) can satisfy the same metric structure (local geometry). For example, the same metric tensor solution to Einstein's field equation can exist on topologically distinct manifolds. Conversely, looking at solutions to the Ising Model, we can say that the same lattice topology can have many different solutions, and when the system is near a critical point, the lattice topology doesn't even matter.

It's only an analogy, but it does suggest at least that the interesting details of the dynamics aren't embedded in the topology of the system. It's more complicated than that.

[colah3]

If you like symmetry, you might enjoy how symmetry falls out of circuit analysis of conv nets here:

https://distill.pub/2020/circuits/equivariance/

[riemannzeta]

Thanks for this additional link, which really underscores for me at least how you're right about patterns in circuits being a better abstraction layer for capturing interesting patterns than topological manifolds.

I wasn't familiar with the term "equivariance" but I "woke up" to this sort of approach to understanding deep neural networks when I read this paper, which shows how restricted boltzman machines have an exact mapping to the renormalization group approach used to study phase transitions in condensed matter and high energy physics:

https://arxiv.org/abs/1410.3831

At high enough energy, everything is symmetric. As energy begins to drain from the system, eventually every symmetry is broken. All fine structure emerges from the breaking of some symmetries.

I'd love to get more in the weeds on this work. I'm in my own local equilibrium of sorts doing much more mundane stuff.