TIL about an apparently notorious #OpenQuestion in #topology: is there a complex structure on the 6-sphere?

The Chern character rules out complex structures on 𝑆ⁿ for n>6. Apparently 𝑆⁴ doesn't even have an almost complex structure, although 𝑆⁶ does (coming as the unit sphere in the pure imaginary #octonions IIUC).

See https://doi.org/10.4153/CMB-1966-003-9 for some overview and refs. Learned from https://twitter.com/CihanPostsThms/status/1632549791036059648

I have been engaging my fellows at Math 4 #Wisdom about working together. This has led to the idea of atomic learning exercises and how they fit within a map of #learning paths. I realized that for the sake of clarity I should distinguish what for me is not part of Math 4 Wisdom because it does not model cognitive frameworks for which I currently have evidence. So I made a map of Math for Fascination or Speculation which includes #octonions, #primes, exceptional Lie groups, Platonic solids.

Octonions.jl v0.2.2 extends all complex analytic functions in the standard library to the #octonions!

Octonions are a type of hypercomplex number whose product is neither associative nor commutative. Besides their normal uses, they are also useful for testing numerical algorithms that are intended to generically work for even weird numbers.

Here we use them to check that the fallback QR and unpivoted LU decompositions in #JuliaLang do the right thing.

https://github.com/JuliaGeometry/Octonions.jl

Interesting post on #wolfram about #hypercomplex numbers (#quaternions, #octonions) using #cayley-dickson construction:

https://community.wolfram.com/groups/-/m/t/1730287