This dude is mind blowing. Big respect.

Some of the most impressive videos are the ones in which he goes and extracts metals from minerals himself all the way.

But God, the typography of the channel name is so insane! Why no space???

Shame the academic system wasn't compatible with him: www.reddit.com/r/codyslab/comments/f5531p/codys_qualifications/ Maybe there were safety issues involved though.

The non-regular version of the hypercube.

Something is useful if it either:

OMG, Ciro Santilli only learned about this in 2021 after: twitter.com/ryancdotorg/status/1375484757916672000

Every Lie algebra corresponds to a single simply connected Lie group.

The Baker-Campbell-Hausdorff formula basically defines how to map an algebra to the group.

Bibliography:

Lie Groups, Physics, and Geometry by Robert Gilmore (2008) 7.2 "The covering problem" gives some amazing intuition on the subject as usual.

Most commonly refers to: exponential map.

Intuition, please? Example? mathoverflow.net/questions/278641/intuition-for-symplectic-groups The key motivation seems to be related to Hamiltonian mechanics. The two arguments of the bilinear form correspond to each set of variables in Hamiltonian mechanics: the generalized positions and generalized momentums, which appear in the same number each.

Seems to be set of matrices that preserve a skew-symmetric bilinear form, which is comparable to the orthogonal group, which preserves a symmetric bilinear form. More precisely, the orthogonal group has:and its generalization the indefinite orthogonal group has:where S is symmetric. So for the symplectic group we have matrices Y such as:where A is antisymmetric. This is explained at: www.ucl.ac.uk/~ucahad0/7302_handout_13.pdf They also explain there that unlike as in the analogous orthogonal group, that definition ends up excluding determinant -1 automatically.

Therefore, just like the special orthogonal group, the symplectic group is also a subgroup of the special linear group.

It's just too charming, and has some deep themes.

We looking at the definition the orthogonal group is the group of all matrices that preserve the dot product, we notice that the dot product is one example of positive definite symmetric bilinear form, which in turn can also be represented by a matrix as shown at: Section "Matrix representation of a symmetric bilinear form".

By looking at this more general point of view, we could ask ourselves what happens to the group if instead of the dot product we took a more general bilinear form, e.g.:The answers to those questions are given by the Sylvester's law of inertia at Section "All indefinite orthogonal groups of matrices of equal metric signature are isomorphic".

This was THE craze thing in Brazil before Pokemon, it was shown from 1994 to 1997. In particular the collectible action figures! It was possibly more popular in Brazil than e.g. in the US: www.quora.com/Why-was-Saint-Seiya-so-popular-in-Brazil

The thing as quite violent, rated for 14-year olds, but no one gave a fuck, 7 yo Ciro was happily watching it. We protect children too much.

That series also had quite a religious feel to it (as obviously suggested by the series English name itself). It must also have been a great motivator to getting young kids into astronomy!

Ciro's favorite character was definitely Andromeda Shun. He was smart and thoughtful, and had the coolest most complex weapon: his chain whips. He's also a bit effeminate, with his pink clothing and a gentle way. Perhaps that is the reason for adult Ciro's mild fascination with the Andromeda Galaxy.

The English name is horrendous... the Portuguese/French name is so much better: Knights of the Zodiac! Saying this in English just reminded Ciro Santilli of the Zodiac Killer. But nevermind.

Ciro Santilli was touched by this, and then watched several other documentaries by Curtis.

Part of Ciro asks if it is all a big conspiracy theory. But it feels and sounds so right.

Notably the part of the governments have lost all power to companies, and can't do anything meaningful anymore to actually represent the people, because globalization reduces the power of governments.