It is harder to measure the impact of nonprofits than of for-profits, since you can't just look at their bank balances.

This is one fundamental difficulty of nonprofit work, how to prove that you deserve the investments and not someone else.

The reason public relations is evil in modern society is because, like discrimination, public relations works by dumb association and not logic or fairness.

If you're the son of the killer, you're fucked.

This is unlike our ideal for law which attempts, though sometimes fails, at isolating cause and effect.

Generally, prizes that pay big lumps of money to well established individuals are a bit useless, it would be better to pay smaller sums to struggling beginners in the field, of which there are aplenty.

The most important part about prizes should not be the money, nor the recognition, but rather explaining better what the laureates did. In this, most prizes fail. Thus Ciro Santilli's project idea: Project to explain each Nobel Prize better.

Ah, this list: en.wikipedia.org/wiki/List_of_prizes_known_as_the_Nobel_or_the_highest_honors_of_a_field

For some reason, Ciro Santilli is mildly obsessed with understanding and visualizing the real projective plane.

To see why this is called a plane, move he center of the sphere to $z=1$, and project each line passing on the center of the sphere on the x-y plane. This works for all points of the sphere, except those at the equator $z=1$. Those are the points at infinity. Note that there is one such point at infinity for each direction in the x-y plane.

Their status is a mess as of 2020s, with several systems ongoing. Long live the "original" collegiate university!

By Zuckerberg. The selection seems decent. And natural sciences only, which is good. A bit more application oriented than the Nobel Prize it seems, e.g. 2022 separates physics and fundamental physics.

Appears to explain award reasoning even worse than the Nobel Foundation.

The Royal Society's Nobel Prize.

royalsociety.org/grants-schemes-awards/awards/copley-medal/ says it is now open to international citizens, but having a quick look at the 2010 awards still suggests that it is very British centric, or at least anglophone centric, much like the society fellowship itself. That's likely the reason why the Nobel prize won, being much more international from the start.

Good background: www.theguardian.com/science/the-h-word/2016/oct/07/prize-science-medal-history-nobel

Nobel Prize of Mathematics.

The Nobel Prize of mathematics!

That 15,000 canadian dollar prize though, what a joke! That's what you get when an impoverished scientist, and not a rich industrialist, creates a prize!

The Nobel Prize of computer software or hardware!

More like a "lifetime achievement" though, rather than the Nobel Prize, which tends to be for more specific achievements.

Like everything else in Lie groups, first start with the matrix as discussed at Section "Lie algebra of a matrix Lie group".

Intuitively, a Lie algebra is a simpler object than a Lie group. Without any extra structure, groups can be very complicated non-linear objects. But a Lie algebra is just an algebra over a field, and one with a restricted bilinear map called the Lie bracket, that has to also be alternating and satisfy the Jacobi identity.

Another important way to think about Lie algebras, is as infinitesimal generators.

Because of the Lie group-Lie algebra correspondence, we know that there is almost a bijection between each Lie group and the corresponding Lie algebra. So it makes sense to try and study the algebra instead of the group itself whenever possible, to try and get insight and proofs in that simpler framework. This is the key reason why people study Lie algebras. One is philosophically reminded of how normal subgroups are a simpler representation of group homomorphisms.

To make things even simpler, because all vector spaces of the same dimension on a given field are isomorphic, the only things we need to specify a Lie group through a Lie algebra are:Note that the Lie bracket can look different under different basis of the Lie algebra however. This is shown for example at Physics from Symmetry by Jakob Schwichtenberg (2015) page 71 for the Lorentz group.

As mentioned at Lie Groups, Physics, and Geometry by Robert Gilmore (2008) Chapter 4 "Lie Algebras", taking the Lie algebra around the identity is mostly a convention, we could treat any other point, and things are more or less equivalent.

Bibliography: