Ciro Santilli @cirosantilli 37

www.marshallscientific.com/MJ-Research-PTC-200-Thermal-Cycler-p/mj-200.htm (archive).

Big goals:

Just art:

Take the group of all Translation in ${\mathbb{R}}^1$.

Let's see how the generator of this group is the derivative operator:

The way to think about this is:

So let's take the exponential map:and we notice that this is exactly the Taylor series of $f(x)$ around the identity element of the translation group, which is 0! Therefore, if $f(x)$ behaves nicely enough, within some radius of convergence around the origin we have for finite $x_0$:

This example shows clearly how the exponential map applied to a (differential) operator can generate finite (non-infinitesimal) Translation!

It is cool how even for such a "simple looking" problem, we were still unable to prove optimality as of 2020!

The most important example is perhaps $SO(3)$ and $SU(2)$, both of which have the same Lie algebra, but are not isomorphic.

Learned readers will ask themselves: so why use an unbalanced tree instead of balanced one, which offers better asymptotic times en.wikipedia.org/wiki/Self-balancing_binary_search_tree?

Likely:

A solution to Laplace's equation.

A device that exhibits the Josephson effect.