Ciro Santilli @cirosantilli 35

By default, we think of polynomials over the real numbers or complex numbers.

However, a polynomial can be defined over any other field just as well, the most notable example being that of a polynomial over a finite field.

For example, given the finite field of order 9, $GP(3)$ and with elements $\{0,1,2\}$, we can denote polynomials over that ring aswhere $x$ is the variable name.

For example, one such polynomial could be:and another one:Note how all the coefficients are members of the finite field we chose.

Given this, we could evaluate the polynomial for any element of the field, e.g.:and so on.

We can also add polynomials as usual over the field:and multiplication works analogously.

Used a lot in quantum mechanics, where the equations are really hard to solve. There's even a dedicated wiki page for it: en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics). Notably, Feynman diagrams are a way to represent perturbation calculations in quantum field theory.

Let's gather some of the best results we come across here:

Physics (like all well done science) is the art of predicting the future by modelling the world with mathematics.

And predicting the future is the first step towards controlling it, i.e.: engineering.

Ciro Santilli doesn't know physics. He writes about it partly to start playing with some scientific content for: OurBigBook.com, partly because this stuff is just amazingly beautiful.

Ciro's main intellectual physics fetishes are to learn quantum electrodynamics (understanding the point of Lie groups being a subpart of that) and condensed matter physics.

Every science is Physics in disguise, but the number of objects in the real world is so large that we can't solve the real equations in practice.

Luckily, due to emergence, we can use uglier higher level approximations of the world to solve many problems, with the complex limits of applicability of those approximations.

Therefore, such higher level approximations are highly specialized, and given different names such as:

As of 2019, all known physics can be described by two theories:

Unifying those two into the theory of everything one of the major goals of modern physics.

Ciro once read that there are two types of mathematicians/scientists (he thinks it was comparing Einstein to some Jack of all trades polymath who didn't do any new discoveries):

TODO long after Ciro forgot where he had read this from originally, someone later pointed him to: www.ams.org/notices/200902/rtx090200212p.pdf Birds and Frogs by Freeman Dyson (2009), which is analogous but about Birds and Frogs. So did Ciro's memory play a trick on him, or is there also a variant; of this metaphor with a gopher?

Ciro is without a doubt the bird type. Perhaps the ultimate scientist is the one who can combine both aspects in the right amount?

Ciro gets bored of things very quickly.

Once he understands the general principles, if the thing is not the next big thing, Ciro considers himself satisfied without all the nitty gritty detail, and moves on to the next attempt.

In the field of mathematics for example, Ciro is generally content with understanding cool theorem statements. More generally, one of Ciro's desires is for example to understand the significance of each physics Nobel Prize.

This is also very clear for example after Ciro achieved Linux Kernel Module Cheat: he now had the perfect setup to learn all the Linux kernel shady details but at the same time after all those years he finally felt that "he could do it, so that was enough", and soon moved to other projects.

If Ciro had become a scientist, he would write the best review papers ever, just like in the current reality he writes amazing programming tutorials on Stack Overflow.

Ciro has in his mind an overly large list of subjects that "he feels he should know the basics of", and whenever he finds something in one of those topics that he does not know enough about, he uncontrollably learns it, even if it is not the most urgent thing to be done. Or at least he puts a mention on his "list of sources" about the subject. Maybe everyone is like that. But Ciro feels that he feels this urge particularly strongly. Correspondingly, if a subject is not in that list, Ciro ignores it without thinking twice.

Ciro believes that high flying birds are the type of people better suited for venture capital investment management: you know a bit of what is hot on several fields to enough depth to decide where to place your bets and how to guide them. But you don't have the patience to actually go deeply into any one of them and deal with each individual shit that comes up.

Cosmos: A Personal Voyage (1980) episode 1 mentions as quoted by the Wikipedia page for Eratosthenes:That's Ciro.

Some related ideas:

See also: having more than one natural language is bad for the world.

Full channel title: "Web of Stories - Life Stories of Remarkable People".

1-2 to hour long interviews, the number of Nobel Prize winners is off-the-charts. The videos have transcripts on the description!

TODO what is their affiliation/who is behind it? There is nothing on the website.

en.wikipedia.org/wiki/Web_of_Stories small wiki with almost no citations.

A polygon is a 2-dimensional polytope, polyhedra is a 3-dimensional polytope.

Related ideas:

Much of this section will be dumped at Section "Website front-end for a mathematical formal proof system" instead.

Most notable example: $L^2$.

This is the future of course, fusion power to generate electricity, and then converting electricity into food somehow.

Hopefully without going through photosynthesis, which feels complicated and wasteful.

Others:

Off-the-shelf techniques to become a teaching superhero.

Customized website idea at: OurBigBook.com.