Cazzandro @KodeGhinn@qoto.org

In 2017, I managed to solve a problem from the “Lviv Scottish book” in https://mathoverflow.net/a/282290/766 . The problem had a prize of “butelka miodu pitnego" (a bottle of honey mead). Today, while I was in Warsaw, some representatives from Lviv, Ukraine came (by train, as the Ukraine airspace is obviously closed) I was very touched and honored to unexpectedly receive the prize in person.

Suppose you were trying to invent a bright orange powder that could easily dye clothes and be hard to wash off. Using your knowledge of quantum mechanics you'd design this symmetrical molecule where an electron's wavefunction can vibrate back and forth along a chain of carbons at the frequency of green light. Absorbing green light makes it look orange! And this molecule doesn't dissolve in water.

Yes: you'd invent turmeric!

Or more precisely 'curcurmin', the molecule that gives turmeric its special properties.

The black atoms are carbons, the white are hydrogens and the red are oxygens.

Read on and check out what pure curcurmin looks like.

(1/n)

One way to view automatic differentiation is to think of it as adjoining an "infinitesimal" element d, such that d²=0, to the reals, ie. forming ℝ[d]/(d²). If f is a polynomial then f(x+d)=f(x)+df'(x) giving a nice way to compute derivatives on a computer - especially as it can be extended to rational and even transcendental functions f. It doesn't form a field though. For example you can't always divide by d.

TIL There is a field, named after Levi-Civita, that generalises ℝ[d]/(d²) quite a bit.
Each element is a "formal" sum ∑aᵢεⁱ where the sum is over some subset S of the rationals which is left-finite, ie. for any z, S has only finitely many elements less than z. Addition and multiplication work in the way you might guess.

This means we can form things like ε^(1/2) or even the "infinite" 1/ε. It's not just a field, it's an ordered field so we have, for example, that 1 > ε^(1/2) > ε > ε² > 0.

You can even construct a Dirac delta-like function δ(x) = ε/π(x²+ε²).

https://en.wikipedia.org/wiki/Levi-Civita_field

The entire Star Wars Franchise was purchased for 40 billion dollars less than Twitter. I think about that sometimes.