TITLE: Is algebraic geometry constructive?
QUESTION [40 upvotes]: Notes: 1) I know next to nothing about algebraic geometry, although I am greatly interested in the field. 2) I realize that "constructive" might be a technical term, here I am using it only in an informal manner.
I hope that this question belongs on this site, since it is not strictly research-level.
As an autodidact (I have an ongoing formal education in physics, but the amount of math we learn here is abysmal, so most of my mathematics knowledge is self-taught), I have noted that algebraic geometry seems to be really impenetrable for somebody who has no formal education in the field, unlike, say, differential geometry or functional analysis, which are areas I can effectively learn on my own.
Pretty much every time I encounter AG-related stuff on sites such as this one or math.se, I see layers and layers of abstractation on top of one another to the point where it makes me wonder, is this field of mathematics constructive, in the sense that can it be used to actually calculate anything or have any use outside mathematics?
The point I am trying to make is that, using differential geometry as an example, is is constructive. No matter how abstractly do I define manifolds, tensor fields, differential forms, connections, etc, they are always resoluble into component functions in some local trivializations, with which one can actually calculate stuff.
Every time I used DG to calculate stellar equilibrium or cosmological evolution or geodesics of some model spacetime, I get actual, direct, palpable, realizable results in terms of real numbers.
I can use differential forms to calculate the volumes of geometric shapes, and every time I use a Lagrangian or Hamiltonian formalism to calculate trajectories for classical mechanical systems, I make use of differential geometry to obtain palpable results.
On top of that, I know that DG is useful outside physics too, I have heard of uses in economy, music theory etc.
I am curious if there is any real-world application of AG where one can use AG to obtain palpable results. I am not curious (for the purpose of this question) about uses to mathematics itself, I know they are numerous. But every time I try to read about AG I get lost in the infinitude of sheaves, stacks, schemes, functors and other highly abstract objects, which often seem so impossible to me to be resolved into calculable numbers.
The final point is, I would like to hear about some interesting applications of algebraic geometry outside mathematics, if there are any.

REPLY [3 votes]: Slightly roughly, a smooth algebraic variety over $\mathbb{R}$ has an open covering by algebraic varieties of the form $U = \{ (x_1 , \ldots , x_n) \in \mathbb{R}^n \ | \ f_1 (x_1 , \ldots , x_n) = 0 , \ldots , f_m (x_1 , \ldots , x_n) = 0 \}$ where the $f_1 , \ldots , f_m$ are polynomials (with some condition on derivatives making this closed submanifold of $\mathbb{R}^n$ smooth). One can then describe the transition function and cocycle condition etc., so this is exactly like a smooth manifold, but in fact of a more restrictive nature (you can only construct from manifolds which are cut out by polynomials).
You can then cover $U$ by open subsets of $\mathbb{R}^{n - m}$, but this will be "not algebraic". An algebraic way of doing this further simplification is called "etale cover".
One then needs a series of Q&A's, to explain why you then see mountains of abstraction, so perhaps this is not the place. Basically and roughly, this is because you can ask for solutions of the same polynomial equations also over $\mathbb{C}$ and over $\mathbb{Z} / p \mathbb{Z}$ and in fact over a lot of other fields and rings, and it turns out that even if you were interested in only one ring, the "manifolds" that you get from others turn out to be related and interconnected, and contain useful information for your original pursuit. Thus, one would like to think of an algebraic manifold as being over all rings at once, and then there are various devices and theories which look quite abstract if one is not used to them, for dealing with this.