TITLE: How to explain to an engineer what algebraic geometry is?
QUESTION [42 upvotes]: This question is similar to this one in that I'm asking about how to introduce a mathematical research topic or activity to a non-mathematician: in this case algebraic geometry, intended as the most classical complex algebraic geometry for simplicity.
Of course some of the difficulties of the present question are a subset of those of the linked question. But I think I want to be more precise here, about what's the pedagogical/heuristic obstacle I want to bypass/remove/etc, which is, after all, a detail.
So, say the engineer is happy with the start

"Algebraic geometry is the study of solutions of systems of polynomial equations in several variables..."

An engineer certainly understands this.

"...with complex coefficients..."

Here she's starting to feel a bit perplexed: why complex numbers and not just reals? But she can feel comfortable again once you tell her it's because you want to have available all the geometry there is, without hiding anything - she can think of roots of one-variable polynomials: in $(x-1)(x^2+1)$ the real solutions are not all there is etcetera.
Happy? Not happy. Because the engineer will inevitably be lead to think that what algebraic geometry consists of is fiddling around with huge systems of polynomial equations trying to actually find its solutions by hand (or by a computer), using maybe tricks that are essentially a sophisticated version of high school concepts like Ruffini's theorem, polynomial division, and various other tricks to explicitly solve systems that are explicitly solvable as you were taught in high school.

Question. How to properly convey that algebraic geometry mostly (yeah, I know, there are also computational aspects but I would contend that the bulk of the area is not about them) doesn't care at all of actually finding the solutions, and that algebraic geometers rarely find themselves busy with manipulating huge polynomial systems, let alone solving them? In other words, how to explain that AG is the study of intrinsic properties of objects described by polynomial systems, without seeming too abstract and far away?
Also, how would you convey that AG's objects are only locally described (or rather, in the light of the previous point, describable) by those polynomial systems in several variables?

REPLY [4 votes]: I would just say that very roughly speaking it's a subject where you are doing geometry and thinking about geometry, but you write about it formally like it is algebra and you use algebra (which can be fairly esoteric).  I have an example from page 119 of 'Introduction to Homological Algebra' by Weibel (the application is originally due to Hartshorne).
Let $R = \mathbb{C}[x_1, x_2, y_1, y_2]$, $P=(x_1,x_2)R$, $Q=(y_1,y_2)R$, and $I = P \: \cap \: Q $.
Since $P$, $Q$, and $m = P+Q = (x_1, x_2, y_1, y_2)R$ are generated by regular sequences, it can be shown that the outside terms in the Mayer-Vietoris sequence
$H^3_P(R) \bigoplus H^3_Q (R) \rightarrow H_I^3(R) \rightarrow H^4_m(R) \rightarrow H^4_P(R) \bigoplus H^4_Q (R) $
vanish, which tells us that $H^3_I(R) \simeq H^4_m(R) \neq 0$.  (The cohomology is local cohomology here).
Esoteric and abstract to the typical engineer?  Perhaps, but leaving aside the words and jargon, this implies that the union of two planes in $\mathbb{C}^4$ which meet at a point cannot be described as the solutions of only two equations $f_1 = f_2 =0$, which is a concrete geometric fact.