As I said earlier, a little knowledge is dangerous. You obviously slept through non-Euclidean geometry. What was the most advanced math class? Algebra I?
You can imagine whatever you might, I have no power to stop your fervid imagination.
I've studied algebra and geometry. Post graduate level. Non-linear, variational calculus, compact manifolds, balls and neighborhoods, epsilons and all that. In seventh grade my teacher, a sweet Chinese woman, btw, taught me non-euclidian geometry, I had already learned set theory and could run through proofs like a hot knife through butter -- very mechanical.
Why do you seperate algebra and geometry? That's very Pythagorean of you, I'd say. You and Ichysaurus both. DesCartes amoung others found systems to join them together, as they actually are joined in the abstract -- human pig-headedness keeps them seperate. Hooray for the Renaissance! In many ways the extreme "evolutionists" have become like that "Holy Mother Church" which nearly offed Galileo Galilea. Algebra and geometry: Symbols are symbols in whatever framework of expression they appear.
"Algebraic geometey" -- most high schoolers know -- do sine and cosine and tangent ring any bells? Sure they do -- they are part of algebraic geometry.