I tend to shy away from questions like this, but for some reason I'm up at a strange hour with nothing better to do. And the news is too depressing. Actually I agree with your friend, "algebraic geometry" does sound a bit dull. We should have come up with a cooler name, but it's too late.

Anyway, here's something. Say you have a compact Riemann surface equipped with a positively curved metric. Then by Gauss-Bonnet, the Euler characteristic is positive. Therefore, modulo basic facts about Riemann surfaces, it must be the Riemann sphere. Now consider a higher dimensional version. Suppose that $X$ is a compact complex manifold with a Kähler metric with positive curvature in a suitable sense (i.e. positive bisectional curvature). Then Frankel conjectured that it must be a projective space. There are two proofs, one due to Siu and Yau uses harmonic maps and another due to Mori using algebraic geometry in positive characteristic. For the second proof, first observe that $X$ is projective algebraic by Kodaira's embedding theorem. Then the curvature condition implies that the tangent bundle is positive in the sense of algebraic geometry (i.e. ample). Mori proved that projective spaces are the only varieties with positive tangent bundle. Scheme theory is needed to move the problem into characteristic $p$, where the main argument takes place.