Some perspectives on cubic Diophantine equations

I will discuss some perspectives, from my thesis work under Peter Sarnak, on counting integral solutions to x_1^3+...+x_6^3 = 0, and on the related problem of "producing" sums of three cubes on average. Some aspects I may discuss are randomness-structure dichotomies, modularity/automorphy, geometry of low-degree varieties, statistics of L-function families, and/or the delta method.