I'm doing my math PhD at Harvard in the same area as Qiaochu. I was also heavily involved in artofproblemsolving and went to MathPath in 2003. I hoped since 2003 that I could stake a manifest destiny in mathematics research.

Qiaochu and I performed similarly in Olympiad competitions, had similar performances in the same undergraduate program, and were both attracted to this website. However, I get the sense that he is driven quite a bit by geometry, or is at least not actively adverse to it. Despite being a homotopy theorist, I find geometry awkward and unmotivated. I cannot form the "vivid" or "bright" images in my mind described in some other article on this website. Qiaochu is also far more social and active in online communities, such as this one and mathoverflow. I wonder about the impact of these differences on our grad school experiences.

Lately I've been feeling particularly incompetent mathematically, to the point that I question how much of a future I have in the subject. Therefore I quite often wonder what mathematical ability is all about, and I look forward to hearing if your perspective gels with my own.

I think it's very important in understanding your first Grothendieck quote to remember that Grothendieck was thrown into Cartan's seminar without requisite training. He was discouraged enough to leave for another institution.