Glad that this approach worked out for you! It’s an amazing feeling when you finally solve or get something that looked so hard initially. I would not deny that this has happened to me, too, but mostly in the cases I knew for sure I could handle with enough effort. I did my PhD in General Relativity, and had to go through a few proofs that required significantly more background than I had at the time. I was able to master the necessary basics of Algebraic Topology enough to add my own small theorem on top of what was already in my research area, yet it was excruciatingly slow and painful to get to that point, and not a lot of fun. I had to abandon any larger ambitions in the area. Similarly, I was used to getting A and A+ in almost all my undergrad and grad classes, yet when I hit advanced grad classes, like String theory and topics in QFT, I was lucky I could get through them, even though other grad students seemed to have had little difficulties there.
In general, I have found that in many areas, especially in math, everyone has their threshold of abilities. Below the threshold the effort required scales basically linearly with the amount of material, Past that threshold any extra learning becomes exponentially more difficult. I mean “exponentially” in the mathematical sense, not in the colloquial one. Gotta know your limits.
Glad that this approach worked out for you! It’s an amazing feeling when you finally solve or get something that looked so hard initially. I would not deny that this has happened to me, too, but mostly in the cases I knew for sure I could handle with enough effort. I did my PhD in General Relativity, and had to go through a few proofs that required significantly more background than I had at the time. I was able to master the necessary basics of Algebraic Topology enough to add my own small theorem on top of what was already in my research area, yet it was excruciatingly slow and painful to get to that point, and not a lot of fun. I had to abandon any larger ambitions in the area. Similarly, I was used to getting A and A+ in almost all my undergrad and grad classes, yet when I hit advanced grad classes, like String theory and topics in QFT, I was lucky I could get through them, even though other grad students seemed to have had little difficulties there.
In general, I have found that in many areas, especially in math, everyone has their threshold of abilities. Below the threshold the effort required scales basically linearly with the amount of material, Past that threshold any extra learning becomes exponentially more difficult. I mean “exponentially” in the mathematical sense, not in the colloquial one. Gotta know your limits.