Recently it has occurred to me that people who I study with fall into two categories. There are those who really understand the mathematics deeply, pouring over every sentence and not leaving a single detail unchecked. There are also those who learn math deeply enough to do the problems in front of them and nearby problems that may come up. For a long time I thought the goal of a mathematics program was to weed out the second type of person. After a year of study though, I have noticed more that to really become a good mathematician you must compromise somewhere. In some sense it is very impractical to know, all of set theory, and analysis if you are going into algebraic geometry, and what is the point of deeply learning the classifications of groups if you don’t need groups for doing much in complex analysis. Of course, this is not to say that we should be ignoring areas of math, and it is important to understand the fundamentals in each area. It is also important to see that we cannot learn it all and that it is probably not a great idea for most of us to try. I think the most humbling time in my year in Budapest has been learning this, and fighting against it, and then finally coming to accept that I will not be able to explore all of these areas thoroughly and deeply, and also move on to begin doing my own research in any reasonable time frame. The narrowing of my scope for what I want to devote my time to has been extremely heart breaking, and the mix and mash of consolidation my interests or cutting them out is one that matured me in a very strange way.
The bright side of the sad aging process is that I will likely have a much stronger application for graduate school, and be more focused in my research once I am done doing all the vetting of my interests and making decisions on which way to take my mathematical life. Also when I have this all sorted out I will have more free time with which I can start developing other areas of my life which have been left neglected when I became consumed by mathematics.