Recently the following article appeared on my Facebook feed: https://www.quantamagazine.org/20160628-peter-scholze-arithmetic-geometry-profile/. Aside from the usual bland mix of singing praises of an archetypal ‘genius’, the article does contain some genuine insights. The most striking of which is Scholze’s description of him learning the proof of Fermat’s Last Theorem by Sir Andrew Wiles: that he “worked backward, figuring out what he needed to learn to make sense of the proof”. Later he also said things like “I never really learned the basic things like linear algebra, actually – I only assimilated it through learning some other stuff.”
If you have experiences learning mathematics at the senior undergraduate level or post-graduate level, you will likely find that these experiences are orthogonal to your own. We spend an inordinate amount of time learning the ‘basics’ for various things, which very much includes linear algebra for example, in order to do research… or so we are told. If you have passed the part of your career where you do more courses than self-learning, then you have likely reached the epiphany that usually it’s not efficient to learn everything there is to know on a subject before actually doing work on the subject.
Some of you have had advisors telling you things like “read these five books (each 300+ pages) before you attempt any research work in the area”. Sometimes this advice comes from highly proficient researchers, which seems odd: if the above ethos is ubiquitous among researchers, why tell your students to do something totally different and entirely more dreadful? I am not sure what the right answer is, but probably part of the reason is a misguided attempt to make research ‘easier’ for students. Perhaps many advisors recall the struggle of trying to understand ‘simple’ phenomena that they encountered in their research careers, that if someone had just told them to read a book or if they were better prepared, would have been trivial to overcome. Perhaps they wish to save their students some time by telling them the shortcut. However, the struggle to understand phenomena on your own is part of what makes research rewarding, and more importantly, it is critical in forging a mind suited to making discoveries.
Of course, I am but a pebble to the avalanche that is Peter Scholze, so my advice may not be worth much. Nevertheless, I feel like I should say this to all prospective and current graduate students: be bold, and give every difficult paper in your field a read. Don’t be intimidated by them. If you don’t understand something, google it until you find what you need to learn the language of the subject. Don’t feel like you need to understand all of Harthshorne before you can read any research papers related to algebraic geometry. Your future self will thank you for this.