My question is probably an odd one here but I would very much like to work in Homological Mirror Symmetry. An example of a course I'd like to be able to take and understand is https://faculty.math.illinois.edu/~jpascale/courses/2018/595/
It uses references like “D. Auroux, A beginner's introduction to Fukaya categories” among others and it is exactly the kind of thing you'd expect as a specialty course for graduate students. Here's my problem though:
Some universities I've seen (notably in Hong Kong and Japan) have undergraduate programs that look very standard compared to the American one and they don't seem to have specialty courses usually taken by undergrads but then these students go into a 2 year master's degree and come out with some serious looking theses on mirror symmetry, Calabi-Yau manifolds and Gauge Theory etc. Bear in mind, the programs are shorter there (PhD is like 3 years or 4 at most...not 5 like in the US) so I feel like I'm missing something here.
That Illinois course is almost entirely out of my reach and I've taken a reasonable amount of algebra (up to Fields and Galois Theory), some basic smooth manifold theory (at the level of Loring Tu's Intro to Manifolds) and a decent amount of the first 3 chapters of Hartshorne's algebraic geometry.
I STILL cannot read Denis Auroux's Fukaya Categories intro and some of the other references so I am seriously wondering what it is these students (in some of the universities I mentioned) are doing that I am not. How are they getting the background required to find these areas of research so accessible?
Any opinions/advice/light shedding would be great.