Letter to a friend in a reputed US grad school, describing my thoughts on whether to apply abroad for a PhD or not.

“I have no interest in coming to the US for grad school because life here is so much more optimized. I never have to worry for making food and I get the best food possible four times a day right where I study. All classes are within two minutes walking distance from my room. I can visit family at home whenever I want. Overall, I can easily study for 12 hours every day without compromising on my mental and social health. The best thing being, I virtually know the who’s who in almost all departments be it professors or students, so If I get stuck, I can just knock on their door and solve stuff together. Its like being given an orchestra. I can go to the Math dept and I know who to ask if I need to rigorously understand Spectral theory for solving fractional dimensional quantum systems. For nonlinear dynamics I recently used help from a guy well known for Fixed point Analysis which in turn helped me link Poincare-Hopf Index theory with bifurcations. You can imagine that there is a disadvantage of this too, because then Im being over reliant and not thinking myself which is true to some extent. But I see the problem with that argument when I stop looking at Som as myself. It doesnt matter what Som thinks if it has already been thought of by humanity as a whole (Union Set). If I think of myself exclusively as Som, then I am stuck with a limit of nearly 100 years that is; I can only see combinations and permutations that could exhaust a hundred years. So if I started to re derive stuff from scratch without considering all external sources I could only get to things that were invented during the Pythagorean era. But given that humanity has already seen these things, it would be redundant. So to reduce redundancy, Im not willing to waste time thinking about stuff that has already been discovered. Id rather spend that energy on stuff thats pertinent.

I have been advised by everyone to stop acting foolish and go to the US for a PhD even if its just for the sake of “it opens more doors”. I understand that there’s many offsets which help the US grad school experience stand out compared to staying back here even if I have to spend a lot of time and enerrgy in handling chores and taking care of myself when I go abroad. You are a prime example of one of those offsets. Ive never had a friend or roommate as strong as you were in defending arguments. Most people just like to get to the end result without much of a fight so they either accept that their point is right and stop arguing with other “petty humans” or let the other guy win by accepting that theyre wrong without seeing why their argument is wrong. Many of your arguments were critical in helping me get to the realizations that I had over the past year. One can know oneself only with reference to others so the mix of students I will be exposed to in the US will ideally offset the other drawbacks. However, I also hate the lengthy visa application procedure that runs over 2 to 3 months but Im hoping that with repeated attempts, it will keep getting shorter than the last time.

Funny that this semester Ive also been caught in with Category theory and eventually (maybe in the next life or after) hope to attack the Langlands Program. You must take courses on Group theory, Homotopy theory, Measure theory and Harmonic Analysis in the math department if possible. Im assuming you’ve done Topology and the electrical engg courses on Group theory for error correcting codes which mostly covers Galois theory, if not check those out too. Im still not sure whether I should go for a math/ pure sciences PhD or stick to engineering. The pure sciences route will be rather tricky because my experience in engineering wont be any competition to those who are pure science undergrads. On the other hand engg will allow me to use abstract ideas and apply it to real world applications. The next year is critical as thats the only time I have left before grad school applications.”

Also find linked Raoul Bott’s interview on his journey in engineering and math Interview