etusblog wrote:I did my undergrad at a university that has its Mathematics Ph.D. program ranked in the top 20s. I got accepted into its Ph.D. program. But I've heard that it is strongly recommended to go elsewhere for graduate studies because one would meet and collaborate with new people. While this may be true, is that the sole reason why it is strongly recommended one should leave his or her undergraduate institution ?
From what I have been told, the biggest reason to attend a different institution for a PhD is because it allows you to meet new people to collaborate with and it diversifies the way which you have learned mathematics, just as you said. There is a lot of value in having a broad and diverse background in the people you have worked with and the institutions you have studied at.
The thing is, this view may impact other factors in your career. Imagine that you are reviewing applications for a tenure track professorship, and you find that one of the applicants attended the same university for their undergraduate and graduate degrees. How would you view this person? For some, this could be a red flag. It could indicate that you don't like change, or that you do not have experience being a "new" member in a department so you may not adapt to the situation well.
However, how this really plays out is dependent on you. If you stay at your current university for your PhD, how will you make connections outside of your university? If you go to conferences, do summer jobs, and otherwise show that you are capable of being a part of the mathematics community, this may make up for going to the same university. Also, what are your other options? Going from a top 20 school to an unranked school seems silly if you have an offer at that top 20 school. But if you have offers from schools ranked close to your current university that you are interested in, that is likely the better idea.