I have encountered several professors who explicitly looked for foreign students so that they could use the student visa as leverage. These were not bad researchers either; one is a very famous mathematician from Brown University. Several of my foreign friends described feeling very scared and depressed because if they did not publish good enough results, they faced being deported (I'm not exaggerating; maybe they perceived the threat to be more plausible than it really was, but even so, do you really want to feel like you could be kicked out of the country?) In a few cases I have specific knowledge that these students fudged research data, left some bugs in their code, etc., just to rush their results to print and submit them to journals. I'm not saying that it's likely this would happen to you. Just don't go to grad school because of visa issues. When I was a high school student, I visited the US Naval Academy for a three-week recruiting program. I thought of it as a great academic institution, lots of access to technology, and it was free. They did a good job of berating that point of view, and making it clear that you should only try to become an officer in the Navy if you want to be an officer in the Navy. It is, in fact, not a free education at all, but rather costly both in terms of your cultural experience and time you will serve in the Navy. It's the same with grad school. Unless you are fully aware of all of the depressing, negative outcomes that most graduate students experience, and you still cannot possibly choose any other career path, then I strongly advise you to consider other options. Work for a financial firm with offices in Hong Kong or Singapore (like GETCO for example) and earn your chance to get transferred back to New York if life in the United States is very important for you. As others have mentioned, getting an H1B visa is not that hard if an employer wants to hire you. Most software companies will do this for a talented employee. You c

This may seem like an odd question, but have you been involved in any high-level math competitions?