What to do after college?

I wish I could give this more than one upvote.

There's not that many legal hoops at all. WIth a bachelor's degree it's pretty simple to get an H-1B visa. I'm from Canada and a ton of my friends get jobs at tech companies after graduation with no problem. Larger companies like Google, Facebook, Microsoft, Zynga are obviously more open to this type of thing, but many of my friends have gotten jobs at early-stage startups as well. You can also start your own tech startup via this "loophole".

You may also want to look at the expected value of being one of the early engineers at a tech startup. My prior is that it is significantly lower than the expected value of starting one.

Also, in my 3 years of graduate school, I have had to maintain LOTS of legacy code left over from previous grad students. This is a good example of what I mean. You are assuming that "software jobs" has a negative factor of managing legacy code, and for some reason you don't think this applies to graduate school. If you study in any applied science field at all, you will have to maintain poorly-written legacy code.

You should view applied science graduate school as a low-paying, low-benefits version of a software development job for places like Yelp or Google or Facebook. You will do Bayesian inference and machine learning on large data sets, no matter what you think you will do or what an adviser says you will do. You will spend less than 10% of your time actually investigating new, original ideas in science, and the other 90% of the time you will do the day-to-day grind of a typical software engineer.

You will have access to good facilities, lots of interesting seminars and colloquia, and the opportunity (/ huge time cost) to take interesting advanced courses. You will also have the freedom to sleep in, arrange your working hours haphazardly, and it will be socially plausible to continue "acting like a college kid" for a few more years. These are definitely good benefits.

But my experience has been that (a) future employers won't care very much about grad school unless you are an amazing programmer; (b) you only earn ~3% more than a person with only a master's degree in your same field, and since they have a ~3 year lead time on you, their lifetime earnings are higher; and (c) the charm of "intellectual student life" wears off pretty fast when you need a good dentist but don't have insurance or when you need to fly home for a family emergency but literally cannot afford the $500 plane ticket, or when you can't travel home around the holidays because there is a journal deadline.

There's just so much to consider that you won't learn if you only talk to faculty members for whom the grad student experience has worked out smoothly.

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 could also consider working in Europe where you have many of the same freedoms and luxuries of life in the US.

The overall point is you need to pretty much independently decide if grad school is right for you first and only then worry about the visa situation. Unless you have some very unusually urgent reasons to stay in the US at almost any cost, then do not just do any old job just to stay here. Find a job you actually want to do.

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

If I don't feel like I am succeeding in the area in which someone is requesting advice, I'm not going to give advice, as it may be destructive.

Thank you for your advice. I wasn't really sure what a grad student's life is like and given the high opportunity cost and limited benefits it seems like a bad idea all around.