I'm an international undergraduate junior at an university in the U.S. whose math PhD program is ranked about 20th in the nation. By the end of my next semester, I will have taken 16 math graduate courses, most of which are for second year graduate students (e.g. Shimura variety, advanced algebraic geometry, and some special topics courses). Thus, in my senior year there will be not many courses new for me. Also, I have to pay $40k/year. Therefore, I'm considering to graduate a year earlier. However, applying directly to the PhD programs of my choice, such as Princeton, is not a good idea, since I don't have as much research experience as the successful applicants of those programs. Therefore, it is preferable if I can concentrate on research without paying too much. One solution is to go to a less competitive PhD program, including the one of my current institution and then to apply to programs of my interest a year later. Is my idea bad? What other opportunities do I have?
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From my personal experience it is unusual for undergrad students to have taken 16 math graduate courses while he/she is still in the junior year. What you have taken clearly exceeds the expectations of most graduate programs. If the professors who taught the classes think highly of you, that could be a strong selling point for your graduate school application. As for research experience, you may not want to take it too seriously as a budding mathematician. As Krantz explained in the book A Mathematician's Survival Guide (which, btw, is a book you may want to read):
Mathematics is a bit different [than other natural sciences]. We don't use test tubes. The hard fact of life is that mathematics is a subject that builds vertically. You are not really qualified to do serious research - the sort of research that mathematicians actually do - until after you pass the qualifying exams.
So research experience is great in exposing you to the world of math research, but your coursework performance may be sufficient to get you to the graduate school you want to be. I think most graduate programs won't mind that you are graduating in three years as long as you complete the same workload as those who graduate in four years.
I should say that I'm not in a math PhD program; so take my advice with a grain of salt.
(Speaking as a math professor.)
For one thing, I recommend working on a research project now. Don't worry too much about graduate courses; instead, do what successful grad students do: go to colloquiua and seminars, chat visitors up about math, try to generate research ideas.
I also recommend applying to Ph.D. programs. Although I haven't been involved with graduate admissions, I suspect there is no harm done in applying this year to Princeton and to any other schools which you are extremely excited about.
That said, your professors will know, and they should ask them. They will know how strong their recommendation letters will be. Either they will think you have a good chance of getting into your top choices of program, or they will know what you should do to improve your chances over the next year and will likely be eager and willing to help you.