I just completed my third year of undergraduate at an unknown prestigious state college. As of now, I am getting a ~3.4 Overall GPA (not as stellar of a GPA as I would like) and a ~3.92 Math GPA. I don't know if I am aiming too high, but here are some of my qualifications in a nutshell: attending an REU this summer, submitted a paper, participated in Putnam Math Competition and placed in top third, TA'd a Honors Calculus class, and tutored a lot. I will be attending a semester program this fall and hopefully a study abroad program this spring. I also gave a math talk in my school and I intend to give one in Mathfest, and I am a member of the Pi Mu Epsilon Honor Society.
I invested most of my time helping my parents with their Chinese business because that is our only source of income but when I do have the time, I explore many areas of mathematics by reading lecture notes online and books. I am currently trying to understand analytical number theory, probability theory, and analysis.
I intend to take the GRE and the GRE Subject Math Test at some point.
Here are the schools that I am considering. I plan on studying analytical number theory in graduate school:
New York University
Bristol University (UK)
University of Wisconsin-Madison
Ohio State University
University of Florida
And tentatively considering two of three choices:
Pennsylvania State University
University of South Carolina
Am I aiming too high? Are there any other reasonable schools that fit my mathematical interests? Do I have a chance with these schools? Thank you in advance!