What are my chances for math grad schools?

[zyxw]

I am a senior who is applying to PhD programs in math. I am almost done with the applications, but should I apply to more schools?

Undergraduate Institution: One the top state schools (ranked top 10 in math)

Overall GPA: 3.88
Math GPA: 3.95

Courses: I have taken graduate level algebra, topology, commutative algebra, real/complex analysis, functional analysis, algebraic geometry, algebraic topology, algebraic number theory, analytic number theory

Math Subject GRE: 820/86%

General GRE scores: 168/156/4.0

Research/Publications:

-REU at William and Mary, which resulted in a paper that was recently published in Finite Fields and Their Applications

- REU at Williams College (SMALL), we just recently submitted a paper for publication

-Research with one of the professors in my school; we have gotten some good results but have not written a paper yet

Other awards:

- Putnam score of 50 (top 200)
- I received the annual undergrad algebra award from our math department
- IMC Third Prize
- 2nd place in the annual math competition at my school

I have applied to:
-Harvard
-Princeton
-Stanford
-MIT
-UC Berkeley
-UCLA
-Columbia
-UChicago
-Northwestern
-Brown
-Wisconsin
-Yale
-Cornell

I am a little worried now because all of the schools that I have applied to are really competitive. Should I apply to a couple of safety schools? Will I be able to transfer to one of the schools listed above if I attend some other school?

 

[Highway]

keep applying places if you can afford it, check out Pitt/CMU. It's def a good to apply to some safety schools that you would care to attend -- even if it was for all of grad school, or just for a masters.

 

[Pengwuino]

Are those typos in your General GRE scores?