<p>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?</p>
<p>Undergraduate Institution: One of the top state schools (ranked top 10 in math)</p>
<p>Overall GPA: 3.88
Math GPA: 3.95</p>
<p>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</p>
<p>Math Subject GRE: 820/86%</p>
<p>General GRE scores: 168/156/4.0</p>
<p>Research/Publications:</p>
<p>-REU at William and Mary, which resulted in a paper that was recently published in Finite Fields and Their Applications</p>
<ul>
<li>REU at Williams College (SMALL), we just recently submitted a paper for publication</li>
</ul>
<p>-Research with one of the professors in my school; we have gotten some good results but have not written a paper yet</p>
<p>Other awards:</p>
<ul>
<li>Putnam score of 50 (top 200)</li>
<li>I received the annual undergrad algebra award from our math department</li>
<li>IMC Third Prize</li>
<li>2nd place in the annual math competition at my school</li>
</ul>
<p>I have applied to:
-Harvard
-Princeton
-Stanford
-MIT
-UC Berkeley
-UCLA
-Columbia
-UChicago
-Northwestern
-Brown
-Wisconsin
-Yale
-Cornell</p>
<p>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?</p>