<p>I cannot speak about substantial aid, but the following have outstanding mathematics departments:</p>
<ul>
<li>University of Rochester</li>
<li>Tulane University</li>
<li>Purdue University</li>
<li>Boston University</li>
<li>Georgia Institute of Technology (“Georgia Tech”)</li>
</ul>
<p>All have competitive admissions, but not nearly as tough as the Ivies. AND as others have said above; If you strike out with the Ivies the first time, then the second time will not likely be better. It is okay to apply to Princton again, but if he wants to get educated in the U.S. he needs to expand beyond the Ivy League. By the way, applying to Stanford, Duke, or the University of Chicago does not count as exanding beyond the Ivy League. He needs to go a solid notch lower. The ones I have listed meet that criteria.</p>
<p>The only way to know what qualifies as a “safety” for this student is for you to tell us his ACT or SAT scores. Otherwise we are just guessing.</p>