<p>CMU is not really known as a “pure math” school. Virtually all the research done by the Math department is applied and a significant part of it is actually related to CS or business. In fact, CMU is strongest precisely in the areas you mentioned at Rose Hulman like discrete mathematics/combinatorics, operations research, probability and statistics, mathematical finance, and logic. There is also a strong group in nonlinear dynamic systems, which could be classified as applied analysis. </p>
<p>If you want to transfer to a “pure math” school, I’d suggest Chicago, NYU, or Berkeley. You could also try the “super reaches” like Caltech, MIT, Princeton, etc., but the odds are low.</p>