<p>Thanks for all the thoughtful responses.</p>
<p>Yeah, I realize that doing well on the Putnam is not a terribly likely thing for me at this point. I'd only have one pass at it. I'm pretty sure I could do reasonably well (better than the average tester), but that doesn't really sound like too positive a boost for an application, so I may as well not put in all the energy that'd be required for that.</p>
<p>It sounds like the MCM might be something worth doing, though, so I'll definitely look into that. My engineering work probably prepares me decently well for that contest, anyway.</p>
<p>I'm already planning on some graduate classes, and I'm pretty sure I'll do fairly well on the GRE subject test (I'm willing to study, and I'm really good at standardized tests). So far so good...</p>
<p>Research is a little tougher. Like I said, I've been "wasting" time doing engineering stuff (I'm actually an engineering major adding a math major at the last minute - oops). I've had tons of internships, done well in some design competitions, and have gotten one random publication in biology (although my work was largely in statistics) in a peer-reviewed but not top-rate publication.</p>
<p>Anyway, none of these experiences really prep me for good research positions in math. I considered some REUs, but the only ones that would take me with my pretty weak math curriculum (up to now) are not too stellar, so I thought I would be better this summer getting some coursework done so I can take some graduate level classes sort of soon. I expect that I'll maintain a good GPA and will be able to get into a good REU by next summer, and hopefully I'll be able to do some math-related research during the school year. </p>
<p>That said, do you think it would look favorable if I got involved in research in a more math-oriented part of engineering, for example control theory? I may be able to line a research position up in that field within the engineering department.</p>
<p>Thanks again for all of your advice... I'm afraid that I do want to go to a "top school," although I am seriously considering programs like the Control and Dynamical Systems PhD at Caltech, which isn't exactly applied math but probably fits my background better, anyway.</p>
<p>So, as a side topic, what do you all think of Caltech's CDS program? How about the University of Texas's Computational and Applied Mathematics program? Both of these programs seem (barely) within the scope of my current resume, and I feel that I can only improve my odds by taking advanced math classes.</p>