Tell me your personal story, but don't overdo it. We'd like to
know what makes to tick, and we'd also like to know why you are
committed to doing a Ph.D. But also keep in mind that not
every story is going to stand out.
Tell us about the research you have done, if any. We want to know that you can describe the problem you confronted, argue that it is important and was challenging to solve, compare (briefly) to prior work, and explain your technical solution and how you validated it at a high-level. Doing this in a short essay is hard, but if you can do so effectively it's a very good sign for research.
Tell us about some problem you are interested in. It doesn't have to be the only problem you want to work on, but it should be a good scientific problem. You should explain what is the scientific question, why it is important, why it is hard, and why it is open. You might give some ideas for an approach to solving it, though characterizing the problem well is often critical. This gives us an idea of what you might do in graduate school and gives us additional insight into your readiness for research.
Tell us the general areas you are interested in. This is your opportunity to show you are interested in more than just the problem above, if you are.
Mention the names of specific faculty if you know you want to work with them. This helps to
ensure the right faculty will read and evaluate your
application. It's OK to mention more than one name, but
naming the whole department is superfluous.
Finally, good luck with your application! CMU is very competitive and unfortunately we do not have the capability to accept all applicants who might succeed here. But professors like myself are excited to see your application, and (in the best case) to have you come here and join us...the best part of our job is working with creative and talented students like you!
- Jonathan Aldrich