Your application essay should therefore highlight what you have done outside of the classroom. Although nobody really expects an undergraduate to be able to propose a specific PhD thesis topic, you should define your general interests and demonstrate your familiarity with current problems in the field. Be aware that, although a potential advisor published a paper on a particular topic five years ago, he or she might no longer be working in that area. Therefore, in your essay be enthusiastic about specific areas of interest, but don't imply that your are inflexible about working on related questions. Remember that you are trying to communicate a seriousness about going into a particular field.
Letters of recommendation can carry a huge amount of weight.
Such an application would rise to the top of the pile immediately, regardless of grades or GRE scores. So think carefully about whom you want to write letters on your behalf, get to know them personally, and impress them with your promise. Alternatively, if your application has some weak spots, it may still rise to the top of the pile if a professor wants you as his or her student. Consequently, you should definitely make contact with prospective advisors as early as possible. Send a personal letter introducing yourself to interesting professors during the spring of your junior year or, at the latest, the fall of your senior year.