planofstudy

I'd be willing to bet that the professor who emailed you is on the selection committee and probably would object to your admission come decision time. Take this person's advice or move on to a program at another school. This professors obviously thinks that you have a better chance of getting into one program over the other. It just so happens that the program that you want to enter is the other.

The fact that you're 20 years old will blow those selection committees away! Because of your age, they'll understand that you're super intelligent, but really haven't had the time nor the opportunity to get published. However, you have a low writing score on your GRE. Being published would offset this, but since you're not I suggest that your essays and SOP's be flawless. Seek the help of professional essay writers at EssayEdge or a similar group to look over your work and offer pointers. I suggest doing this rather than having your professors read over your essays. Find someone who does it for a living to help you out. Good luck!

If they ask for 500 words, give them 500 words exactly. Just in case someone's counting, follow their instructions. What if there was 1 slot left to fill and they had to decide between you and one other person with similar qualifications? Who's to say that the number of words in your SOP won't be the deciding factor?

Make a good old-fashioned phone call to the graduate admissions offices that you've applied to and simply ask them if they've received your GRE scores. This is probably a common occurrence for those schools considering that other graduate applicants have come before you with the very same issue.