This is a two parter, feel free to respond to one, two or no parts: 1)I'll be taking a grad level sequence in mathematical logic and possibly another in analysis with measure theory. I also have taken a couple independent studies in algebraic geometry ( my university does not normally offer them). I was curious about how much benefit achieving A's in graduate level courses can improve your chances of being accepted to a good grad school. Intuitively, it seems like it should at least demonstrate that I can handle the coursework required, so that won't be a problem. I would also hope that it would show some dedication to a particular area or to a couple of them (specifically my interest lies in logic and algebraic geometry, specifically applying topos theory and category theory to logic and computer science, unfortunately that is a little tangential to what I have on paper). I doubt it would convince anyone that I can handle creative research (I'm working on getting some more of that in), but will it give me a significant leg up? Note: I am not basing whether I will take these courses on anyone's advice, I am/will/ have already taken them. I just want to see a few opinions regarding how they will affect my chances for getting into a, say, top 40 school. 2) If there is a professor who's research seems interesting to me (even if I don't have a full understanding, but at least have the gist of it), and I would like to possibly have him advise me for my PhD, should I just send him an email? Should I try to be fluent with his some of his research first or is that overkill (it may be a possibility for some of the people who I am interested in)? Is it enough just to be interested in his subarea but not be totally familiar with his very focused work?