I have a slightly different approach towards the question of research. In my statement of purpose and other essays, and during my interviews, I talked about the aspects of my field that I was interested in and the kind of ideas I had. After my acceptance, I talked to my professors about these ideas, I looked into their recent research and found where there was an overlap. I did not really delve into the stats methods used but tried to get a general high-level overview where their interests lay. Where there was an overlap, I sent an email in and set up times to talk on the phone/skype. I asked them to list some papers that I could read in the meantime to develop a better understanding of the material (the bibliographies in the recent papers helped a lot too). I've already started working on my project and am aiming for a conference in September. My program believes that the #1 thing they can do for their students is guide them towards publication. The professor I am working with on this paper has been instrumental in "helping me write for academia", and has constantly provided motivation, mentorship, and guidance throughout this process.
As far as teaching responsibilities go, we are expected to teach 1 class a semester for 2 semesters starting the second year. The rest of the time the departments expects us to be in classes and working towards improving our understanding of the literature, and advancing it.
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