Depth of Research? Choice 1: I prefer to do research in an area where the problems I solve are problems that anybody can understand and appreciate. My proofs may be very difficult to understand, but at least anyone can understand what I've achieved and relate to it. Not too much prerequisite knowledge will be required for me to gain the background knowledge to do the research, but the problems I attempt to solve will nevertheless require great technique. Choice 2: I prefer to do research in a very advanced area where very few people have gone, thus exploring an essentially new world that I will know more about than anyone else. This will require me to gain tremendous prerequisite knowledge first, but such exploration into unknown realms will leave me intrigued every time I do my research. Only mathematicians in my very specific, extremely advanced area (and there probably won't be too many of them) can appreciate and understand my work. Choice 3: The middle ground between 1 and 2. A good deal of prerequisite knowledge will be required before I do my research but not so much that I lose my audience or will require me too many years to gain the background knowledge. Though the entire general public may not understand what I'm researching in, most math students will understand what I've achieved and have some appreciation. Choice 4: Between 1 and 3. Choice 5: Between 2 and 3. I realize that some math students may not desire to become mathematicians, and those who do may not even know their own answer yet. Nevertheless, I'm just trying to get a feel for what others may think. Such long-term planning never hurts.