@gaokaomaster : I don’t know, it seems a little late to uproot yourself for prestige and very particular jobs in one field…you need to carefully consider those job applications and whether or not your location will improve your chances. Certainly, whether you stay at UC Davis or transfer, pump up your experience more. Like get fellowships and internships (academic departments offer these…you do math…perhaps if you combine that interest with something else, you can connect with Emory’s QTM department in some capacity or take a few business school classes, as I believe you can take about 4…so you may also want to consider flexibility of curriculum and access to courses that will help at each) that make you more competitive as you are ultimately competing with people who have such experiences and these experiences were typically made more accessible by the school. I would beg you, for the sake of enhancing that essay, and actually convincing yourself of solid reasons to transfer, to seriously look for academic, social and opportunity differences offered by each of these schools vs. UC Davis. The transition can be rough if you don’t know what you really want out of it other than the degree and some supposed job offer. And again, keep in mind, that students who have attended these schools for 2-3 years (your colleagues and competitors in some cases) will have been “groomed” by the school through certain programs and opps. You’ll need to know what you can get involved in early so that you can position yourself well.
BTW, I don’t know about class-sizes, I just know that at Vanderbilt and Emory, they are pretty small in math. Pace and depth are purely instructor dependent at any school and it usually reflects itself in how they write homework problems and exams. If the UC Davis instructors you took (or could have taken) all seem to be “taking some fire out of” (read: watering down) the exams and HW, it may be because of class-size. But this effect doesn’t come from them pitching it “to the masses” so much as not wanting to grade several complex items for a large section. By slowing the pace and depth, they ensure that they can avoid this.
ND or Vanderbilt may be a bit more friendly toward “conservative” views. Emory is extremely diverse demographically and certainly is perhaps the most liberal (perhaps even more so than Duke) among the “southern Ivies” (Rice, Vanderbilt, Emory, Duke) which is not a surprise due to its location and the lack of big time sports fervor seen at 2 of the others. Admittedly, if you are worried about location, it seems Emory and JHU hold the advantages but there are the trade-offs I mention above. If you want the type of social atmosphere seen at schools with a legitimate sports fervor, they may not be it.
Don’t just go to the prestigious schools to take up space and earn grades in classes, figure out what you plan to do differently at them than you would be doing if at UC Davis. I haven’t seen you say much about this. So far, it looks like you plan to just co-exist with the student body, get the degree, and hope it is enough. In which case, maybe stick with Davis.
Other posters may be flattered and excited that someone is considering leaving their current school for the “awesome” school they attend(ed) so may swear everything is perfect and you should just leave because the grass must be greener. The reality is that it becomes your responsibility to make it greener if you go through it, so think and actually plan carefully. Being at a prestigious school does not make things suddenly just work.