Transfer or stay: what's the best for PhD? Serious suggestions needed.

Dear friends,
I am a sophomore at UNC chapel hill studying pure mathematics and am planning to apply for PhD in math. I have transferred twice due to my major change (business to mathematics) Now I still have the opportunity to transfer again to a more prestigious university (Cornell, Columbia etc,) and I really don’t know if I should apply or not. Here are some of my thoughts on the pros and cons:

  1. I have already built strong relationship with one great professor and am very likely to continue doing undergraduate research with him till graduation. Transferring means the loss of this relationship and probably a gap of undergraduate research.
  2. I believe I am qualified for a more prestigious school since I am much qualified then the time of acceptance to UNC. (My application to UNC only used my low SAT grades and one semester GPA at Indiana University on easy courses, but now I have more activities and GPA for higher level courses). Thus, it makes me feel bad not attending one of the greatest universities.
  3. I don’t know if I could find a professor that would like to supervise my reading and my research.

In conclusion, I don’t know which path would make me a more competitive candidate for PhD programs?
Your opinions matter! Thank you!

Carolina is a great school; it will not hold you back from Grad school. Given that you have great recs and research available, staying is a no-brainer.

You will come across as a prestige hound who is only looking for a fancy name on their diploma. That will be a negative for many graduate schools.

Don’t transfer again. Focus on other things. Build on what you have now, it will be far more important. Letters of recommendation carry a lot of weight. Try to publish, try to take a graduate class or two in math. Keep your GPA high. Start thinking about the GRE, particularly the Math subject test. The math subject test is very important at the top schools, it isn’t given as frequently or in as many locations as the general GRE. How do I know this? My daughter’s boyfriend has applied to PhD programs in math this year. The competition is fierce.