I am having a situation. I am applying to America at the end of the year. Now I have 3 LoR from my department, 1 of which is not so strong (the professor barely knows me).
Anyway, this is why I want another LoR. Days ago I contacted a professor from one of the universities I am going to apply, sending him my thesis and asking to work under him in some project he may assign. He turned me down, but wanted me to contact him when I apply (could be a good sign).
Obviously he did, since he barely knows me and I am even not sure to accept the admission from his university (if I luckily get accepted). The fact that he turned me down makes me think. Should I contact more professors? I mean if he accepts me, gives me some project and ultimately I do not go to his university, would it make him angry? And is the chance of I having a LoR from a famous professor in this way high?
This leads me to another option. I can go to my current university and strengthen my research with a professor there to get a stronger LoR. One drawback is that he is not as good as professors I may contact via email and he works in a different area, so basically I have to learn from scratch (my area and his are both algebra, though).
The most important question I want to ask in this post is that: what should I choose between two options?
I hope you guys can read all and answer to all the questions here, I would really appreciate it