phd - Value of recommendation letters from professors in a different field

I will be a second year math under-grad in September, and currently doing an internship at applied math research institute. However, my interest after my bachelors is to do PhD in pure math at a top math school. My question is:

How important would the recommendation letters from my current supervisors be while I apply to grad school? (Note: My supervisors are very active in their field of research, but are not quite famous.)

Answer

Take what I suggest with a bucket of salt and a sprinkling of bbq sauce.

1. 
   

   The recommendation letters would indeed be very important, it certainly can't hurt to have the letters on hand, particularly as your supervisors are active researchers.

   
   


2. 
   

   Is hard to answer, as it depends on the university. But, having said that, I would say that the admissions people would look on any research favourably, as well as the internship.

   
   


3. 
   

   Is it possible to write or co-author a paper in analysis? Perhaps you could speak with your supervisors about how to combine the two, or to see if they can recommend another academic in that field that you may co-author with.