If you know that you want to work on areas like Motivic Homotopy Theory, K-Theory, Derived Algebraic Geometry etc I would want to believe that you have spent some time with these subjects, and would automatically know the names of people who work on these problems. If these niche subjects catch you fascination but you do not have experience then the best thing would be to apply to programs with strong generic math backgrounds and learn the mathematics and see where you go from there. For that your teachers would be best judge. You do seem very inexperienced with background material requesting for groups applying these techniques to physics etc, which though they exist but are very niche and very intertwined.
An experienced person would know that for these niche areas especially in the US, you would rather want to apply to people to work with rather than top places. Eg. Princeton or Columbia or MIT for that matter does not have anyone really doing K-theory, or Motivic Homotopy theory. But they are all TOP schools. Likewise Chicago or UCLA i think do not have any Kollar type birational algebraic geometer or Gromov Witten theory - correct me if i am wrong (I am pretty sure about UCLA) . But these are among the top 10 places i can think of, for doing algebraic geometry in america, I would throw in Harvard, Michigan and Berkley.
Because if you know that you want to do there are people at many places which are fantastic per say but there are hardly any groups unlike some places in Germany and France.
So the best way is to learn these subjects if you can before you apply (by learn i mean atleast get a feel).
I hope i did not come across as rude.