I am a Masters' student applying from India to US graduate schools for a Ph.D. in Mathematics(some of them are top departments like Stanford, MIT, UCLA, Berkeley, Illinois Urbana-Champaign). I wish to work at the interface of Analysis and Differential Geometry. Hence, I wanted to explore Geometric Analysis and took extensive differential geometry courses, a winter course on Geometric Measure Theory and did self-study on Geometric Analysis. However, as we did not have anyone in our department working in these areas, I did my Masters' thesis (ongoing) on noncommutative geometry. For this, I studied operator algebras, followed by compact quantum groups and am currently working with quantum isometries of noncommutative manifolds.
I am a little confused about what I should include or exclude from my Statement of Purpose(SOP). I have written that I am interested in both Analysis and Geometry and have mentioned: (1) Geometric Analysis, and (2) Operator Algebras and Noncommutative Geometry, as the fields that interest me. Then I explain my background in both. Is this a valid approach? Or will it give the impression that I don't have a well formed research interest and work against my selection at the graduate school?
While I was towards the end of my school life, I had a publication modelling a small physical problem. I am currently developing it into pedagogic material with a retired professor in the USA. Also, during my undergraduate studies I had two more publications in applied mathematics (control theory). I am including the first one in my SOP to illustrate my motivation in research from younger days. But should I include the other publications? I am afraid that I might run the risk of giving the committee the false impression that I am too inclined in applied mathematics.