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.

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    But should I include the other publications? — Yes. You should include all your publications — giving the committee the false impression — Giving an accurate impression of your research interests is the whole point of the statement of purpose. Just spell it out. – JeffE Nov 28 '15 at 17:50
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    Take a look at this -- I think this will give you the guidance you need; if not, explain further: academia.stackexchange.com/questions/58857/… – aparente001 Nov 29 '15 at 15:34
  • Thank you for your replies. Regarding my first question : (in the following, (1) = Geometric Analysis, (2)= Operator Algebras & Noncommutative Geometry, as I had written in my question ) My primary interest was (1) but I did my Masters' thesis on (2) as faculty working on (1) was not available at the institute. But in many of the places that I am applying, (1) is a very active area while (2) has not been taken up that strongly. Will writing about my preparation for (1) and also talking about my research exposure in (2) make a strong SOP in terms of maturity, flexibility and motivation? – user252334 Dec 3 '15 at 4:56

even someone who does have their PhD may have relatively prominent publications in fields tangential to their core interests. make your statement of purpose concise - a single statement about your transition between fields can prevent a misunderstanding - and cite all the best work you've done.

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