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Related: Choosing research ideas to include in a statement of purpose

I know I must describe my past and current research in my personal/research statement or statement of purpose. But do I use technical terms?

I was given the advice that since the people reading my statement are mathematicians (I am applying to US for applied mathematics), they should know what I am talking about.

Although I am applying to applied mathematics, it would also be nice to hear from other fields (biology, computer science, engineering etc) about this.

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At least in computer science: Yes, but.

What admissions committees are looking for in your application is strong evidence of research potential. One of the indicators of research potential is the ability to write about your chosen field in its native language; not just technical terms, but their proper usage and context. In particular, if you are describing your past research results, or a particular future research problem, you should describe it just as any professional researcher would.

However, there is technical language and technical language, and you want the former. The vast majority of your statements should be understandable to a general audience of (in your case) applied mathematicians, not just to experts in your chosen subsubsubfield. For the really technical details, cite the literature or point to your own work on your web page. (The ability to effectively explain your research to non-experts, demonstrated knowledge of the literature, and willingness to embarrass yourself in public by posting your work to the web are also strong indicators of research potential.)

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I believe there are two types of documents: statement of purpose and research statement. The research statement describes what are you going to do in the PhD.. am I correct ?..I saw this requirement in the CS @ Oxford. –  seteropere Nov 20 '12 at 19:25
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@seteropere - UK application process is different from the US. For the UK, you need the statement I mentioned and a research proposal (pick a project, details of how you plan to do the project) + supervisor before applying. For the US, you don't need to pick a project, supervisor or have a detailed research proposal. Just the general statement of purpose. –  Legendre Nov 21 '12 at 0:43
    
In the US, the research statement is called a "statement of purpose". But if you want admission to a top department, you need to write a research statement. –  JeffE Nov 21 '12 at 1:12
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@JeffE - Thanks for taking time to answer my question. To clarify: when you say "research statement" you are referring to a description of the applicant's past, present, future research like what you described in an answer for the related question. For the UK, this is also the best way to meet the statement of purpose requirement. However, in the UK, they also require a detailed research proposal (can be ~6 pages) outlining step by step how you plan to e.g. acquire data, analyze the data, what kind of reports you will produce when for a specific research topic agreed by a supervisor. –  Legendre Nov 21 '12 at 12:35

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