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I need to write a statement of research interest for internships in applied mathematics and computational science that I am applying for this summer.

Having just finished my first semester of graduate school, I feel that I have not fully defined my research interests. However, I have participated in several ongoing research projects and have already interned at a national laboratory.

How can I write a successful statement of research interest at this point in my education? Should I write about what I have already done, or focus on what I want to research in the future?

If I focus on the future, I feel that I will have trouble articulating where I want my research to go... because I really don't have an answer yet!

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I would write about both.

I usually think the writing of a statement of purpose as a story I want to tell. Write about your past projects to introduce you and your research interests, giving the reader some context on why you are applying for the position and the reasons your profile/know-how fits well in the job description.

Even if you are not sure about the exactly subject of your next project, you must know the topics that are interesting for you and the ones you definitely are not willing to work ever. You are an early master's student, no one is expecting you to tell exactly to where your research will lead you. However, they expect you to give some clues in order to assign you with the appropriate supervisor and in a project that is best suitable for your ambitions.

I would suggest you to read this text where they summarise the main characteristics of a good statement of purpose.

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