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I am currently writing my Statement of Purpose for graduate admission in mathematics. I am wondering how specific I should be when describing my own research.

Should I merely include the (broad) area of my research? Should I explicitly state what I am working on, but not more? Or should I also loosely describe the main idea, and explain why this is interesting (possibly also mentioning some results)?

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Two equally successful statements of purpose can look very different from one another. Some graduate students enter with broad interests and others enter already knowing a specific question they want to work on. Both types of students can be good applicants and will have very different personal statements, potentially with very different levels detail when describing past research. Your advisor will give you much better, and more specific advise than the general advice you would get from academia.stackexchange.

Write a draft of your Personal Statement and then show it to your advisor [or other math professor who knows you well] to get feedback. After editing it based on this feedback, it is also a good idea to ask for feedback from professor(s) you intend on asking for a letter of recommendation. This serves two purposes (1) likely these letter writers will be less knowledgeable of your research area and give you a different perspective than your advisor (one that may mimic the academic diversity, to some degree, on an admissions committee) (2) Them giving you feedback on your Personal Statement will allow them to get to know you better and hence write you a better reference letter.

Generally speaking, if you can go into some detail, this is a good thing, and will set you apart from other applicants. In pure math, it is quite rare for students to have a very specific research direction before graduate school. Of course don't go so far into detail that only specialists can understand it.

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The Statement of Purpose (SoP) literally means a short eassy of your career path till date. So, mostly, the selection committee look into your SoP to know who you really are and what is your research plan. So start your SoP with few lines how did you get motivated to mathematics through accademic achievements. Then provide your current reseach interests, relevant results with some achievements if any (Conference Presentations/publications) and future plan. All of your mentioned points should be in SoP in proper way- give more stress about your main idea and explain why this is interesting and why in this particular grad school/ university.

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    In pure math it might be somewhat rare to have publications or even conference presentations (with the exception of the very best students). In such a case you might describe some topics of interest you explored as part of advanced course work [preferably something that shows you went above and beyond what was explicitly assigned], or and independent study with a professor, or anything else that is research related.
    – WetlabStudent
    Commented Nov 9, 2015 at 6:14
  • @WetLabStudent.. I am fully agree with you. It is really rare to have conference presentations and publications in Mathematics, especially for a grad school applicant. Yes, as you suggested the applicant can describe advances courseworks along with some programming skills.
    – Kay
    Commented Nov 9, 2015 at 8:07

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