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I have several questions regarding the writing of SoP (for math Ph.D. admission to some US universities), which are written down sequentially. There may be some vague questions. I will be helpful in any comments. Thanks in advance.

  1. One of the US university ask only SoP describing background and interests, no CV at all. Should I write all necessary details like my past achievements, which course I have attended(sort intro to the key topics of these syllabi), my CGPA, or percentage of marks in my SoP?
  2. Should I include any professor's(of the university where I am going to apply) name and his work(some articles and books) that I used in my past study?
  3. Is there any rule of decorating the SoP, I mean use of margin, font size, space between paragraphs. I asked this question because, in some lines, I have to write some mathematical notation(contained in the name of a topic), and in this case, I have to use Latex. So is there any problem If I submit it as a .pdf file made by latex language?
  4. Should I include any sentence like, "I have plan to take some courses on..... in the upcoming semester''? Here upcoming means, if a grad school selects me, then the gap between the joining and current semester.
  5. If some grad school offers an independent study program, should I write a sentence like " the independent study program of your university is also a flexible way of enhancing my knowledge required for research"? I know that I am intended to join as a research student, but still, not all requirements can not be completed just after passing the MS degree, at least in case myself.
  6. After completing the MS program, I have joined some university in the home country as a regular Ph.D. student. So there is a one year gap between passing MS degree and applying for US university. So should I write what I have done in this gap year of that particular university? I have done some credit courses (in this gap year) that are helpful for research in my domain.
  • Too many questions, actually. – Buffy Dec 27 '19 at 18:27
  • @Buffy, another question, Should I write the above in two different questions? If you agree, then I can do it. Actually I do not know, how many can be asked once at a time in this site? – Sumanta Dec 27 '19 at 18:30
  • Leave out what isn't essential. – Buffy Dec 27 '19 at 18:31
  • This is not a very good fit as a question. Could you focus on just one aspect? I would remove Q about styling. Also your Q6 about 1y of PhD program is separate question, not about writing SoP. – aaaaa says reinstate Monica Dec 27 '19 at 19:08
  • Have you seen sample SoP on the internet? there are website that can provide you with good starting point, right now you literally starting with blank sheet of paper. Get a draft, find a friend in twitter (AcademicChatter?) to review it – aaaaa says reinstate Monica Dec 27 '19 at 19:09
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As others have noted, we discourage long lists of questions. It's better to post separate questions that are meaty enough to stand on their own. But I'll take a stab at it (disclaimer: not a mathematician).

One of the US university ask only SoP describing background and interests, no CV at all. Should I write all necessary details like my past achievements, which course I have attended(sort intro to the key topics of these syllabi), my CGPA, or percentage of marks in my SoP?

Usually they will request your transcripts, which contain this information. Still, describing past achievements is usually appropriate, as this describes the foundation for your future plans.

Should I include any professors(of the university where I am going to apply) name and his work(some articles and books) that I used in my past study?

There are a lot of posts on this site about how and when to mention specific professors that you might hope to work with. Being familiar with a professor's textbook or article might be worth discussing with that professor, if you reach out to them. But it's a bit strange to mention it in an SoP, unless you work it in as a "fun fact" rather than a serious topic.

Is there any rule of decorating the SoP, I mean use of margin, font size, space between paragraphs. I asked this question because, in some lines, I have to write some mathematical notation(contained in the name of a topic), and in this case, I have to use Latex. So is there any problem If I submit it as a .pdf file made by latex language?

This is entirely down to your university's rules. I would expect that most would accept a PDF. But I would also suggest that mathematical notation should be kept to a minimum (and Microsoft Word is able to handle fairly sophisticated mathematical notation, though it is painful).

Should I include any sentence like, "I have plan to take some courses on..... in the upcoming semester''? Here upcoming means, if a grad school selects me, then the gap between the joining and current semester.

It can't hurt, but again, this should not be the focus of your SoP. Your SoP should focus on your achievements, interests, and plans. I suggests reading some other posts about SoPs on this site -- there are lots of common traps (e.g., far too many SoPs talk about inane childhood stories; this is a professional document and should focus on your professional interests and plans).

If some grad school offers an independent study program, should I write a sentence like " the independent study program of your university is also a flexible way of enhancing my knowledge required for research"? I know that I am intended to join as a research student, but still, not all requirements can not be completed just after passing the MS degree, at least in case myself.

I don't really understand why a PhD program would offer independent study. In any case, mentioning your familiarity and approval of any university's particularities is a good thing. But I would work on the writing -- your sentence is very long and contains no real information other than your familiarity with this requirement's existence.

After completing the MS program, I have joined some university in the home country as a regular Ph.D. student. So there is a one year gap between passing MS degree and applying for US university. So should I write what I have done in this gap year of that particular university? I have done some credit courses (in this gap year) that are helpful for research in my domain.

Sure, this is part of your narrative. But I would expect them to require a transcript from this school, so you don't need to dwell on it.

Finally, one unsolicited comment: I would seriously consider hiring an editing service (ideally one familiar with academic norms) to help you revise your document. In my experience (and basing this off the writing in your question), a good editor would be able to take 4-5 pages from you and help you cut it down to a very well-written 1.5-2 pages, just by deleting and rewording. And you will learn a lot from watching them do this; this is a very valuable skill.

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  • I have a question regarding the second answer. Suppose I have mentioned only the article's name and corresponding professor name in a sentence like, "I am also impressed with the works of Prof. ABCD in the XYZW domain (I have used some papers related to EFGH of him while reading PQRS), where I also find my most current penchant," then is it right? Or I have to take permission from the professor. – Sumanta Dec 30 '19 at 6:21
  • Here I am just talking about precisely one professor. Also, I have said that I have done the course PQRS in recent months and what I have done and what is the impact of the course on me. – Sumanta Dec 30 '19 at 6:29
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I think your overarching question is "What do I include in a statement of purpose for US PhD applications". You can find several great answers to that question if you search this forum. Further there are many, many sample SoPs available online. If you review previous forum answers as well as sample SoPs, you will be able to answer these very specific questions on your own.

And please do have a native English speaker proofread your SoP, your CV, and all other application material before you submit.

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I won't deal with the specific questions, but give general advice.

The primary thing that your application must show is that you are a great candidate for success in whatever you are applying to. For application to degree programs, there are normally three components:

The CV is intended to show your past accomplishments. This brings you to NOW, but details the past.

The SoP is intended for you to mention your goals and how your past enables you to achieve them. It should be entirely forward looking.

The Letters of Recommendation give an independent view of people who should know of the likelihood of your success.

But if you can't submit a CV for some application then the SoP has to do double duty. Hopefully it can be long enough to detail your past, but it should still be focused on your future and how you intend to achieve it and can be expected to.

But you don't need to list everything from your past in such a combination document. List, primarily, those things that are enablers of your goals and show how they make you a good fit for the program you apply to.

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