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In my SOP for PhD math in US grad schools, I am mentioning a specific instance where I came up with an easier solution to a problem using elementary theorems, instead of using more complex theorems. Here I'm trying to mention the names of theorems that I could have used and one of them is a non-famous/named theorem (Theorem 16.4c in Willard's General Topology) from a famous Topology text. What is the best way to mention this result? Even though there is no word limit, I don't think quoting the entire thing would be wise. I could just mention its number and mention the book in the references or I could try to summarize it as a "Separability criteria for the product space of non singleton hausdorff spaces". Or should I avoid mentioning any theorems at all?

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    What is an SOP? It might be obvious to someone from the US but to anyone here, it is/would-be meaningless?
    – CrimsonDark
    Dec 26, 2022 at 10:47
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    I may not advise on SoP for US PhD applications (so not a full answer), but a proposed scenario where the committee has to look up a somewhat obscure reference can not be good. Summarizing seems like a much better choice here, but even then stating the problem you were able to solve and not going into great detail about the solution might be preferable. Unless it was a particularly elegant/strong result, in which case you should be able to summarize it well ;)
    – Lodinn
    Dec 26, 2022 at 10:56
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    @CrimsonDark My bad, SOP=Statement of Purpose. I've edited it in the title.
    – ZSMJ
    Dec 26, 2022 at 11:13
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    Was any of this published? If not, what assurance do you have that it is correct?
    – Buffy
    Dec 26, 2022 at 13:34
  • @Buffy Oh, this wasn't an open problem, just a tough exercise that I wanted to mention since having multiple ways of solving problems is one of the things that attracted me to topology. I read in SOP guides that one should mention specific events from one's past that made them interested in their field, instead of generic sentences that could apply to anyone so I thought including specific names would be helpful. My professor and posts on Math Stack Exchange confirmed that its correct.
    – ZSMJ
    Dec 26, 2022 at 16:04

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Other questions imply that you are not from the US. You don't have a good understanding of the SoP here. It is future focused. It is about your goals for both study and thereafter. Why do you want to study this subject at this place and what do you hope to do thereafter.

If you focus the SoP on the past you are wasting an opportunity. Especially focusing it on "successes" in coursework is a poor use of the words you have - even if there is no specific limit. People will tend to stop reading if it is too long in any case.

However, if you want to study in the specific area of these math topics then say that and mention, in a phrase, that you have experience there already and have produced some simpler proofs than what you find in the literature.

But, IMO, more important than the specifics of those proofs is what insight you have gained in proving them that (this is the important part) makes you especially suited for further study in this area. "This is what I want to do, I have some skill here, and have had a bit of success already".

Prepare to be asked for details, but the details shouldn't be in the SoP. The CV is the place for that, though I realize that for one application at least you aren't being asked for one.

But, focus on short and long term goals and how you hope to achieve them. Statement of Purpose.

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