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I am a recent mathematics graduate from the UK who will apply to math PhD programs in the United States.

I completed a master's in pure mathematics at a top program in the UK focusing on geometry. I am thinking about what to write in my statement of purpose. I want to write about my academic journey and include how I have thought about the subject more broadly. For example one of the paragraphs in the statement of purpose is: "I have seen an interesting phenomenon where for smooth manifolds its global geometry is related to its local geometry. For instance, in my third-year project, I wrote about the Chern-Gauss-Bonnet theorem. This theorem states that for even-dimensional manifolds the Euler class (a characteristic class calculated from the curvature of a connection on the tangent bundle) is related to the Euler characteristic (as defined by the Betti numbers). This is a fascinating result."

I am unsure if I am making a mistake talking about this.

There are other such paragraphs in there. There are also the usual statement of purpose paragraphs like explaining contextual information, what my plans are, why I want to do a PhD...

Thank you.

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  • If you can relate these "thoughts" to publications, then certainly they are worth mentioning. Also, if you can say "this professor has also done research in this area, and I'm interested in working with them" then that could also be helpful (assuming they are interested in taking you on) Dec 3, 2023 at 5:16
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    (Math professor here) In general, talking about the actual subject matter you want to study is probably a good idea. This could do with some rewriting though, as it reads somewhat like a rewriting of the first sentence of the Wikipedia article. Instead of "This is a fascinating result", "I was fascinated by...", and convince the reader you genuinely learned something without getting bogged down in technicalities. Good luck!
    – academic
    Dec 3, 2023 at 10:08

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It would depend on what else you say in the SoP. If you are intending (hoping, actually) to continue to follow those threads in your doctoral research then they are germane to your "purpose"; i.e. goals. Otherwise they are just artifacts from the past that might be mentioned in the CV as work in progress or similar.

Keep in mind that the SoP is about the future, your goals for doctoral study and thereafter. It isn't about the past, nor is it a place to brag. But if you use it to indicate that you are already prepared to move ahead on your goals then it is a positive, provided that you have room to say the most important things.

Also, don't phrase the SoP as if you intend to do one thing and one thing only in doctoral study. That will make it harder to get accepted and to find an advisor. You need to be somewhat flexible.

Your last paragraph seems to indicate that you have the right idea.

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