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I'm applying for Ph.D. in Mathematics and I'm writing a statement of purpose (or SOP) where I plan to express my interest in geometric representation theory and categorification. To me, an intriguing aspect of the field is that a mature understanding of mathematical objects sometimes results in their categorification, i.e. discovery of certain categories containing strictly more abundant information than the previous objects.

I only have advanced undergraduate knowledge of mathematics and the above statement is solely based on what I tried to understand from folklore and a few examples, such as Khovanov homology, which I heard of because they are relevant to my BS degree(physics). Technically, I understand the basics of category theory but not more.

Of course, I will be as honest about my current knowledge as I can. Is it favorable to state my attention to such topics when I have neither deep nor working knowledge of them?

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Yes.

A major role of the statement of purpose is to allow whoever is reading your application to determine if there are faculty whose interests match yours and could serve as a suitable advisor. I think it can only be to your benefit to accurately convey this information.

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    But keep it short and focus on your goals, which is the main purpose. Short and long term. It is more important if this is the thread you want to follow in grad study than if it is just "intriguing".
    – Buffy
    Sep 21 at 16:41
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    One issue is that if you present your interests as too specialized, and no one in the department works on them, they will think their department is not a good fit for you.
    – Kimball
    Sep 22 at 11:34
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    Indeed so. But if your interests are that specialized, they will be correct. So, convey your interests accurately. Sep 22 at 11:57

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