In general, are papers published by mathematicians the same thing as PhD theses? In other words, if a mathematician submitted a paper for a PhD would it get accepted? Or are papers of higher quality than PhD theses?
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Papers published in the field of Mathematics are no different than any other field. In PhD thesis what you usually do is to usually put all your related work together and defend it against a committee. For example, if you have published three papers improving/related to your previous work then you combine your work together and defend it as your PhD thesis but if it is just one that you have published at all then you defend that alone as your PhD thesis. The question should be is the Conference/Journal you're publishing your work is of higher quality/ranked ? Higher the rank better the recognition.