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What are some characteristics of top quality research work in math? What do papers in top notch math journals have in common?

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    Good luck getting mathematicians to agree on what the top quality research work is. Feb 4, 2019 at 6:12
  • As I write this there are two answers. One (mine) is more about how you create great math. The other is how you might recognize that an existing paper might be great math. Answers to quite different questions, actually.
    – Buffy
    Feb 4, 2019 at 15:07
  • Yes that's true. Well, publishing in Annals,JAMS and other top journals of that caliber certainly helps with social status in math. But not all the great good quality papers are published in those journals . Also not all papers in top journals that don't deal with open questions nor open a new theory. For instance, they give simple proofs for an already known results. And typically those papers dont get cited as one expects
    – BigM
    Feb 4, 2019 at 15:16

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Since you ask about top quality research, I'll say some things about the extremes. Perhaps you can extrapolate a bit from the extremes to come to an understanding.

The first kind of superlative mathematical work is one that settles an old problem that many have worked on unsuccessfully in the past.

The second sort, though it may take a while to recognize it as such, is a paper that opens an entirely new field of mathematics. Sometimes the originator may not even recognize his/her work as a fundamental advance.

So, really good math papers are those that, perhaps, approach one of these extremes in some way. An old, settled, result proved with a new technique might be interesting if the new way of proving something lets others think in a new way about other problems.

Non mathematicians often think of mathematics as a bunch of facts. Early learners in mathematics think of it as proving theorems. But before you can have a statement of a theorem you need the insight to see what might be true and provable from what is already accepted. Some of those insights turn out to be valid, others not. But it isn't about the facts, nor about the proofs of the facts, but an exploration of what might also be true and provable. If you can do that, you are doing real mathematics.

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In my opinion the main factor of a good research paper is the high number of citations. It shows that the paper has been read and further developed by other mathematicians.

This is my true story. When I was near the end of my undergraduate and looking for a supervisor for my Phd study. At that time there was a guy, who advertised himself and by many other people, having high-profile publications. He had papers in Inventiones of Maths, Advances of Maths, Pacific Journal of Maths,... These are considered as high-quality journals for many mathematicians. However, there was one thing that bothered me at that time, no one cited his papers, except for himself.

I took me two years to figure out that all of his papers are false. Believe me or not. He worked on an area that was old and not many people are working on this. More importantly, his papers all contained mistakes, and be incurable. This is a good lesson for me. The first time I met him, I asked him some basic questions about distribution theory. It was easy for me to reliaze that he is not sure about even some basic maths, which is standard for any analytics.

During my Phd, I had a good time to spend time reading other people's work. And I come to realize that so-called high-ranked journals might not be parallel with the quality of their papers.

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  • It appears that perhaps you inadvertently left off the end of the story? Feb 4, 2019 at 14:51

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