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I am a postdoc in math who defended his Ph.D. a year ago. I have 5 papers where three of them were written by me(only author). I have always made some (minor) problems that the most referees (except one referee who read my paper and asked me to revise it; the paper is submitted international Mathematics Research Notices)  reject my paper at the first, which make me really sad, then I fix them and submit other journals. On the other hand, I have two papers with some great mathematicians that the referees wrote nice reports. I would appreciate it if one could give me some advice for the following questions:

1)When I wrote a paper, then I put it in arxiv, and then I submitted it to some journal. I was wondering that I don't put the paper in arxiv anymore and submit it first and then if it is accepted, then I put it in arxiv.

2)  When I was noticed that I made a mistake in my paper (by some referee or colleagues), I felt so bad as my paper has been in arxiv for a while and some mathematicians read it and they found out that I made a mistake. I was thinking that affects my reputation, e.g., I have this thing in my mind that they don't count me anymore as a good mathematician and never read my future paper because they think this guy always makes mistakes. This thought is killing me, in particular, I think since I made some mistakes in my papers, no one will read my future papers. I read my papers several times and checked, but there are still some problems, even I decided to ask some good to read my paper and then I add their name, but they said no.

I have neither a mentor nor a supervisor to help me. I would be grateful if I could hear some advice for my career.

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First, everyone makes mistakes. The people that you fear are judging you also make mistakes. It is part of the human condition.

It is sad that when you put a paper on arXiv that those finding errors don't contact you so that you can fix them. It would be better if they did, but that might also make you sad.

The best solution, actually, is to find some other people to work with and bounce ideas off of. If they are trustworthy, as most are, then they can give you feedback without fear of and bad effects.

Another solution, if you really have no one to work with, is to submit your work first and depend on the reviewers to give you the feedback that you didn't get otherwise. And note that those reviewers also make mistakes. But if you take their reviews to heart, then the first look at your paper will see a more refined product than you could produce without feedback.

Collaboration in math is now recognized as a good thing. In the past it wasn't used so much, but the internet has changed a lot of things.

Don't overthink it too much. Everyone makes mistakes. I just read that Einstein's first estimate of gravitational effects in general relativity was off by a factor of two.

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    "Einstein's first estimate of gravitational effects in general relativity was off by a factor of two" - just two?! That's considered "absolutely nailed it" in physics... Two orders of magnitude is when it's kinda off!
    – Lodinn
    Oct 22, 2021 at 5:11
  • I think that's a slightly misleading statement: in general relativity, the angle of deflection of light by a massive object is twice that in Newtonian gravity. This was predicted by Einstein and verified by Eddington. So it was Newton who was off by a factor of two (although he never computed the deflection angle), not Einstein. Oct 22, 2021 at 10:07
  • @astronat: See en.wikipedia.org/wiki/…. It isn't my field and I'm only trying to get a feel for how his insights came together.
    – Buffy
    Oct 22, 2021 at 10:23
  • @Buffy : Thank you very much for your advice. I agree with you that I should find some other people to work with and bounce ideas off, but the expert mathematicians wouldn't usually like to work with young mathematicians like me. I have been trying to find some good collaborations.
    – Adam
    Oct 24, 2021 at 16:37
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You need to chill out. Rejection of papers is a normal part of research work, and something that academics at all levels experience regularly. Making mistakes in published work is also something that unfortunately happens occasionally; researchers are not infallible, and neither are peer-reviewers. When you find a mistake in your published work (or when someone else finds it for you), you are generally expected to correct your paper in a timely manner (unless the mistake is so small that a correction is not warranted), but it is unlikely there will be any reputational damage if you do this.

Sometimes we make mistakes in research work, and it is entirely possible that mistakes will make it through the peer-review process for a journal without being discovered. That is why journals provide processes for errata, corrigendum and retraction. When academics point out mistakes in each other's work, this is a collaborative service that is a healthy part of the research environment. It means that there are extra sets of eyes on your work, helping you to ensure that mistakes are corrected as soon as possible. Sensible academics do not search for mistakes in the work of others with a view to "discrediting" them --- they do this with a view to helping improve the published literature in their field and helping ensure that mistakes are not repeated.

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    Thank you very much for your advice that makes me feel better.
    – Adam
    Oct 24, 2021 at 16:38

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