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I'm starting my fifth year in math PhD. For two years I've been thinking on a well known problem (well known enough to have a Wikipedia page) in my field with no success and all of a sudden I think I did solve it with a very short solution. I haven't written it down yet but I can do so in 1 week. The bottom line is that I'm in a situation that there is nothing positive about my CV except solving the conjecture. I do not have any other publications and I'm very unknown,no professional mathematician in my field is aware of my existence. I'm planning to apply for postdoc positions this coming fall but what I'm struggling with is the letters of recommendation. Let's say I'll write my arxiv version of the paper in 1 week and about 1 week it will take it for me to explain it to my adviser and proofread it. My adviser hopefully will help me to give some talks at some places (I haven't given any talks outside of my university during my PhD). I think the standard procedure is that after giving the talk I should start asking for the LOR from the people who attended my talk until I can get 2 LORs. Problem is that this procedure will potentially take a lot of time and as people require at least a month to write LORs by that time the deadlines will be passed. I wanted to ask for some ideas on how to get LORs in my situation?

I was thinking maybe I can send emails directly to the people in my field with my preprint and directly ask for the LOR but someone told me this is very unusual and might not be a good idea. What are your opinions on this?

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    Go over your proposed solution in detail with your adviser before making any plans based on the assumption that you've solved a well known problem. Most of the rest of this will frankly be also better served by then talking to your adviser about what you should be doing.
    – JoshuaZ
    Sep 1 '19 at 18:00
  • Is staying one more year an option? (I.e. has your funding run out or do you have personal reasons you need a higher salary.). If so, in your situation it’s very much in your interest to wait one more year. Sep 1 '19 at 19:40
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    no professional mathematician in my field is aware of my existence — Surely you have an advisor, no?
    – JeffE
    Sep 1 '19 at 20:30
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    Huh? If you solved the problem this year, how does staying for another year change whether you solved the problem in a timely manner? Sep 1 '19 at 22:29
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    @lincclay Your reasoning strikes me as strange -- indeed, so much so that I'd discourage you from sharing it when job searching. It might make a negative impression. In any case, if you indeed solved a well known problem then your problem-solving skills are definitely good enough. Are your collaboration skills? Your speaking skills? I'd recommend taking the next year to do the following: submit your work to a top journal; give talks everywhere you can; talk to senior scientists in your field; strike up a collaboration or two. Good luck.
    – academic
    Sep 2 '19 at 13:14
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Things have changed since I was about to finish a PhD a long time ago, but the fact that you are unknown outside your university would have been common then, perhaps even now.

What you suggest seems to me to have two downsides. First, you are putting all your eggs into one basket (the conjecture) and second, you are entering into a time consuming process where others will need to verify your work. But beyond that, those others don't really know much about you beyond this one thing.

I would suggest, instead, that you depend on your advisor for a very strong letter and on others (professors and such) with whom you have interacted and who can speak now to your general mathematical ability and future potential. That can all be done quickly and can get you into what may be just the first round of applications.

But don't abandon your stated idea here to reach out to those others who have seen your work and might also help with an application if the more common approach seems not to be working as well as you hope.

You can get started early and if it doesn't work out, then the longer term solution may start to fall into place for the next round.

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Congrats, you are excited about this breakthrough and that's understandable. Nevertheless the only way to have your work properly evaluated and validated is to submit it to a reputable journal. Yes, it takes more time, but that's how you maximize the impact of your work.

You indicate in the comments that you can do one more year but you don't want to because you set yourself the ambitious goal of "solving one of the conjectures in [your] field in a timely manner". Well good news: you did solve the conjecture in a timely manner! You reached your goal, now you should make sure that your work gets the credit it deserves. Trying to cut corners for the short term goal of getting LORs for a postdoc position is a big risk and a potential waste of opportunity:

  • If you are interested in a career in academia, it's worth taking the time to do things carefully and reap all the benefits
  • If not, then why do you care about getting a postdoc position in the first place?
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  • So are you suggesting that if I'm interested in academia I shouldn't bother applying this year?
    – lincclay
    Sep 1 '19 at 23:54
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    I think you can apply this year, but imho you shouldn't rush the process on your discovery in order to get LORs. This might make your application less convincing for this year, but it's worth risking not getting a postdoc this year in order to have much better opportunities later.
    – Erwan
    Sep 2 '19 at 0:16

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