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At some point in the evolution of a project, one has to decide when to start talking about it in seminars and at conferences. I am interested in hearing about how other mathematicians make this decision. Of course the decision to give a talk on work in progress may depend on the state of the project, on the venue and the audience, and on other factors.

My specific questions are:

  1. At what stage of a project do you consider it to be ready to talk about?
  2. How does the venue (departmental seminar/colloquium/conference/other) affect your decision?
  3. What other factors do you consider when deciding whether to talk about work in progress?

I would be interested in hearing about relevant experiences that have informed your decision process, but please keep anecdotes anonymous. Mainly I want to hear about people's criteria for evaluating when their work is ready to be publicly presented.

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Relevant to this question are the remarks of Grothendieck on "speculation":

http://pages.bangor.ac.uk/~mas010/Grothendieck-speculation.html

I agree with him on the general negative view of speculation. I have tried out some speculative ideas, and you can get some funny looks from superior persons, who can be easily inclined to use the word "nonsense" or "ridiculous"! I have had this with regard to my higher dimensional ideas. Or even over the years with regard to the use of groupoids at all. On the other hand, by giving a speculative talk in 1975 I got a very helpful idea from a later discussion.

On the other hand, I have found talking about these ideas has helped to make them real, and suggest they should be pursued, even how they should be pursued. If someone says "That can't work because ..." then that is useful information. So is "If your idea was any good, it should do .....". That could focus the mind; what would it need to do that? This turned out to be a key!

I would advise writing for yourself as speculatively as you can, taking ideas as far or beyond as you can, trying to catch ideas before they vanish. Then they should be looked at in the cold light of day, looking carefully for obstructions to them working. If these obstructions exist, that would be interesting. If they can be gradually be overcome, that is even more interesting! In either way, you may be doing something different. I expect we all want to find something new rather than solving already formulated problems. How to go about that, and how much effort to put into it, especially for career purposes, is worth discussion, especially in terms of balance of effort and probabilities of success, given one has to produce papers.

There are also dangers about talking of new ideas on a famous problem: I have been told of one well known example of this (elementary proof of the Prime Number Theorem). Unfortunately, there is as far as I know no systematic discussion of such ethical issues, and of attention to due scholarship.

Hope that helps.

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I think this is field-specific and very much depends on what is valued most: the statement of the result or the proof. This especially goes for solutions of well known open problems, so as to avoid these kind of stories.

For example, in Enumerative Combinatorics and Discrete Probability, two areas close to me, these priorities are sort of opposite. In the former, there are very few open problems. A nice new formula or a new bijection construction, even if only conjectured and checked by a computer, is already a lot of progress. Once you convince yourself that you can finish the proof, you can start giving talks - people will trust your judgement.

However, in Discrete Probability, there are lots of open problems and conjectures, often delicate and technically difficult. I would advise NOT to speak about your results until the proofs are fully written and carefully checked by somebody. This might work once or twice, but eventually there will be a seemingly trivial mistake which you overlooked in the first draft. Unfortunately, often enough such mistakes can completely destroy your proof.

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  1. When I have carefully checked all proofs and preferaby have either have a rough latex draft of the proofs or reliable handwritten notes. But I am cautious. If you have not checked something carefully you should make it clear that you believe you have a proof but details need to be checked.

  2. Venue doesn't really matter.

  3. If I think that others are very close to proving the result I would probably make slides available with the date of the talk to cover priority.

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    $\begingroup$ 3 - interesting idea... do you think slides can really prove your priority? If you have not posted your proofs, why cannot others have their own proofs "almost ready"? $\endgroup$
    – Yulia Kuznetsova
    Commented Jan 9, 2012 at 17:25
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    $\begingroup$ I would defer on #2: I'd probably feel more relaxed about talking about work in progress at my home institution's departmental seminar than at another department than a conference. As for colloquia, work in progress as the main topic off the colloquium is totally silly unless you're talking about something really big. $\endgroup$
    – Sándor Kovács
    Commented Jan 9, 2012 at 17:30
  • $\begingroup$ Slides with a date are not a perfect solution (ArXiv is better) but if they give some indication of the proof the may hold up in a court of law. All these things depend on where you are in your career. At the beginning you have more chance of somebody developing your ideas faster than you. $\endgroup$
    – Benjamin Steinberg
    Commented Jan 10, 2012 at 1:25
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    $\begingroup$ I feel very uncomfortable with this "slides priority". I have in mind truly independent work on the same problem, not "developing your ideas". I would not at all be happy to see someone pretending to be the first while he has only an idea of the proof and me (imagine) am in the process of writing down the proof accurately. But Igor's right that this depends on the field. $\endgroup$
    – Yulia Kuznetsova
    Commented Jan 10, 2012 at 11:23
  • $\begingroup$ I did say in 1 one should honestly state what one has not verified in detail. I am not suggesting slides as a way to cheat about priority, but as a way to make a somewhat public record of what you have spoken about at a talk so that someone in the audience cannot claim not to remember you said such and such in a talk. I don't see how slides are any better or worse than published research announcements sketching a proof without details (this has also been done in ArXiv). $\endgroup$
    – Benjamin Steinberg
    Commented Jan 10, 2012 at 13:32
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My advisor had some comments on this question and I am posting a paraphrased version of them, with his permission. He says (roughly):

Before talking about work in progress, one should carefully consider what the message of the talk is going to be. If the message is to communicate an idea rather than a complete proof, then this may be one situation where it is appropriate to talk about work in progress.

However, once you speak publicly about your work, other people are free to use your ideas. If you indicate what you specifically plan to do with your ideas, then it is not considered appropriate for others to work in that specific direction. But if they see ways to adapt your ideas or your results in order to apply them to something you did not say you intend to work on, then that is alright.

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(1) If I've thought carefully enough about a project that I have something to teach my audience by presenting what I know, I'll talk about it (qualifying the parts that I'm unsure of). I prefer to give talks about what I'm most excited by at the moment; this isn't always what is most complete.

(2) The venue doesn't matter, but if there are fancy-pants mathematicians in the audience I will be more nervous about presenting partial results

(3) If I have something that is complete, that I'm excited about, and that will be new to the audience I'm talking to I'll usually choose that over something that is in progress.

Maybe I should be more worried about being scooped, but (a) I'm not presenting a proof of the Riemann Hypothesis (if I were it would be wrong) (b) Its hard to scoop someone based on a talk that they gave (c) Publish-or-perish pressure aside, most of the mathematicians that I've met are people of integrity (d) if someone uses my ideas to do something I might have done but haven't yet done this is a credit to my work (and moreso if they cite me).

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